7238:
1867:
6739:
693:
XYZ each have a delta of 0.25 (= 25%), it will gain or lose value just like 2,500 shares of XYZ as the price changes for small price movements (100 option contracts covers 10,000 shares). The sign and percentage are often dropped – the sign is implicit in the option type (negative for put, positive for call) and the percentage is understood. The most commonly quoted are 25 delta put, 50 delta put/50 delta call, and 25 delta call. 50 Delta put and 50 Delta call are not quite identical, due to spot and forward differing by the discount factor, but they are often conflated.
7233:{\displaystyle {\begin{aligned}d_{1}&={\frac {\ln(S/K)+\left(r-q+{\frac {1}{2}}\sigma ^{2}\right)\tau }{\sigma {\sqrt {\tau }}}}\\d_{2}&={\frac {\ln(S/K)+\left(r-q-{\frac {1}{2}}\sigma ^{2}\right)\tau }{\sigma {\sqrt {\tau }}}}=d_{1}-\sigma {\sqrt {\tau }}\\\varphi (x)&={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}\\\Phi (x)&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-{\frac {1}{2}}y^{2}}\,dy=1-{\frac {1}{\sqrt {2\pi }}}\int _{x}^{\infty }e^{-{\frac {1}{2}}y^{2}}\,dy\end{aligned}}}
12478:
12468:
12458:
1909:, the sensitivity of the option delta with respect to change in volatility; or alternatively, the partial of vega with respect to the underlying instrument's price. Vanna can be a useful sensitivity to monitor when maintaining a delta- or vega-hedged portfolio as vanna will help the trader to anticipate changes to the effectiveness of a delta-hedge as volatility changes or the effectiveness of a vega-hedge against change in the underlying spot price.
12488:
12165:
11947:
462:— and are very useful for derivatives traders, especially those who seek to hedge their portfolios from adverse changes in market conditions. For this reason, those Greeks which are particularly useful for hedging—such as delta, theta, and vega—are well-defined for measuring changes in the parameters spot price, time and volatility. Although rho (the partial derivative with respect to the
9165:
6019:
1333:, time to expiry, is defined in years. Practitioners commonly prefer to view theta in terms of change in number of days to expiry rather than number of years to expiry. Therefore, reported theta is usually divided by number of days in a year. (Whether to count calendar days or business days varies by personal choice, with arguments for both.)
10026:
677:(or a short call); depending on price, a call option behaves as if one owns 1 share of the underlying stock (if deep in the money), or owns nothing (if far out of the money), or something in between, and conversely for a put option. The difference between the delta of a call and the delta of a put at the same strike is equal to one. By
914:
2866:
2022:
8947:
2150:
9754:
2875:. Zomma is the third derivative of the option value, twice to underlying asset price and once to volatility. Zomma can be a useful sensitivity to monitor when maintaining a gamma-hedged portfolio as zomma will help the trader to anticipate changes to the effectiveness of the hedge as volatility changes.
492:
The players in the market make competitive trades involving many billions (of $ , £ or €) of underlying every day, so it is important to get the sums right. In practice they will use more sophisticated models which go beyond the simplifying assumptions used in the Black-Scholes model and hence in the
2993:
The mathematical result of the formula for color (see below) is expressed in gamma per year. It is often useful to divide this by the number of days per year to arrive at the change in gamma per day. This use is fairly accurate when the number of days remaining until option expiration is large. When
1308:
value in a long option. Sometimes deep-in-the-money options will gain more from increasing discount factors than they lose from decreasing extrinsic value, and reported theta will be a positive value for a long option instead of a more typical negative value (and the option will be an early exercise
763:
Given a
European call and put option for the same underlying, strike price and time to maturity, and with no dividend yield, the sum of the absolute values of the delta of each option will be 1 – more precisely, the delta of the call (positive) minus the delta of the put (negative) equals 1. This is
1266:
As time passes, with decreasing time to expiry and all else being equal, an option's extrinsic value decreases. Typically (but see below), this means an option loses value with time, which is conventionally referred to as long options typically having short (negative) theta. In fact, typically, the
692:
These numbers are commonly presented as a percentage of the total number of shares represented by the option contract(s). This is convenient because the option will (instantaneously) behave like the number of shares indicated by the delta. For example, if a portfolio of 100 American call options on
1862:
Most long options have positive gamma and most short options have negative gamma. Long options have a positive relationship with gamma because as price increases, Gamma increases as well, causing Delta to approach 1 from 0 (long call option) and 0 from −1 (long put option). The inverse is true for
706:
2.75 shares of the underlying. This portfolio will then retain its total value regardless of which direction the price of XYZ moves. (Albeit for only small movements of the underlying, a short amount of time and not-withstanding changes in other market conditions such as volatility and the rate of
10280:
An asset has a Beta of zero if its returns change independently of changes in the market's returns. A positive beta means that the asset's returns generally follow the market's returns, in the sense that they both tend to be above their respective averages together, or both tend to be below their
696:
Delta is always positive for long calls and negative for long puts (unless they are zero). The total delta of a complex portfolio of positions on the same underlying asset can be calculated by simply taking the sum of the deltas for each individual position – delta of a portfolio is linear in the
2290:
increases and short vega as it decreases, which can be scalped in a way analogous to long gamma. And an initially vega-neutral, long-vomma position can be constructed from ratios of options at different strikes. Vomma is positive for long options away from the money, and initially increases with
10213:
of the price of the bond with respect to interest rates (duration is the first derivative); it is then analogous to gamma. In general, the higher the convexity, the more sensitive the bond price is to the change in interest rates. Bond convexity is one of the most basic and widely used forms of
2166:
The mathematical result of the formula for charm (see below) is expressed in delta/year. It is often useful to divide this by the number of days per year to arrive at the delta decay per day. This use is fairly accurate when the number of days remaining until option expiration is large. When an
2533:
The word 'Vera' was coined by R. Naryshkin in early 2012 when this sensitivity needed to be used in practice to assess the impact of volatility changes on rho-hedging, but no name yet existed in the available literature. 'Vera' was picked to sound similar to a combination of Vega and Rho, its
5281:
1303:
While extrinsic value is decreasing with time passing, sometimes a countervailing factor is discounting. For deep-in-the-money options of some types (for puts in Black-Scholes, puts and calls in Black's), as discount factors increase towards 1 with the passage of time, that is an element of
5112:
6183:
750:
calls and puts have a delta of approximately 0.5 and −0.5 respectively with a slight bias towards higher deltas for ATM calls since the risk-free rate introduces some offset to the delta. The negative discounted probability of an option ending up in the money at expiry is called the
8603:
4807:
10237:
are introduced. These values are typically calculated using a tree-based model, built for the entire yield curve (as opposed to a single yield to maturity), and therefore capturing exercise behavior at each point in the option's life as a function of both time and interest rates; see
5840:
2989:
Color is a third-order derivative of the option value, twice to underlying asset price and once to time. Color can be an important sensitivity to monitor when maintaining a gamma-hedged portfolio as it can help the trader to anticipate the effectiveness of the hedge as time passes.
