Black–Scholes model

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an implied volatility from it in a Black–Scholes valuation model. This has been described as using "the wrong number in the wrong formula to get the right price". This approach also gives usable values for the hedge ratios (the Greeks). Even when more advanced models are used, traders prefer to think in terms of Black–Scholes implied volatility as it allows them to evaluate and compare options of different maturities, strikes, and so on. For a discussion as to the various alternative approaches developed here, see

2870:(long an asset-or-nothing call, short a cash-or-nothing call). A call option exchanges cash for an asset at expiry, while an asset-or-nothing call just yields the asset (with no cash in exchange) and a cash-or-nothing call just yields cash (with no asset in exchange). The Black–Scholes formula is a difference of two terms, and these two terms are equal to the values of the binary call options. These binary options are less frequently traded than vanilla call options, but are easier to analyze. 8294: 14071: 14061: 14051: 1133: 10621: 10921:, said that Black–Scholes had "underpinned massive economic growth" and the "international financial system was trading derivatives valued at one quadrillion dollars per year" by 2007. He said that the Black–Scholes equation was the "mathematical justification for the trading"—and therefore—"one ingredient in a rich stew of financial irresponsibility, political ineptitude, perverse incentives and lax regulation" that contributed to the 14081: 7952: 13758: 10737:(the change in option value for a change in these parameters, or equivalently the partial derivatives with respect to these variables), and hedging these Greeks mitigates the risk caused by the non-constant nature of these parameters. Other defects cannot be mitigated by modifying the model, however, notably tail risk and liquidity risk, and these are instead managed outside the model, chiefly by minimizing these risks and by 13540: 1394: 8289:{\displaystyle {\begin{aligned}\lambda _{1}&={-\left(r-q-{1 \over {2}}\sigma ^{2}\right)+{\sqrt {\left(r-q-{1 \over {2}}\sigma ^{2}\right)^{2}+2\sigma ^{2}r}} \over {\sigma ^{2}}}\\\lambda _{2}&={-\left(r-q-{1 \over {2}}\sigma ^{2}\right)-{\sqrt {\left(r-q-{1 \over {2}}\sigma ^{2}\right)^{2}+2\sigma ^{2}r}} \over {\sigma ^{2}}}\end{aligned}}} 10807:, implied volatility is substantially higher for low strikes, and slightly lower for high strikes. Currencies tend to have more symmetrical curves, with implied volatility lowest at-the-money, and higher volatilities in both wings. Commodities often have the reverse behavior to equities, with higher implied volatility for higher strikes. 1955: 10235: 1622: 6950: 2510: 10810:

Despite the existence of the volatility smile (and the violation of all the other assumptions of the Black–Scholes model), the Black–Scholes PDE and Black–Scholes formula are still used extensively in practice. A typical approach is to regard the volatility surface as a fact about the market, and use

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One of the attractive features of the Black–Scholes model is that the parameters in the model other than the volatility (the time to maturity, the strike, the risk-free interest rate, and the current underlying price) are unequivocally observable. All other things being equal, an option's theoretical

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The assumptions of the Black–Scholes model are not all empirically valid. The model is widely employed as a useful approximation to reality, but proper application requires understanding its limitations – blindly following the model exposes the user to unexpected risk. Among the most

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the FOR/DOM exchange rate (i.e., 1 unit of foreign currency is worth S units of domestic currency) one can observe that paying out 1 unit of the domestic currency if the spot at maturity is above or below the strike is exactly like a cash-or nothing call and put respectively. Similarly, paying out 1

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In fact, the Black–Scholes formula for the price of a vanilla call option (or put option) can be interpreted by decomposing a call option into an asset-or-nothing call option minus a cash-or-nothing call option, and similarly for a put—the binary options are easier to analyze, and correspond to the

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The model is widely used, although often with some adjustments, by options market participants. The model's assumptions have been relaxed and generalized in many directions, leading to a plethora of models that are currently used in derivative pricing and risk management. The insights of the model,

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which may be interpolated to pick an appropriate rate to use in the Black–Scholes formula. Another consideration is that interest rates vary over time. This volatility may make a significant contribution to the price, especially of long-dated options. This is simply like the interest rate and bond

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With these assumptions, suppose there is a derivative security also trading in this market. It is specified that this security will have a certain payoff at a specified date in the future, depending on the values taken by the stock up to that date. Even though the path the stock price will take in

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With the lone exception of out of the money options with less than ninety days to expiration, the extent to which the B-S model underprices (overprices) an in the money (out of the money) option increases with the extent to which the option is in the money (out of the money), and decreases as the

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In practice, some sensitivities are usually quoted in scaled-down terms, to match the scale of likely changes in the parameters. For example, rho is often reported divided by 10,000 (1 basis point rate change), vega by 100 (1 vol point change), and theta by 365 or 252 (1 day decay based on either

336:, consisting of a long position in the stock and a short position in the option, whose value will not depend on the price of the stock". Their dynamic hedging strategy led to a partial differential equation which governs the price of the option. Its solution is given by the Black–Scholes formula. 10907:

wrote: "I believe the Black–Scholes formula, even though it is the standard for establishing the dollar liability for options, produces strange results when the long-term variety are being valued... The Black–Scholes formula has approached the status of holy writ in finance ... If the formula is

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The Black–Scholes formula has only one parameter that cannot be directly observed in the market: the average future volatility of the underlying asset, though it can be found from the price of other options. Since the option value (whether put or call) is increasing in this parameter, it can be

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is more complicated, as the probability of expiring in the money and the value of the asset at expiry are not independent. More precisely, the value of the asset at expiry is variable in terms of cash, but is constant in terms of the asset itself (a fixed quantity of the asset), and thus these

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Delta is the most important Greek since this usually confers the largest risk. Many traders will zero their delta at the end of the day if they are not speculating on the direction of the market and following a delta-neutral hedging approach as defined by Black–Scholes. When a trader seeks to

12613:—Companion Web site to a Nova episode originally broadcast on February 8, 2000. "The film tells the fascinating story of the invention of the Black–Scholes Formula, a mathematical Holy Grail that forever altered the world of finance and earned its creators the 1997 Nobel Prize in Economics." 9774:

In the standard Black–Scholes model, one can interpret the premium of the binary option in the risk-neutral world as the expected value = probability of being in-the-money * unit, discounted to the present value. The Black–Scholes model relies on symmetry of distribution and ignores the

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Useful approximation: although volatility is not constant, results from the model are often helpful in setting up hedges in the correct proportions to minimize risk. Even when the results are not completely accurate, they serve as a first approximation to which adjustments can be made.

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By computing the implied volatility for traded options with different strikes and maturities, the Black–Scholes model can be tested. If the Black–Scholes model held, then the implied volatility for a particular stock would be the same for all strikes and maturities. In practice, the

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for their work, the committee citing their discovery of the risk neutral dynamic revision as a breakthrough that separates the option from the risk of the underlying security. Although ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the

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The above model can be extended for variable (but deterministic) rates and volatilities. The model may also be used to value European options on instruments paying dividends. In this case, closed-form solutions are available if the dividend is a known proportion of the stock price.

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is obtained. Rather than quoting option prices in terms of dollars per unit (which are hard to compare across strikes, durations and coupon frequencies), option prices can thus be quoted in terms of implied volatility, which leads to trading of volatility in option markets.

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in their trades. In 1970, they decided to return to the academic environment. After three years of efforts, the formula—named in honor of them for making it public—was finally published in 1973 in an article titled "The Pricing of Options and Corporate Liabilities", in the

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applied to extended time periods, however, it can produce absurd results. In fairness, Black and Scholes almost certainly understood this point well. But their devoted followers may be ignoring whatever caveats the two men attached when they first unveiled the formula."

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Barone-Adesi and Whaley is a further approximation formula. Here, the stochastic differential equation (which is valid for the value of any derivative) is split into two components: the European option value and the early exercise premium. With some assumptions, a

10831:. As the bond reaches its maturity date, all of the prices involved with the bond become known, thereby decreasing its volatility, and the simple Black–Scholes model does not reflect this process. A large number of extensions to Black–Scholes, beginning with the 8805: 8556: 1633: 7916: 10611:

Since a binary call is a mathematical derivative of a vanilla call with respect to strike, the price of a binary call has the same shape as the delta of a vanilla call, and the delta of a binary call has the same shape as the gamma of a vanilla call.

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Bjerksund and Stensland provide an approximation based on an exercise strategy corresponding to a trigger price. Here, if the underlying asset price is greater than or equal to the trigger price it is optimal to exercise, and the value must equal

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In general this inequality does not have a closed form solution, though an American call with no dividends is equal to a European call and the Roll–Geske–Whaley method provides a solution for an American call with one dividend; see also

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The Greeks are important not only in the mathematical theory of finance, but also for those actively trading. Financial institutions will typically set (risk) limit values for each of the Greeks that their traders must not exceed.

6328: 1225: 1116: 10868:. In either case, this can be treated as a continuous dividend for the purposes of a Black–Scholes valuation, provided that there is no glaring asymmetry between the short stock borrowing cost and the long stock lending income. 10246: 958: 3826: 8651: 291:, and it is assumed that the drift and volatility of the motion are constant. If drift and volatility are time-varying, a suitably modified Black–Scholes formula can be deduced, as long as the volatility is not random. 10705:

reversible, as the model's original output, price, can be used as an input and one of the other variables solved for; the implied volatility calculated in this way is often used to quote option prices (that is, as a

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Results using the Black–Scholes model differ from real world prices because of simplifying assumptions of the model. One significant limitation is that in reality security prices do not follow a strict stationary

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problem of finding the time to execute the option. Since the American option can be exercised at any time before the expiration date, the Black–Scholes equation becomes a variational inequality of the form:

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and options on stocks paying a known cash dividend (in the short term, more realistic than a proportional dividend) are more difficult to value, and a choice of solution techniques is available (for example

2018: 8878: 8656: 7185:. This approximation is computationally inexpensive and the method is fast, with evidence indicating that the approximation may be more accurate in pricing long dated options than Barone-Adesi and Whaley. 3753: 11781: 6134: 10501: 8391: 7181:

call option… and (ii) a rebate that is received at the knock-out date if the option is knocked out prior to the maturity date". The formula is readily modified for the valuation of a put option, using

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Despite the lack of a general analytical solution for American put options, it is possible to derive such a formula for the case of a perpetual option – meaning that the option never expires (i.e.,

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the future is unknown, the derivative's price can be determined at the current time. For the special case of a European call or put option, Black and Scholes showed that "it is possible to create a

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argue that the Black–Scholes model merely recasts existing widely used models in terms of practically impossible "dynamic hedging" rather than "risk", to make them more compatible with mainstream

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This interpretation is incorrect because either both binaries expire in the money or both expire out of the money (either cash is exchanged for the asset or it is not), but the probabilities

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Note that from the formulae, it is clear that the gamma is the same value for calls and puts and so too is the vega the same value for calls and puts options. This can be seen directly from

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probability measure (numéraire=stock) and the equivalent martingale probability measure (numéraire=risk free asset), respectively. The risk neutral probability density for the stock price

1950:{\displaystyle {\begin{aligned}C(S_{t},t)&=N(d_{+})S_{t}-N(d_{-})Ke^{-r(T-t)}\\d_{+}&={\frac {1}{\sigma {\sqrt {T-t}}}}\left\\d_{-}&=d_{+}-\sigma {\sqrt {T-t}}\\\end{aligned}}} 2608: 5474:

For options on indices, it is reasonable to make the simplifying assumption that dividends are paid continuously, and that the dividend amount is proportional to the level of the index.

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The book gives a series of historical references supporting the theory that option traders use much more robust hedging and pricing principles than the Black, Scholes and Merton model.

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It is also possible to extend the Black–Scholes framework to options on instruments paying discrete proportional dividends. This is useful when the option is struck on a single stock.

9686: 9611: 9391: 9125: 5855: 5716: 3042: 10029: 9536: 7793: 5932: 4634: 7222: 3863: 3694: 3581:, is correct, as the value of the cash is independent of movements of the underlying asset, and thus can be interpreted as a simple product of "probability times value", while the 7217: 6695: 4057: 1381:

asset and the bank account asset (cash) in such a way as to "eliminate risk". This implies that there is a unique price for the option given by the Black–Scholes formula (see the

4363:. To calculate the probability under the real ("physical") probability measure, additional information is required—the drift term in the physical measure, or equivalently, the 4004: 2818: 10917: 339:

Several of these assumptions of the original model have been removed in subsequent extensions of the model. Modern versions account for dynamic interest rates (Merton, 1976),

7003: 2708: 4220: 10230:{\displaystyle C=-{\frac {dC_{v}(K,\sigma (K))}{dK}}=-{\frac {\partial C_{v}}{\partial K}}-{\frac {\partial C_{v}}{\partial \sigma }}{\frac {\partial \sigma }{\partial K}}} 8296:

In order to have a finite solution for the perpetual put, since the boundary conditions imply upper and lower finite bounds on the value of the put, it is necessary to set

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of the price with respect to the parameter values. One Greek, "gamma" (as well as others not listed here) is a partial derivative of another Greek, "delta" in this case.

4444:" measure the sensitivity of the value of a derivative product or a financial portfolio to changes in parameter values while holding the other parameters fixed. They are 3532: 2010: 9826: 5556: 4562: 3225: 3183: 3129: 3087: 994: 4334: 3618: 3579: 3389: 3350: 3271: 787: 4295: 4259: 3943: 3907: 3657: 3492: 3465: 3429: 3307: 2957: 2853: 10052: 9797: 8327: 7655: 6360: 1617:{\displaystyle {\begin{aligned}&C(0,t)=0{\text{ for all }}t\\&C(S,t)\rightarrow S-K{\text{ as }}S\rightarrow \infty \\&C(S,T)=\max\{S-K,0\}\end{aligned}}} 575: 1177: 711: 674: 629: 12391: 9868: 9835:

A binary call option is, at long expirations, similar to a tight call spread using two vanilla options. One can model the value of a binary cash-or-nothing option,

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process, nor is the risk-free interest actually known (and is not constant over time). The variance has been observed to be non-constant leading to models such as

7032: 6945:{\displaystyle {\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV\leq 0} 6554: 5438: 2641: 845: 529: 476: 7175: 2505:{\displaystyle {\begin{aligned}C(F,\tau )&=D\left\\d_{+}&={\frac {1}{\sigma {\sqrt {\tau }}}}\left\\d_{-}&=d_{+}-\sigma {\sqrt {\tau }}\end{aligned}}} 401: 155:(thanks to continuous revision). Further, the Black–Scholes equation, a partial differential equation that governs the price of the option, enables pricing using 11785: 10544:

is sometimes called the "skew slope" or just "skew". If the skew is typically negative, the value of a binary call will be higher when taking skew into account.

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unit of the foreign currency if the spot at maturity is above or below the strike is exactly like an asset-or nothing call and put respectively. Hence by taking

8561: 7145: 10689:, corresponding to extreme price changes; such events would be very rare if returns were lognormally distributed, but are observed much more often in practice. 10072: 7052: 6574: 5579: 1435: 1415: 1217: 1197: 809: 733: 553: 423: 375: 13377: 5516: 1363:{\displaystyle {\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV=0} 11889: 10803:

The typical shape of the implied volatility curve for a given maturity depends on the underlying instrument. Equities tend to have skewed curves: compared to

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Introducing auxiliary variables allows for the formula to be simplified and reformulated in a form that can be more convenient (this is a special case of the

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theory. They also assert that Boness in 1964 had already published a formula that is "actually identical" to the Black–Scholes call option pricing equation.

10450:{\displaystyle -{\frac {\partial C_{v}}{\partial K}}=-{\frac {\partial (SN(d_{1})-Ke^{-r(T-t)}N(d_{2}))}{\partial K}}=e^{-r(T-t)}N(d_{2})=C_{\text{no skew}}} 7388: 6145: 10892:

and Taleb have also criticized dynamic hedging and state that a number of researchers had put forth similar models prior to Black and Scholes. In response,

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The Black–Scholes model assumes that the market consists of at least one risky asset, usually called the stock, and one riskless asset, usually called the

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of an option at given prices, durations and exercise prices. Solving for volatility over a given set of durations and strike prices, one can construct an

4912: 860: 9020:{\displaystyle V(S)={K \over {1-\lambda _{2}}}\left({\lambda _{2}-1 \over {\lambda _{2}}}\right)^{\lambda _{2}}\left({S \over {K}}\right)^{\lambda _{2}}} 14697: 10972: 4340:

the asset price at expiration is above the exercise price. For related discussion – and graphical representation – see

1006: 7504: 3765: 11769: 12117: 10932:, the model does not work directly. When dealing with options whose underlying can go negative, practitioners may use a different model such as the 14521: 9779:

of the distribution of the asset. Market makers adjust for such skewness by, instead of using a single standard deviation for the underlying asset

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MacKenzie, Donald; Yuval Millo (2003). "Constructing a Market, Performing Theory: The Historical Sociology of a Financial Derivatives Exchange".

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to model volatility changes. Pricing discrepancies between empirical and the Black–Scholes model have long been observed in options that are far

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in that it can be adjusted to deal with some of its failures. Rather than considering some parameters (such as volatility or interest rates) as

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the option by buying and selling the underlying asset in a specific way to eliminate risk. This type of hedging is called "continuously revised

15124: 14117: 10983: 10550: 14654: 14634: 4396:. Thus the option price is the expected value of the discounted payoff of the option. Computing the option price via this expectation is the 11681:

Breeden, D. T., & Litzenberger, R. H. (1978). Prices of state-contingent claims implicit in option prices. Journal of business, 621-651.

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was the first to publish a paper expanding the mathematical understanding of the options pricing model, and coined the term "Black–Scholes

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as a boundary condition, one ends up with the pricing of options that pay one unit above some predefined strike price and nothing below.

2236:{\displaystyle {\begin{aligned}P(S_{t},t)&=Ke^{-r(T-t)}-S_{t}+C(S_{t},t)\\&=N(-d_{-})Ke^{-r(T-t)}-N(-d_{+})S_{t}\end{aligned}}\,} 12516:

Ajay Shah. Black, Merton and Scholes: Their work and its consequences. Economic and Political Weekly, XXXII(52):3337–3342, December 1997

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position, as inherent in the derivation, is not typically free of cost; equivalently, it is possible to lend out a long stock position

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Although the original model assumed no dividends, trivial extensions to the model can accommodate a continuous dividend yield factor.

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Yalincak, Hakan (2012). "Criticism of the Black–Scholes Model: But Why Is It Still Used? (The Answer is Simpler than the Formula".

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Macbeth, James D.; Merville, Larry J. (December 1979). "An Empirical Examination of the Black-Scholes Call Option Pricing Model".

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It is possible to have intuitive interpretations of the Black–Scholes formula, with the main subtlety being the interpretation of

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In the case of a digital call (this is a call FOR/put DOM) paying out one unit of the domestic currency gotten as present value:

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In the case of a digital call (this is a call FOR/put DOM) paying out one unit of the foreign currency gotten as present value:

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In the case of a digital put (this is a put FOR/call DOM) paying out one unit of the domestic currency gotten as present value:

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In the case of a digital put (this is a put FOR/call DOM) paying out one unit of the foreign currency gotten as present value:

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Pricing the Future: Finance, Physics, and the 300-Year Journey to the Black–Scholes Equation; A Story of Genius and Discovery

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is correctly interpreted as the present value, using the risk-free interest rate, of the expected asset price at expiration,

10510: 4772: 4421: 15094: 15058: 11881: 15362: 15099: 13245: 11616: 10967: 8800:{\displaystyle V_{S}(S_{-})=\lambda _{2}{K-S_{-} \over {S_{-}}}=-1\implies S_{-}={\lambda _{2}K \over {\lambda _{2}-1}}} 215:. Black and Scholes then attempted to apply the formula to the markets, but incurred financial losses, due to a lack of 168: 104:

price given the risk of the security and its expected return (instead replacing the security's expected return with the

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also claims to have guessed the Black–Scholes formula in 1967 but kept it to himself to make money for his investors.

5184: 4874: 4492: 15418: 15164: 12981: 12441: 12407: 12301: 12287: 12256: 11811: 11666: 11581: 11495: 11173: 7219:). In this case, the time decay of the option is equal to zero, which leads to the Black–Scholes PDE becoming an ODE: 351:

The notation used in the analysis of the Black-Scholes model is defined as follows (definitions grouped by subject):

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a useful approximation, particularly when analyzing the direction in which prices move when crossing critical points

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the assumption of continuous time and continuous trading, yielding gap risk, which can be hedged with Gamma hedging;

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is the probability that the call will be exercised provided one assumes that the asset drift is the risk-free rate.

15209: 15015: 14999: 14914: 14742: 14712: 14134: 13483: 12651: 8551:{\displaystyle V(S_{-})=A_{2}(S_{-})^{\lambda _{2}}=K-S_{-}\implies A_{2}={K-S_{-} \over {(S_{-})^{\lambda _{2}}}}} 5306: 1627:

The value of a call option for a non-dividend-paying underlying stock in terms of the Black–Scholes parameters is:

851: 10812: 7911:{\displaystyle {1 \over {2}}\sigma ^{2}\lambda _{i}^{2}+\left(r-q-{1 \over {2}}\sigma ^{2}\right)\lambda _{i}-r=0} 7385:

denote the lower exercise boundary, below which it is optimal to exercise the option. The boundary conditions are:

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which can be interpreted as a drift factor (in the risk-neutral measure for appropriate numéraire). The use of

238: 9622: 9547: 9317: 9056: 5727: 5594: 3870: 2968: 147:, are frequently used by market participants, as distinguished from the actual prices. These insights include 15367: 15144: 14980: 14879: 14864: 14403: 14276: 14192: 14103: 12336: 11872:

Boness, A James, 1964, Elements of a theory of stock-option value, Journal of Political Economy, 72, 163–175.

9981: 9475: 5866: 4568: 4468: 15139: 15019: 10925:. He clarified that "the equation itself wasn't the real problem", but its abuse in the financial industry. 15149: 14074: 13909: 12624: 4456:

establish an effective delta-hedge for a portfolio, the trader may also seek to neutralize the portfolio's

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The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the

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into account. The skew matters because it affects the binary considerably more than the regular options.

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Note that "Vega" is not a letter in the Greek alphabet; the name arises from misreading the Greek letter

4460:, as this will ensure that the hedge will be effective over a wider range of underlying price movements. 15084: 15408: 15403: 15089: 15074: 14717: 14687: 14254: 14152: 13608: 13314: 13255: 13077: 12556: 5588:

Under this formulation the arbitrage-free price implied by the Black–Scholes model can be shown to be:

4382: 3951: 2719: 1127: 250: 86: 12309:"An Equation and its Worlds: Bricolage, Exemplars, Disunity and Performativity in Financial Economics" 11690:

Gatheral, J. (2006). The volatility surface: a practitioner's guide (Vol. 357). John Wiley & Sons.

195:. They based their thinking on work previously done by market researchers and practitioners including 15398: 15393: 15169: 14970: 14884: 14869: 14800: 14376: 14259: 14157: 14036: 13980: 13699: 13626: 13570: 13359: 13170: 12143: 10865: 10761: 6958: 5409:(so a forward has zero gamma and zero vega). N' is the standard normal probability density function. 2651: 288: 12369: 11364: 10665:

the model tends to underprice deep out-of-the-money options and overprice deep in-the-money options.

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This pays out one unit of asset if the spot is below the strike at maturity. Its value is given by:

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This pays out one unit of asset if the spot is above the strike at maturity. Its value is given by:

4176: 15003: 14889: 14391: 14366: 14311: 13873: 13743: 13478: 13473: 12244: 12090:"The Great Switch – Negative Prices Are Forcing Traders To Change Their Derivatives Pricing Models" 11559: 10999: 9135:

This pays out one unit of cash if the spot is below the strike at maturity. Its value is given by:

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This pays out one unit of cash if the spot is above the strike at maturity. Its value is given by:

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The normality assumption of the Black–Scholes model does not capture extreme movements such as

7351:{\displaystyle {1 \over {2}}\sigma ^{2}S^{2}{d^{2}V \over {dS^{2}}}+(r-q)S{dV \over {dS}}-rV=0} 4464: 4300: 3873:. In addition, another way to see that the naive interpretation is incorrect is that replacing 3584: 3545: 3355: 3316: 3237: 760: 433: 272:

The following assumptions are made about the assets (which relate to the names of the assets):

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In practice, interest rates are not constant—they vary by tenor (coupon frequency), giving an

4264: 4228: 3912: 3876: 3635: 3470: 3434: 3398: 3276: 2879: 2831: 15413: 15269: 15251: 15231: 15226: 14945: 14777: 14757: 14604: 14547: 14386: 14296: 14010: 13668: 13640: 13621: 13438: 13205: 13103: 12782: 11852: 11014: 11009: 10977: 10877: 10752:

and computing prices from it, one can use the model to solve for volatility, which gives the

10037: 9782: 8299: 7626: 6345: 4356: 3135:

factor is for discounting, because the expiration date is in future, and removing it changes

2863: 2858:

The formula can be interpreted by first decomposing a call option into the difference of two

560: 9466:, the domestic interest rate, and the rest as above, the following results can be obtained: 7501:

The solutions to the ODE are a linear combination of any two linearly independent solutions:

1147: 681: 644: 599: 15344: 15299: 15289: 14975: 14950: 14919: 14899: 14737: 14659: 14644: 14511: 14064: 13915: 13827: 13663: 13645: 13563: 13392: 13349: 13339: 13329: 13324: 13050: 12991: 12926: 12880: 12875: 12749: 12709: 12676: 12171:

Black, Fischer; Scholes, Myron (1973). "The Pricing of Options and Corporate Liabilities".