466:) is a primary input into the Black–Scholes model, the overall impact on the value of a short-term option corresponding to changes in the risk-free interest rate is generally insignificant and therefore higher-order derivatives involving the risk-free interest rate are not common.
2425:
8156:
4275:
701:
his entire position in the underlying by buying or shorting the number of shares indicated by the total delta. For example, if the delta of a portfolio of options in XYZ (expressed as shares of the underlying) is +2.75, the trader would be able to delta-hedge the portfolio by
8934:
7981:
5422:
4100:
745:
call option has a delta of 0.15, the trader might estimate that the option has approximately a 15% chance of expiring in-the-money. Similarly, if a put contract has a delta of −0.25, the trader might expect the option to have a 25% probability of expiring in-the-money.
3090:
2984:
2281:
2528:
9761:
5825:
1300:, time to expiry). However, by convention, practitioners usually prefer to refer to theta exposure ("decay") of a long option as negative (instead of the passage of time as negative), and so theta is usually reported as -1 times the first derivative, as above.
778:
919:
For example, if the delta of a call is 0.42 then one can compute the delta of the corresponding put at the same strike price by 0.42 − 1 = −0.58. To derive the delta of a call from a put, one can similarly take −0.58 and add 1 to get 0.42.
2719:
2757:
1915:
7786:
3906:
1857:
10268:
is a number describing the volatility of an asset in relation to the volatility of the benchmark that said asset is being compared to. This benchmark is generally the overall financial market and is often estimated via the use of representative
9160:{\displaystyle {\begin{aligned}d_{1}&={\frac {\ln(F/K)+{\frac {1}{2}}\sigma ^{2}\tau }{\sigma {\sqrt {\tau }}}}\\d_{2}&={\frac {\ln(F/K)-{\frac {1}{2}}\sigma ^{2}\tau }{\sigma {\sqrt {\tau }}}}=d_{1}-\sigma {\sqrt {\tau }}\end{aligned}}}
8793:
4944:
771:
If the value of delta for an option is known, one can calculate the value of the delta of the option of the same strike price, underlying and maturity but opposite right by subtracting 1 from a known call delta or adding 1 to a known put delta.
3133:
measures the rate of change of vega in one underlying due to a change in the level of another underlying. Equivalently, it measures the rate of change of delta in the second underlying due to a change in the volatility of the first underlying.
1881:
When a trader seeks to establish an effective delta-hedge for a portfolio, the trader may also seek to neutralize the portfolio's gamma, as this will ensure that the hedge will be effective over a wider range of underlying price movements.
2044:
9593:
2159:. Charm can be an important Greek to measure/monitor when delta-hedging a position over a weekend. Charm is a second-order derivative of the option value, once to price and once to the passage of time. It is also then the derivative of
6372:
1556:
1166:
Vega can be an important Greek to monitor for an option trader, especially in volatile markets, since the value of some option strategies can be particularly sensitive to changes in volatility. The value of an at-the-money
6726:
1415:
Except under extreme circumstances, the value of an option is less sensitive to changes in the risk-free interest rate than to changes in other parameters. For this reason, rho is the least used of the first-order Greeks.
429:. Rho, lambda, epsilon, and vera are left out as they are not as important as the rest. Three places in the table are not occupied, because the respective quantities have not yet been defined in the financial literature.
5118:
443:
of the value of a portfolio to a small change in a given underlying parameter, so that component risks may be treated in isolation, and the portfolio rebalanced accordingly to achieve a desired exposure; see for example
5507:
4955:
6030:
9350:
8465:
4669:
525:; the use of that name might have led to confusion) and "zomma" are invented, but sound similar to Greek letters. The names "color" and "charm" presumably derive from the use of these terms for exotic properties of
6014:{\displaystyle -e^{-q\tau }{\frac {\varphi (d_{1})}{S^{2}\sigma {\sqrt {\tau }}}}\left({\frac {d_{1}}{\sigma {\sqrt {\tau }}}}+1\right)=-{\frac {\Gamma }{S}}\left({\frac {d_{1}}{\sigma {\sqrt {\tau }}}}+1\right)\,}
8661:
3622:
987:
1419:
Rho is typically expressed as the amount of money, per share of the underlying, that the value of the option will gain or lose as the risk-free interest rate rises or falls by 1.0% per annum (100 basis points).
9586:
9426:
3518:
1727:
5685:
2344:
2611:
1672:
yield, a measure of the dividend risk. The dividend yield impact is in practice determined using a 10% increase in those yields. Obviously, this sensitivity can only be applied to derivative instruments of
7987:
4106:
1604:
1261:
8808:
7818:
5296:
3937:
6523:
9509:
3011:
2900:
10021:{\displaystyle =e^{d_{1}x-{\frac {1}{2}}x^{2}}\cdot e^{{\frac {1}{2}}\cdot {(d_{1}-x)}^{2}-{\frac {1}{2}}\cdot {d_{1}}^{2}}=e^{d_{1}x-{\frac {1}{2}}x^{2}+{\frac {1}{2}}\cdot (2d_{1}-x)(-x)}=e^{0}=1.}
2200:
659:
10281:
respective averages together. A negative beta means that the asset's returns generally move opposite the market's returns: one will tend to be above its average when the other is below its average.
8459:
2451:
1410:
8952:
6744:
5714:
10093:
9232:
4426:
909:{\displaystyle \Delta ({\text{call}})-\Delta ({\text{put}})=1,{\text{ therefore: }}\Delta ({\text{call}})=\Delta ({\text{put}})+1{\text{ and }}\Delta ({\text{put}})=\Delta ({\text{call}})-1.}
4360:
4577:
4515:
1874:
Gamma is greatest approximately at-the-money (ATM) and diminishes the further out you go either in-the-money (ITM) or out-of-the-money (OTM). Gamma is important because it corrects for the
7645:
3765:
6646:
6589:
7585:
3705:
9271:
4634:
7531:
2861:{\displaystyle {\text{Zomma}}={\frac {\partial \Gamma }{\partial \sigma }}={\frac {\partial {\text{vanna}}}{\partial S}}={\frac {\partial ^{3}V}{\partial S^{2}\,\partial \sigma }}}
2642:
489:, a second-order derivative of the value function. The remaining sensitivities in this list are common enough that they have common names, but this list is by no means exhaustive.
8377:
8277:
2017:{\displaystyle {\text{Vanna}}={\frac {\partial \Delta }{\partial \sigma }}={\frac {\partial {\mathcal {V}}}{\partial S}}={\frac {\partial ^{2}V}{\partial S\,\partial \sigma }}.}
2194:. Vomma is the second derivative of the option value with respect to the volatility, or, stated another way, vomma measures the rate of change to vega as volatility changes.
7680:
7673:
3800:
3793:
1782:
1043:
304:
190:
8667:
7466:
7372:
4818:
2430:
It is common practice to divide the mathematical result of veta by 100 times the number of days per year to reduce the value to the percentage change in vega per one day.
1640:
3358:
11025:
10332:
4451:
3382:
10299:
is the expected time to exercise an
American or Bermudan option. Fugit is usefully computed for hedging purposes — for example, one can represent flows of an American
4601:
3188:
2338:
measures the rate of change in the vega with respect to the passage of time. Veta is the second derivative of the value function; once to volatility and once to time.
2145:{\displaystyle {\text{Charm}}=-{\frac {\partial \Delta }{\partial \tau }}={\frac {\partial \Theta }{\partial S}}=-{\frac {\partial ^{2}V}{\partial \tau \,\partial S}}}
1452:
1366:, measures sensitivity to the interest rate: it is the derivative of the option value with respect to the risk-free interest rate (for the relevant outstanding term).
1154:
9749:{\displaystyle {\frac {F}{K}}\cdot {\frac {\varphi (d_{1})}{\varphi (d_{2})}}={\frac {F}{K}}\cdot e^{{\frac {1}{2}}\cdot {d_{2}}^{2}-{\frac {1}{2}}\cdot {d_{1}}^{2}}}
8405:
7811:
7494:
5708:
4662:
3930:
3650:
1770:
1479:
1331:
1298:
1207:
1106:
569:
381:
344:
263:
241:
214:
162:
131:
8180:
4299:
3215:
2445:) measures the rate of change in rho with respect to volatility. Vera is the second derivative of the value function; once to volatility and once to interest rate.
1661:
1364:
1130:
3322:
3298:
3274:
3250:
1732:
Numerically, all first-order sensitivities can be interpreted as spreads in expected returns. Information geometry offers another (trigonometric) interpretation.
6194:
7281:
3419:
1019:
613:
593:
11784:
1499:
12461:
11696:
5276:{\displaystyle -qe^{-q\tau }\Phi (-d_{1})-e^{-q\tau }\varphi (d_{1}){\frac {2(r-q)\tau -d_{2}\sigma {\sqrt {\tau }}}{2\tau \sigma {\sqrt {\tau }}}}\,}
10767:
6657:
5107:{\displaystyle qe^{-q\tau }\Phi (d_{1})-e^{-q\tau }\varphi (d_{1}){\frac {2(r-q)\tau -d_{2}\sigma {\sqrt {\tau }}}{2\tau \sigma {\sqrt {\tau }}}}\,}
1159:
Vega is typically expressed as the amount of money per underlying share that the option's value will gain or lose as volatility rises or falls by 1
6178:{\displaystyle e^{-q\tau }{\frac {\varphi (d_{1})\left(d_{1}d_{2}-1\right)}{S\sigma ^{2}{\sqrt {\tau }}}}=\Gamma {\frac {d_{1}d_{2}-1}{\sigma }}\,}
8598:{\displaystyle e^{-r\tau }{\frac {\varphi (d_{1})}{F\sigma {\sqrt {\tau }}}}=Ke^{-r\tau }{\frac {\varphi (d_{2})}{F^{2}\sigma {\sqrt {\tau }}}}\,}
4802:{\displaystyle e^{-q\tau }{\frac {\varphi (d_{1})}{S\sigma {\sqrt {\tau }}}}=Ke^{-r\tau }{\frac {\varphi (d_{2})}{S^{2}\sigma {\sqrt {\tau }}}}\,}
5433:
9277:
11490:
11065:
8613:
3524:
945:
752:
421:
Definition of Greeks as the sensitivity of an option's price and risk (in the first row) to the underlying parameter (in the first column).