12017: 11909: 11451: 11225:

Black, Fischer; Scholes, Myron (1973). "The Pricing of Options and Corporate Liabilities".

9846: 9442: 9415: 7361: 4425: 4409: 4364: 2867: 965: 740: 586: 485: 152: 78: 12515: 11857:

Option Traders Use (very) Sophisticated Heuristics, Never the Black–Scholes–Merton Formula

7008: 6530: 5423: 2617: 821: 514: 452: 8: 15339: 15179: 15104: 14909: 14669: 14579: 14469: 14084: 13990: 13970: 13797: 13772: 13397: 13185: 13108: 12931: 12616: 12550: 11412: 7182: 7154: 5398: 4457: 4445: 4405: 4392:

says that the solution to this type of PDE, when discounted appropriately, is actually a

1961: 1461: 582: 426: 380: 324: 148: 12021: 11956: 9961:{\displaystyle C=\lim _{\epsilon \to 0}{\frac {C_{v}(K-\epsilon )-C_{v}(K)}{\epsilon }}} 7127: 1397:

A European call valued using the Black–Scholes pricing equation for varying asset price

276:

Riskless rate: The rate of return on the riskless asset is constant and thus called the

15388: 15309: 15274: 15189: 15159: 14990: 14929: 14924: 14747: 14584: 14249: 14187: 14126: 13995: 13940: 13852: 13448: 13433: 13402: 13387: 13354: 13220: 13011: 12976: 12739: 12704: 12667: 12633:

Black–Scholes: The maths formula linked to the financial crash (April 27, 2012 article)

12397: 12382: 12350: 12341: 12273: 12230: 12188: 11736: 11540: 11358:

Don Chance (June 3, 2011). "Derivation and Interpretation of the Black–Scholes Model".

11287: 11242: 11163: 10900: 10789: 10757: 10753: 10625: 10057: 9035: 7120:

that approximates the solution for the latter is then obtained. This solution involves

7117: 7037: 6559: 6323:{\displaystyle d_{2}=d_{1}-\sigma {\sqrt {T-t}}={\frac {1}{\sigma {\sqrt {T-t}}}}\left} 5564: 4376: 4359:

sense, and neither of these is the true probability of expiring in-the-money under the

1420: 1400: 1202: 1182: 794: 718: 578: 538: 408: 360: 164: 70: 12598: 5480: 15329: 14542: 14459: 14428: 14321: 14301: 14291: 14147: 14142: 13718: 13694: 13678: 13453: 13443: 13382: 13369: 13344: 13230: 13016: 12812: 12489: 12469: 12447: 12437: 12417: 12403: 12386: 12297: 12283: 12277: 12252: 12192: 12063: 12033: 11985: 11939: 11807: 11799: 11701: 11662: 11633: 11388: 11246: 11169: 11085: 10669:

In short, while in the Black–Scholes model one can perfectly hedge options by simply

7919: 7178: 5168:{\displaystyle -{\frac {SN'(d_{+})\sigma }{2{\sqrt {T-t}}}}+rKe^{-r(T-t)}N(-d_{-})\,} 3865:

factor – is due to the difference between the median and mean of the

438: 313:

Ability to borrow and lend any amount, even fractional, of cash at the riskless rate.

156: 15134: 14785: 12345: 11053: 10928:

The Black–Scholes model assumes positive underlying prices; if the underlying has a

5033:{\displaystyle -{\frac {SN'(d_{+})\sigma }{2{\sqrt {T-t}}}}-rKe^{-r(T-t)}N(d_{-})\,} 3227:

is the future value of a cash-or-nothing call. In risk-neutral terms, these are the

953:{\displaystyle N(x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-z^{2}/2}\,dz.} 15349: 15236: 15119: 14995: 14732: 14489: 14464: 14413: 14264: 14217: 13887: 13822: 13334: 13273: 13268: 13250: 13180: 12946: 12941: 12913: 12865: 12744: 12684: 12588: 12486:

Lecturing Birds on Flying: Can Mathematical Theories Destroy the Financial Markets?

12374: 12331: 12323: 12220: 12212: 12180: 12025: 11728: 11532: 11277: 11269: 11234: 10949: 10836: 10797: 10782: 10734: 10686: 10642: 10504: 6797: 4441: 4417: 3621:

quantities are independent if one changes numéraire to the asset rather than cash.

340: 227: 117: 97: 74: 25: 14341: 12583: 12144:"Switch to Bachelier Options Pricing Model - Effective April 22, 2020 - CME Group" 4412:, which differs from the real world measure. For the underlying logic see section 15314: 15214: 15199: 14960: 14894: 14572: 14516: 14499: 14244: 14026: 13930: 13920: 13901: 13868: 13792: 13787: 13723: 13708: 13544: 13514: 13509: 13463: 13299: 13294: 13240: 13150: 13058: 13031: 12971: 12966: 12936: 12885: 12870: 12787: 12767: 11979: 11916: 11804:

Volatility and correlation in the pricing of equity, FX and interest-rate options

11190:"The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997" 10933: 10885: 10861: 10656: 9829: 8653:

The second boundary condition yields the location of the lower exercise boundary:

6793: 4397: 3821:{\textstyle m={\frac {1}{\sigma {\sqrt {\tau }}}}\ln \left({\frac {F}{K}}\right)} 1965: 1442: 1374: 1111:{\displaystyle N'(x)={\frac {dN(x)}{dx}}={\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}.} 848: 333: 316:

Ability to buy and sell any amount, even fractional, of the stock (this includes

283:

Random walk: The instantaneous log return of the stock price is an infinitesimal

231: 216: 208: 200: 188: 132: 124: 15129: 14361: 12112: 12110: 4381:

A standard derivation for solving the Black–Scholes PDE is given in the article

1456:

with the Black–Scholes equation. This follows since the formula can be obtained

131:" and is the basis of more complicated hedging strategies such as those used by 120:, who first wrote an academic paper on the subject, is sometimes also credited. 15319: 15284: 15204: 14810: 14557: 14474: 14443: 14438: 14418: 14408: 14351: 14326: 14306: 14271: 14239: 14222: 13925: 13905: 13673: 13519: 13504: 13304: 13215: 13165: 13142: 13123: 12951: 12893: 12860: 12855: 12835: 12759: 12630: 12538: 11952: 11901: 11827: 10994: 10929: 10904: 10889: 10824: 10738: 10714:

The first point is self-evidently useful. The others can be further discussed:

10692:

Nevertheless, Black–Scholes pricing is widely used in practice, because it is:

10649: 7147:, such that one is indifferent between early exercise and holding to maturity. 5582: 4401: 3228: 266: 196: 14346: 12573: 12225: 11282: 10800:(the 3D graph of implied volatility against strike and maturity) is not flat. 15382: 15221: 14762: 14594: 14552: 14494: 14316: 14232: 14172: 14000: 13965: 13891: 13499: 13468: 13309: 13235: 13195: 13190: 13026: 12898: 12845: 12840: 12822: 12719: 12699: 12620: 12610: 12426: 12327: 12107: 12067: 12037: 12005: 11658: 11560:"A quadratic approximation to American prices due to Barone-Adesi and Whaley" 11384: 10988: 10670: 3231:

of the asset and the expected value of the cash in the risk-neutral measure.

2859: 2611: 317: 212: 204: 184: 180: 128: 113: 109: 11782:"Science in Finance X: Dynamic hedging and further defence of Black-Scholes" 5401:, since the difference of a put and a call is a forward, which is linear in 15279: 15241: 14795: 14727: 14616: 14611: 14423: 14356: 14331: 14167: 14005: 13960: 13847: 13832: 13713: 13319: 13093: 13021: 13001: 12961: 12830: 12802: 12792: 12734: 12578: 12528: 12416:

Mandelbrot & Hudson, "The (Mis)Behavior of Markets" Basic Books, 2006.

11905: 11777: 11761: 11630:

Heard on the Street: Quantitative Questions from Wall Street Job Interviews

10961: 10893: 10804: 10240:

The first term is equal to the premium of the binary option ignoring skew:

8646:{\displaystyle V(S)=(K-S_{-})\left({S \over {S_{-}}}\right)^{\lambda _{2}}} 5454: 812: 262: 105: 94: 12008:(2012). "In Pursuit of the Unknown: 17 Equations That Changed the World". 3945:

in the formula yields a negative value for out-of-the-money call options.

3539: 15324: 14859: 14843: 14838: 14833: 14823: 14626: 14567: 14562: 14526: 14286: 14177: 13837: 13777: 13200: 13068: 13039: 13035: 12986: 12777: 12772: 12560: 12464:

Haug, E. G (2007). "Option Pricing and Hedging from Theory to Practice".

10955: 10848: 10832: 10828: 10596:{\displaystyle C=C_{\text{no skew}}-{\text{Vega}}_{v}\cdot {\text{Skew}}} 5417: 3542:, as discussed below. Simply put, the interpretation of the cash option, 2252: 1453: 1449: 284: 12296:

Derman, Emanuel. "My Life as a Quant" John Wiley & Sons, Inc. 2004.

4297:, however, does not lend itself to a simple probability interpretation. 1393: 187:

demonstrated in 1968 that a dynamic revision of a portfolio removes the

15334: 14874: 14818: 14702: 14655:

Generalized autoregressive conditional heteroskedasticity (GARCH) model

14095: 13586: 13524: 13160: 13155: 12921: 12807: 12239: 12234: 12197: 11740: 11544: 11291: 11003: 9972: 7494:{\displaystyle V(S_{-})=K-S_{-},\quad V_{S}(S_{-})=-1,\quad V(S)\leq K} 3234:

A naive, and slightly incorrect, interpretation of these terms is that

1445: 1378: 1132: 532: 136: 12500:

The book takes a critical look at the Black, Scholes and Merton model.

12029: 1373:

A key financial insight behind the equation is that one can perfectly

14828: 13603: 12714: 12636: 12147: 11077: 10638: 10620: 6334: 4163:{\displaystyle p(S,T)={\frac {N^{\prime }}{S_{T}\sigma {\sqrt {T}}}}} 3495: 1136:

Simulated geometric Brownian motions with parameters from market data

307: 13757: 12535:

When You Cannot Hedge Continuously: The Corrections to Black–Scholes

12216: 11882:

A Perspective on Quantitative Finance: Models for Beating the Market

11856: 11732: 11536: 11518: 11273: 13284: 13006: 12903: 12724: 12436:

Taleb, Nassim. "Dynamic Hedging" John Wiley & Sons, Inc. 1997.

12378: 12184: 11238: 9776: 295: 11935:

Derman and Taleb's The Illusions of Dynamic Replication: A Comment

12551:

Solution of the Black–Scholes Equation Using the Green's Function

9828:

where volatility depends on strike price, thus incorporating the

4347:

The equivalent martingale probability measure is also called the

631:

is the price of the option as a function of the underlying asset

11766:"Science in Finance IX: In defence of Black, Scholes and Merton" 7583:{\displaystyle V(S)=A_{1}S^{\lambda _{1}}+A_{2}S^{\lambda _{2}}} 6517:{\displaystyle S_{t}=S_{0}(1-\delta )^{n(t)}e^{ut+\sigma W_{t}}} 5469: 13539: 12203:

Merton, Robert C. (1973). "Theory of Rational Option Pricing".

8873:{\textstyle S\geq S_{-}={\lambda _{2}K \over {\lambda _{2}-1}}} 13555: 11981:

In Pursuit of the Unknown: 17 Equations That Changed the World

11582:"Approximation Of American Option Values: Barone-Adesi-Whaley" 10918:

In Pursuit of the Unknown: 17 Equations That Changed the World

3748:{\textstyle \left(r\pm {\frac {1}{2}}\sigma ^{2}\right)\tau ,} 310:

opportunity (i.e., there is no way to make a riskless profit).

12118:"Traders Rewriting Risk Models After Oil's Plunge Below Zero" 11496:"Closed-Form American Call Option Pricing: Roll-Geske-Whaley" 10682: 9975:

of the price of a vanilla call with respect to strike price:

4400:

approach and can be done without knowledge of PDEs. Note the

2645:

Given put–call parity, which is expressed in these terms as:

323:

The above transactions do not incur any fees or costs (i.e.,

12594:

Black–Scholes–Merton Implied Volatility Surface Model (Java)

12170: 11521:"Efficient analytic approximation of American option values" 11260:

Merton, Robert (1973). "Theory of Rational Option Pricing".

11224: 11076: 9034:

By solving the Black–Scholes differential equation with the

6129:{\displaystyle d_{1}={\frac {1}{\sigma {\sqrt {T-t}}}}\left} 4756:{\displaystyle {\frac {N'(d_{+})}{S\sigma {\sqrt {T-t}}}}\,} 12399:

An Engine, not a Camera: How Financial Models Shape Markets

12279:

Capital Ideas: The Improbable Origins of Modern Wall Street

12049: 12047: 11957:"2008 Letter to the Shareholders of Berkshire Hathaway Inc" 11165:

An Engine, Not a Camera: How Financial Models Shape Markets

1437:. In this particular example, the strike price is set to 1. 51: 14635:

Autoregressive conditional heteroskedasticity (ARCH) model

12056:"The mathematical equation that caused the banks to crash" 11932:

See also: Doriana Ruffinno and Jonathan Treussard (2006).

11338:): Risk-Adjusted Probabilities in the Black–Scholes Model" 10721:

Basis for more refined models: The Black–Scholes model is

10606: 10496:{\displaystyle {\frac {\partial C_{v}}{\partial \sigma }}} 48: 11519:

Giovanni Barone-Adesi & Robert E Whaley (June 1987).

10733:

and thus added sources of risk. This is reflected in the

108:

rate). The equation and model are named after economists

37: 14163:

Independent and identically distributed random variables

12521:

The mathematical equation that caused the banks to crash

12044: 11972: 10744:

Explicit modeling: this feature means that, rather than

9971:

Thus, the value of a binary call is the negative of the

4686:{\displaystyle {\frac {\partial ^{2}V}{\partial S^{2}}}} 167:" that is then used to calibrate other models, e.g. for 12446:

Thorp, Ed. "A Man for all Markets" Random House, 2017.

10648:

the assumption of instant, cost-less trading, yielding

6556:

is the number of dividends that have been paid by time

6362:

of the stock price is paid out at pre-determined times

5937:

is the modified forward price that occurs in the terms

3352:

is the probability of the option expiring in the money

3273:

is the probability of the option expiring in the money

581:

of the stock's returns. This is the square root of the

14640:

Autoregressive integrated moving average (ARIMA) model

12266: 10537:{\displaystyle {\frac {\partial \sigma }{\partial K}}} 8813: 8388:. From the first boundary condition, it is known that: 7177:, otherwise the option "boils down to: (i) a European 4799:{\displaystyle {\frac {\partial V}{\partial \sigma }}} 4404:

of the option payoff is not done under the real world

3834: 3828: – in other words, the reason for the 3768: 3702: 3665: 3309:, multiplied by the value of the underlying at expiry 287:

with drift; more precisely, the stock price follows a

11006:

in the valuation of options with complicated features

10958:, a variant of the Black–Scholes option pricing model 10553: 10513: 10465: 10249: 10083: 10060: 10040: 9984: 9879: 9849: 9805: 9785: 9700: 9625: 9550: 9478: 9445: 9418: 9320: 9232: 9144: 9059: 8886: 8659: 8564: 8394: 8335: 8302: 7955: 7928: 7801: 7663: 7629: 7596: 7507: 7391: 7364: 7225: 7199: 7157: 7130: 7060: 7040: 7011: 6961: 6810: 6709: 6588: 6579:

The price of a call option on such a stock is again:

6562: 6533: 6430: 6368: 6348: 6148: 5986: 5943: 5869: 5730: 5597: 5567: 5527: 5483: 5426: 5309: 5225: 5187: 5047: 4915: 4877: 4813: 4775: 4700: 4648: 4571: 4533: 4495: 4303: 4267: 4231: 4179: 4068: 4020: 3954: 3915: 3879: 3762:

for moneyness rather than the standardized moneyness

3638: 3587: 3548: 3504: 3473: 3437: 3401: 3358: 3319: 3279: 3240: 3191: 3149: 3095: 3089:

is the present value of an asset-or-nothing call and

3053: 2971: 2882: 2834: 2722: 2654: 2620: 2564: 2523: 2264: 2021: 1973: 1636: 1473: 1423: 1403: 1228: 1205: 1185: 1150: 1009: 968: 863: 824: 797: 763: 743: 721: 684: 647: 602: 563: 541: 517: 488: 455: 411: 383: 363: 93:, which gives a theoretical estimate of the price of 57: 40: 28: 12354: 10760:. In this application of the Black–Scholes model, a 10673:, in practice there are many other sources of risk. 6784:

is the forward price for the dividend paying stock.

3185:

is the future value of an asset-or-nothing call and

3131:

is the present value of a cash-or-nothing call. The

54: 34: 31: 11417: 11072: 11070: 10991:, to which the Black–Scholes PDE can be transformed 10835:, have been used to deal with this phenomenon. See 10813:

Financial economics § Challenges and criticism

585:of the stock's log price process, a measure of its 45: 11756: 11754: 11632:(16th ed.). Timothy Crack. pp. 159–162. 10595: 10536: 10495: 10449: 10229: 10066: 10046: 10023: 9960: 9862: 9820: 9791: 9758: 9680: 9605: 9530: 9458: 9431: 9385: 9295: 9207: 9119: 9019: 8872: 8799: 8645: 8558:Therefore, the value of the perpetual put becomes: 8550: 8380: 8321: 8288: 7941: 7910: 7787: 7649: 7615: 7582: 7493: 7377: 7350: 7211: 7169: 7139: 7096: 7046: 7026: 6997: 6944: 6773: 6689: 6568: 6548: 6516: 6413: 6354: 6335:Instruments paying discrete proportional dividends 6322: 6128: 5969: 5926: 5849: 5710: 5573: 5550: 5510: 5432: 5384: 5294: 5210: 5167: 5032: 4900: 4857: 4798: 4755: 4685: 4628: 4556: 4518: 4328: 4289: 4253: 4214: 4162: 4051: 3998: 3937: 3901: 3857: 3820: 3747: 3688: 3651: 3612: 3573: 3526: 3486: 3459: 3423: 3383: 3344: 3301: 3265: 3219: 3177: 3123: 3081: 3036: 2951: 2847: 2812: 2702: 2635: 2602: 2548: 2504: 2235: 2004: 1949: 1616: 1441:The Black–Scholes formula calculates the price of 1429: 1409: 1362: 1211: 1191: 1171: 1110: 988: 952: 839: 803: 781: 749: 727: 705: 668: 623: 569: 547: 523: 501: 470: 417: 395: 369: 343:and taxes (Ingersoll, 1976), and dividend payout. 12337:20.500.11820/835ab5da-2504-4152-ae5b-139da39595b8 11798: 11452:"The Black–Scholes equation for American options" 10655:the assumption of a stationary process, yielding 9799:across all strikes, incorporating a variable one 7623:, substitution of this solution into the ODE for 6774:{\displaystyle F=S_{0}(1-\delta )^{n(T)}e^{rT}\,} 4008:probabilities of the option expiring in-the-money 1960:The price of a corresponding put option based on 15380: 14522:Stochastic chains with memory of variable length 12205:Bell Journal of Economics and Management Science 11409:Basic Concepts and Techniques of Risk Management 11262:Bell Journal of Economics and Management Science 11067: 9887: 5211:{\displaystyle {\frac {\partial V}{\partial r}}} 4901:{\displaystyle {\frac {\partial V}{\partial t}}} 4519:{\displaystyle {\frac {\partial V}{\partial S}}} 3871:Itō's lemma applied to geometric Brownian motion 1586: 1460:the equation for the corresponding terminal and 815:of the option, also known as the exercise price. 12589:Chicago Option Pricing Model (Graphing Version) 11751: 11718: 10936:or simply add a constant offset to the prices. 10852:price relationship which is inversely related. 10637:the underestimation of extreme moves, yielding 6342:A typical model is to assume that a proportion 5477:The dividend payment paid over the time period 11604:Petter Bjerksund and Gunnar Stensland, 2002. 11557: 11471: 10984:Fuzzy pay-off method for real option valuation 10973:Datar–Mathews method for real option valuation 10659:, which can be hedged with volatility hedging; 6421:. The price of the stock is then modelled as: 4342:Datar–Mathews method for real option valuation 3391:multiplied by the value of the cash at expiry 14111: 13571: 12652: 11861:Journal of Economic Behavior and Organization 11084:(7th ed.). New York: McGraw-Hill/Irwin. 10034:When one takes volatility skew into account, 8880:, the perpetual American put option is worth: 5470:Instruments paying continuous yield dividends 5385:{\displaystyle -K(T-t)e^{-r(T-t)}N(-d_{-})\,} 478:is the price of the underlying asset at time 12579:Black–Scholes in Java -moving to link below- 11351: 10823:Black–Scholes cannot be applied directly to 10615: 9843:, as an infinitesimally tight spread, where 2603:{\displaystyle F=e^{r\tau }S={\frac {S}{D}}} 1607: 1589: 241:and other options markets around the world. 14055:Alternative investment management companies 14032:Standards Board for Alternative Investments 12567: 11253: 5295:{\displaystyle K(T-t)e^{-r(T-t)}N(d_{-})\,} 14650:Autoregressive–moving-average (ARMA) model 14118: 14104: 14080: 13936:Taxation of private equity and hedge funds 13578: 13564: 12659: 12645: 12544: 12509: 12463: 11579: 11493: 11357: 11309: 11307: 11305: 11303: 11301: 10899:In his 2008 letter to the shareholders of 8746: 8742: 8480: 8476: 8381:{\displaystyle V(S)=A_{2}S^{\lambda _{2}}} 5447: 4463:The Greeks for Black–Scholes are given in 2246: 1382: 676:is the price of a European call option and 159:when an explicit formula is not possible. 144: 123:The main principle behind the model is to 12395: 12368: 12335: 12306: 12272: 12224: 11998: 11606:Closed Form Valuation of American Options 11551: 11363: 11281: 11158: 11131:Mandelbrot & Hudson, 2006. pp. 72–75. 11027: 10871: 9759:{\displaystyle P=Se^{-r_{f}T}N(-d_{1})\,} 9755: 9677: 9602: 9527: 9296:{\displaystyle C=Se^{-q(T-t)}N(d_{1}).\,} 9292: 9208:{\displaystyle P=e^{-r(T-t)}N(-d_{2}).\,} 9204: 9116: 6770: 6686: 6414:{\displaystyle t_{1},t_{2},\ldots ,t_{n}} 5923: 5846: 5707: 5541: 5413:calendar days or trading days per year). 5381: 5291: 5164: 5029: 4858:{\displaystyle SN'(d_{+}){\sqrt {T-t}}\,} 4854: 4752: 4625: 4553: 2232: 1121: 940: 256: 246:Nobel Memorial Prize in Economic Sciences 14125: 11984:. New York: Basic Books. 13 March 2012. 11699: 11465: 11152: 11113:Mandelbrot & Hudson, 2006. pp. 9–10. 10619: 9681:{\displaystyle C=Se^{-r_{f}T}N(d_{1})\,} 9606:{\displaystyle P=e^{-r_{d}T}N(-d_{2})\,} 9396: 9386:{\displaystyle P=Se^{-q(T-t)}N(-d_{1}),} 9218: 9120:{\displaystyle C=e^{-r(T-t)}N(d_{2}).\,} 9042:two terms in the Black–Scholes formula. 5850:{\displaystyle P(S_{t},t)=e^{-r(T-t)}\,} 5711:{\displaystyle C(S_{t},t)=e^{-r(T-t)}\,} 3037:{\displaystyle C=DN(d_{+})F-DN(d_{-})K,} 1392: 1388: 1131: 403:generally representing the present year. 13484:Power reverse dual-currency note (PRDC) 13424:Constant proportion portfolio insurance 12249:Options, Futures, and Other Derivatives 12053: 11951: 11832:"The Problem with Black, Scholes et al" 11826: 11776: 11760: 11573: 11487: 11313: 11298: 10818: 10776: 10607:Relationship to vanilla options' Greeks 10024:{\displaystyle C=-{\frac {dC_{v}}{dK}}} 9531:{\displaystyle C=e^{-r_{d}T}N(d_{2})\,} 9306: 9045: 7788:{\displaystyle \leftS^{\lambda _{i}}=0} 6792:The problem of finding the price of an 5927:{\displaystyle F=S_{t}e^{(r-q)(T-t)}\,} 4629:{\displaystyle -N(-d_{+})=N(d_{+})-1\,} 2855:and why there are two different terms. 1142:parabolic partial differential equation 83:parabolic partial differential equation 16:Mathematical model of financial markets 15381: 14956:Doob's martingale convergence theorems 12666: 12483: 12202: 12087: 11875: 11655:Options, Futures and Other Derivatives 11449: 11381:Options, Futures and Other Derivatives 11259: 10842: 10702:a robust basis for more refined models 9130: 3858:{\textstyle {\frac {1}{2}}\sigma ^{2}} 3689:{\textstyle {\frac {1}{2}}\sigma ^{2}} 713:is the price of a European put option. 302:The assumptions about the market are: 14708:Constant elasticity of variance (CEV) 14698:Chan–Karolyi–Longstaff–Sanders (CKLS) 14099: 13636:fixed-income relative-value investing 13559: 12640: 12164: 12004: 11923:, Vol. 5, No. 4, August 2005, 323–326 11627: 11474:"Extending the Black Scholes formula" 11425:"Extending the Black Scholes formula" 11401: 10980:(contains a list of related articles) 4435: 1377:the option by buying and selling the 244:Merton and Scholes received the 1997 13419:Collateralized debt obligation (CDO) 12559:Expository article by mathematician 12243: 12211:(1). The RAND Corporation: 141–183. 12199:(Black and Scholes' original paper.) 11910:The illusions of dynamic replication 11652: 11378: 11372: 11122:Mandelbrot & Hudson, 2006. p. 74 11080:; Alex Kane; Alan J. Marcus (2008). 7212:{\displaystyle T\rightarrow \infty } 6690:{\displaystyle C(S_{0},T)=e^{-rT}\,} 4052:{\displaystyle S_{T}\in (0,\infty )} 191:of the security, thus inventing the 12574:Black–Scholes in Multiple Languages 12267:Historical and sociological aspects 10968:Brownian model of financial markets 10915:, author of the 2012 book entitled 10855: 6787: 1199:is the price of the underlying and 13: 15195:Skorokhod's representation theorem 14976:Law of large numbers (weak/strong) 12457: 12054:Stewart, Ian (February 12, 2012). 10525: 10517: 10484: 10469: 10375: 10289: 10271: 10256: 10218: 10210: 10198: 10183: 10168: 10153: 7206: 7034:denotes the payoff at stock price 6918: 6910: 6882: 6868: 6822: 6814: 5199: 5191: 4889: 4881: 4787: 4779: 4667: 4653: 4507: 4499: 4098: 4043: 3534:as probabilities of expiring ITM ( 3494:can be interpreted as measures of 1557: 1336: 1328: 1300: 1286: 1240: 1232: 902: 14: 15435: 15165:Martingale representation theorem 12504: 12088:Duncan, Felicity (22 July 2020). 11206:(Press release). October 14, 1997 9029: 4422:"Derivatives pricing: the Q world 4010:under the equivalent exponential 3999:{\displaystyle N(d_{+}),N(d_{-})} 2823: 2813:{\displaystyle P(F,\tau )=D\left} 735:is the time of option expiration. 81:investment instruments. From the 15210:Stochastic differential equation 15100:Doob's optional stopping theorem 15095:Doob–Meyer decomposition theorem 14079: 14070: 14069: 14060: 14059: 14050: 14049: 13756: 13538: 12619:A TV-programme on the so-called 11104:Taleb, 1997. pp. 91 and 110–111. 11054:"Scholes on merriam-webster.com" 7188: 4349:risk-neutral probability measure 1140:The Black–Scholes equation is a 852:cumulative distribution function 100:and shows that the option has a 24: 15080:Convergence of random variables 14966:Fisher–Tippett–Gnedenko theorem 13585: 12599:Online Black–Scholes Calculator 12433:(New York: Basic, 2011) 298 pp. 12136: 12081: 11945: 11926: 11895: 11866: 11845: 11820: 11792: 11712: 11693: 11684: 11675: 11646: 11621: 11610: 11598: 11512: 11443: 11218: 11196: 11182: 7472: 7433: 6998:{\displaystyle V(S,t)\geq H(S)} 4467:below. They can be obtained by 4428:; for details, once again, see 2703:{\displaystyle C-P=D(F-K)=S-DK} 14678:Binomial options pricing model 13246:Year-on-year inflation-indexed 11774:; And the subsequent article: 11628:Crack, Timothy Falcon (2015). 11204:"Nobel Prize Foundation, 1997" 11143: 11134: 11125: 11116: 11107: 11098: 11046: 10652:, which is difficult to hedge; 10431: 10418: 10410: 10398: 10370: 10367: 10354: 10346: 10334: 10314: 10301: 10292: 10130: 10127: 10121: 10109: 9949: 9943: 9927: 9915: 9894: 9815: 9809: 9752: 9736: 9674: 9661: 9599: 9583: 9524: 9511: 9377: 9361: 9353: 9341: 9286: 9273: 9265: 9253: 9198: 9182: 9174: 9162: 9110: 9097: 9089: 9077: 8896: 8890: 8743: 8683: 8670: 8599: 8580: 8574: 8568: 8529: 8515: 8477: 8441: 8427: 8411: 8398: 8345: 8339: 7738: 7726: 7720: 7701: 7517: 7511: 7482: 7476: 7457: 7444: 7408: 7395: 7307: 7295: 7203: 7091: 7085: 7076: 7064: 7021: 7015: 6992: 6986: 6977: 6965: 6752: 6746: 6739: 6726: 6683: 6680: 6667: 6655: 6642: 6633: 6611: 6592: 6543: 6537: 6480: 6474: 6467: 6454: 6312: 6300: 6118: 6106: 5918: 5906: 5903: 5891: 5843: 5840: 5824: 5812: 5796: 5787: 5782: 5770: 5753: 5734: 5704: 5701: 5688: 5676: 5663: 5654: 5649: 5637: 5620: 5601: 5505: 5484: 5378: 5362: 5354: 5342: 5328: 5316: 5288: 5275: 5267: 5255: 5241: 5229: 5161: 5145: 5137: 5125: 5078: 5065: 5026: 5013: 5005: 4993: 4946: 4933: 4838: 4825: 4725: 4712: 4616: 4603: 4594: 4578: 4550: 4537: 4471:of the Black–Scholes formula. 4370: 4351:. Note that both of these are 4323: 4310: 4284: 4271: 4248: 4235: 4215:{\displaystyle d_{-}=d_{-}(K)} 4209: 4203: 4132: 4129: 4116: 4103: 4084: 4072: 4046: 4034: 3993: 3980: 3971: 3958: 3932: 3919: 3896: 3883: 3869:; it is the same factor as in 3604: 3591: 3565: 3552: 3521: 3508: 3454: 3441: 3418: 3405: 3375: 3362: 3336: 3323: 3296: 3283: 3257: 3244: 3208: 3195: 3166: 3153: 3143:value (value at expiry). Thus 3115: 3102: 3073: 3060: 3025: 3012: 2997: 2984: 2938: 2925: 2913: 2900: 2799: 2783: 2771: 2755: 2738: 2726: 2713:the price of a put option is: 2682: 2670: 2343: 2330: 2318: 2305: 2284: 2272: 2215: 2199: 2188: 2176: 2159: 2143: 2127: 2108: 2084: 2072: 2048: 2029: 1997: 1985: 1885: 1873: 1750: 1738: 1721: 1708: 1689: 1676: 1663: 1644: 1580: 1568: 1554: 1534: 1531: 1519: 1494: 1482: 1166: 1154: 1045: 1039: 1024: 1018: 983: 977: 873: 867: 834: 828: 700: 688: 663: 651: 618: 606: 465: 459: 239:Chicago Board Options Exchange 1: 15145:Kolmogorov continuity theorem 14981:Law of the iterated logarithm 13256:Zero-coupon inflation-indexed 12604: 12466:Derivatives: Models on Models 12357:American Journal of Sociology 11746:time to expiration decreases. 11040: 10952:for calculating option prices 10790:monotonic increasing function 10633:significant limitations are: 9439:, the foreign interest rate, 3498:(in standard deviations) and 2614:of the underlying asset, and 2549:{\displaystyle D=e^{-r\tau }} 15150:Kolmogorov extension theorem 14829:Generalized queueing network 14337:Interacting particle systems 13910:security characteristic line 12625:Long-Term Capital Management 12173:Journal of Political Economy 11227:Journal of Political Economy 11168:. Cambridge, MA: MIT Press. 10837:Bond option § Valuation 9870:is a vanilla European call: 7942:{\displaystyle \lambda _{i}} 7795:Rearranging the terms gives: 7054:and the terminal condition: 998:probability density function 996:denotes the standard normal 757:is the time until maturity: 354:General and market related: 223:Journal of Political Economy 7: 14282:Continuous-time random walk 13898:Capital asset pricing model 13617:Capital structure arbitrage 13459:Foreign exchange derivative 12851:Callable bull/bear contract 11886:Quantitative Finance Review 11314:Nielsen, Lars Tyge (1993). 10939: 10923:financial crisis of 2007–08 10641:, which can be hedged with 9403:Foreign exchange derivative 7616:{\displaystyle S_{-}\leq S} 7097:{\displaystyle V(S,T)=H(S)} 5970:{\displaystyle d_{1},d_{2}} 3527:{\displaystyle N(d_{\pm })} 2005:{\displaystyle e^{-r(T-t)}} 346: 85:in the model, known as the 10: 15440: 15290:Extreme value theory (EVT) 15090:Doob decomposition theorem 14382:Ornstein–Uhlenbeck process 14153:Chinese restaurant process 13700:Commodity trading advisors 12557:The Black–Scholes Equation 12396:MacKenzie, Donald (2006). 12307:MacKenzie, Donald (2003). 11942:- Department of Economics. 11140:Derman, 2004. pp. 143–147. 10780: 10758:implied volatility surface 9821:{\displaystyle \sigma (K)} 9400: 8329:, leading to the solution 7122:finding the critical value 5551:{\displaystyle qS_{t}\,dt} 4557:{\displaystyle N(d_{+})\,} 4374: 3220:{\displaystyle N(d_{-})~K} 3178:{\displaystyle N(d_{+})~F} 3124:{\displaystyle DN(d_{-})K} 3082:{\displaystyle DN(d_{+})F} 1125: 174: 67:Black–Scholes–Merton model 15358: 15262: 15170:Optional stopping theorem 15067: 15029: 14971:Large deviation principle 14938: 14852: 14809: 14776: 14723:Heath–Jarrow–Morton (HJM) 14668: 14660:Moving-average (MA) model 14645:Autoregressive (AR) model 14625: 14535: 14470:Hidden Markov model (HMM) 14452: 14404:Schramm–Loewner evolution 14208: 14133: 14045: 14037:Managed Funds Association 14019: 13981:High-net-worth individual 13953: 13861: 13815: 13806: 13765: 13754: 13732: 13687: 13654: 13602: 13593: 13533: 13492: 13411: 13368: 13360:Stock market index future 13264: 13141: 13049: 12912: 12821: 12758: 12692: 12683: 12674: 12316:Social Studies of Science 11149:Thorp, 2017. pp. 183–189. 10762:coordinate transformation 10616:Black–Scholes in practice 4807: 4694: 4476: 4329:{\displaystyle SN(d_{+})} 3613:{\displaystyle N(d_{+})F} 3574:{\displaystyle N(d_{-})K} 3384:{\displaystyle N(d_{-}),} 3345:{\displaystyle N(d_{-})K} 3266:{\displaystyle N(d_{+})F} 1144:that describes the price 782:{\displaystyle \tau =T-t} 377:is a time in years; with 294:The stock does not pay a 289:geometric Brownian motion 15419:1973 in economic history 15085:Doléans-Dade exponential 14915:Progressively measurable 14713:Cox–Ingersoll–Ross (CIR) 13874:Arbitrage pricing theory 13479:Mortgage-backed security 13474:Interest rate derivative 13449:Equity-linked note (ELN) 13434:Credit-linked note (CLN) 12568:Computer implementations 12553:, Prof. Dennis Silverman 12328:10.1177/0306312703336002 11020: 11000:Monte Carlo option model 10896:has defended the model. 4414:"risk neutral valuation" 4361:real probability measure 4290:{\displaystyle N(d_{+})} 4254:{\displaystyle N(d_{-})} 3938:{\displaystyle N(d_{-})} 3902:{\displaystyle N(d_{+})} 3652:{\displaystyle d_{\pm }} 3487:{\displaystyle d_{\pm }} 3467:are not equal. In fact, 3460:{\displaystyle N(d_{-})} 3424:{\displaystyle N(d_{+})} 3302:{\displaystyle N(d_{+})} 2952:{\displaystyle C=D\left} 2848:{\displaystyle d_{\pm }} 15305:Mathematical statistics 15295:Large deviations theory 15125:Infinitesimal generator 14986:Maximal ergodic theorem 14905:Piecewise-deterministic 14507:Random dynamical system 14372:Markov additive process 13986:Institutional investors 13879:Assets under management 13704:managed futures account 13429:Contract for difference 12730:Risk-free interest rate 12545:Derivation and solution 12510:Discussion of the model 11851:Espen Gaarder Haug and 11558:Bernt Ødegaard (2003). 11472:Bernt Ødegaard (2003). 10964:, a financial art piece 10876:Espen Gaarder Haug and 10792:of implied volatility. 10047:{\displaystyle \sigma } 9792:{\displaystyle \sigma } 9769: 8322:{\displaystyle A_{1}=0} 7650:{\displaystyle i={1,2}} 6355:{\displaystyle \delta } 5448:Extensions of the model 5420:(variously rendered as 3867:log-normal distribution 2556:is the discount factor 2247:Alternative formulation 570:{\displaystyle \sigma } 434:continuously compounded 430:risk-free interest rate 278:risk-free interest rate 163:inverted to produce a " 15424:Non-Newtonian calculus 15140:Karhunen–Loève theorem 15075:Cameron–Martin formula 15039:Burkholder–Davis–Gundy 14434:Variance gamma process 14011:Sovereign wealth funds 13783:High-frequency trading 13632:Fixed income arbitrage 13211:Forward Rate Agreement 12623:and the bankruptcy of 12484:Triana, Pablo (2009). 11863:, Vol. 77, No. 2, 2011 11721:The Journal of Finance 11653:Hull, John C. (2005). 11379:Hull, John C. (2008). 