2420:{\displaystyle {\text{Veta}}={\frac {\partial {\mathcal {V}}}{\partial \tau }}={\frac {\partial ^{2}V}{\partial \sigma \,\partial \tau }}}
12438:
10229:
based calculations here do not consider how changes in interest rates will alter the cash flows due to option exercise. To address this,
9515:
9356:
3426:
2549:
This partial derivative has a fundamental role in the
Breeden–Litzenberger formula, which uses quoted call option prices to estimate the
1683:
5518:
3117:
measures the sensitivity of the derivative's value to a change in the correlation between the underlyings. It is also commonly known as
12342:
11495:
8151:{\displaystyle -{\frac {Fe^{-r\tau }\varphi (d_{1})\sigma }{2{\sqrt {\tau }}}}+rKe^{-r\tau }\Phi (-d_{2})-rFe^{-r\tau }\Phi (-d_{1})\,}
4270:{\displaystyle -e^{-q\tau }{\frac {S\varphi (d_{1})\sigma }{2{\sqrt {\tau }}}}+rKe^{-r\tau }\Phi (-d_{2})-qSe^{-q\tau }\Phi (-d_{1})\,}
2559:
8929:{\displaystyle Fe^{-r\tau }\varphi (d_{1}){\sqrt {\tau }}{\frac {d_{1}d_{2}}{\sigma }}={\mathcal {V}}{\frac {d_{1}d_{2}}{\sigma }}\,}
7976:{\displaystyle -{\frac {Fe^{-r\tau }\varphi (d_{1})\sigma }{2{\sqrt {\tau }}}}-rKe^{-r\tau }\Phi (d_{2})+rFe^{-r\tau }\Phi (d_{1})\,}
5417:{\displaystyle Se^{-q\tau }\varphi (d_{1}){\sqrt {\tau }}{\frac {d_{1}d_{2}}{\sigma }}={\mathcal {V}}{\frac {d_{1}d_{2}}{\sigma }}\,}
4095:{\displaystyle -e^{-q\tau }{\frac {S\varphi (d_{1})\sigma }{2{\sqrt {\tau }}}}-rKe^{-r\tau }\Phi (d_{2})+qSe^{-q\tau }\Phi (d_{1})\,}
741:). For this reason some option traders use the absolute value of delta as an approximation for percent moneyness. For example, if an
571:, measures the rate of change of the theoretical option value with respect to changes in the underlying asset's price. Delta is the
1564:
1220:
10883:
3085:{\displaystyle {\text{Ultima}}={\frac {\partial {\text{vomma}}}{\partial \sigma }}={\frac {\partial ^{3}V}{\partial \sigma ^{3}}}}
2979:{\displaystyle {\text{Color}}={\frac {\partial \Gamma }{\partial \tau }}={\frac {\partial ^{3}V}{\partial S^{2}\,\partial \tau }}}
11830:
6387:
2276:{\displaystyle {\text{Vomma}}={\frac {\partial {\mathcal {V}}}{\partial \sigma }}={\frac {\partial ^{2}V}{\partial \sigma ^{2}}}}
2736:
is the third derivative of the value function with respect to the underlying spot price. Speed can be important to monitor when
11520:
10700:
9432:
2523:{\displaystyle {\text{Vera}}={\frac {\partial \rho }{\partial \sigma }}={\frac {\partial ^{2}V}{\partial \sigma \,\partial r}}}
518:
621:
455:(a relatively simple idealised model of certain financial markets) are relatively easy to calculate — a desirable property of
12042:
11984:
11017:
10409:
8410:
5820:{\displaystyle Se^{-q\tau }\varphi (d_{1}){\frac {1}{K^{2}\sigma {\sqrt {\tau }}}}=\left({\frac {S}{K}}\right)^{2}\Gamma \,}
1372:
10904:
10239:
10150:
11652:
10303:
like the flows of a swap starting at the fugit multiplied by delta, and then use these to compute other sensitivities.
10034:
9173:
4366:
2994:
an option nears expiration, color itself may change quickly, rendering full day estimates of gamma change inaccurate.
11388:
10618:
10578:
10547:
10500:
10444:
10375:
10230:
4306:
4521:
4459:
11890:
11058:
3221:
2167:
option nears expiration, charm itself may change quickly, rendering full day estimates of delta decay inaccurate.
7591:
3711:
17:
10675:
10180:
change in the yield (i.e. it is equivalent to DV01 divided by the market price). Unlike the lambda, which is an
6595:
6538:
1772:, measures the rate of change in the delta with respect to changes in the underlying price. Gamma is the second
11825:
11470:
10833:
7537:
3657:
9243:
2714:{\displaystyle {\text{Speed}}={\frac {\partial \Gamma }{\partial S}}={\frac {\partial ^{3}V}{\partial S^{3}}}}
12512:
10401:
4608:
7499:
2291:
distance from the money (but drops off as vega drops off). (Specifically, vomma is positive where the usual
995:
is not the name of any Greek letter. The glyph used is a non-standard majuscule version of the Greek letter
12481:
12316:
3194:
3140:
measures the rate of change of vega in one underlying to a change in the volatility of another underlying.
12304:
11865:
11403:
11257:
11051:
8283:
8187:
12522:
12015:
11721:
11662:
11484:
10780:
7781:{\displaystyle Fe^{-r\tau }\varphi (d_{1}){\sqrt {\tau }}=Ke^{-r\tau }\varphi (d_{2}){\sqrt {\tau }}\,}
3901:{\displaystyle Se^{-q\tau }\varphi (d_{1}){\sqrt {\tau }}=Ke^{-r\tau }\varphi (d_{2}){\sqrt {\tau }}\,}
3127:
measures the rate of change of delta in one underlying to a change in the level of another underlying.
1852:{\displaystyle \Gamma ={\frac {\partial \Delta }{\partial S}}={\frac {\partial ^{2}V}{\partial S^{2}}}}
423:
First-order Greeks are in blue, second-order Greeks are in green, and third-order Greeks are in yellow.
10819:
How to
Calculate Options Prices and Their Greeks: Exploring the Black Scholes Model from Delta to Vega
8788:{\displaystyle -e^{-r\tau }\varphi (d_{1}){\frac {d_{2}}{\sigma }}\,={\frac {\mathcal {V}}{F}}\left\,}
4939:{\displaystyle -e^{-q\tau }\varphi (d_{1}){\frac {d_{2}}{\sigma }}\,={\frac {\mathcal {V}}{S}}\left\,}
768:: a long call plus a short put (a call minus a put) replicates a forward, which has delta equal to 1.
12517:
12443:
12387:
12106:
12033:
11977:
11766:
11577:
10327:
10184:(a percentage change in output for a percentage change in input), the modified duration is instead a
436:
10984:
10952:
7654:
3774:
1609:
It is similar to the concept of delta but expressed in percentage terms rather than absolute terms.
1024:
285:
171:
12280:
12150:
11885:
11880:
10517:
10492:
10432:
10367:
10234:
7378:
7288:
10927:
10856:
10337:
1625:
12382:
12290:
12285:
12249:
12146:
12110:
11835:
11535:
11505:
11480:
11363:
11204:
11136:
10342:
3330:
3167:
3149:
463:
452:
10436:
10426:
10364:
The options applications handbook: hedging and speculating techniques for professional investors
7247:(commonly used for commodities and options on futures) the Greeks can be calculated as follows:
12392:
12189:
12062:
12038:
12023:
11632:
11617:
11582:
11525:
10181:
4436:
3366:
10845:
10800:
10660:
4586:
3173:
1437:
1139:
12417:
12075:
12047:
12028:
11845:
11612:
11510:
11189:
10971:
8390:
7796:
7479:
5693:
4647:
3915:
3635:
1755:
1464:
1315:
1282:
1192:
1091:
554:
366:
329:
248:
226:
199:
147:
116:
66:
is dependent. The name is used because the most common of these sensitivities are denoted by
2534:
respective first-order Greeks. This name is now in a wider use, including, for example, the
12471:
12322:
12234:
12070:
12052:
11970:
11799:
11756:
11746:
11736:
11731:
11457:
11398:
11333:
11287:
11282:
11156:
11116:
11083:
10610:
10158:
10104:
8165:
6367:{\displaystyle -e^{-q\tau }{\frac {\varphi (d_{1})}{2S\tau \sigma {\sqrt {\tau }}}}\left\,}
4284:
3200:
2550:
2191:
1646:
1349:
1115:
936:
932:
738:
514:
510:
426:
63:
47:
31:
11037:
2538:
computer algebra software (which has 'BlackScholesVera' function in its
Finance package).
8:
12491:
12397:
12377:
12204:
12179:
11804:
11592:
11515:
11338:
10265:
10215:
3306:
3282:
3258:
3234:
2617:
1875:
765:
678:
59:
12402:
12347:
12259:
11855:
11840:
11809:
11794:
11761:
11627:
11418:
11383:
11146:
11111:
11074:
10646:
10564:
7266:
3404:
2287:
1490:
598:
578:
572:
521:. Several names such as "vega" (whose symbol is similar to the lower-case Greek letter
501:
The use of Greek letter names is presumably by extension from the common finance terms
440:
43:
2636:
measures the rate of change in Gamma with respect to changes in the underlying price.
1551:{\displaystyle \lambda =\Omega ={\frac {\partial V}{\partial S}}\times {\frac {S}{V}}}
1210:, measures the sensitivity of the value of the derivative to the passage of time (see
1004:
12125:
12101:
12085:
11860:
11850:
11789:
11776:
11751:
11637:
11423:
11219:
10761:
10656:
10614:
10574:
10543:
10496:
10440:
10405:
10395:
10371:
10226:
10210:
10196:
10169:
10146:
10137:
are used analogously to the delta of an option. The closest analogue to the delta is
1309:
candidate, if exercisable, and a
European option may become worth less than parity).
1211:
459:
3005:
measures the sensitivity of the option vomma with respect to change in volatility.
1866:
12229:
11741:
11680:
11675:
11657:
11587:
11353:
11348:
11320:
11272:
11091:
10959:
10844:
Breeden, Litzenberger, Prices of State-Contingent Claims
Implicit in Option Prices
10749:
3155:
2535:
1674:
1489:
change in option value per percentage change in the underlying price, a measure of
1160:
742:
530:
456:
70:(as are some other finance measures). Collectively these have also been called the
51:
11014:
697:
constituents. Since the delta of underlying asset is always 1.0, the trader could
12433:
12337:
12327:
12308:
12275:
12199:
12194:
12130:
12115:
11951:
11921:
11916:
11870:
11706:
11701:
11647:
11557:
11465:
11438:
11378:
11373:
11343:
11292:
11277:
11194:
11174:
11021:
10312:
10270:
10222:
10185:
3218:
3191:
1172:
734:
703:
502:
425:
Vanna, charm and veta appear twice, since partial cross derivatives are equal by
10593:
3111:, its Greeks are extended to include the cross-effects between the underlyings.
12332:
12312:
12080:
11926:
11911:
11711:
11622:
11572:
11549:
11530:
11358:
11300:
11267:
11262:
11242:
11166:
10928:"Rotating Mountain Range Options, Valuation & risks / Performance analysis"
10317:
10251:
10202:
10126:
10120:
6721:{\displaystyle e^{-r\tau }{\frac {\varphi (d_{2})}{K\sigma {\sqrt {\tau }}}}\,}
1748:
1184:
721:
The (absolute value of) Delta is close to, but not identical with, the percent
547:
506:
67:
11007:
10738:
12506:
12407:
12372:
12298:
11906:
11875:
11716:
11642:
11602:
11597:
11433:
11305:
11252:
11247:
11229:
11126:
11106:
10753:
10570:
10322:
10206:
10141:, which is the reduction in price (in currency units) for an increase of one
10134:
10116:
2737:
2038:
measures the instantaneous rate of change of delta over the passage of time.
698:
445:
2751:
measures the rate of change of gamma with respect to changes in volatility.
1163:. All options (both calls and puts) will gain value with rising volatility.
669:
For a vanilla option, delta will be a number between 0.0 and 1.0 for a long
12412:
12367:
12254:
12239:
12120:
11726:
11500:
11428:
11408:
11368:
11237:
11209:
11199:
11141:
10130:
5502:{\displaystyle -K\tau e^{-r\tau }\varphi (d_{2}){\frac {d_{1}}{\sigma }}\,}
2616:
For call options, it can be approximated using infinitesimal portfolios of
2541:
1901:, is a second-order derivative of the option value, once to the underlying
1342:
747:
730:
10607:
The
Handbook of Convertible Bonds: Pricing, Strategies and Risk Management
12244:
12184:
11607:
11475:
11446:
11442:
11393:
11184:
11179:
11006:
Delta, Gamma, GammaP, Gamma symmetry, Vanna, Speed, Charm, Saddle Gamma:
10958:. Humboldt-Universität zu Berlin, Wirtschaftswissenschaftliche Fakultät.
10142:
9345:{\displaystyle d_{1}={\frac {\ln {\frac {F}{K}}+{\frac {1}{2}}x^{2}}{x}}}
7244:
3224:. Note that the gamma and vega formulas are the same for calls and puts.
3159:
996:
670:
11040:, R package to compute Greeks for European-, American- and Asian options
3099:. Ultima is a third-order derivative of the option value to volatility.
1912:
If the underlying value has continuous second partial derivatives, then
755:, which is the first derivative of option price with respect to strike.