10946:Binomial options model 10911:British mathematician 10872:Criticism and comments 10729:one considers them as 10629: 10597: 10538: 10497: 10451: 10231: 10068: 10048: 10025: 9962: 9864: 9822: 9793: 9760: 9682: 9607: 9532: 9460: 9433: 9387: 9297: 9209: 9121: 9021: 8874: 8801: 8647: 8552: 8382: 8323: 8290: 7943: 7912: 7789: 7651: 7617: 7584: 7495: 7379: 7352: 7213: 7171: 7141: 7098: 7048: 7028: 6999: 6946: 6775: 6691: 6570: 6550: 6518: 6415: 6356: 6324: 6130: 5971: 5928: 5851: 5712: 5575: 5552: 5512: 5434: 5386: 5296: 5212: 5169: 5034: 4902: 4859: 4800: 4757: 4687: 4630: 4558: 4520: 4383:Black–Scholes equation 4330: 4291: 4255: 4216: 4164: 4053: 4000: 3939: 3903: 3859: 3822: 3749: 3690: 3653: 3614: 3575: 3528: 3488: 3461: 3425: 3385: 3346: 3303: 3267: 3221: 3179: 3125: 3083: 3038: 2953: 2849: 2814: 2704: 2637: 2604: 2550: 2506: 2237: 2006: 1951: 1618: 1438: 1431: 1411: 1364: 1213: 1193: 1173: 1172:{\displaystyle V(S,t)} 1137: 1128:Black–Scholes equation 1122:Black–Scholes equation 1112: 990: 954: 841: 805: 783: 751: 729: 707: 706:{\displaystyle P(S,t)} 670: 669:{\displaystyle C(S,t)} 625: 624:{\displaystyle V(S,t)} 571: 549: 525: 503: 472: 419: 397: 371: 257:Fundamental hypotheses 143:as exemplified by the 87:Black–Scholes equation 73:for the dynamics of a 15270:Actuarial mathematics 15232:Uniform integrability 15227:Stratonovich integral 15155:Lévy–Prokhorov metric 15059:Marcinkiewicz–Zygmund 14946:Central limit theorem 14548:Gaussian random field 14377:McKean–Vlasov process 14297:Dyson Brownian motion 14158:Galton–Watson process 13853:Structured securities 13669:Distressed securities 13641:Statistical arbitrage 13627:Equity market neutral 13622:Convertible arbitrage 13439:Credit default option 12783:Employee stock option 12584:Black–Scholes in Java 11853:Nassim Nicholas Taleb 11407:Martin Haugh (2016). 11015:Stochastic volatility 11010:Real options analysis 10978:Financial mathematics 10882:neoclassical economic 10878:Nassim Nicholas Taleb 10623: 10598: 10539: 10507:of the vanilla call; 10498: 10452: 10232: 10069: 10049: 10026: 9963: 9865: 9863:{\displaystyle C_{v}} 9823: 9794: 9761: 9683: 9608: 9533: 9461: 9459:{\displaystyle r_{d}} 9434: 9432:{\displaystyle r_{f}} 9401:Further information: 9397:Foreign Exchange (FX) 9388: 9298: 9219:Asset-or-nothing call 9210: 9122: 9022: 8875: 8802: 8648: 8553: 8383: 8324: 8291: 7944: 7913: 7790: 7652: 7618: 7585: 7496: 7380: 7378:{\displaystyle S_{-}} 7353: 7214: 7172: 7142: 7110:Black's approximation 7099: 7049: 7029: 7000: 6947: 6776: 6692: 6571: 6551: 6519: 6416: 6357: 6325: 6131: 5972: 5929: 5852: 5713: 5576: 5553: 5518:is then modelled as: 5513: 5435: 5387: 5297: 5213: 5170: 5035: 4903: 4860: 4801: 4758: 4688: 4631: 4559: 4521: 4331: 4292: 4256: 4222:is defined as above. 4217: 4165: 4054: 4001: 3948:In detail, the terms 3940: 3904: 3860: 3823: 3750: 3691: 3654: 3615: 3576: 3538:), in the respective 3529: 3489: 3462: 3426: 3386: 3347: 3304: 3268: 3222: 3180: 3126: 3084: 3039: 2954: 2864:asset-or-nothing call 2850: 2815: 2705: 2638: 2605: 2551: 2507: 2238: 2007: 1952: 1619: 1432: 1412: 1396: 1389:Black–Scholes formula 1365: 1214: 1194: 1179:of the option, where 1174: 1135: 1113: 991: 989:{\displaystyle N'(x)} 955: 842: 806: 784: 752: 750:{\displaystyle \tau } 730: 708: 671: 626: 572: 550: 526: 504: 502:{\displaystyle S_{t}} 473: 420: 398: 372: 193:risk neutral argument 145:Black–Scholes formula 91:Black–Scholes formula 89:, one can deduce the 15345:Time series analysis 15300:Mathematical finance 15185:Reflection principle 14512:Regenerative process 14312:Fleming–Viot process 14127:Stochastic processes 13971:Financial endowments 13916:Fundamental analysis 13664:Shareholder activism 13646:Volatility arbitrage 13393:Inflation derivative 13378:Commodity derivative 13350:Single-stock futures 13340:Normal backwardation 13330:Interest rate future 13171:Conditional variance 12677:Derivative (finance) 11921:Quantitative Finance 11890:Option Theory Part 1 11839:Derivatives Strategy 10819:Valuing bond options 10777:The volatility smile 10626:stock market crashes 10551: 10511: 10463: 10247: 10081: 10058: 10038: 9982: 9877: 9847: 9803: 9783: 9698: 9623: 9548: 9476: 9443: 9416: 9318: 9307:Asset-or-nothing put 9230: 9142: 9057: 9046:Cash-or-nothing call 8884: 8811: 8657: 8562: 8392: 8333: 8300: 7953: 7926: 7922:, the solutions for 7799: 7661: 7627: 7594: 7505: 7389: 7362: 7223: 7197: 7155: 7128: 7058: 7038: 7027:{\displaystyle H(S)} 7009: 6959: 6808: 6707: 6586: 6560: 6549:{\displaystyle n(t)} 6531: 6428: 6366: 6346: 6146: 5984: 5941: 5867: 5728: 5595: 5565: 5525: 5481: 5433:{\displaystyle \nu } 5424: 5307: 5223: 5185: 5045: 4913: 4875: 4811: 4773: 4698: 4646: 4569: 4531: 4493: 4426:Mathematical finance 4410:risk-neutral measure 4408:, but an artificial 4365:market price of risk 4301: 4265: 4229: 4177: 4066: 4018: 3952: 3913: 3877: 3832: 3766: 3700: 3663: 3636: 3585: 3546: 3502: 3471: 3435: 3399: 3356: 3317: 3277: 3238: 3189: 3147: 3093: 3051: 2969: 2880: 2868:cash-or-nothing call 2832: 2720: 2652: 2636:{\displaystyle S=DF} 2618: 2562: 2521: 2262: 2019: 1971: 1634: 1471: 1421: 1401: 1226: 1203: 1183: 1148: 1007: 966: 861: 840:{\displaystyle N(x)} 822: 795: 761: 741: 719: 682: 645: 600: 561: 539: 524:{\displaystyle \mu } 515: 486: 471:{\displaystyle S(t)} 453: 409: 381: 361: 153:risk-neutral pricing 15340:Stochastic analysis 15180:Quadratic variation 15175:Prokhorov's theorem 15110:Feynman–Kac formula 14580:Markov random field 14228:Birth–death process 14085:List of hedge funds 14075:Hedge fund managers 13991:Insurance companies 13976:Fund of hedge funds 13884:Black–Scholes model 13798:Proprietary trading 13773:Algorithmic trading 13740:Fund of hedge funds 13545:Business portal 13398:Property derivative 12611:Trillion Dollar Bet 12531:, February 12, 2012 12022:2012PhT....65i..52N 11580:Don Chance (2008). 11494:Don Chance (2008). 11413:Columbia University 10849:interest rate curve 10843:Interest rate curve 9131:Cash-or-nothing put 7838: 7170:{\displaystyle S-X} 5405:and independent of 4446:partial derivatives 4420:as well as section 4406:probability measure 4390:Feynman–Kac formula 3628:instead of forward 1505: for all 1462:boundary conditions 1417:and time-to-expiry 911: 583:quadratic variation 436:(also known as the 396:{\displaystyle t=0} 325:frictionless market 149:no-arbitrage bounds 15310:Probability theory 15190:Skorokhod integral 15160:Malliavin calculus 14743:Korn-Kreer-Lenssen 14627:Time series models 14590:Pitman–Yor process 13941:Technical analysis 13403:Weather derivative 13388:Freight derivative 13370:Exotic derivatives 13290:Commodities future 12977:Intermarket spread 12740:Synthetic position 12668:Derivatives market 12282:. The Free Press. 12226:10338.dmlcz/135817 12165:Primary references 11953:Buffett, Warren E. 11915:2008-07-03 at the 11525:Journal of Finance 11431:. October 22, 2003 11283:10338.dmlcz/135817 10901:Berkshire Hathaway 10798:volatility surface 10754:implied volatility 10708:quoting convention 10630: 10593: 10534: 10493: 10447: 10227: 10064: 10044: 10021: 9958: 9901: 9860: 9818: 9789: 9756: 9678: 9603: 9528: 9456: 9429: 9383: 9293: 9205: 9117: 9036:Heaviside function 9017: 8870: 8797: 8643: 8548: 8378: 8319: 8286: 8284: 7939: 7908: 7824: 7785: 7647: 7613: 7580: 7491: 7375: 7348: 7209: 7167: 7140:{\displaystyle s*} 7137: 7118:quadratic equation 7094: 7044: 7024: 6995: 6942: 6796:is related to the 6771: 6687: 6566: 6546: 6514: 6411: 6352: 6320: 6126: 5967: 5924: 5847: 5708: 5571: 5561:for some constant 5548: 5508: 5430: 5382: 5292: 5208: 5165: 5030: 4898: 4855: 4796: 4753: 4683: 4626: 4554: 4516: 4436:The Options Greeks 4377:Martingale pricing 4326: 4287: 4251: 4212: 4160: 4049: 3996: 3935: 3899: 3855: 3818: 3745: 3686: 3649: 3610: 3571: 3524: 3484: 3457: 3421: 3381: 3342: 3299: 3263: 3217: 3175: 3121: 3079: 3034: 2949: 2873:Thus the formula: 2845: 2810: 2700: 2633: 2600: 2546: 2502: 2500: 2233: 2230: 2002: 1947: 1945: 1614: 1612: 1439: 1427: 1407: 1360: 1209: 1189: 1169: 1138: 1108: 986: 950: 894: 837: 801: 779: 747: 725: 703: 666: 621: 579:standard deviation 567: 545: 521: 499: 482:, also denoted as 468: 415: 393: 367: 165:volatility surface 71:mathematical model 15409:Stochastic models 15404:Options (finance) 15376: 15375: 15330:Signal processing 15049:Doob's upcrossing 15044:Doob's martingale 15008:Engelbert–Schmidt 14951:Donsker's theorem 14885:Feller-continuous 14753:Rendleman–Bartter 14543:Dirichlet process 14460:Branching process 14429:Telegraph process 14322:Geometric process 14302:Empirical process 14292:Diffusion process 14148:Branching process 14143:Bernoulli process 14093: 14092: 13949: 13948: 13752: 13751: 13719:Long/short equity 13695:Convergence trade 13679:Special situation 13553: 13552: 13454:Equity derivative 13444:Credit derivative 13412:Other derivatives 13383:Energy derivative 13345:Perpetual futures 13226:Overnight indexed 13176:Constant maturity 13137: 13136: 13084:Finite difference 13017:Protective option 12631:BBC News Magazine 12495:978-0-470-40675-5 12475:978-0-470-01322-9 12452:978-1-4000-6796-1 12427:Szpiro, George G. 12422:978-0-465-04355-2 12251:. Prentice Hall. 12030:10.1063/PT.3.1720 11991:978-1-84668-531-6 11940:Boston University 11888:, 2003. Also see 11830:(November 1995). 11800:Riccardo Rebonato 11639:978-0-9941182-5-7 11394:978-0-13-505283-9 11160:MacKenzie, Donald 11091:978-0-07-326967-2 10770:volatility domain 10696:easy to calculate 10591: 10577: 10567: 10532: 10491: 10444: 10382: 10278: 10225: 10205: 10175: 10142: 10067:{\displaystyle K} 10054:is a function of 10019: 9956: 9886: 8998: 8964: 8925: 8868: 8807:To conclude, for 8795: 8731: 8624: 8546: 8280: 8265: 8223: 8173: 8117: 8102: 8060: 8010: 7920:quadratic formula 7869: 7812: 7679: 7331: 7290: 7236: 7047:{\displaystyle S} 6925: 6896: 6842: 6829: 6569:{\displaystyle t} 6283: 6249: 6217: 6214: 6189: 6089: 6055: 6023: 6020: 5574:{\displaystyle q} 5444:, and ν) as a V. 5395: 5394: 5206: 5103: 5100: 4971: 4968: 4896: 4852: 4794: 4750: 4747: 4681: 4514: 4357:measure theoretic 4158: 4155: 3843: 3812: 3792: 3789: 3722: 3674: 3624:If one uses spot 3536:percent moneyness 3213: 3171: 2598: 2496: 2434: 2417: 2392: 2389: 2253:Black '76 formula 1941: 1866: 1831: 1799: 1796: 1549: 1506: 1430:{\displaystyle T} 1410:{\displaystyle S} 1343: 1314: 1260: 1247: 1212:{\displaystyle t} 1192:{\displaystyle S} 1075: 1074: 1057: 892: 891: 804:{\displaystyle K} 728:{\displaystyle T} 548:{\displaystyle S} 439:force of interest 418:{\displaystyle r} 370:{\displaystyle t} 341:transaction costs 157:numerical methods 15431: 15399:Finance theories 15394:Financial models 15350:Machine learning 15237:Usual hypotheses 15120:Girsanov theorem 15105:Dynkin's formula 14870:Continuous paths 14778:Actuarial models 14718:Garman–Kohlhagen 14688:Black–Karasinski 14683:Black–Derman–Toy 14670:Financial models 14536:Fields and other 14465:Gaussian process 14414:Sigma-martingale 14218:Additive process 14120: 14113: 14106: 14097: 14096: 14083: 14082: 14073: 14072: 14063: 14062: 14053: 14052: 13996:Investment banks 13843:Foreign exchange 13813: 13812: 13760: 13600: 13599: 13580: 13573: 13566: 13557: 13556: 13543: 13542: 13315:Forwards pricing 13089:Garman–Kohlhagen 12690: 12689: 12661: 12654: 12647: 12638: 12637: 12499: 12479: 12413: 12390: 12372: 12349: 12339: 12313: 12293: 12274:Bernstein, Peter 12262: 12238: 12228: 12196: 12159: 12158: 12156: 12154: 12140: 12134: 12133: 12131: 12129: 12114: 12105: 12104: 12102: 12100: 12085: 12079: 12078: 12076: 12074: 12062:. The Observer. 12051: 12042: 12041: 12002: 11996: 11995: 11976: 11970: 11969: 11967: 11966: 11961: 11949: 11943: 11930: 11924: 11899: 11893: 11892:by Edward Thorpe 11879: 11873: 11870: 11864: 11849: 11843: 11842: 11836: 11824: 11818: 11817: 11796: 11790: 11789: 11784:. Archived from 11773: 11768:. Archived from 11758: 11749: 11748: 11727:(5): 1173–1186. 11716: 11710: 11709: 11697: 11691: 11688: 11682: 11679: 11673: 11672: 11650: 11644: 11643: 11625: 11619: 11617:American options 11614: 11608: 11602: 11596: 11595: 11593: 11591: 11586: 11577: 11571: 11570: 11568: 11566: 11555: 11549: 11548: 11516: 11510: 11509: 11507: 11505: 11500: 11491: 11485: 11484: 11482: 11480: 11469: 11463: 11462: 11460: 11458: 11447: 11441: 11440: 11438: 11436: 11421: 11415: 11405: 11399: 11398: 11383:(7th ed.). 11376: 11370: 11369: 11367: 11355: 11349: 11348: 11342: 11311: 11296: 11295: 11285: 11257: 11251: 11250: 11222: 11216: 11215: 11213: 11211: 11200: 11194: 11193: 11186: 11180: 11179: 11156: 11150: 11147: 11141: 11138: 11132: 11129: 11123: 11120: 11114: 11111: 11105: 11102: 11096: 11095: 11074: 11065: 11064: 11062: 11060: 11050: 11034: 11031: 10950:numerical method 10856:Short stock rate 10783:Volatility smile 10687:out-of-the-money 10643:out-of-the-money 10602: 10600: 10599: 10594: 10592: 10589: 10584: 10583: 10578: 10575: 10569: 10568: 10565: 10543: 10541: 10540: 10535: 10533: 10531: 10523: 10515: 10502: 10500: 10499: 10494: 10492: 10490: 10482: 10481: 10480: 10467: 10456: 10454: 10453: 10448: 10446: 10445: 10442: 10430: 10429: 10414: 10413: 10383: 10381: 10373: 10366: 10365: 10350: 10349: 10313: 10312: 10287: 10279: 10277: 10269: 10268: 10267: 10254: 10236: 10234: 10233: 10228: 10226: 10224: 10216: 10208: 10206: 10204: 10196: 10195: 10194: 10181: 10176: 10174: 10166: 10165: 10164: 10151: 10143: 10141: 10133: 10108: 10107: 10094: 10073: 10071: 10070: 10065: 10053: 10051: 10050: 10045: 10030: 10028: 10027: 10022: 10020: 10018: 10010: 10009: 10008: 9995: 9967: 9965: 9964: 9959: 9957: 9952: 9942: 9941: 9914: 9913: 9903: 9900: 9869: 9867: 9866: 9861: 9859: 9858: 9827: 9825: 9824: 9819: 9798: 9796: 9795: 9790: 9765: 9763: 9762: 9757: 9751: 9750: 9732: 9731: 9727: 9726: 9687: 9685: 9684: 9679: 9673: 9672: 9657: 9656: 9652: 9651: 9612: 9610: 9609: 9604: 9598: 9597: 9579: 9578: 9574: 9573: 9537: 9535: 9534: 9529: 9523: 9522: 9507: 9506: 9502: 9501: 9465: 9463: 9462: 9457: 9455: 9454: 9438: 9436: 9435: 9430: 9428: 9427: 9392: 9390: 9389: 9384: 9376: 9375: 9357: 9356: 9302: 9300: 9299: 9294: 9285: 9284: 9269: 9268: 9214: 9212: 9211: 9206: 9197: 9196: 9178: 9177: 9126: 9124: 9123: 9118: 9109: 9108: 9093: 9092: 9026: 9024: 9023: 9018: 9016: 9015: 9014: 9013: 9003: 8999: 8997: 8989: 8982: 8981: 8980: 8979: 8969: 8965: 8963: 8962: 8961: 8951: 8944: 8943: 8933: 8926: 8924: 8923: 8922: 8903: 8879: 8877: 8876: 8871: 8869: 8867: 8860: 8859: 8849: 8845: 8844: 8834: 8829: 8828: 8806: 8804: 8803: 8798: 8796: 8794: 8787: 8786: 8776: 8772: 8771: 8761: 8756: 8755: 8732: 8730: 8729: 8728: 8718: 8717: 8716: 8700: 8698: 8697: 8682: 8681: 8669: 8668: 8652: 8650: 8649: 8644: 8642: 8641: 8640: 8639: 8629: 8625: 8623: 8622: 8621: 8608: 8598: 8597: 8557: 8555: 8554: 8549: 8547: 8545: 8544: 8543: 8542: 8541: 8527: 8526: 8513: 8512: 8511: 8495: 8490: 8489: 8475: 8474: 8456: 8455: 8454: 8453: 8439: 8438: 8426: 8425: 8410: 8409: 8387: 8385: 8384: 8379: 8377: 8376: 8375: 8374: 8360: 8359: 8328: 8326: 8325: 8320: 8312: 8311: 8295: 8293: 8292: 8287: 8285: 8281: 8279: 8278: 8277: 8267: 8266: 8261: 8260: 8245: 8244: 8239: 8235: 8234: 8233: 8224: 8222: 8214: 8194: 8189: 8185: 8184: 8183: 8174: 8172: 8164: 8141: 8132: 8131: 8118: 8116: 8115: 8114: 8104: 8103: 8098: 8097: 8082: 8081: 8076: 8072: 8071: 8070: 8061: 8059: 8051: 8031: 8026: 8022: 8021: 8020: 8011: 8009: 8001: 7978: 7969: 7968: 7948: 7946: 7945: 7940: 7938: 7937: 7917: 7915: 7914: 7909: 7895: 7894: 7885: 7881: 7880: 7879: 7870: 7868: 7860: 7837: 7832: 7823: 7822: 7813: 7811: 7803: 7794: 7792: 7791: 7786: 7778: 7777: 7776: 7775: 7761: 7757: 7750: 7749: 7713: 7712: 7700: 7699: 7690: 7689: 7680: 7678: 7670: 7656: 7654: 7653: 7648: 7646: 7622: 7620: 7619: 7614: 7606: 7605: 7589: 7587: 7586: 7581: 7579: 7578: 7577: 7576: 7562: 7561: 7549: 7548: 7547: 7546: 7532: 7531: 7500: 7498: 7497: 7492: 7456: 7455: 7443: 7442: 7429: 7428: 7407: 7406: 7384: 7382: 7381: 7376: 7374: 7373: 7357: 7355: 7354: 7349: 7332: 7330: 7322: 7314: 7291: 7289: 7288: 7287: 7274: 7270: 7269: 7259: 7257: 7256: 7247: 7246: 7237: 7235: 7227: 7218: 7216: 7215: 7210: 7176: 7174: 7173: 7168: 7146: 7144: 7143: 7138: 7103: 7101: 7100: 7095: 7053: 7051: 7050: 7045: 7033: 7031: 7030: 7025: 7004: 7002: 7001: 6996: 6951: 6949: 6948: 6943: 6926: 6924: 6916: 6908: 6897: 6895: 6894: 6893: 6880: 6876: 6875: 6865: 6863: 6862: 6853: 6852: 6843: 6835: 6830: 6828: 6820: 6812: 6798:optimal stopping 6788:American options 6780: 6778: 6777: 6772: 6769: 6768: 6756: 6755: 6725: 6724: 6696: 6694: 6693: 6688: 6679: 6678: 6654: 6653: 6632: 6631: 6604: 6603: 6575: 6573: 6572: 6567: 6555: 6553: 6552: 6547: 6523: 6521: 6520: 6515: 6513: 6512: 6511: 6510: 6484: 6483: 6453: 6452: 6440: 6439: 6420: 6418: 6417: 6412: 6410: 6409: 6391: 6390: 6378: 6377: 6361: 6359: 6358: 6353: 6329: 6327: 6326: 6321: 6319: 6315: 6299: 6295: 6294: 6293: 6284: 6276: 6254: 6250: 6245: 6244: 6235: 6218: 6216: 6215: 6204: 6195: 6190: 6179: 6171: 6170: 6158: 6157: 6135: 6133: 6132: 6127: 6125: 6121: 6105: 6101: 6100: 6099: 6090: 6082: 6060: 6056: 6051: 6050: 6041: 6024: 6022: 6021: 6010: 6001: 5996: 5995: 5976: 5974: 5973: 5968: 5966: 5965: 5953: 5952: 5933: 5931: 5930: 5925: 5922: 5921: 5885: 5884: 5856: 5854: 5853: 5848: 5839: 5838: 5811: 5810: 5786: 5785: 5746: 5745: 5717: 5715: 5714: 5709: 5700: 5699: 5675: 5674: 5653: 5652: 5613: 5612: 5580: 5578: 5577: 5572: 5557: 5555: 5554: 5549: 5540: 5539: 5517: 5515: 5514: 5511:{\displaystyle } 5509: 5455:American options 5443: 5439: 5437: 5436: 5431: 5391: 5389: 5388: 5383: 5377: 5376: 5358: 5357: 5301: 5299: 5298: 5293: 5287: 5286: 5271: 5270: 5217: 5215: 5214: 5209: 5207: 5205: 5197: 5189: 5174: 5172: 5171: 5166: 5160: 5159: 5141: 5140: 5104: 5102: 5101: 5090: 5084: 5077: 5076: 5064: 5052: 5039: 5037: 5036: 5031: 5025: 5024: 5009: 5008: 4972: 4970: 4969: 4958: 4952: 4945: 4944: 4932: 4920: 4907: 4905: 4904: 4899: 4897: 4895: 4887: 4879: 4864: 4862: 4861: 4856: 4853: 4842: 4837: 4836: 4824: 4805: 4803: 4802: 4797: 4795: 4793: 4785: 4777: 4762: 4760: 4759: 4754: 4751: 4749: 4748: 4737: 4728: 4724: 4723: 4711: 4702: 4692: 4690: 4689: 4684: 4682: 4680: 4679: 4678: 4665: 4661: 4660: 4650: 4635: 4633: 4632: 4627: 4615: 4614: 4593: 4592: 4563: 4561: 4560: 4555: 4549: 4548: 4525: 4523: 4522: 4517: 4515: 4513: 4505: 4497: 4474: 4473: 4418:Rational pricing 4335: 4333: 4332: 4327: 4322: 4321: 4296: 4294: 4293: 4288: 4283: 4282: 4260: 4258: 4257: 4252: 4247: 4246: 4221: 4219: 4218: 4213: 4202: 4201: 4189: 4188: 4169: 4167: 4166: 4161: 4159: 4157: 4156: 4151: 4146: 4145: 4135: 4128: 4127: 4115: 4114: 4102: 4101: 4091: 4058: 4056: 4055: 4050: 4030: 4029: 4005: 4003: 4002: 3997: 3992: 3991: 3970: 3969: 3944: 3942: 3941: 3936: 3931: 3930: 3908: 3906: 3905: 3900: 3895: 3894: 3864: 3862: 3861: 3856: 3854: 3853: 3844: 3836: 3827: 3825: 3824: 3819: 3817: 3813: 3805: 3793: 3791: 3790: 3785: 3776: 3754: 3752: 3751: 3746: 3738: 3734: 3733: 3732: 3723: 3715: 3695: 3693: 3692: 3687: 3685: 3684: 3675: 3667: 3658: 3656: 3655: 3650: 3648: 3647: 3619: 3617: 3616: 3611: 3603: 3602: 3580: 3578: 3577: 3572: 3564: 3563: 3533: 3531: 3530: 3525: 3520: 3519: 3493: 3491: 3490: 3485: 3483: 3482: 3466: 3464: 3463: 3458: 3453: 3452: 3430: 3428: 3427: 3422: 3417: 3416: 3390: 3388: 3387: 3382: 3374: 3373: 3351: 3349: 3348: 3343: 3335: 3334: 3308: 3306: 3305: 3300: 3295: 3294: 3272: 3270: 3269: 3264: 3256: 3255: 3226: 3224: 3223: 3218: 3211: 3207: 3206: 3184: 3182: 3181: 3176: 3169: 3165: 3164: 3130: 3128: 3127: 3122: 3114: 3113: 3088: 3086: 3085: 3080: 3072: 3071: 3043: 3041: 3040: 3035: 3024: 3023: 2996: 2995: 2958: 2956: 2955: 2950: 2948: 2944: 2937: 2936: 2912: 2911: 2854: 2852: 2851: 2846: 2844: 2843: 2819: 2817: 2816: 2811: 2809: 2805: 2798: 2797: 2770: 2769: 2709: 2707: 2706: 2701: 2642: 2640: 2639: 2634: 2609: 2607: 2606: 2601: 2599: 2591: 2583: 2582: 2555: 2553: 2552: 2547: 2545: 2544: 2511: 2509: 2508: 2503: 2501: 2497: 2492: 2484: 2483: 2467: 2466: 2453: 2449: 2445: 2444: 2435: 2427: 2422: 2418: 2410: 2393: 2391: 2390: 2385: 2376: 2367: 2366: 2353: 2349: 2342: 2341: 2317: 2316: 2242: 2240: 2239: 2234: 2231: 2227: 2226: 2214: 2213: 2192: 2191: 2158: 2157: 2133: 2120: 2119: 2101: 2100: 2088: 2087: 2041: 2040: 2011: 2009: 2008: 2003: 2001: 2000: 1956: 1954: 1953: 1948: 1946: 1942: 1931: 1923: 1922: 1906: 1905: 1892: 1888: 1872: 1868: 1867: 1862: 1861: 1852: 1836: 1832: 1827: 1826: 1817: 1800: 1798: 1797: 1786: 1777: 1768: 1767: 1754: 1753: 1720: 1719: 1701: 1700: 1688: 1687: 1656: 1655: 1623: 1621: 1620: 1615: 1613: 1563: 1550: 1547: 1514: 1507: 1504: 1477: 1452:. This price is 1436: 1434: 1433: 1428: 1416: 1414: 1413: 1408: 1369: 1367: 1366: 1361: 1344: 1342: 1334: 1326: 1315: 1313: 1312: 1311: 1298: 1294: 1293: 1283: 1281: 1280: 1271: 1270: 1261: 1253: 1248: 1246: 1238: 1230: 1218: 1216: 1215: 1210: 1198: 1196: 1195: 1190: 1178: 1176: 1175: 1170: 1117: 1115: 1114: 1109: 1104: 1103: 1099: 1094: 1093: 1076: 1067: 1063: 1058: 1056: 1048: 1031: 1017: 995: 993: 992: 987: 976: 959: 957: 956: 951: 939: 938: 934: 929: 928: 910: 905: 893: 884: 880: 846: 844: 843: 838: 810: 808: 807: 802: 788: 786: 785: 780: 756: 754: 753: 748: 734: 732: 731: 726: 712: 710: 709: 704: 675: 673: 672: 667: 630: 628: 627: 622: 593:Option related: 576: 574: 573: 568: 554: 552: 551: 546: 530: 528: 527: 522: 508: 506: 505: 500: 498: 497: 477: 475: 474: 469: 424: 422: 421: 416: 402: 400: 399: 394: 376: 374: 373: 368: 228:Robert C. Merton 133:investment banks 118:Robert C. Merton 75:financial market 64: 63: 60: 59: 56: 53: 50: 47: 43: 42: 39: 36: 33: 30: 15439: 15438: 15434: 15433: 15432: 15430: 15429: 15428: 15379: 15378: 15377: 15372: 15354: 15315:Queueing theory 15258: 15200:Skorokhod space 15063: 15054:Kunita–Watanabe 15025: 14991:Sanov's theorem 14961:Ergodic theorem 14934: 14930:Time-reversible 14848: 14811:Queueing models 14805: 14801:Sparre–Anderson 14791:Cramér–Lundberg 14772: 14758:SABR volatility 14664: 14621: 14573:Boolean network 14531: 14517:Renewal process 14448: 14397:Non-homogeneous 14387:Poisson process 14277:Contact process 14240:Brownian motion 14210:Continuous time 14204: 14198:Maximal entropy 14129: 14124: 14094: 14089: 14041: 14027:Fund governance 14015: 13945: 13869:Absolute return 13857: 13808: 13802: 13793:Program trading 13788:Prime brokerage 13761: 13748: 13728: 13724:Trend following 13709:Dedicated short 13683: 13650: 13607: 13595: 13589: 13584: 13554: 13549: 13537: 13529: 13515:Great Recession 13510:Government debt 13488: 13464:Fund derivative 13407: 13364: 13325:Futures pricing 13300:Dividend future 13295:Currency future 13278: 13260: 13133: 13109:Put–call parity 13045: 13032:Vertical spread 12967:Diagonal spread 12937:Calendar spread 12908: 12817: 12754: 12679: 12670: 12665: 12607: 12570: 12547: 12512: 12507: 12496: 12476: 12460: 12458:Further reading 12410: 12370:10.1.1.461.4099 12311: 12290: 12269: 12259: 12217:10.2307/3003143 12167: 12162: 12152: 12150: 12142: 12141: 12137: 12127: 12125: 12124:. 21 April 2020 12116: 12115: 12108: 12098: 12096: 12086: 12082: 12072: 12070: 12052: 12045: 12003: 11999: 11992: 11978: 11977: 11973: 11964: 11962: 11959: 11950: 11946: 11931: 11927: 11917:Wayback Machine 11900: 11896: 11880: 11876: 11871: 11867: 11850: 11846: 11834: 11828:Kalotay, Andrew 11825: 11821: 11814: 11797: 11793: 11775: 11759: 11752: 11733:10.2307/2327242 11717: 11713: 11698: 11694: 11689: 11685: 11680: 11676: 11669: 11651: 11647: 11640: 11626: 11622: 11615: 11611: 11603: 11599: 11589: 11587: 11584: 11578: 11574: 11564: 11562: 11556: 11552: 11537:10.2307/2328254 11517: 11513: 11503: 11501: 11498: 11492: 11488: 11478: 11476: 11470: 11466: 11456: 11454: 11448: 11444: 11434: 11432: 11423: 11422: 11418: 11406: 11402: 11395: 11377: 11373: 11365:10.1.1.363.2491 11356: 11352: 11340: 11337: 11326: 11316:"Understanding 11312: 11299: 11274:10.2307/3003143 11258: 11254: 11223: 11219: 11209: 11207: 11202: 11201: 11197: 11188: 11187: 11183: 11176: 11157: 11153: 11148: 11144: 11139: 11135: 11130: 11126: 11121: 11117: 11112: 11108: 11103: 11099: 11092: 11075: 11068: 11058: 11056: 11052: 11051: 11047: 11043: 11038: 11037: 11032: 11028: 11023: 10942: 10934:Bachelier model 10874: 10866:for a small fee 10858: 10845: 10825:bond securities 10821: 10785: 10779: 10657:volatility risk 10618: 10609: 10588: 10579: 10574: 10573: 10564: 10560: 10552: 10549: 10548: 10524: 10516: 10514: 10512: 10509: 10508: 10483: 10476: 10472: 10468: 10466: 10464: 10461: 10460: 10441: 10437: 10425: 10421: 10391: 10387: 10374: 10361: 10357: 10327: 10323: 10308: 10304: 10288: 10286: 10270: 10263: 10259: 10255: 10253: 10248: 10245: 10244: 10217: 10209: 10207: 10197: 10190: 10186: 10182: 10180: 10167: 10160: 10156: 10152: 10150: 10134: 10103: 10099: 10095: 10093: 10082: 10079: 10078: 10059: 10056: 10055: 10039: 10036: 10035: 10011: 10004: 10000: 9996: 9994: 9983: 9980: 9979: 9937: 9933: 9909: 9905: 9904: 9902: 9890: 9878: 9875: 9874: 9854: 9850: 9848: 9845: 9844: 9830:volatility skew 9804: 9801: 9800: 9784: 9781: 9780: 9772: 9746: 9742: 9722: 9718: 9714: 9710: 9699: 9696: 9695: 9668: 9664: 9647: 9643: 9639: 9635: 9624: 9621: 9620: 9593: 9589: 9569: 9565: 9561: 9557: 9549: 9546: 9545: 9518: 9514: 9497: 9493: 9489: 9485: 9477: 9474: 9473: 9450: 9446: 9444: 9441: 9440: 9423: 9419: 9417: 9414: 9413: 9405: 9399: 9371: 9367: 9334: 9330: 9319: 9316: 9315: 9309: 9280: 9276: 9246: 9242: 9231: 9228: 9227: 9221: 9192: 9188: 9155: 9151: 9143: 9140: 9139: 9133: 9104: 9100: 9070: 9066: 9058: 9055: 9054: 9048: 9032: 9009: 9005: 9004: 8993: 8988: 8984: 8983: 8975: 8971: 8970: 8957: 8953: 8952: 8939: 8935: 8934: 8932: 8928: 8927: 8918: 8914: 8907: 8902: 8885: 8882: 8881: 8855: 8851: 8850: 8840: 8836: 8835: 8833: 8824: 8820: 8812: 8809: 8808: 8782: 8778: 8777: 8767: 8763: 8762: 8760: 8751: 8747: 8724: 8720: 8719: 8712: 8708: 8701: 8699: 8693: 8689: 8677: 8673: 8664: 8660: 8658: 8655: 8654: 8635: 8631: 8630: 8617: 8613: 8612: 8607: 8603: 8602: 8593: 8589: 8563: 8560: 8559: 8537: 8533: 8532: 8528: 8522: 8518: 8514: 8507: 8503: 8496: 8494: 8485: 8481: 8470: 8466: 8449: 8445: 8444: 8440: 8434: 8430: 8421: 8417: 8405: 8401: 8393: 8390: 8389: 8370: 8366: 8365: 8361: 8355: 8351: 8334: 8331: 8330: 8307: 8303: 8301: 8298: 8297: 8283: 8282: 8273: 8269: 8268: 8256: 8252: 8240: 8229: 8225: 8218: 8213: 8200: 8196: 8195: 8193: 8179: 8175: 8168: 8163: 8150: 8146: 8142: 8140: 8133: 8127: 8123: 8120: 8119: 8110: 8106: 8105: 8093: 8089: 8077: 8066: 8062: 8055: 8050: 8037: 8033: 8032: 8030: 8016: 8012: 8005: 8000: 7987: 7983: 7979: 7977: 7970: 7964: 7960: 7956: 7954: 7951: 7950: 7933: 7929: 7927: 7924: 7923: 7890: 7886: 7875: 7871: 7864: 7859: 7846: 7842: 7833: 7828: 7818: 7814: 7807: 7802: 7800: 7797: 7796: 7771: 7767: 7766: 7762: 7745: 7741: 7708: 7704: 7695: 7691: 7685: 7681: 7674: 7669: 7668: 7664: 7662: 7659: 7658: 7636: 7628: 7625: 7624: 7601: 7597: 7595: 7592: 7591: 7572: 7568: 7567: 7563: 7557: 7553: 7542: 7538: 7537: 7533: 7527: 7523: 7506: 7503: 7502: 7451: 7447: 7438: 7434: 7424: 7420: 7402: 7398: 7390: 7387: 7386: 7369: 7365: 7363: 7360: 7359: 7323: 7315: 7313: 7283: 7279: 7275: 7265: 7261: 7260: 7258: 7252: 7248: 7242: 7238: 7231: 7226: 7224: 7221: 7220: 7198: 7195: 7194: 7191: 7183:put–call parity 7156: 7153: 7152: 7129: 7126: 7125: 7059: 7056: 7055: 7039: 7036: 7035: 7010: 7007: 7006: 6960: 6957: 6956: 6917: 6909: 6907: 6889: 6885: 6881: 6871: 6867: 6866: 6864: 6858: 6854: 6848: 6844: 6834: 6821: 6813: 6811: 6809: 6806: 6805: 6794:American option 6790: 6761: 6757: 6742: 6738: 6720: 6716: 6708: 6705: 6704: 6674: 6670: 6649: 6645: 6621: 6617: 6599: 6595: 6587: 6584: 6583: 6561: 6558: 6557: 6532: 6529: 6528: 6506: 6502: 6489: 6485: 6470: 6466: 6448: 6444: 6435: 6431: 6429: 6426: 6425: 6405: 6401: 6386: 6382: 6373: 6369: 6367: 6364: 6363: 6347: 6344: 6343: 6337: 6289: 6285: 6275: 6262: 6258: 6240: 6236: 6234: 6230: 6223: 6219: 6203: 6199: 6194: 6178: 6166: 6162: 6153: 6149: 6147: 6144: 6143: 6095: 6091: 6081: 6068: 6064: 6046: 6042: 6040: 6036: 6029: 6025: 6009: 6005: 6000: 5991: 5987: 5985: 5982: 5981: 5961: 5957: 5948: 5944: 5942: 5939: 5938: 5890: 5886: 5880: 5876: 5868: 5865: 5864: 5834: 5830: 5806: 5802: 5763: 5759: 5741: 5737: 5729: 5726: 5725: 5695: 5691: 5670: 5666: 5630: 5626: 5608: 5604: 5596: 5593: 5592: 5566: 5563: 5562: 5535: 5531: 5526: 5523: 5522: 5482: 5479: 5478: 5472: 5450: 5441: 5425: 5422: 5421: 5399:put–call parity 5372: 5368: 5335: 5331: 5308: 5305: 5304: 5282: 5278: 5248: 5244: 5224: 5221: 5220: 5198: 5190: 5188: 5186: 5183: 5182: 5155: 5151: 5118: 5114: 5089: 5085: 5072: 5068: 5057: 5053: 5051: 5046: 5043: 5042: 5020: 5016: 4986: 4982: 4957: 4953: 4940: 4936: 4925: 4921: 4919: 4914: 4911: 4910: 4888: 4880: 4878: 4876: 4873: 4872: 4841: 4832: 4828: 4817: 4812: 4809: 4808: 4786: 4778: 4776: 4774: 4771: 4770: 4736: 4729: 4719: 4715: 4704: 4703: 4701: 4699: 4696: 4695: 4674: 4670: 4666: 4656: 4652: 4651: 4649: 4647: 4644: 4643: 4610: 4606: 4588: 4584: 4570: 4567: 4566: 4544: 4540: 4532: 4529: 4528: 4506: 4498: 4496: 4494: 4491: 4490: 4469:differentiation 4438: 4398:risk neutrality 4379: 4373: 4317: 4313: 4302: 4299: 4298: 4278: 4274: 4266: 4263: 4262: 4242: 4238: 4230: 4227: 4226: 4197: 4193: 4184: 4180: 4178: 4175: 4174: 4150: 4141: 4137: 4136: 4123: 4119: 4110: 4106: 4097: 4093: 4092: 4090: 4067: 4064: 4063: 4025: 4021: 4019: 4016: 4015: 3987: 3983: 3965: 3961: 3953: 3950: 3949: 3926: 3922: 3914: 3911: 3910: 3890: 3886: 3878: 3875: 3874: 3849: 3845: 3835: 3833: 3830: 3829: 3804: 3800: 3784: 3780: 3775: 3767: 3764: 3763: 3761: 3728: 3724: 3714: 3707: 3703: 3701: 3698: 3697: 3680: 3676: 3666: 3664: 3661: 3660: 3659:instead of the 3643: 3639: 3637: 3634: 3633: 3598: 3594: 3586: 3583: 3582: 3559: 3555: 3547: 3544: 3543: 3515: 3511: 3503: 3500: 3499: 3478: 3474: 3472: 3469: 3468: 3448: 3444: 3436: 3433: 3432: 3412: 3408: 3400: 3397: 3396: 3369: 3365: 3357: 3354: 3353: 3330: 3326: 3318: 3315: 3314: 3290: 3286: 3278: 3275: 3274: 3251: 3247: 3239: 3236: 3235: 3202: 3198: 3190: 3187: 3186: 3160: 3156: 3148: 3145: 3144: 3109: 3105: 3094: 3091: 3090: 3067: 3063: 3052: 3049: 3048: 3019: 3015: 2991: 2987: 2970: 2967: 2966: 2932: 2928: 2907: 2903: 2896: 2892: 2881: 2878: 2877: 2839: 2835: 2833: 2830: 2829: 2826: 2793: 2789: 2765: 2761: 2751: 2747: 2721: 2718: 2717: 2653: 2650: 2649: 2619: 2616: 2615: 2590: 2575: 2571: 2563: 2560: 2559: 2534: 2530: 2522: 2519: 2518: 2499: 2498: 2491: 2479: 2475: 2468: 2462: 2458: 2455: 2454: 2440: 2436: 2426: 2409: 2405: 2398: 2394: 2384: 2380: 2375: 2368: 2362: 2358: 2355: 2354: 2337: 2333: 2312: 2308: 2301: 2297: 2287: 2265: 2263: 2260: 2259: 2249: 2229: 2228: 2222: 2218: 2209: 2205: 2169: 2165: 2153: 2149: 2131: 2130: 2115: 2111: 2096: 2092: 2065: 2061: 2051: 2036: 2032: 2022: 2020: 2017: 2016: 1978: 1974: 1972: 1969: 1968: 1966:discount factor 1962:put–call parity 1944: 1943: 1930: 1918: 1914: 1907: 1901: 1897: 1894: 1893: 1857: 1853: 1851: 1844: 1840: 1822: 1818: 1816: 1812: 1805: 1801: 1785: 1781: 1776: 1769: 1763: 1759: 1756: 1755: 1731: 1727: 1715: 1711: 1696: 1692: 1683: 1679: 1666: 1651: 1647: 1637: 1635: 1632: 1631: 1611: 1610: 1561: 1560: 1546: 1512: 1511: 1503: 1474: 1472: 1469: 1468: 1422: 1419: 1418: 1402: 1399: 1398: 1391: 1335: 1327: 1325: 1307: 1303: 1299: 1289: 1285: 1284: 1282: 1276: 1272: 1266: 1262: 1252: 1239: 1231: 1229: 1227: 1224: 1223: 1204: 1201: 1200: 1184: 1181: 1180: 1149: 1146: 1145: 1130: 1124: 1095: 1089: 1085: 1081: 1077: 1062: 1049: 1032: 1030: 1010: 1008: 1005: 1004: 969: 967: 964: 963: 930: 924: 920: 916: 912: 906: 898: 879: 862: 859: 858: 849:standard normal 823: 820: 819: 796: 793: 792: 762: 759: 758: 742: 739: 738: 720: 717: 716: 683: 680: 679: 646: 643: 642: 601: 598: 597: 562: 559: 558: 540: 537: 536: 516: 513: 512: 493: 489: 487: 484: 483: 454: 451: 450: 446:Asset related: 410: 407: 406: 382: 379: 378: 362: 359: 358: 349: 334:hedged position 259: 251:Swedish Academy 232:options pricing 217:risk management 209:Edward O. Thorp 201:Louis Bachelier 189:expected return 177: 169:OTC derivatives 44: 27: 23: 17: 12: 11: 5: 15437: 15427: 15426: 15421: 15416: 15411: 15406: 15401: 15396: 15391: 15374: 15373: 15371: 15370: 15365: 15363:List of topics 15359: 15356: 15355: 15353: 15352: 15347: 15342: 15337: 15332: 15327: 15322: 15320:Renewal theory 15317: 15312: 15307: 15302: 15297: 15292: 15287: 15285:Ergodic theory 15282: 15277: 15275:Control theory 15272: 15266: 15264: 15260: 15259: 15257: 15256: 15255: 15254: 15249: 15239: 15234: 15229: 15224: 15219: 15218: 15217: 15207: 15205:Snell envelope 15202: 15197: 15192: 15187: 15182: 15177: 15172: 15167: 15162: 15157: 15152: 15147: 15142: 15137: 15132: 15127: 15122: 15117: 15112: 15107: 15102: 15097: 15092: 15087: 15082: 15077: 15071: 15069: 15065: 15064: 15062: 15061: 15056: 15051: 15046: 15041: 15035: 15033: 15027: 15026: 15024: 15023: 15004:Borel–Cantelli 14993: 14988: 14983: 14978: 14973: 14968: 14963: 14958: 14953: 14948: 14942: 14940: 14939:Limit theorems 14936: 14935: 14933: 14932: 14927: 14922: 14917: 14912: 14907: 14902: 14897: 14892: 14887: 14882: 14877: 14872: 14867: 14862: 14856: 14854: 14850: 14849: 14847: 14846: 14841: 14836: 14831: 14826: 14821: 14815: 14813: 14807: 14806: 14804: 14803: 14798: 14793: 14788: 14782: 14780: 14774: 14773: 14771: 14770: 14765: 14760: 14755: 14750: 14745: 14740: 14735: 14730: 14725: 14720: 14715: 14710: 14705: 14700: 14695: 14690: 14685: 14680: 14674: 14672: 14666: 14665: 14663: 14662: 14657: 14652: 14647: 14642: 14637: 14631: 14629: 14623: 14622: 14620: 14619: 14614: 14609: 14608: 14607: 14602: 14592: 14587: 14582: 14577: 14576: 14575: 14570: 14560: 14558:Hopfield model 14555: 14550: 14545: 14539: 14537: 14533: 14532: 14530: 14529: 14524: 14519: 14514: 14509: 14504: 14503: 14502: 14497: 14492: 14487: 14477: 14475:Markov process 14472: 14467: 14462: 14456: 14454: 14450: 14449: 14447: 14446: 14444:Wiener sausage 14441: 14439:Wiener process 14436: 14431: 14426: 14421: 14419:Stable process 14416: 14411: 14409:Semimartingale 14406: 14401: 14400: 14399: 14394: 14384: 14379: 14374: 14369: 14364: 14359: 14354: 14352:Jump diffusion 14349: 14344: 14339: 14334: 14329: 14327:Hawkes process 14324: 14319: 14314: 14309: 14307:Feller process 14304: 14299: 14294: 14289: 14284: 14279: 14274: 14272:Cauchy process 14269: 14268: 14267: 14262: 14257: 14252: 14247: 14237: 14236: 14235: 14225: 14223:Bessel process 14220: 14214: 14212: 14206: 14205: 14203: 14202: 14201: 14200: 14195: 14190: 14185: 14175: 14170: 14165: 14160: 14155: 14150: 14145: 14139: 14137: 14131: 14130: 14123: 14122: 14115: 14108: 14100: 14091: 14090: 14088: 14087: 14077: 14067: 14057: 14046: 14043: 14042: 14040: 14039: 14034: 14029: 14023: 14021: 14017: 14016: 14014: 14013: 14008: 14003: 14001:Merchant banks 13998: 13993: 13988: 13983: 13978: 13973: 13968: 13966:Family offices 13963: 13957: 13955: 13951: 13950: 13947: 13946: 13944: 13943: 13938: 13933: 13928: 13926:Securitization 13923: 13918: 13913: 13895: 13881: 13876: 13871: 13865: 13863: 13859: 13858: 13856: 13855: 13850: 13845: 13840: 13835: 13830: 13825: 13819: 13817: 13810: 13804: 13803: 13801: 13800: 13795: 13790: 13785: 13780: 13775: 13769: 13767: 13763: 13762: 13755: 13753: 13750: 13749: 13747: 13746: 13736: 13734: 13730: 13729: 13727: 13726: 13721: 13716: 13711: 13706: 13697: 13691: 13689: 13685: 13684: 13682: 13681: 13676: 13674:Risk arbitrage 13671: 13666: 13660: 13658: 13652: 13651: 13649: 13648: 13643: 13638: 13629: 13624: 13619: 13613: 13611: 13609:relative value 13597: 13591: 13590: 13583: 13582: 13575: 13568: 13560: 13551: 13550: 13548: 13547: 13534: 13531: 13530: 13528: 13527: 13522: 13520:Municipal debt 13517: 13512: 13507: 13505:Corporate debt 13502: 13496: 13494: 13490: 13489: 13487: 13486: 13481: 13476: 13471: 13466: 13461: 13456: 13451: 13446: 13441: 13436: 13431: 13426: 13421: 13415: 13413: 13409: 13408: 13406: 13405: 13400: 13395: 13390: 13385: 13380: 13374: 13372: 13366: 13365: 13363: 13362: 13357: 13352: 13347: 13342: 13337: 13332: 13327: 13322: 13317: 13312: 13307: 13305:Forward market 13302: 13297: 13292: 13287: 13281: 13279: 13277: 13276: 13271: 13265: 13262: 13261: 13259: 13258: 13253: 13248: 13243: 13238: 13233: 13228: 13223: 13218: 13213: 13208: 13203: 13198: 13193: 13188: 13186:Credit default 13183: 13178: 13173: 13168: 13163: 13158: 13153: 13147: 13145: 13139: 13138: 13135: 13134: 13132: 13131: 13126: 13121: 13116: 13111: 13106: 13101: 13096: 13091: 13086: 13081: 13071: 13066: 13061: 13055: 13053: 13047: 13046: 13044: 13043: 13029: 13024: 13019: 13014: 13009: 13004: 12999: 12994: 12989: 12984: 12982:Iron butterfly 12979: 12974: 12969: 12964: 12959: 12954: 12952:Covered option 12949: 12944: 12939: 12934: 12929: 12924: 12918: 12916: 12910: 12909: 12907: 12906: 12901: 12896: 12891: 12890:Mountain range 12888: 12883: 12878: 12873: 12868: 12863: 12858: 12853: 12848: 12843: 12838: 12833: 12827: 12825: 12819: 12818: 12816: 12815: 12810: 12805: 12800: 12795: 12790: 12785: 12780: 12775: 12770: 12764: 12762: 12756: 12755: 12753: 12752: 12747: 12742: 12737: 12732: 12727: 12722: 12717: 12712: 12707: 12702: 12696: 12694: 12687: 12681: 12680: 12675: 12672: 12671: 12664: 12663: 12656: 12649: 12641: 12635: 12634: 12628: 12614: 12606: 12603: 12602: 12601: 12596: 12591: 12586: 12581: 12576: 12569: 12566: 12565: 12564: 12554: 12546: 12543: 12542: 12541: 12539:Emanuel Derman 12532: 12518: 12511: 12508: 12506: 12505:External links 12503: 12502: 12501: 12494: 12481: 12474: 12459: 12456: 12455: 12454: 12444: 12434: 12424: 12414: 12408: 12393: 12379:10.1086/374404 12363:(1): 107–145. 