689:
with unit factor, so the derivative dF/dS is 1. See the formulas below.
46:; first-order or higher) representing the sensitivity of the price of a
11993:
11931:
11567:
11562:
11328:
11214:
10274:
3163:
3108:
1902:
1773:
1665:
1486:
674:
12010:
11121:
11043:
10637:
Soklakov, A. N. (2023). "Information
Geometry of Risks and Returns".
8656:{\displaystyle ={\frac {\partial ^{2}V}{\partial F\partial \sigma }}}
3617:{\displaystyle e^{-r\tau }K\Phi (-d_{2})-Se^{-q\tau }\Phi (-d_{1})\,}
982:{\displaystyle {\mathcal {V}}={\frac {\partial V}{\partial \sigma }}}
722:
716:
55:
12164:
1171:, for example, is extremely dependent on changes to volatility. See
11691:
11413:
11310:
11131:
10651:
10300:
9581:{\displaystyle {\frac {F}{K}}=e^{d_{1}\cdot x-{\frac {1}{2}}x^{2}}}
9421:{\displaystyle d_{1}\cdot x=\ln {\frac {F}{K}}+{\frac {1}{2}}x^{2}}
3513:{\displaystyle Se^{-q\tau }\Phi (d_{1})-e^{-r\tau }K\Phi (d_{2})\,}
1722:{\displaystyle \varepsilon =\psi ={\frac {\partial V}{\partial q}}}
1669:
1168:
39:
10963:
5680:{\displaystyle -Se^{-q\tau }\varphi (d_{1}){\sqrt {\tau }}\left\,}
10151:
Bond duration § Risk – duration as interest rate sensitivity
1618:
1271:
number. The change in option value is typically negative because
935:. Vega is the derivative of the option value with respect to the
2606:{\displaystyle \varpi ={\frac {\partial ^{2}V}{\partial K^{2}}}}
595:
of the option with respect to the underlying instrument's price
11946:
2894:
measures the rate of change of gamma over the passage of time.
1429:
1267:
literal first derivative w.r.t. time of an option's value is a
1133:
526:
469:
The most common of the Greeks are the first order derivatives:
11962:
10834:
Derivatives – Second-Order Greeks – The
Financial Encyclopedia
10205:
is a measure of the sensitivity of the duration to changes in
10296:
10290:
10261:
1599:{\displaystyle \lambda =\Omega =\Delta \times {\frac {S}{V}}}
1457:
1256:{\displaystyle \Theta =-{\frac {\partial V}{\partial \tau }}}
1084:
1776:
of the value function with respect to the underlying price.
10333:
Greek letters used in mathematics, science, and engineering
10154:
10138:
6518:{\displaystyle {\frac {-{\mathcal {V}}}{\sigma ^{2}}}\left}
1905:
and once to volatility. It is mathematically equivalent to
93:
758:
9504:{\displaystyle \ln(F/K)=d_{1}\cdot x-{\frac {1}{2}}x^{2}}
3107:
If the value of a derivative is dependent on two or more
2286:
With positive vomma, a position will become long vega as
681:, long a call and short a put is equivalent to a forward
10953:"Correlation Risk Premia for Multi-Asset Equity Options"
10240:
Lattice model (finance) § Interest rate derivatives
654:{\displaystyle \Delta ={\frac {\partial V}{\partial S}}}
2542:
Second-order partial derivative with respect to strike
10781:
Derivatives – Delta Decay – The Financial Encyclopedia
10604:
8454:{\displaystyle ={\partial ^{2}V \over \partial F^{2}}}
3360:(represented as a unit-less fraction of one year), and
3143:
1405:{\displaystyle \rho ={\frac {\partial V}{\partial r}}}
1007:
10037:
9764:
9596:
9518:
9435:
9359:
9280:
9246:
9176:
8950:
8811:
8670:
8616:
8468:
8413:
8393:
8286:
8190:
8168:
7990:
7821:
7799:
7683:
7657:
7594:
7540:
7502:
7482:
7381:
7291:
7269:
6742:
6660:
6598:
6541:
6390:
6197:
6033:
5843:
5717:
5696:
5521:
5436:
5299:
5121:
4958:
4821:
4672:
4650:
4611:
4589:
4524:
4462:
4439:
4369:
4309:
4287:
4109:
3940:
3918:
3803:
3777:
3714:
3660:
3638:
3527:
3429:
3407:
3369:
3333:
3309:
3285:
3261:
3237:
3203:
3176:
3014:
2903:
2760:
2645:
2562:
2454:
2347:
2203:
2047:
1918:
1785:
1758:
1686:
1649:
1628:
1567:
1502:
1467:
1440:
1375:
1352:
1318:
1285:
1223:
1195:
1142:
1118:
1094:
1054:
1027:
948:
781:
624:
601:
581:
557:
369:
332:
288:
251:
229:
202:
174:
150:
119:
11038:
greeks: Sensitivities of Prices of Financial Options
10632:
10630:
10482:
10480:
10478:
10476:
513:(the standard deviation of logarithmic returns) and
10474:
10472:
10470:
10468:
10466:
10464:
10462:
10460:
10458:
10456:
10087:
10020:
9748:
9580:
9503:
9420:
9344:
9265:
9226:
9159:
8928:
8787:
8655:
8597:
8453:
8399:
8371:
8271:
8174:
8150:
7975:
7805:
7780:
7667:
7639:
7579:
7525:
7488:
7460:
7366:
7275:
7232:
6720:
6640:
6583:
6517:
6366:
6177:
6013:
5819:
5702:
5679:
5501:
5416:
5275:
5106:
4938:
4801:
4656:
4628:
4595:
4571:
4509:
4445:
4420:
4354:
4293:
4269:
4094:
3924:
3900:
3787:
3759:
3699:
3644:
3616:
3512:
3413:
3376:
3352:
3316:
3292:
3268:
3244:
3209:
3182:
3084:
2978:
2860:
2713:
2605:
2522:
2419:
2275:
2144:
2016:
1851:
1764:
1721:
1655:
1634:
1598:
1550:
1473:
1446:
1404:
1358:
1325:
1292:
1255:
1201:
1148:
1124:
1100:
1037:
1013:
981:
908:
653:
607:
587:
563:
375:
338:
298:
257:
235:
208:
184:
156:
125:
10950:
10627:
10566:Options, Futures, and Other Derivative Securities
10088:{\displaystyle F\varphi (d_{1})=K\varphi (d_{2})}
9227:{\displaystyle F\varphi (d_{1})=K\varphi (d_{2})}
4421:{\displaystyle -K\tau e^{-r\tau }\Phi (-d_{2})\,}
3102:
12504:
10453:
10176:change in the market price of the bond(s) for a
1664:), is the percentage change in option value per
10428:Black–Scholes and beyond: option pricing models
4355:{\displaystyle K\tau e^{-r\tau }\Phi (d_{2})\,}
2305:terms are of the same sign, which is true when
1870:Long option delta, underlying price, and gamma.