12352: 12322:(6): 831–868. 12304: 12294: 12288: 12268: 12265: 12264: 12263: 12257: 12241: 12200: 12185:10.1086/260062 12179:(3): 637–654. 12166: 12163: 12161: 12160: 12135: 12106: 12080: 12043: 12006:Nahin, Paul J. 11997: 11990: 11971: 11955:(2009-02-27). 11944: 11938:, WP2006-019, 11925: 11902:Emanuel Derman 11894: 11874: 11865: 11844: 11819: 11812: 11791: 11788:on 2008-11-20. 11780:(2008-07-23). 11772:on 2008-07-24. 11764:(2008-04-29). 11750: 11711: 11692: 11683: 11674: 11667: 11645: 11638: 11620: 11609: 11597: 11572: 11550: 11511: 11486: 11464: 11442: 11416: 11400: 11393: 11371: 11350: 11335: 11324: 11297: 11268:(1): 141–183. 11252: 11239:10.1086/260062 11233:(3): 637–654. 11217: 11195: 11181: 11174: 11151: 11142: 11133: 11124: 11115: 11106: 11097: 11090: 11066: 11044: 11042: 11039: 11036: 11035: 11025: 11024: 11022: 11019: 11018: 11017: 11012: 11007: 10997: 10995:Jump diffusion 10992: 10986: 10981: 10975: 10970: 10965: 10959: 10953: 10941: 10938: 10930:negative price 10905:Warren Buffett 10890:Emanuel Derman 10873: 10870: 10857: 10854: 10844: 10841: 10820: 10817: 10781:Main article: 10778: 10775: 10739:stress testing 10712: 10711: 10703: 10700: 10697: 10667: 10666: 10663: 10660: 10653: 10650:liquidity risk 10646: 10617: 10614: 10608: 10605: 10604: 10603: 10587: 10582: 10572: 10563: 10559: 10556: 10530: 10527: 10522: 10519: 10489: 10486: 10479: 10475: 10471: 10458: 10457: 10440: 10436: 10433: 10428: 10424: 10420: 10417: 10412: 10409: 10406: 10403: 10400: 10397: 10394: 10390: 10386: 10380: 10377: 10372: 10369: 10364: 10360: 10356: 10353: 10348: 10345: 10342: 10339: 10336: 10333: 10330: 10326: 10322: 10319: 10316: 10311: 10307: 10303: 10300: 10297: 10294: 10291: 10285: 10282: 10276: 10273: 10266: 10262: 10258: 10252: 10238: 10237: 10223: 10220: 10215: 10212: 10203: 10200: 10193: 10189: 10185: 10179: 10173: 10170: 10163: 10159: 10155: 10149: 10146: 10140: 10137: 10132: 10129: 10126: 10123: 10120: 10117: 10114: 10111: 10106: 10102: 10098: 10092: 10089: 10086: 10063: 10043: 10032: 10031: 10017: 10014: 10007: 10003: 9999: 9993: 9990: 9987: 9969: 9968: 9955: 9951: 9948: 9945: 9940: 9936: 9932: 9929: 9926: 9923: 9920: 9917: 9912: 9908: 9899: 9896: 9893: 9889: 9885: 9882: 9857: 9853: 9817: 9814: 9811: 9808: 9788: 9771: 9768: 9767: 9766: 9754: 9749: 9745: 9741: 9738: 9735: 9730: 9725: 9721: 9717: 9713: 9709: 9706: 9703: 9689: 9688: 9676: 9671: 9667: 9663: 9660: 9655: 9650: 9646: 9642: 9638: 9634: 9631: 9628: 9614: 9613: 9601: 9596: 9592: 9588: 9585: 9582: 9577: 9572: 9568: 9564: 9560: 9556: 9553: 9539: 9538: 9526: 9521: 9517: 9513: 9510: 9505: 9500: 9496: 9492: 9488: 9484: 9481: 9453: 9449: 9426: 9422: 9398: 9395: 9394: 9393: 9382: 9379: 9374: 9370: 9366: 9363: 9360: 9355: 9352: 9349: 9346: 9343: 9340: 9337: 9333: 9329: 9326: 9323: 9308: 9305: 9304: 9303: 9291: 9288: 9283: 9279: 9275: 9272: 9267: 9264: 9261: 9258: 9255: 9252: 9249: 9245: 9241: 9238: 9235: 9220: 9217: 9216: 9215: 9203: 9200: 9195: 9191: 9187: 9184: 9181: 9176: 9173: 9170: 9167: 9164: 9161: 9158: 9154: 9150: 9147: 9132: 9129: 9128: 9127: 9115: 9112: 9107: 9103: 9099: 9096: 9091: 9088: 9085: 9082: 9079: 9076: 9073: 9069: 9065: 9062: 9047: 9044: 9031: 9030:Binary options 9028: 9012: 9008: 9002: 8996: 8992: 8987: 8978: 8974: 8968: 8960: 8956: 8950: 8947: 8942: 8938: 8931: 8921: 8917: 8913: 8910: 8906: 8901: 8898: 8895: 8892: 8889: 8866: 8863: 8858: 8854: 8848: 8843: 8839: 8832: 8827: 8823: 8819: 8816: 8793: 8790: 8785: 8781: 8775: 8770: 8766: 8759: 8754: 8750: 8745: 8741: 8738: 8735: 8727: 8723: 8715: 8711: 8707: 8704: 8696: 8692: 8688: 8685: 8680: 8676: 8672: 8667: 8663: 8638: 8634: 8628: 8620: 8616: 8611: 8606: 8601: 8596: 8592: 8588: 8585: 8582: 8579: 8576: 8573: 8570: 8567: 8540: 8536: 8531: 8525: 8521: 8517: 8510: 8506: 8502: 8499: 8493: 8488: 8484: 8479: 8473: 8469: 8465: 8462: 8459: 8452: 8448: 8443: 8437: 8433: 8429: 8424: 8420: 8416: 8413: 8408: 8404: 8400: 8397: 8373: 8369: 8364: 8358: 8354: 8350: 8347: 8344: 8341: 8338: 8318: 8315: 8310: 8306: 8276: 8272: 8264: 8259: 8255: 8251: 8248: 8243: 8238: 8232: 8228: 8221: 8217: 8212: 8209: 8206: 8203: 8199: 8192: 8188: 8182: 8178: 8171: 8167: 8162: 8159: 8156: 8153: 8149: 8145: 8139: 8136: 8134: 8130: 8126: 8122: 8121: 8113: 8109: 8101: 8096: 8092: 8088: 8085: 8080: 8075: 8069: 8065: 8058: 8054: 8049: 8046: 8043: 8040: 8036: 8029: 8025: 8019: 8015: 8008: 8004: 7999: 7996: 7993: 7990: 7986: 7982: 7976: 7973: 7971: 7967: 7963: 7959: 7958: 7936: 7932: 7907: 7904: 7901: 7898: 7893: 7889: 7884: 7878: 7874: 7867: 7863: 7858: 7855: 7852: 7849: 7845: 7841: 7836: 7831: 7827: 7821: 7817: 7810: 7806: 7784: 7781: 7774: 7770: 7765: 7760: 7756: 7753: 7748: 7744: 7740: 7737: 7734: 7731: 7728: 7725: 7722: 7719: 7716: 7711: 7707: 7703: 7698: 7694: 7688: 7684: 7677: 7673: 7667: 7645: 7642: 7639: 7635: 7632: 7612: 7609: 7604: 7600: 7575: 7571: 7566: 7560: 7556: 7552: 7545: 7541: 7536: 7530: 7526: 7522: 7519: 7516: 7513: 7510: 7490: 7487: 7484: 7481: 7478: 7475: 7471: 7468: 7465: 7462: 7459: 7454: 7450: 7446: 7441: 7437: 7432: 7427: 7423: 7419: 7416: 7413: 7410: 7405: 7401: 7397: 7394: 7372: 7368: 7347: 7344: 7341: 7338: 7335: 7329: 7326: 7321: 7318: 7312: 7309: 7306: 7303: 7300: 7297: 7294: 7286: 7282: 7278: 7273: 7268: 7264: 7255: 7251: 7245: 7241: 7234: 7230: 7208: 7205: 7202: 7190: 7187: 7166: 7163: 7160: 7136: 7133: 7093: 7090: 7087: 7084: 7081: 7078: 7075: 7072: 7069: 7066: 7063: 7043: 7023: 7020: 7017: 7014: 6994: 6991: 6988: 6985: 6982: 6979: 6976: 6973: 6970: 6967: 6964: 6955:together with 6953: 6952: 6941: 6938: 6935: 6932: 6929: 6923: 6920: 6915: 6912: 6906: 6903: 6900: 6892: 6888: 6884: 6879: 6874: 6870: 6861: 6857: 6851: 6847: 6841: 6838: 6833: 6827: 6824: 6819: 6816: 6789: 6786: 6782: 6781: 6767: 6764: 6760: 6754: 6751: 6748: 6745: 6741: 6737: 6734: 6731: 6728: 6723: 6719: 6715: 6712: 6698: 6697: 6685: 6682: 6677: 6673: 6669: 6666: 6663: 6660: 6657: 6652: 6648: 6644: 6641: 6638: 6635: 6630: 6627: 6624: 6620: 6616: 6613: 6610: 6607: 6602: 6598: 6594: 6591: 6565: 6545: 6542: 6539: 6536: 6525: 6524: 6509: 6505: 6501: 6498: 6495: 6492: 6488: 6482: 6479: 6476: 6473: 6469: 6465: 6462: 6459: 6456: 6451: 6447: 6443: 6438: 6434: 6408: 6404: 6400: 6397: 6394: 6389: 6385: 6381: 6376: 6372: 6351: 6336: 6333: 6332: 6331: 6318: 6314: 6311: 6308: 6305: 6302: 6298: 6292: 6288: 6282: 6279: 6274: 6271: 6268: 6265: 6261: 6257: 6253: 6248: 6243: 6239: 6233: 6229: 6226: 6222: 6213: 6210: 6207: 6202: 6198: 6193: 6188: 6185: 6182: 6177: 6174: 6169: 6165: 6161: 6156: 6152: 6137: 6136: 6124: 6120: 6117: 6114: 6111: 6108: 6104: 6098: 6094: 6088: 6085: 6080: 6077: 6074: 6071: 6067: 6063: 6059: 6054: 6049: 6045: 6039: 6035: 6032: 6028: 6019: 6016: 6013: 6008: 6004: 5999: 5994: 5990: 5964: 5960: 5956: 5951: 5947: 5935: 5934: 5920: 5917: 5914: 5911: 5908: 5905: 5902: 5899: 5896: 5893: 5889: 5883: 5879: 5875: 5872: 5858: 5857: 5845: 5842: 5837: 5833: 5829: 5826: 5823: 5820: 5817: 5814: 5809: 5805: 5801: 5798: 5795: 5792: 5789: 5784: 5781: 5778: 5775: 5772: 5769: 5766: 5762: 5758: 5755: 5752: 5749: 5744: 5740: 5736: 5733: 5719: 5718: 5706: 5703: 5698: 5694: 5690: 5687: 5684: 5681: 5678: 5673: 5669: 5665: 5662: 5659: 5656: 5651: 5648: 5645: 5642: 5639: 5636: 5633: 5629: 5625: 5622: 5619: 5616: 5611: 5607: 5603: 5600: 5583:dividend yield 5570: 5559: 5558: 5547: 5544: 5538: 5534: 5530: 5507: 5504: 5501: 5498: 5495: 5492: 5489: 5486: 5471: 5468: 5449: 5446: 5429: 5393: 5392: 5380: 5375: 5371: 5367: 5364: 5361: 5356: 5353: 5350: 5347: 5344: 5341: 5338: 5334: 5330: 5327: 5324: 5321: 5318: 5315: 5312: 5302: 5290: 5285: 5281: 5277: 5274: 5269: 5266: 5263: 5260: 5257: 5254: 5251: 5247: 5243: 5240: 5237: 5234: 5231: 5228: 5218: 5204: 5201: 5196: 5193: 5180: 5176: 5175: 5163: 5158: 5154: 5150: 5147: 5144: 5139: 5136: 5133: 5130: 5127: 5124: 5121: 5117: 5113: 5110: 5107: 5099: 5096: 5093: 5088: 5083: 5080: 5075: 5071: 5067: 5063: 5060: 5056: 5050: 5040: 5028: 5023: 5019: 5015: 5012: 5007: 5004: 5001: 4998: 4995: 4992: 4989: 4985: 4981: 4978: 4975: 4967: 4964: 4961: 4956: 4951: 4948: 4943: 4939: 4935: 4931: 4928: 4924: 4918: 4908: 4894: 4891: 4886: 4883: 4870: 4866: 4865: 4851: 4848: 4845: 4840: 4835: 4831: 4827: 4823: 4820: 4816: 4806: 4792: 4789: 4784: 4781: 4768: 4764: 4763: 4746: 4743: 4740: 4735: 4732: 4727: 4722: 4718: 4714: 4710: 4707: 4693: 4677: 4673: 4669: 4664: 4659: 4655: 4641: 4637: 4636: 4624: 4621: 4618: 4613: 4609: 4605: 4602: 4599: 4596: 4591: 4587: 4583: 4580: 4577: 4574: 4564: 4552: 4547: 4543: 4539: 4536: 4526: 4512: 4509: 4504: 4501: 4488: 4484: 4483: 4480: 4477: 4437: 4434: 4372: 4369: 4325: 4320: 4316: 4312: 4309: 4306: 4286: 4281: 4277: 4273: 4270: 4250: 4245: 4241: 4237: 4234: 4225:Specifically, 4211: 4208: 4205: 4200: 4196: 4192: 4187: 4183: 4171: 4170: 4154: 4149: 4144: 4140: 4134: 4131: 4126: 4122: 4118: 4113: 4109: 4105: 4100: 4096: 4089: 4086: 4083: 4080: 4077: 4074: 4071: 4048: 4045: 4042: 4039: 4036: 4033: 4028: 4024: 3995: 3990: 3986: 3982: 3979: 3976: 3973: 3968: 3964: 3960: 3957: 3934: 3929: 3925: 3921: 3918: 3898: 3893: 3889: 3885: 3882: 3852: 3848: 3842: 3839: 3816: 3811: 3808: 3803: 3799: 3796: 3788: 3783: 3779: 3774: 3771: 3759: 3744: 3741: 3737: 3731: 3727: 3721: 3718: 3713: 3710: 3706: 3696:term there is 3683: 3679: 3673: 3670: 3646: 3642: 3609: 3606: 3601: 3597: 3593: 3590: 3570: 3567: 3562: 3558: 3554: 3551: 3523: 3518: 3514: 3510: 3507: 3481: 3477: 3456: 3451: 3447: 3443: 3440: 3420: 3415: 3411: 3407: 3404: 3380: 3377: 3372: 3368: 3364: 3361: 3341: 3338: 3333: 3329: 3325: 3322: 3298: 3293: 3289: 3285: 3282: 3262: 3259: 3254: 3250: 3246: 3243: 3229:expected value 3216: 3210: 3205: 3201: 3197: 3194: 3174: 3168: 3163: 3159: 3155: 3152: 3120: 3117: 3112: 3108: 3104: 3101: 3098: 3078: 3075: 3070: 3066: 3062: 3059: 3056: 3045: 3044: 3033: 3030: 3027: 3022: 3018: 3014: 3011: 3008: 3005: 3002: 2999: 2994: 2990: 2986: 2983: 2980: 2977: 2974: 2962:breaks up as: 2960: 2959: 2947: 2943: 2940: 2935: 2931: 2927: 2924: 2921: 2918: 2915: 2910: 2906: 2902: 2899: 2895: 2891: 2888: 2885: 2860:binary options 2842: 2838: 2825: 2824:Interpretation 2822: 2821: 2820: 2808: 2804: 2801: 2796: 2792: 2788: 2785: 2782: 2779: 2776: 2773: 2768: 2764: 2760: 2757: 2754: 2750: 2746: 2743: 2740: 2737: 2734: 2731: 2728: 2725: 2711: 2710: 2699: 2696: 2693: 2690: 2687: 2684: 2681: 2678: 2675: 2672: 2669: 2666: 2663: 2660: 2657: 2632: 2629: 2626: 2623: 2597: 2594: 2589: 2586: 2581: 2578: 2574: 2570: 2567: 2543: 2540: 2537: 2533: 2529: 2526: 2513: 2512: 2495: 2490: 2487: 2482: 2478: 2474: 2471: 2469: 2465: 2461: 2457: 2456: 2452: 2448: 2443: 2439: 2433: 2430: 2425: 2421: 2416: 2413: 2408: 2404: 2401: 2397: 2388: 2383: 2379: 2374: 2371: 2369: 2365: 2361: 2357: 2356: 2352: 2348: 2345: 2340: 2336: 2332: 2329: 2326: 2323: 2320: 2315: 2311: 2307: 2304: 2300: 2296: 2293: 2290: 2288: 2286: 2283: 2280: 2277: 2274: 2271: 2268: 2267: 2248: 2245: 2244: 2243: 2225: 2221: 2217: 2212: 2208: 2204: 2201: 2198: 2195: 2190: 2187: 2184: 2181: 2178: 2175: 2172: 2168: 2164: 2161: 2156: 2152: 2148: 2145: 2142: 2139: 2136: 2134: 2132: 2129: 2126: 2123: 2118: 2114: 2110: 2107: 2104: 2099: 2095: 2091: 2086: 2083: 2080: 2077: 2074: 2071: 2068: 2064: 2060: 2057: 2054: 2052: 2050: 2047: 2044: 2039: 2035: 2031: 2028: 2025: 2024: 1999: 1996: 1993: 1990: 1987: 1984: 1981: 1977: 1958: 1957: 1940: 1937: 1934: 1929: 1926: 1921: 1917: 1913: 1910: 1908: 1904: 1900: 1896: 1895: 1891: 1887: 1884: 1881: 1878: 1875: 1871: 1865: 1860: 1856: 1850: 1847: 1843: 1839: 1835: 1830: 1825: 1821: 1815: 1811: 1808: 1804: 1795: 1792: 1789: 1784: 1780: 1775: 1772: 1770: 1766: 1762: 1758: 1757: 1752: 1749: 1746: 1743: 1740: 1737: 1734: 1730: 1726: 1723: 1718: 1714: 1710: 1707: 1704: 1699: 1695: 1691: 1686: 1682: 1678: 1675: 1672: 1669: 1667: 1665: 1662: 1659: 1654: 1650: 1646: 1643: 1640: 1639: 1625: 1624: 1609: 1606: 1603: 1600: 1597: 1594: 1591: 1588: 1585: 1582: 1579: 1576: 1573: 1570: 1567: 1564: 1562: 1559: 1556: 1553: 1548: as 1545: 1542: 1539: 1536: 1533: 1530: 1527: 1524: 1521: 1518: 1515: 1513: 1510: 1502: 1499: 1496: 1493: 1490: 1487: 1484: 1481: 1478: 1476: 1426: 1406: 1390: 1387: 1371: 1370: 1359: 1356: 1353: 1350: 1347: 1341: 1338: 1333: 1330: 1324: 1321: 1318: 1310: 1306: 1302: 1297: 1292: 1288: 1279: 1275: 1269: 1265: 1259: 1256: 1251: 1245: 1242: 1237: 1234: 1208: 1188: 1168: 1165: 1162: 1159: 1156: 1153: 1126:Main article: 1123: 1120: 1119: 1118: 1107: 1102: 1098: 1092: 1088: 1084: 1080: 1073: 1070: 1066: 1061: 1055: 1052: 1047: 1044: 1041: 1038: 1035: 1029: 1026: 1023: 1020: 1016: 1013: 985: 982: 979: 975: 972: 961: 960: 949: 946: 943: 937: 933: 927: 923: 919: 915: 909: 904: 901: 897: 890: 887: 883: 878: 875: 872: 869: 866: 836: 833: 830: 827: 817: 816: 800: 790: 778: 775: 772: 769: 766: 746: 736: 724: 714: 702: 699: 696: 693: 690: 687: 677: 665: 662: 659: 656: 653: 650: 640: 639:in particular: 620: 617: 614: 611: 608: 605: 591: 590: 566: 556: 544: 520: 510: 496: 492: 467: 464: 461: 458: 444: 443: 414: 404: 392: 389: 386: 366: 348: 345: 329: 328: 321: 314: 311: 300: 299: 292: 281: 258: 255: 197:Paul Samuelson 176: 173: 95:European-style 15: 9: 6: 4: 3: 2: 15436: 15425: 15422: 15420: 15417: 15415: 15412: 15410: 15407: 15405: 15402: 15400: 15397: 15395: 15392: 15390: 15387: 15386: 15384: 15369: 15366: 15364: 15361: 15360: 15357: 15351: 15348: 15346: 15343: 15341: 15338: 15336: 15333: 15331: 15328: 15326: 15323: 15321: 15318: 15316: 15313: 15311: 15308: 15306: 15303: 15301: 15298: 15296: 15293: 15291: 15288: 15286: 15283: 15281: 15278: 15276: 15273: 15271: 15268: 15267: 15265: 15261: 15253: 15250: 15248: 15245: 15244: 15243: 15240: 15238: 15235: 15233: 15230: 15228: 15225: 15223: 15222:Stopping time 15220: 15216: 15213: 15212: 15211: 15208: 15206: 15203: 15201: 15198: 15196: 15193: 15191: 15188: 15186: 15183: 15181: 15178: 15176: 15173: 15171: 15168: 15166: 15163: 15161: 15158: 15156: 15153: 15151: 15148: 15146: 15143: 15141: 15138: 15136: 15133: 15131: 15128: 15126: 15123: 15121: 15118: 15116: 15113: 15111: 15108: 15106: 15103: 15101: 15098: 15096: 15093: 15091: 15088: 15086: 15083: 15081: 15078: 15076: 15073: 15072: 15070: 15066: 15060: 15057: 15055: 15052: 15050: 15047: 15045: 15042: 15040: 15037: 15036: 15034: 15032: 15028: 15021: 15017: 15013: 15012:Hewitt–Savage 15009: 15005: 15001: 14997: 14996:Zero–one laws 14994: 14992: 14989: 14987: 14984: 14982: 14979: 14977: 14974: 14972: 14969: 14967: 14964: 14962: 14959: 14957: 14954: 14952: 14949: 14947: 14944: 14943: 14941: 14937: 14931: 14928: 14926: 14923: 14921: 14918: 14916: 14913: 14911: 14908: 14906: 14903: 14901: 14898: 14896: 14893: 14891: 14888: 14886: 14883: 14881: 14878: 14876: 14873: 14871: 14868: 14866: 14863: 14861: 14858: 14857: 14855: 14851: 14845: 14842: 14840: 14837: 14835: 14832: 14830: 14827: 14825: 14822: 14820: 14817: 14816: 14814: 14812: 14808: 14802: 14799: 14797: 14794: 14792: 14789: 14787: 14784: 14783: 14781: 14779: 14775: 14769: 14766: 14764: 14761: 14759: 14756: 14754: 14751: 14749: 14746: 14744: 14741: 14739: 14736: 14734: 14731: 14729: 14726: 14724: 14721: 14719: 14716: 14714: 14711: 14709: 14706: 14704: 14701: 14699: 14696: 14694: 14693:Black–Scholes 14691: 14689: 14686: 14684: 14681: 14679: 14676: 14675: 14673: 14671: 14667: 14661: 14658: 14656: 14653: 14651: 14648: 14646: 14643: 14641: 14638: 14636: 14633: 14632: 14630: 14628: 14624: 14618: 14615: 14613: 14610: 14606: 14603: 14601: 14598: 14597: 14596: 14595:Point process 14593: 14591: 14588: 14586: 14583: 14581: 14578: 14574: 14571: 14569: 14566: 14565: 14564: 14561: 14559: 14556: 14554: 14553:Gibbs measure 14551: 14549: 14546: 14544: 14541: 14540: 14538: 14534: 14528: 14525: 14523: 14520: 14518: 14515: 14513: 14510: 14508: 14505: 14501: 14498: 14496: 14493: 14491: 14488: 14486: 14483: 14482: 14481: 14478: 14476: 14473: 14471: 14468: 14466: 14463: 14461: 14458: 14457: 14455: 14451: 14445: 14442: 14440: 14437: 14435: 14432: 14430: 14427: 14425: 14422: 14420: 14417: 14415: 14412: 14410: 14407: 14405: 14402: 14398: 14395: 14393: 14390: 14389: 14388: 14385: 14383: 14380: 14378: 14375: 14373: 14370: 14368: 14365: 14363: 14360: 14358: 14355: 14353: 14350: 14348: 14345: 14343: 14342:Itô diffusion 14340: 14338: 14335: 14333: 14330: 14328: 14325: 14323: 14320: 14318: 14317:Gamma process 14315: 14313: 14310: 14308: 14305: 14303: 14300: 14298: 14295: 14293: 14290: 14288: 14285: 14283: 14280: 14278: 14275: 14273: 14270: 14266: 14263: 14261: 14258: 14256: 14253: 14251: 14248: 14246: 14243: 14242: 14241: 14238: 14234: 14231: 14230: 14229: 14226: 14224: 14221: 14219: 14216: 14215: 14213: 14211: 14207: 14199: 14196: 14194: 14191: 14189: 14188:Self-avoiding 14186: 14184: 14181: 14180: 14179: 14176: 14174: 14173:Moran process 14171: 14169: 14166: 14164: 14161: 14159: 14156: 14154: 14151: 14149: 14146: 14144: 14141: 14140: 14138: 14136: 14135:Discrete time 14132: 14128: 14121: 14116: 14114: 14109: 14107: 14102: 14101: 14098: 14086: 14078: 14076: 14068: 14066: 14058: 14056: 14048: 14047: 14044: 14038: 14035: 14033: 14030: 14028: 14025: 14024: 14022: 14018: 14012: 14009: 14007: 14006:Pension funds 14004: 14002: 13999: 13997: 13994: 13992: 13989: 13987: 13984: 13982: 13979: 13977: 13974: 13972: 13969: 13967: 13964: 13962: 13961:Vulture funds 13959: 13958: 13956: 13952: 13942: 13939: 13937: 13934: 13932: 13929: 13927: 13924: 13922: 13919: 13917: 13914: 13911: 13907: 13903: 13899: 13896: 13893: 13892:delta neutral 13889: 13885: 13882: 13880: 13877: 13875: 13872: 13870: 13867: 13866: 13864: 13860: 13854: 13851: 13849: 13848:Money markets 13846: 13844: 13841: 13839: 13836: 13834: 13831: 13829: 13826: 13824: 13821: 13820: 13818: 13814: 13811: 13805: 13799: 13796: 13794: 13791: 13789: 13786: 13784: 13781: 13779: 13776: 13774: 13771: 13770: 13768: 13764: 13759: 13745: 13744:Multi-manager 13741: 13738: 13737: 13735: 13731: 13725: 13722: 13720: 13717: 13715: 13712: 13710: 13707: 13705: 13701: 13698: 13696: 13693: 13692: 13690: 13686: 13680: 13677: 13675: 13672: 13670: 13667: 13665: 13662: 13661: 13659: 13657: 13653: 13647: 13644: 13642: 13639: 13637: 13633: 13630: 13628: 13625: 13623: 13620: 13618: 13615: 13614: 13612: 13610: 13605: 13601: 13598: 13592: 13588: 13581: 13576: 13574: 13569: 13567: 13562: 13561: 13558: 13546: 13541: 13536: 13535: 13532: 13526: 13523: 13521: 13518: 13516: 13513: 13511: 13508: 13506: 13503: 13501: 13500:Consumer debt 13498: 13497: 13495: 13493:Market issues 13491: 13485: 13482: 13480: 13477: 13475: 13472: 13470: 13469:Fund of funds 13467: 13465: 13462: 13460: 13457: 13455: 13452: 13450: 13447: 13445: 13442: 13440: 13437: 13435: 13432: 13430: 13427: 13425: 13422: 13420: 13417: 13416: 13414: 13410: 13404: 13401: 13399: 13396: 13394: 13391: 13389: 13386: 13384: 13381: 13379: 13376: 13375: 13373: 13371: 13367: 13361: 13358: 13356: 13353: 13351: 13348: 13346: 13343: 13341: 13338: 13336: 13333: 13331: 13328: 13326: 13323: 13321: 13318: 13316: 13313: 13311: 13310:Forward price 13308: 13306: 13303: 13301: 13298: 13296: 13293: 13291: 13288: 13286: 13283: 13282: 13280: 13275: 13272: 13270: 13267: 13266: 13263: 13257: 13254: 13252: 13249: 13247: 13244: 13242: 13239: 13237: 13234: 13232: 13229: 13227: 13224: 13222: 13221:Interest rate 13219: 13217: 13214: 13212: 13209: 13207: 13204: 13202: 13199: 13197: 13194: 13192: 13189: 13187: 13184: 13182: 13179: 13177: 13174: 13172: 13169: 13167: 13164: 13162: 13159: 13157: 13154: 13152: 13149: 13148: 13146: 13144: 13140: 13130: 13127: 13125: 13122: 13120: 13117: 13115: 13114:MC Simulation 13112: 13110: 13107: 13105: 13102: 13100: 13097: 13095: 13092: 13090: 13087: 13085: 13082: 13079: 13075: 13074:Black–Scholes 13072: 13070: 13067: 13065: 13062: 13060: 13057: 13056: 13054: 13052: 13048: 13041: 13037: 13033: 13030: 13028: 13027:Risk reversal 13025: 13023: 13020: 13018: 13015: 13013: 13010: 13008: 13005: 13003: 13000: 12998: 12995: 12993: 12990: 12988: 12985: 12983: 12980: 12978: 12975: 12973: 12970: 12968: 12965: 12963: 12960: 12958: 12957:Credit spread 12955: 12953: 12950: 12948: 12945: 12943: 12940: 12938: 12935: 12933: 12930: 12928: 12925: 12923: 12920: 12919: 12917: 12915: 12911: 12905: 12902: 12900: 12897: 12895: 12892: 12889: 12887: 12884: 12882: 12881:Interest rate 12879: 12877: 12876:Forward start 12874: 12872: 12869: 12867: 12864: 12862: 12859: 12857: 12854: 12852: 12849: 12847: 12844: 12842: 12839: 12837: 12834: 12832: 12829: 12828: 12826: 12824: 12820: 12814: 12811: 12809: 12806: 12804: 12803:Option styles 12801: 12799: 12796: 12794: 12791: 12789: 12786: 12784: 12781: 12779: 12776: 12774: 12771: 12769: 12766: 12765: 12763: 12761: 12757: 12751: 12748: 12746: 12743: 12741: 12738: 12736: 12733: 12731: 12728: 12726: 12723: 12721: 12720:Open interest 12718: 12716: 12713: 12711: 12708: 12706: 12703: 12701: 12700:Delta neutral 12698: 12697: 12695: 12691: 12688: 12686: 12682: 12678: 12673: 12669: 12662: 12657: 12655: 12650: 12648: 12643: 12642: 12639: 12632: 12629: 12626: 12622: 12621:Midas formula 12618: 12615: 12612: 12609: 12608: 12600: 12597: 12595: 12592: 12590: 12587: 12585: 12582: 12580: 12577: 12575: 12572: 12571: 12562: 12558: 12555: 12552: 12549: 12548: 12540: 12536: 12533: 12530: 12526: 12522: 12519: 12517: 12514: 12513: 12497: 12491: 12487: 12482: 12477: 12471: 12467: 12462: 12461: 12453: 12449: 12445: 12443: 12442:0-471-15280-3 12439: 12435: 12432: 12428: 12425: 12423: 12419: 12415: 12411: 12409:0-262-13460-8 12405: 12402:. MIT Press. 12401: 12400: 12394: 12392: 12388: 12384: 12380: 12376: 12371: 12366: 12362: 12358: 12353: 12351: 12347: 12343: 12338: 12333: 12329: 12325: 12321: 12317: 12310: 12305: 12303: 12302:0-471-39420-3 12299: 12295: 12291: 12289:0-02-903012-9 12285: 12281: 12280: 12275: 12271: 12270: 12260: 12258:0-13-601589-1 12254: 12250: 12246: 12245:Hull, John C. 12242: 12240: 12236: 12232: 12227: 12222: 12218: 12214: 12210: 12206: 12201: 12198: 12194: 12190: 12186: 12182: 12178: 12174: 12169: 12168: 12149: 12145: 12139: 12123: 12122:Bloomberg.com 12119: 12113: 12111: 12095: 12091: 12084: 12069: 12065: 12061: 12057: 12050: 12048: 12039: 12035: 12031: 12027: 12023: 12019: 12015: 12011: 12010:Physics Today 12007: 12001: 11993: 11987: 11983: 11982: 11975: 11958: 11954: 11948: 11941: 11937: 11936: 11929: 11922: 11918: 11914: 11911: 11907: 11903: 11898: 11891: 11887: 11883: 11878: 11869: 11862: 11858: 11854: 11848: 11840: 11833: 11829: 11823: 11815: 11813:0-471-89998-4 11809: 11805: 11801: 11795: 11787: 11783: 11779: 11778:Wilmott, Paul 11771: 11767: 11763: 11762:Wilmott, Paul 11757: 11755: 11747: 11742: 11738: 11734: 11730: 11726: 11722: 11715: 11707: 11703: 11696: 11687: 11678: 11670: 11668:0-13-149908-4 11664: 11660: 11659:Prentice Hall 11656: 11649: 11641: 11635: 11631: 11624: 11618: 11613: 11607: 11601: 11583: 11576: 11561: 11554: 11546: 11542: 11538: 11534: 11531:(2): 301–20. 11530: 11526: 11522: 11515: 11497: 11490: 11475: 11468: 11453: 11446: 11430: 11429:finance.bi.no 11426: 11420: 11414: 11410: 11404: 11396: 11390: 11386: 11385:Prentice Hall 11382: 11375: 11366: 11361: 11354: 11346: 11339: 11334: 11330: 11323: 11319: 11310: 11308: 11306: 11304: 11302: 11293: 11289: 11284: 11279: 11275: 11271: 11267: 11263: 11256: 11248: 11244: 11240: 11236: 11232: 11228: 11221: 11205: 11199: 11191: 11185: 11177: 11175:0-262-13460-8 11171: 11167: 11166: 11161: 11155: 11146: 11137: 11128: 11119: 11110: 11101: 11093: 11087: 11083: 11079: 11073: 11071: 11055: 11049: 11045: 11030: 11026: 11016: 11013: 11011: 11008: 11005: 11001: 10998: 10996: 10993: 10990: 10989:Heat equation 10987: 10985: 10982: 10979: 10976: 10974: 10971: 10969: 10966: 10963: 10960: 10957: 10954: 10951: 10948:, a discrete 10947: 10944: 10943: 10937: 10935: 10931: 10926: 10924: 10920: 10919: 10914: 10909: 10906: 10902: 10897: 10895: 10891: 10887: 10883: 10879: 10869: 10867: 10863: 10853: 10850: 10840: 10838: 10834: 10830: 10826: 10816: 10814: 10808: 10806: 10801: 10799: 10793: 10791: 10784: 10774: 10771: 10767: 10763: 10759: 10755: 10751: 10748:a volatility 10747: 10742: 10740: 10736: 10732: 10728: 10724: 10719: 10715: 10709: 10704: 10701: 10698: 10695: 10694: 10693: 10690: 10688: 10684: 10680: 10674: 10672: 10671:Delta hedging 10664: 10661: 10658: 10654: 10651: 10647: 10644: 10640: 10636: 10635: 10634: 10627: 10622: 10613: 10585: 10580: 10570: 10561: 10557: 10554: 10547: 10546: 10545: 10528: 10520: 10506: 10487: 10477: 10473: 10438: 10434: 10426: 10422: 10415: 10407: 10404: 10401: 10395: 10392: 10388: 10384: 10378: 10362: 10358: 10351: 10343: 10340: 10337: 10331: 10328: 10324: 10320: 10317: 10309: 10305: 10298: 10295: 10283: 10280: 10274: 10264: 10260: 10250: 10243: 10242: 10241: 10221: 10213: 10201: 10191: 10187: 10177: 10171: 10161: 10157: 10147: 10144: 10138: 10135: 10124: 10118: 10115: 10112: 10104: 10100: 10096: 10090: 10087: 10084: 10077: 10076: 10075: 10061: 10041: 10015: 10012: 10005: 10001: 9997: 9991: 9988: 9985: 9978: 9977: 9976: 9974: 9953: 9946: 9938: 9934: 9930: 9924: 9921: 9918: 9910: 9906: 9897: 9891: 9883: 9880: 9873: 9872: 9871: 9855: 9851: 9842: 9838: 9833: 9831: 9812: 9806: 9786: 9778: 9747: 9743: 9739: 9733: 9728: 9723: 9719: 9715: 9711: 9707: 9704: 9701: 9694: 9693: 9692: 9669: 9665: 9658: 9653: 9648: 9644: 9640: 9636: 9632: 9629: 9626: 9619: 9618: 9617: 9594: 9590: 9586: 9580: 9575: 9570: 9566: 9562: 9558: 9554: 9551: 9544: 9543: 9542: 9519: 9515: 9508: 9503: 9498: 9494: 9490: 9486: 9482: 9479: 9472: 9471: 9470: 9467: 9451: 9447: 9424: 9420: 9410: 9404: 9380: 9372: 9368: 9364: 9358: 9350: 9347: 9344: 9338: 9335: 9331: 9327: 9324: 9321: 9314: 9313: 9312: 9289: 9281: 9277: 9270: 9262: 9259: 9256: 9250: 9247: 9243: 9239: 9236: 9233: 9226: 9225: 9224: 9201: 9193: 9189: 9185: 9179: 9171: 9168: 9165: 9159: 9156: 9152: 9148: 9145: 9138: 9137: 9136: 9113: 9105: 9101: 9094: 9086: 9083: 9080: 9074: 9071: 9067: 9063: 9060: 9053: 9052: 9051: 9043: 9039: 9037: 9027: 9010: 9006: 9000: 8994: 8990: 8985: 8976: 8972: 8966: 8958: 8954: 8948: 8945: 8940: 8936: 8929: 8919: 8915: 8911: 8908: 8904: 8899: 8893: 8887: 8864: 8861: 8856: 8852: 8846: 8841: 8837: 8830: 8825: 8821: 8817: 8814: 8791: 8788: 8783: 8779: 8773: 8768: 8764: 8757: 8752: 8748: 8739: 8736: 8733: 8725: 8721: 8713: 8709: 8705: 8702: 8694: 8690: 8686: 8678: 8674: 8665: 8661: 8636: 8632: 8626: 8618: 8614: 8609: 8604: 8594: 8590: 8586: 8583: 8577: 8571: 8565: 8538: 8534: 8523: 8519: 8508: 8504: 8500: 8497: 8491: 8486: 8482: 8471: 8467: 8463: 8460: 8457: 8450: 8446: 8435: 8431: 8422: 8418: 8414: 8406: 8402: 8395: 8371: 8367: 8362: 8356: 8352: 8348: 8342: 8336: 8316: 8313: 8308: 8304: 8274: 8270: 8262: 8257: 8253: 8249: 8246: 8241: 8236: 8230: 8226: 8219: 8215: 8210: 8207: 8204: 8201: 8197: 8190: 8186: 8180: 8176: 8169: 8165: 8160: 8157: 8154: 8151: 8147: 8143: 8137: 8135: 8128: 8124: 8111: 8107: 8099: 8094: 8090: 8086: 8083: 8078: 8073: 8067: 8063: 8056: 8052: 8047: 8044: 8041: 8038: 8034: 8027: 8023: 8017: 8013: 8006: 8002: 7997: 7994: 7991: 7988: 7984: 7980: 7974: 7972: 7965: 7961: 7934: 7930: 7921: 7905: 7902: 7899: 7896: 7891: 7887: 7882: 7876: 7872: 7865: 7861: 7856: 7853: 7850: 7847: 7843: 7839: 7834: 7829: 7825: 7819: 7815: 7808: 7804: 7782: 7779: 7772: 7768: 7763: 7758: 7754: 7751: 7746: 7742: 7735: 7732: 7729: 7723: 7717: 7714: 7709: 7705: 7696: 7692: 7686: 7682: 7675: 7671: 7665: 7643: 7640: 7637: 7633: 7630: 7610: 7607: 7602: 7598: 7573: 7569: 7564: 7558: 7554: 7550: 7543: 7539: 7534: 7528: 7524: 7520: 7514: 7508: 7488: 7485: 7479: 7473: 7469: 7466: 7463: 7460: 7452: 7448: 7439: 7435: 7430: 7425: 7421: 7417: 7414: 7411: 7403: 7399: 7392: 7370: 7366: 7345: 7342: 7339: 7336: 7333: 7327: 7324: 7319: 7316: 7310: 7304: 7301: 7298: 7292: 7284: 7280: 7276: 7271: 7266: 7262: 7253: 7249: 7243: 7239: 7232: 7228: 7200: 7189:Perpetual put 7186: 7184: 7180: 7164: 7161: 7158: 7148: 7134: 7131: 7123: 7119: 7113: 7111: 7105: 7088: 7082: 7079: 7073: 7070: 7067: 7061: 7041: 7018: 7012: 6989: 6983: 6980: 6974: 6971: 6968: 6962: 6939: 6936: 6933: 6930: 6927: 6921: 6913: 6904: 6901: 6898: 6890: 6886: 6877: 6872: 6859: 6855: 6849: 6845: 6839: 6836: 6831: 6825: 6817: 6804: 6803: 6802: 6799: 6795: 6785: 6765: 6762: 6758: 6749: 6743: 6735: 6732: 6729: 6721: 6717: 6713: 6710: 6703: 6702: 6701: 6675: 6671: 6664: 6661: 6658: 6650: 6646: 6639: 6636: 6628: 6625: 6622: 6618: 6614: 6608: 6605: 6600: 6596: 6589: 6582: 6581: 6580: 6577: 6563: 6540: 6534: 6507: 6503: 6499: 6496: 6493: 6490: 6486: 6477: 6471: 6463: 6460: 6457: 6449: 6445: 6441: 6436: 6432: 6424: 6423: 6422: 6406: 6402: 6398: 6395: 6392: 6387: 6383: 6379: 6374: 6370: 6349: 6340: 6316: 6309: 6306: 6303: 6296: 6290: 6286: 6280: 6277: 6272: 6269: 6266: 6263: 6259: 6255: 6251: 6246: 6241: 6237: 6231: 6227: 6224: 6220: 6211: 6208: 6205: 6200: 6196: 6191: 6186: 6183: 6180: 6175: 6172: 6167: 6163: 6159: 6154: 6150: 6142: 6141: 6140: 6122: 6115: 6112: 6109: 6102: 6096: 6092: 6086: 6083: 6078: 6075: 6072: 6069: 6065: 6061: 6057: 6052: 6047: 6043: 6037: 6033: 6030: 6026: 6017: 6014: 6011: 6006: 6002: 5997: 5992: 5988: 5980: 5979: 5978: 5962: 5958: 5954: 5949: 5945: 5915: 5912: 5909: 5900: 5897: 5894: 5887: 5881: 5877: 5873: 5870: 5863: 5862: 5861: 5835: 5831: 5827: 5821: 5818: 5815: 5807: 5803: 5799: 5793: 5790: 5779: 5776: 5773: 5767: 5764: 5760: 5756: 5750: 5747: 5742: 5738: 5731: 5724: 5723: 5722: 5696: 5692: 5685: 5682: 5679: 5671: 5667: 5660: 5657: 5646: 5643: 5640: 5634: 5631: 5627: 5623: 5617: 5614: 5609: 5605: 5598: 5591: 5590: 5589: 5586: 5584: 5568: 5545: 5542: 5536: 5532: 5528: 5521: 5520: 5519: 5502: 5499: 5496: 5493: 5490: 5487: 5475: 5467: 5465: 5461: 5456: 5445: 5427: 5419: 5414: 5410: 5408: 5404: 5400: 5373: 5369: 5365: 5359: 5351: 5348: 5345: 5339: 5336: 5332: 5325: 5322: 5319: 5313: 5310: 5303: 5283: 5279: 5272: 5264: 5261: 5258: 5252: 5249: 5245: 5238: 5235: 5232: 5226: 5219: 5202: 5194: 5181: 5178: 5177: 5156: 5152: 5148: 5142: 5134: 5131: 5128: 5122: 5119: 5115: 5111: 5108: 5105: 5097: 5094: 5091: 5086: 5081: 5073: 5069: 5061: 5058: 5054: 5048: 5041: 5021: 5017: 5010: 5002: 4999: 4996: 4990: 4987: 4983: 4979: 4976: 4973: 4965: 4962: 4959: 4954: 4949: 4941: 4937: 4929: 4926: 4922: 4916: 4909: 4892: 4884: 4871: 4868: 4867: 4849: 4846: 4843: 4833: 4829: 4821: 4818: 4814: 4790: 4782: 4769: 4766: 4765: 4744: 4741: 4738: 4733: 4730: 4720: 4716: 4708: 4705: 4675: 4671: 4662: 4657: 4642: 4639: 4638: 4622: 4619: 4611: 4607: 4600: 4597: 4589: 4585: 4581: 4575: 4572: 4565: 4545: 4541: 4534: 4527: 4510: 4502: 4489: 4486: 4485: 4481: 4478: 4475: 4472: 4470: 4466: 4461: 4459: 4453: 4449: 4447: 4443: 4433: 4431: 4427: 4423: 4419: 4415: 4411: 4407: 4403: 4399: 4395: 4391: 4386: 4384: 4378: 4368: 4366: 4362: 4358: 4354: 4353:probabilities 4350: 4345: 4343: 4339: 4318: 4314: 4307: 4304: 4279: 4275: 4268: 4243: 4239: 4232: 4223: 4206: 4198: 4194: 4190: 4185: 4181: 4152: 4147: 4142: 4138: 4124: 4120: 4111: 4107: 4094: 4087: 4081: 4078: 4075: 4069: 4062: 4061: 4060: 4040: 4037: 4031: 4026: 4022: 4013: 4009: 3988: 3984: 3977: 3974: 3966: 3962: 3955: 3946: 3927: 3923: 3916: 3891: 3887: 3880: 3872: 3868: 3850: 3846: 3840: 3837: 3814: 3809: 3806: 3801: 3797: 3794: 3786: 3781: 3777: 3772: 3769: 3758: 3742: 3739: 3735: 3729: 3725: 3719: 3716: 3711: 3708: 3704: 3681: 3677: 3671: 3668: 3644: 3640: 3631: 3627: 3622: 3607: 3599: 3595: 3588: 3568: 3560: 3556: 3549: 3541: 3537: 3516: 3512: 3505: 3497: 3479: 3475: 3449: 3445: 3438: 3413: 3409: 3402: 3394: 3378: 3370: 3366: 3359: 3339: 3331: 3327: 3320: 3312: 3291: 3287: 3280: 3260: 3252: 3248: 3241: 3232: 3230: 3214: 3203: 3199: 3192: 3172: 3161: 3157: 3150: 3142: 3138: 3134: 3118: 3110: 3106: 3099: 3096: 3076: 3068: 3064: 3057: 3054: 3031: 3028: 3020: 3016: 3009: 3006: 3003: 3000: 2992: 2988: 2981: 2978: 2975: 2972: 2965: 2964: 2963: 2945: 2941: 2933: 2929: 2922: 2919: 2916: 2908: 2904: 2897: 2893: 2889: 2886: 2883: 2876: 2875: 2874: 2871: 2869: 2865: 2861: 2856: 2840: 2836: 2806: 2802: 2794: 2790: 2786: 2780: 2777: 2774: 2766: 2762: 2758: 2752: 2748: 2744: 2741: 2735: 2732: 2729: 2723: 2716: 2715: 2714: 2697: 2694: 2691: 2688: 2685: 2679: 2676: 2673: 2667: 2664: 2661: 2658: 2655: 2648: 2647: 2646: 2643: 2630: 2627: 2624: 2621: 2613: 2612:forward price 2595: 2592: 2587: 2584: 2579: 2576: 2572: 2568: 2565: 2557: 2541: 2538: 2535: 2531: 2527: 2524: 2516: 2493: 2488: 2485: 2480: 2476: 2472: 2470: 2463: 2459: 2450: 2446: 2441: 2437: 2431: 2428: 2423: 2419: 2414: 2411: 2406: 2402: 2399: 2395: 2386: 2381: 2377: 2372: 2370: 2363: 2359: 2350: 2346: 2338: 2334: 2327: 2324: 2321: 2313: 2309: 2302: 2298: 2294: 2291: 2289: 2281: 2278: 2275: 2269: 2258: 2257: 2256: 2254: 2223: 2219: 2210: 2206: 2202: 2196: 2193: 2185: 2182: 2179: 2173: 2170: 2166: 2162: 2154: 2150: 2146: 2140: 2137: 2135: 2124: 2121: 2116: 2112: 2105: 2102: 2097: 2093: 2089: 2081: 2078: 2075: 2069: 2066: 2062: 2058: 2055: 2053: 2045: 2042: 2037: 2033: 2026: 2015: 2014: 2013: 1994: 1991: 1988: 1982: 1979: 1975: 1967: 1963: 1938: 1935: 1932: 1927: 1924: 1919: 1915: 1911: 1909: 1902: 1898: 1889: 1882: 1879: 1876: 1869: 1863: 1858: 1854: 1848: 1845: 1841: 1837: 1833: 1828: 1823: 1819: 1813: 1809: 1806: 1802: 1793: 1790: 1787: 1782: 1778: 1773: 1771: 1764: 1760: 1747: 1744: 1741: 1735: 1732: 1728: 1724: 1716: 1712: 1705: 1702: 1697: 1693: 1684: 1680: 1673: 1670: 1668: 1660: 1657: 1652: 1648: 1641: 1630: 1629: 1628: 1604: 1601: 1598: 1595: 1592: 1583: 1577: 1574: 1571: 1565: 1551: 1543: 1540: 1537: 1528: 1525: 1522: 1516: 1508: 1500: 1497: 1491: 1488: 1485: 1479: 1467: 1466: 1465: 1463: 1459: 1455: 1451: 1447: 1444: 1424: 1404: 1395: 1386: 1384: 1380: 1376: 1357: 1354: 1351: 1348: 1345: 1339: 1331: 1322: 1319: 1316: 1308: 1304: 1295: 1290: 1277: 1273: 1267: 1263: 1257: 1254: 1249: 1243: 1235: 1222: 1221: 1220: 1206: 1186: 1163: 1160: 1157: 1151: 1143: 1134: 1129: 1105: 1100: 1096: 1090: 1086: 1082: 1078: 1071: 1068: 1064: 1059: 1053: 1050: 1042: 1036: 1033: 1027: 1021: 1014: 1011: 1003: 1002: 1001: 999: 980: 973: 970: 947: 944: 941: 935: 931: 925: 921: 917: 913: 907: 899: 895: 888: 885: 881: 876: 870: 864: 857: 856: 855: 853: 850: 831: 825: 814: 798: 791: 776: 773: 770: 767: 764: 744: 737: 722: 715: 697: 694: 691: 685: 678: 660: 657: 654: 648: 641: 638: 634: 615: 612: 609: 603: 596: 595: 594: 588: 584: 580: 564: 557: 555:, annualized. 542: 534: 518: 511: 494: 490: 481: 462: 456: 449: 448: 447: 441: 440: 435: 431: 428: 412: 405: 390: 387: 384: 364: 357: 356: 355: 352: 344: 342: 337: 335: 326: 322: 319: 318:short selling 315: 312: 309: 305: 304: 303: 297: 293: 290: 286: 282: 279: 275: 274: 273: 270: 268: 264: 254: 252: 247: 242: 240: 235: 233: 229: 225: 224: 218: 214: 213:Case Sprenkle 210: 206: 205:Sheen Kassouf 202: 198: 194: 190: 186: 185:Myron Scholes 182: 181:Fischer Black 172: 170: 166: 160: 158: 154: 150: 146: 140: 138: 134: 130: 129:delta hedging 126: 121: 119: 115: 114:Myron Scholes 111: 110:Fischer Black 107: 103: 99: 96: 92: 88: 84: 80: 76: 72: 68: 62: 22: 21:Black–Scholes 15414:Stock market 15280:Econometrics 15242:Wiener space 15130:Itô integral 15031:Inequalities 14920:Self-similar 14890:Gauss–Markov 14880:Exchangeable 14860:Càdlàg paths 14796:Risk process 14748:LIBOR market 14692: 14617:Random graph 14612:Random field 14424:Superprocess 14362:Lévy process 14357:Jump process 14332:Hunt process 14168:Markov chain 13883: 13838:Fixed income 13714:Global macro 13656:Event-driven 13320:Forward rate 13231:Total return 13119:Real options 13073: 13022:Ratio spread 13002:Naked option 12962:Debit spread 12793:Fixed income 12735:Strike price 12529:The Observer 12485: 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Retrieved 12138: 12126:. Retrieved 12121: 12097:. Retrieved 12093: 12083: 12071:. Retrieved 12060:The Guardian 12059: 12016:(9): 52–53. 