10878:
10876:
10149:, where yield is the underlying variable; see
10110:
4572:{\displaystyle S\tau e^{-q\tau }\Phi (-d_{1})}
4510:{\displaystyle -S\tau e^{-q\tau }\Phi (d_{1})}
1108:, is sometimes used (by academics) instead of
11978:
11059:
10951:Fengler, Matthias; Schwendner, Peter (2003).
10766:: CS1 maint: DOI inactive as of March 2024 (
10489:The Complete Guide to Option Pricing Formulas
10389:
10387:
1312:By convention in options valuation formulas,
710:
673:(or a short put) and 0.0 and −1.0 for a long
12462:Alternative investment management companies
12439:Standards Board for Alternative Investments
10873:
10605:De Spiegeleer, Jan; Schoutens, Wim (2015).
10361:
7640:{\displaystyle -e^{-r\tau }\Phi (-d_{1})\,}
3760:{\displaystyle -e^{-q\tau }\Phi (-d_{1})\,}
12487:
12343:Taxation of private equity and hedge funds
11985:
11971:
11066:
11052:
10384:
6641:{\displaystyle e^{-r\tau }\Phi (-d_{2})\,}
6584:{\displaystyle -e^{-r\tau }\Phi (d_{2})\,}
10650:
10393:
10338:PnL explained § Sensitivities method
8925:
8784:
8726:
8594:
8368:
8147:
7972:
7777:
7636:
7580:{\displaystyle e^{-r\tau }\Phi (d_{1})\,}
7576:
7457:
7219:
7143:
6717:
6637:
6580:
6363:
6174:
6010:
5816:
5676:
5498:
5413:
5272:
5103:
4935:
4877:
4798:
4625:
4417:
4351:
4266:
4091:
3897:
3756:
3700:{\displaystyle e^{-q\tau }\Phi (d_{1})\,}
3696:
3613:
3509:
3373:
3349:
3313:
3289:
3265:
3241:
2966:
2848:
2510:
2407:
2132:
2001:
1322:
1289:
10698:
10673:
10636:
9266:{\displaystyle x=\sigma {\sqrt {\tau }}}
2163:with respect to the underlying's price.
1865:
729:probability that the option will expire
27:Model parameters in mathematical finance
11891:Power reverse dual-currency note (PRDC)
11831:Constant proportion portfolio insurance
10946:
10944:
10794:
10792:
10790:
10788:
10732:
10730:
10728:
10726:
10724:
10722:
10720:
10537:
10518:"Options Greeks: Delta Risk and Reward"
4629:{\displaystyle \Delta {\frac {S}{V}}\,}
759:Relationship between call and put delta
58:on which the value of an instrument or
14:
12505:
11073:
10676:"Understanding How Gamma Affect Delta"
10424:
7526:{\displaystyle =\partial V/\partial F}
2724:This is also sometimes referred to as
1735:
12043:fixed-income relative-value investing
11966:
11047:
10829:
10827:
10556:
10191:—a percentage change in output for a
2623:
2190:measures second-order sensitivity to
1049:was adopted because the Greek letter
536:
54:to changes in one or more underlying
11826:Collateralized debt obligation (CDO)
10941:
10798:
10785:
10736:
10717:
10562:
10486:
3095:Ultima has also been referred to as
1079:are pronounced in American English.
707:return for a risk-free investment).
85:
11013:Volga, Vanna, Speed, Charm, Color:
10701:"Why is Long Option Gamma Positive"
10098:
8372:{\displaystyle -\tau e^{-r\tau }\,}
8272:{\displaystyle -\tau e^{-r\tau }\ }
3144:Formulae for European option Greeks
2871:Zomma has also been referred to as
24:
10838:
10824:
10362:Banks, Erik; Siegel, Paul (2006).
8891:
8733:
8644:
8638:
8624:
8435:
8421:
8394:
8343:
8315:
8244:
8219:
8125:
8078:
7953:
7909:
7800:
7660:
7614:
7557:
7517:
7506:
7483:
7432:
7404:
7339:
7314:
7184:
7103:
7061:
6615:
6561:
6399:
6136:
5960:
5813:
5379:
5144:
4978:
4884:
4651:
4612:
4547:
4488:
4395:
4332:
4244:
4197:
4072:
4028:
3919:
3780:
3734:
3677:
3639:
3591:
3547:
3490:
3449:
3204:
3066:
3052:
3036:
3026:
2967:
2953:
2939:
2923:
2918:
2915:
2849:
2835:
2821:
2805:
2795:
2780:
2775:
2772:
2695:
2681:
2665:
2660:
2657:
2587:
2573:
2511:
2504:
2490:
2474:
2466:
2408:
2401:
2387:
2371:
2364:
2359:
2257:
2243:
2227:
2220:
2215:
2133:
2126:
2112:
2093:
2088:
2085:
2070:
2065:
2062:
2002:
1995:
1981:
1965:
1958:
1953:
1938:
1933:
1930:
1833:
1819:
1803:
1798:
1795:
1786:
1759:
1710:
1702:
1580:
1574:
1526:
1518:
1509:
1468:
1393:
1385:
1244:
1236:
1224:
1196:
1143:
1030:
970:
962:
951:
886:
869:
844:
827:
799:
782:
642:
634:
625:
558:
519:Black–Scholes option pricing model
370:
333:
291:
252:
230:
203:
177:
151:
25:
12534:
10995:
10397:Options as a Strategic Investment
3170:are calculated as follows, where
12486:
12477:
12476:
12467:
12466:
12457:
12456:
12163:
11945:
10884:"Greeks for Multi-Asset Options"
10540:Mastering Financial Calculations
10515:
3222:cumulative distribution function
664:
11992:
10920:
10897:
10849:
10811:
10774:
10692:
10667:
10394:Macmillan, Lawrence G. (1993).