12013: 12009: 12000: 11980: 11974: 11963:. Retrieved 11947: 11934: 11928: 11920: 11906:Nassim Taleb 11897: 11885: 11877: 11868: 11860: 11847: 11838: 11822: 11803: 11794: 11786:the original 11770:the original 11744: 11724: 11720: 11714: 11695: 11686: 11677: 11654: 11648: 11629: 11623: 11612: 11600: 11588:. Retrieved 11575: 11563:. Retrieved 11553: 11528: 11524: 11514: 11502:. Retrieved 11489: 11477:. Retrieved 11467: 11455:. Retrieved 11450:André Jaun. 11445: 11433:. Retrieved 11428: 11419: 11403: 11380: 11374: 11353: 11344: 11332: 11328: 11321: 11317: 11265: 11261: 11255: 11230: 11226: 11220: 11208:. Retrieved 11198: 11184: 11164: 11154: 11145: 11136: 11127: 11118: 11109: 11100: 11081: 11057:. Retrieved 11048: 11029: 10962:Black Shoals 10927: 10916: 10910: 10898: 10894:Paul Wilmott 10886:Edward Thorp 10875: 10859: 10846: 10822: 10809: 10805:at-the-money 10802: 10794: 10786: 10769: 10766:price domain 10765: 10749: 10745: 10743: 10730: 10726: 10722: 10720: 10716: 10713: 10707: 10691: 10675: 10668: 10631: 10610: 10459: 10239: 10033: 9970: 9840: 9839:, at strike 9836: 9834: 9773: 9690: 9615: 9540: 9468: 9408: 9407:Denoting by 9406: 9310: 9222: 9134: 9049: 9040: 9033: 7192: 7149: 7114: 7106: 6954: 6791: 6783: 6699: 6578: 6526: 6341: 6338: 6138: 5936: 5859: 5720: 5587: 5560: 5476: 5473: 5451: 5415: 5411: 5406: 5402: 5396: 4462: 4454: 4450: 4439: 4387: 4380: 4352: 4346: 4224: 4172: 4007: 3947: 3756: 3629: 3625: 3623: 3535: 3392: 3310: 3233: 3140: 3136: 3132: 3046: 2961: 2872: 2857: 2827: 2712: 2644: 2558: 2517: 2514: 2250: 1959: 1626: 1450:call options 1440: 1383:next section 1372: 1139: 962: 847:denotes the 818: 813:strike price 636: 632: 592: 479: 445: 437: 353: 350: 338: 330: 301: 271: 263:money market 260: 243: 236: 221: 192: 178: 161: 141: 122: 106:risk-neutral 101: 90: 66: 20: 18: 15325:Ruin theory 15263:Disciplines 15135:Itô's lemma 14910:Predictable 14585:Percolation 14568:Potts model 14563:Ising model 14527:White noise 14485:Differences 14347:Itô process 14287:Cox process 14183:Loop-erased 14178:Random walk 14065:Hedge funds 13828:Derivatives 13823:Commodities 13778:Day trading 13688:Directional 13587:Hedge funds 13251:Zero Coupon 13181:Correlation 13129:Vanna–Volga 12987:Iron condor 12773:Bond option 12617:BBC Horizon 12561:Terence Tao 12525:Ian Stewart 11082:Investments 10956:Black model 10913:Ian Stewart 10862:short stock 10833:Black model 10829:pull-to-par 10827:because of 10788:value is a 4465:closed form 4402:expectation 4371:Derivations 285:random walk 265:, cash, or 179:Economists 137:hedge funds 77:containing 15383:Categories 15335:Statistics 15115:Filtration 15016:Kolmogorov 15000:Blumenthal 14925:Stationary 14865:Continuous 14853:Properties 14738:Hull–White 14480:Martingale 14367:Local time 14255:Fractional 14233:pure birth 14020:Governance 13594:Investment 13525:Tax policy 13241:Volatility 13151:Amortising 12992:Jelly roll 12927:Box spread 12922:Backspread 12914:Strategies 12750:Volatility 12745:the Greeks 12710:Expiration 12605:Historical 12012:. Review. 11965:2024-02-29 11345:LT Nielsen 11078:Bodie, Zvi 11041:References 11004:simulation 10731:variables, 10679:log-normal 9973:derivative 7918:Using the 7179:up-and-out 6700:where now 5860:where now 4442:The Greeks 4394:martingale 4375:See also: 4338:given that 4012:martingale 1458:by solving 1454:consistent 1379:underlying 587:volatility 533:drift rate 427:annualized 79:derivative 15389:Equations 15247:Classical 14260:Geometric 14250:Excursion 13954:Investors 13604:Arbitrage 13216:Inflation 13166:Commodity 13124:Trinomial 13059:Bachelier 13051:Valuation 12932:Butterfly 12866:Commodore 12715:Moneyness 12488:. Wiley. 12468:. Wiley. 12387:145805302 12365:CiteSeerX 12193:154552078 12148:CME Group 12094:Intuition 12073:April 29, 12068:0029-7712 12038:0031-9228 11806:. Wiley. 11360:CiteSeerX 11247:154552078 11210:March 26, 11059:March 26, 10860:Taking a 10764:from the 10727:constant, 10639:tail risk 10586:⋅ 10571:− 10526:∂ 10521:σ 10518:∂ 10488:σ 10485:∂ 10470:∂ 10405:− 10393:− 10376:∂ 10341:− 10329:− 10318:− 10290:∂ 10284:− 10272:∂ 10257:∂ 10251:− 10219:∂ 10214:σ 10211:∂ 10202:σ 10199:∂ 10184:∂ 10178:− 10169:∂ 10154:∂ 10148:− 10119:σ 10091:− 10042:σ 9992:− 9954:ϵ 9931:− 9925:ϵ 9922:− 9895:→ 9892:ϵ 9807:σ 9787:σ 9740:− 9716:− 9641:− 9587:− 9563:− 9491:− 9365:− 9348:− 9336:− 9260:− 9248:− 9186:− 9169:− 9157:− 9084:− 9072:− 9007:λ 8973:λ 8955:λ 8946:− 8937:λ 8916:λ 8912:− 8862:− 8853:λ 8838:λ 8826:− 8818:≥ 8789:− 8780:λ 8765:λ 8753:− 8744:⟹ 8737:− 8726:− 8714:− 8706:− 8691:λ 8679:− 8633:λ 8619:− 8595:− 8587:− 8535:λ 8524:− 8509:− 8501:− 8478:⟹ 8472:− 8464:− 8447:λ 8436:− 8407:− 8368:λ 8271:σ 8254:σ 8227:σ 8211:− 8205:− 8191:− 8177:σ 8161:− 8155:− 8144:− 8125:λ 8108:σ 8091:σ 8064:σ 8048:− 8042:− 8014:σ 7998:− 7992:− 7981:− 7962:λ 7931:λ 7897:− 7888:λ 7873:σ 7857:− 7851:− 7826:λ 7816:σ 7769:λ 7752:− 7743:λ 7733:− 7715:− 7706:λ 7693:λ 7683:σ 7608:≤ 7603:− 7570:λ 7540:λ 7486:≤ 7464:− 7453:− 7426:− 7418:− 7404:− 7371:− 7334:− 7302:− 7240:σ 7207:∞ 7204:→ 7162:− 7135:∗ 6981:≥ 6937:≤ 6928:− 6919:∂ 6911:∂ 6883:∂ 6869:∂ 6846:σ 6823:∂ 6815:∂ 6736:δ 6733:− 6659:− 6623:− 6500:σ 6464:δ 6461:− 6396:… 6350:δ 6307:− 6287:σ 6273:− 6267:− 6228:⁡ 6209:− 6201:σ 6184:− 6176:σ 6173:− 6113:− 6093:σ 6073:− 6034:⁡ 6015:− 6007:σ 5913:− 5898:− 5828:− 5816:− 5800:− 5777:− 5765:− 5680:− 5644:− 5632:− 5428:ν 5374:− 5366:− 5349:− 5337:− 5323:− 5311:− 5284:− 5262:− 5250:− 5236:− 5200:∂ 5192:∂ 5157:− 5149:− 5132:− 5120:− 5095:− 5082:σ 5049:− 5022:− 5000:− 4988:− 4974:− 4963:− 4950:σ 4917:− 4890:∂ 4882:∂ 4847:− 4791:σ 4788:∂ 4780:∂ 4742:− 4734:σ 4668:∂ 4654:∂ 4620:− 4582:− 4573:− 4508:∂ 4500:∂ 4244:− 4199:− 4186:− 4148:σ 4112:− 4099:′ 4044:∞ 4032:∈ 3989:− 3928:− 3847:σ 3798:⁡ 3787:τ 3782:σ 3740:τ 3726:σ 3712:± 3678:σ 3645:± 3561:− 3540:numéraire 3517:± 3496:moneyness 3480:± 3450:− 3371:− 3332:− 3204:− 3139:value to 3111:− 3021:− 3004:− 2934:− 2920:− 2841:± 2787:− 2778:− 2767:− 2759:− 2736:τ 2692:− 2677:− 2659:− 2580:τ 2542:τ 2536:− 2494:τ 2489:σ 2486:− 2464:− 2447:τ 2438:σ 2403:⁡ 2387:τ 2382:σ 2339:− 2325:− 2282:τ 2203:− 2194:− 2183:− 2171:− 2155:− 2147:− 2090:− 2079:− 2067:− 1992:− 1980:− 1936:− 1928:σ 1925:− 1903:− 1880:− 1855:σ 1810:⁡ 1791:− 1783:σ 1745:− 1733:− 1717:− 1703:− 1596:− 1558:∞ 1555:→ 1541:− 1535:→ 1346:− 1337:∂ 1329:∂ 1301:∂ 1287:∂ 1264:σ 1241:∂ 1233:∂ 1219:is time: 1083:− 1072:π 918:− 903:∞ 900:− 896:∫ 889:π 774:− 765:τ 745:τ 565:σ 519:μ 308:arbitrage 15368:Category 15252:Abstract 14786:Bühlmann 14392:Compound 13596:strategy 13355:Slippage 13285:Contango 13269:Forwards 13236:Variance 13196:Dividend 13191:Currency 13104:Margrabe 13099:Lattices 13078:equation 13064:Binomial 13012:Strangle 13007:Straddle 12904:Swaption 12886:Lookback 12871:Compound 12813:Warrants 12788:European 12768:American 12760:Vanillas 12725:Pin risk 12705:Exercise 12346:15524084 12276:(1992). 12247:(1997). 11913:Archived 11908:(2005). 11855:(2011). 11802:(1999). 11590:June 25, 11565:June 25, 11435:July 21, 11162:(2006). 11002:, using 10940:See also 10750:a priori 10746:assuming 10645:options; 9777:skewness 5460:lattices 5062:′ 4930:′ 4822:′ 4709:′ 4424:" under 4006:are the 2866:minus a 1443:European 1015:′ 974:′ 635:at time 347:Notation 296:dividend 234:model". 14875:Ergodic 14763:Vašíček 14605:Poisson 14265:Meander 13816:Markets 13807:Related 13766:Trading 13274:Futures 12894:Rainbow 12861:Cliquet 12856:Chooser 12836:Barrier 12823:Exotics 12685:Options 12235:3003143 12153:3 April 12128:3 April 12099:2 April 12018:Bibcode 11741:2327242 11706:2115141 11545:2328254 11504:May 16, 11292:3003143 10768:to the 10566:no skew 10503:is the 10443:no skew 7657:yields: 3137:present 2610:is the 2515:where: 811:is the 577:is the 531:is the 425:is the 175:History 98:options 15215:Tanaka 14900:Mixing 14895:Markov 14768:Wilkie 14733:Ho–Lee 14728:Heston 14500:Super- 14245:Bridge 14193:Biased 13888:Greeks 13833:Equity 13335:Margin 13201:Equity 13094:Heston 12997:Ladder 12947:Condor 12942:Collar 12899:Spread 12846:Binary 12841:Basket 12627:(LTCM) 12492: 12472: 12450: 12440: 12420: 12406: 12385: 12367: 12344: 12300: 12286: 12255: 12233: 12191: 12066: 12036: 11988: 11810: 11739: 11704: 11665: 11636: 11543: 11479:May 5, 11457:May 5, 11391: 11362: 11327:) and 11290: 11245: 11172: 11088: 10735:Greeks 10723:robust 7005:where 6527:where 4416:under 4173:where 3313:while 3212: 3170: 3141:future 3047:where 102:unique 15068:Tools 14844:M/M/c 14839:M/M/1 14834:M/G/1 14824:Fluid 14490:Local 13931:Short 13921:Hedge 13902:alpha 13809:terms 13733:Other 13206:Forex 13161:Basis 13156:Asset 13143:Swaps 13069:Black 12972:Fence 12831:Asian 12693:Terms 12383:S2CID 12342:S2CID 12312:(PDF) 12231:JSTOR 12189:S2CID 11960:(PDF) 11835:(PDF) 11737:JSTOR 11585:(PDF) 11541:JSTOR 11499:(PDF) 11341:(PDF) 11288:JSTOR 11243:S2CID 11021:Notes 10683:GARCH 5581:(the 5464:grids 4869:Theta 4640:Gamma 4487:Delta 4458:gamma 4355:in a 2862:: an 1964:with 1375:hedge 125:hedge 69:is a 15020:Lévy 14819:Bulk 14703:Chen 14495:Sub- 14453:Both 13906:beta 13862:Misc 13040:Bull 13036:Bear 12778:Call 12490:ISBN 12470:ISBN 12448:ISBN 12438:ISBN 12418:ISBN 12404:ISBN 12298:ISBN 12284:ISBN 12253:ISBN 12155:2021 12130:2021 12101:2021 12075:2020 12064:ISSN 12034:ISSN 11986:ISBN 11904:and 11808:ISBN 11702:SSRN 11663:ISBN 11634:ISBN 11592:2012 11567:2012 11506:2012 11481:2012 11459:2012 11437:2017 11389:ISBN 11212:2012 11170:ISBN 11086:ISBN 11061:2012 10590:Skew 10576:Vega 10505:Vega 9770:Skew 7949:are: 7590:For 7358:Let 6139:and 5721:and 5462:and 4767:Vega 4482:Put 4479:Call 4430:Hull 4388:The 3431:and 2012:is: 1448:and 267:bond 211:and 183:and 151:and 135:and 112:and 19:The 14600:Cox 12808:Put 12527:in 12523:by 12375:doi 12361:109 12332:hdl 12324:doi 12221:hdl 12213:doi 12181:doi 12026:doi 11729:doi 11533:doi 11278:hdl 11270:doi 11235:doi 9888:lim 5585:). 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10162:v 10158:C 10145:= 10139:K 10136:d 10131:) 10128:) 10125:K 10122:( 10116:, 10113:K 10110:( 10105:v 10101:C 10097:d 10088:= 10085:C 10062:K 10016:K 10013:d 10006:v 10002:C 9998:d 9989:= 9986:C 9950:) 9947:K 9944:( 9939:v 9935:C 9928:) 9919:K 9916:( 9911:v 9907:C 9898:0 9884:= 9881:C 9856:v 9852:C 9841:K 9837:C 9816:) 9813:K 9810:( 9753:) 9748:1 9744:d 9737:( 9734:N 9729:T 9724:f 9720:r 9712:e 9708:S 9705:= 9702:P 9675:) 9670:1 9666:d 9662:( 9659:N 9654:T 9649:f 9645:r 9637:e 9633:S 9630:= 9627:C 9600:) 9595:2 9591:d 9584:( 9581:N 9576:T 9571:d 9567:r 9559:e 9555:= 9552:P 9525:) 9520:2 9516:d 9512:( 9509:N 9504:T 9499:d 9495:r 9487:e 9483:= 9480:C 9452:d 9448:r 9425:f 9421:r 9409:S 9381:, 9378:) 9373:1 9369:d 9362:( 9359:N 9354:) 9351:t 9345:T 9342:( 9339:q 9332:e 9328:S 9325:= 9322:P 9290:. 9287:) 9282:1 9278:d 9274:( 9271:N 9266:) 9263:t 9257:T 9254:( 9251:q 9244:e 9240:S 9237:= 9234:C 9202:. 9199:) 9194:2 9190:d 9183:( 9180:N 9175:) 9172:t 9166:T 9163:( 9160:r 9153:e 9149:= 9146:P 9114:. 9111:) 9106:2 9102:d 9098:( 9095:N 9090:) 9087:t 9081:T 9078:( 9075:r 9068:e 9064:= 9061:C 9011:2 9001:) 8995:K 8991:S 8986:( 8977:2 8967:) 8959:2 8949:1 8941:2 8930:( 8920:2 8909:1 8905:K 8900:= 8897:) 8894:S 8891:( 8888:V 8865:1 8857:2 8847:K 8842:2 8831:= 8822:S 8815:S 8792:1 8784:2 8774:K 8769:2 8758:= 8749:S 8740:1 8734:= 8722:S 8710:S 8703:K 8695:2 8687:= 8684:) 8675:S 8671:( 8666:S 8662:V 8637:2 8627:) 8615:S 8610:S 8605:( 8600:) 8591:S 8584:K 8581:( 8578:= 8575:) 8572:S 8569:( 8566:V 8539:2 8530:) 8520:S 8516:( 8505:S 8498:K 8492:= 8487:2 8483:A 8468:S 8461:K 8458:= 8451:2 8442:) 8432:S 8428:( 8423:2 8419:A 8415:= 8412:) 8403:S 8399:( 8396:V 8372:2 8363:S 8357:2 8353:A 8349:= 8346:) 8343:S 8340:( 8337:V 8317:0 8314:= 8309:1 8305:A 8275:2 8263:r 8258:2 8250:2 8247:+ 8242:2 8237:) 8231:2 8220:2 8216:1 8208:q 8202:r 8198:( 8187:) 8181:2 8170:2 8166:1 8158:q 8152:r 8148:( 8138:= 8129:2 8112:2 8100:r 8095:2 8087:2 8084:+ 8079:2 8074:) 8068:2 8057:2 8053:1 8045:q 8039:r 8035:( 8028:+ 8024:) 8018:2 8007:2 8003:1 7995:q 7989:r 7985:( 7975:= 7966:1 7935:i 7906:0 7903:= 7900:r 7892:i 7883:) 7877:2 7866:2 7862:1 7854:q 7848:r 7844:( 7840:+ 7835:2 7830:i 7820:2 7809:2 7805:1 7783:0 7780:= 7773:i 7764:S 7759:] 7755:r 7747:i 7739:) 7736:q 7730:r 7727:( 7724:+ 7721:) 7718:1 7710:i 7702:( 7697:i 7687:2 7676:2 7672:1 7666:[ 7644:2 7641:, 7638:1 7634:= 7631:i 7611:S 7599:S 7574:2 7565:S 7559:2 7555:A 7551:+ 7544:1 7535:S 7529:1 7525:A 7521:= 7518:) 7515:S 7512:( 7509:V 7489:K 7483:) 7480:S 7477:( 7474:V 7470:, 7467:1 7461:= 7458:) 7449:S 7445:( 7440:S 7436:V 7431:, 7422:S 7415:K 7412:= 7409:) 7400:S 7396:( 7393:V 7367:S 7346:0 7343:= 7340:V 7337:r 7328:S 7325:d 7320:V 7317:d 7311:S 7308:) 7305:q 7299:r 7296:( 7293:+ 7285:2 7281:S 7277:d 7272:V 7267:2 7263:d 7254:2 7250:S 7244:2 7233:2 7229:1 7201:T 7165:X 7159:S 7132:s 7092:) 7089:S 7086:( 7083:H 7080:= 7077:) 7074:T 7071:, 7068:S 7065:( 7062:V 7042:S 7022:) 7019:S 7016:( 7013:H 6993:) 6990:S 6987:( 6984:H 6978:) 6975:t 6972:, 6969:S 6966:( 6963:V 6940:0 6934:V 6931:r 6922:S 6914:V 6905:S 6902:r 6899:+ 6891:2 6887:S 6878:V 6873:2 6860:2 6856:S 6850:2 6840:2 6837:1 6832:+ 6826:t 6818:V 6766:T 6763:r 6759:e 6753:) 6750:T 6747:( 6744:n 6740:) 6730:1 6727:( 6722:0 6718:S 6714:= 6711:F 6684:] 6681:) 6676:2 6672:d 6668:( 6665:N 6662:K 6656:) 6651:1 6647:d 6643:( 6640:N 6637:F 6634:[ 6629:T 6626:r 6619:e 6615:= 6612:) 6609:T 6606:, 6601:0 6597:S 6593:( 6590:C 6564:t 6544:) 6541:t 6538:( 6535:n 6508:t 6504:W 6497:+ 6494:t 6491:u 6487:e 6481:) 6478:t 6475:( 6472:n 6468:) 6458:1 6455:( 6450:0 6446:S 6442:= 6437:t 6433:S 6407:n 6403:t 6399:, 6393:, 6388:2 6384:t 6380:, 6375:1 6371:t 6330:. 6317:] 6313:) 6310:t 6304:T 6301:( 6297:) 6291:2 6281:2 6278:1 6270:q 6264:r 6260:( 6256:+ 6252:) 6247:K 6242:t 6238:S 6232:( 6221:[ 6212:t 6206:T 6197:1 6192:= 6187:t 6181:T 6168:1 6164:d 6160:= 6155:2 6151:d 6123:] 6119:) 6116:t 6110:T 6107:( 6103:) 6097:2 6087:2 6084:1 6079:+ 6076:q 6070:r 6066:( 6062:+ 6058:) 6053:K 6048:t 6044:S 6038:( 6027:[ 6018:t 6012:T 6003:1 5998:= 5993:1 5989:d 5963:2 5959:d 5955:, 5950:1 5946:d 5919:) 5916:t 5910:T 5907:( 5904:) 5901:q 5895:r 5892:( 5888:e 5882:t 5878:S 5874:= 5871:F 5844:] 5841:) 5836:1 5832:d 5825:( 5822:N 5819:F 5813:) 5808:2 5804:d 5797:( 5794:N 5791:K 5788:[ 5783:) 5780:t 5774:T 5771:( 5768:r 5761:e 5757:= 5754:) 5751:t 5748:, 5743:t 5739:S 5735:( 5732:P 5705:] 5702:) 5697:2 5693:d 5689:( 5686:N 5683:K 5677:) 5672:1 5668:d 5664:( 5661:N 5658:F 5655:[ 5650:) 5647:t 5641:T 5638:( 5635:r 5628:e 5624:= 5621:) 5618:t 5615:, 5610:t 5606:S 5602:( 5599:C 5569:q 5546:t 5543:d 5537:t 5533:S 5529:q 5506:] 5503:t 5500:d 5497:+ 5494:t 5491:, 5488:t 5485:[ 5442:ν 5407:σ 5403:S 5379:) 5370:d 5363:( 5360:N 5355:) 5352:t 5346:T 5343:( 5340:r 5333:e 5329:) 5326:t 5320:T 5317:( 5314:K 5289:) 5280:d 5276:( 5273:N 5268:) 5265:t 5259:T 5256:( 5253:r 5246:e 5242:) 5239:t 5233:T 5230:( 5227:K 5203:r 5195:V 5162:) 5153:d 5146:( 5143:N 5138:) 5135:t 5129:T 5126:( 5123:r 5116:e 5112:K 5109:r 5106:+ 5098:t 5092:T 5087:2 5079:) 5074:+ 5070:d 5066:( 5059:N 5055:S 5027:) 5018:d 5014:( 5011:N 5006:) 5003:t 4997:T 4994:( 4991:r 4984:e 4980:K 4977:r 4966:t 4960:T 4955:2 4947:) 4942:+ 4938:d 4934:( 4927:N 4923:S 4893:t 4885:V 4850:t 4844:T 4839:) 4834:+ 4830:d 4826:( 4819:N 4815:S 4783:V 4745:t 4739:T 4731:S 4726:) 4721:+ 4717:d 4713:( 4706:N 4676:2 4672:S 4663:V 4658:2 4623:1 4617:) 4612:+ 4608:d 4604:( 4601:N 4598:= 4595:) 4590:+ 4586:d 4579:( 4576:N 4551:) 4546:+ 4542:d 4538:( 4535:N 4511:S 4503:V 4440:" 4324:) 4319:+ 4315:d 4311:( 4308:N 4305:S 4285:) 4280:+ 4276:d 4272:( 4269:N 4249:) 4240:d 4236:( 4233:N 4210:) 4207:K 4204:( 4195:d 4191:= 4182:d 4153:T 4143:T 4139:S 4133:] 4130:) 4125:T 4121:S 4117:( 4108:d 4104:[ 4095:N 4088:= 4085:) 4082:T 4079:, 4076:S 4073:( 4070:p 4047:) 4041:, 4038:0 4035:( 4027:T 4023:S 3994:) 3985:d 3981:( 3978:N 3975:, 3972:) 3967:+ 3963:d 3959:( 3956:N 3933:) 3924:d 3920:( 3917:N 3897:) 3892:+ 3888:d 3884:( 3881:N 3851:2 3841:2 3838:1 3815:) 3810:K 3807:F 3802:( 3778:1 3773:= 3770:m 3760:− 3757:d 3743:, 3736:) 3730:2 3720:2 3717:1 3709:r 3705:( 3682:2 3672:2 3669:1 3641:d 3626:S 3608:F 3605:) 3600:+ 3596:d 3592:( 3589:N 3569:K 3566:) 3557:d 3553:( 3550:N 3522:) 3513:d 3509:( 3506:N 3476:d 3455:) 3446:d 3442:( 3439:N 3419:) 3414:+ 3410:d 3406:( 3403:N 3379:, 3376:) 3367:d 3363:( 3360:N 3340:K 3337:) 3328:d 3324:( 3321:N 3297:) 3292:+ 3288:d 3284:( 3281:N 3261:F 3258:) 3253:+ 3249:d 3245:( 3242:N 3215:K 3209:) 3200:d 3196:( 3193:N 3173:F 3167:) 3162:+ 3158:d 3154:( 3151:N 3133:D 3119:K 3116:) 3107:d 3103:( 3100:N 3097:D 3077:F 3074:) 3069:+ 3065:d 3061:( 3058:N 3055:D 3032:, 3029:K 3026:) 3017:d 3013:( 3010:N 3007:D 3001:F 2998:) 2993:+ 2989:d 2985:( 2982:N 2979:D 2976:= 2973:C 2946:] 2942:K 2939:) 2930:d 2926:( 2923:N 2917:F 2914:) 2909:+ 2905:d 2901:( 2898:N 2894:[ 2890:D 2887:= 2884:C 2837:d 2807:] 2803:F 2800:) 2795:+ 2791:d 2784:( 2781:N 2775:K 2772:) 2763:d 2756:( 2753:N 2749:[ 2745:D 2742:= 2739:) 2733:, 2730:F 2727:( 2724:P 2698:K 2695:D 2689:S 2686:= 2683:) 2680:K 2674:F 2671:( 2668:D 2665:= 2662:P 2656:C 2631:F 2628:D 2625:= 2622:S 2596:D 2593:S 2588:= 2585:S 2577:r 2573:e 2569:= 2566:F 2539:r 2532:e 2528:= 2525:D 2481:+ 2477:d 2473:= 2460:d 2451:] 2442:2 2432:2 2429:1 2424:+ 2420:) 2415:K 2412:F 2407:( 2396:[ 2378:1 2373:= 2364:+ 2360:d 2351:] 2347:K 2344:) 2335:d 2331:( 2328:N 2322:F 2319:) 2314:+ 2310:d 2306:( 2303:N 2299:[ 2295:D 2292:= 2285:) 2279:, 2276:F 2273:( 2270:C 2224:t 2220:S 2216:) 2211:+ 2207:d 2200:( 2197:N 2189:) 2186:t 2180:T 2177:( 2174:r 2167:e 2163:K 2160:) 2151:d 2144:( 2141:N 2138:= 2128:) 2125:t 2122:, 2117:t 2113:S 2109:( 2106:C 2103:+ 2098:t 2094:S 2085:) 2082:t 2076:T 2073:( 2070:r 2063:e 2059:K 2056:= 2049:) 2046:t 2043:, 2038:t 2034:S 2030:( 2027:P 1998:) 1995:t 1989:T 1986:( 1983:r 1976:e 1939:t 1933:T 1920:+ 1916:d 1912:= 1899:d 1890:] 1886:) 1883:t 1877:T 1874:( 1870:) 1864:2 1859:2 1849:+ 1846:r 1842:( 1838:+ 1834:) 1829:K 1824:t 1820:S 1814:( 1803:[ 1794:t 1788:T 1779:1 1774:= 1765:+ 1761:d 1751:) 1748:t 1742:T 1739:( 1736:r 1729:e 1725:K 1722:) 1713:d 1709:( 1706:N 1698:t 1694:S 1690:) 1685:+ 1681:d 1677:( 1674:N 1671:= 1664:) 1661:t 1658:, 1653:t 1649:S 1645:( 1642:C 1608:} 1605:0 1602:, 1599:K 1593:S 1590:{ 1584:= 1581:) 1578:T 1575:, 1572:S 1569:( 1566:C 1552:S 1544:K 1538:S 1532:) 1529:t 1526:, 1523:S 1520:( 1517:C 1509:t 1501:0 1498:= 1495:) 1492:t 1489:, 1486:0 1483:( 1480:C 1425:T 1405:S 1358:0 1355:= 1352:V 1349:r 1340:S 1332:V 1323:S 1320:r 1317:+ 1309:2 1305:S 1296:V 1291:2 1278:2 1274:S 1268:2 1258:2 1255:1 1250:+ 1244:t 1236:V 1207:t 1187:S 1167:) 1164:t 1161:, 1158:S 1155:( 1152:V 1106:. 1101:2 1097:/ 1091:2 1087:x 1079:e 1069:2 1065:1 1060:= 1054:x 1051:d 1046:) 1043:x 1040:( 1037:N 1034:d 1028:= 1025:) 1022:x 1019:( 1012:N 984:) 981:x 978:( 971:N 948:. 945:z 942:d 936:2 932:/ 926:2 922:z 914:e 908:x 886:2 882:1 877:= 874:) 871:x 868:( 865:N 835:) 832:x 829:( 826:N 799:K 789:. 777:t 771:T 768:= 723:T 701:) 698:t 695:, 692:S 689:( 686:P 664:) 661:t 658:, 655:S 652:( 649:C 633:S 619:) 616:t 613:, 610:S 607:( 604:V 589:. 543:S 509:. 495:t 491:S 480:t 466:) 463:t 460:( 457:S 413:r 391:0 388:= 385:t 365:t 298:. 280:. 61:/ 58:z 55:l 49:ʃ 46:ˈ 41:k 38:æ 35:l 32:b 29:ˌ 26:/

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Knowledge

Black–Scholes model

Index