11653:Year-on-year inflation-indexed
10821:. John Wiley & Sons. 2015.
10598:
10587:
10531:
10509:
10418:
10355:
10103:Some related risk measures of
10082:
10069:
10057:
10044:
9994:
9985:
9982:
9960:
9853:
9834:
9650:
9637:
9629:
9616:
9456:
9442:
9221:
9208:
9196:
9183:
9083:
9069:
8995:
8981:
8847:
8834:
8706:
8693:
8566:
8553:
8504:
8491:
8365:
8362:
8346:
8334:
8318:
8309:
8263:
8260:
8247:
8235:
8222:
8213:
8144:
8128:
8097:
8081:
8032:
8019:
7969:
7956:
7925:
7912:
7863:
7850:
7767:
7754:
7719:
7706:
7668:{\displaystyle {\mathcal {V}}}
7633:
7617:
7573:
7560:
7454:
7451:
7435:
7423:
7407:
7398:
7358:
7355:
7342:
7330:
7317:
7308:
7070:
7064:
7002:
6996:
6897:
6883:
6787:
6773:
6696:
6683:
6634:
6618:
6577:
6564:
6471:
6442:
6304:
6292:
6236:
6223:
6069:
6056:
5882:
5869:
5753:
5740:
5560:
5547:
5478:
5465:
5335:
5322:
5222:
5210:
5201:
5188:
5163:
5147:
5053:
5041:
5032:
5019:
4994:
4981:
4857:
4844:
4770:
4757:
4708:
4695:
4566:
4550:
4504:
4491:
4414:
4398:
4348:
4335:
4263:
4247:
4216:
4200:
4151:
4138:
4088:
4075:
4044:
4031:
3982:
3969:
3887:
3874:
3839:
3826:
3788:{\displaystyle {\mathcal {V}}}
3753:
3737:
3693:
3680:
3610:
3594:
3566:
3550:
3506:
3493:
3465:
3452:
3103:Greeks for multi-asset options
2740:or gamma-hedging a portfolio.
1038:{\displaystyle {\mathcal {V}}}
897:
889:
880:
872:
855:
847:
838:
830:
810:
802:
793:
785:
685:, which is linear in the spot
435:The Greeks are vital tools in
299:{\displaystyle {\mathcal {V}}}
185:{\displaystyle {\mathcal {V}}}
13:
1:
11663:Zero-coupon inflation-indexed
10699:Willette, Jeff (2014-05-28).
10674:Willette, Jeff (2014-05-28).
10487:Haug, Espen Gaardner (2007).
10402:New York Institute of Finance
10348:
7461:{\displaystyle e^{-r\tau }\,}
7367:{\displaystyle e^{-r\tau }\ }
38:are the quantities (known in
12317:security characteristic line
11026:Vanilla Options - Uwe Wystup
11015:Vanilla Options - Uwe Wystup
11008:Vanilla Options - Espen Haug
10799:Haug, Espen Gaarder (2003),
10737:Haug, Espen Gaarder (2003),
10542:(3rd ed.). Pearson UK.
3195:probability density function
1635:{\displaystyle \varepsilon }
1493:, sometimes called gearing.
7:
12305:Capital asset pricing model
12024:Capital structure arbitrage
11866:Foreign exchange derivative
11258:Callable bull/bear contract
10306:
10165:
10157:, measuring sensitivity to
10111:Bond duration and convexity
3353:{\displaystyle \tau =T-t\,}
2155:Charm has also been called
1156:), though these are rare).
733:(if the market moves under
408:
10:
12539:
12107:Commodity trading advisors
10801:"Know Your Weapon, Part 2"
10739:"Know Your Weapon, Part 1"
10288:
10249:
10195:change in input. See also
10145:(i.e. 0.01% per annum) in
10114:
3147:
2551:risk-neutral probabilities
2160:
1612:
714:
711:As a proxy for probability
486:
478:
470:
439:. Each Greek measures the
396:
391:
386:
349:
314:
309:
275:
270:
265:
216:
164:
12452:
12444:Managed Funds Association
12426:
12388:High-net-worth individual
12360:
12268:
12222:
12213:
12172:
12161:
12139:
12094:
12061:
12009:
12000:
11940:
11899:
11818:
11775:
11767:Stock market index future
11671:
11548:
11456:
11319:
11228:
11165:
11099:
11090:
11081:
10613:. pp. 255, 269–270.
10328:Financial risk management
9170:(*) It can be shown that
8805:
8798:
8664:
8462:
8382:
7677:
7471:
6654:
6528:
6384:
6377:
6191:
6027:
5837:
5830:
5711:
5515:
5430:
5293:
5286:
4815:
4666:
4639:
4605:
4446:{\displaystyle \epsilon }
3797:
3627:
3377:{\displaystyle \sigma \,}
2997:
1668:change in the underlying
1423:
939:of the underlying asset.
474:
354:
319:
192:
101:
96:
12281:Arbitrage pricing theory
11886:Mortgage-backed security
11881:Interest rate derivative
11856:Equity-linked note (ELN)
11841:Credit-linked note (CLN)
10861:Investment & Finance
10754:10.1002/wilm.42820030313
10493:McGraw-Hill Professional
10433:McGraw-Hill Professional
10368:McGraw-Hill Professional
10284:
4596:{\displaystyle \lambda }
3183:{\displaystyle \varphi }
2878:
2743:
2628:
2553:implied by such prices.
2170:
2026:
1885:
1742:
1447:{\displaystyle \lambda }
1275:is a negative number (a
1178:
1149:{\displaystyle \Lambda }
931:measures sensitivity to
725:of an option, i.e., the
541:
517:(time to expiry) in the
496:
482:
12393:Institutional investors
12286:Assets under management
12111:managed futures account
11836:Contract for difference
11137:Risk-free interest rate
10245:
10131:fixed income securities
8400:{\displaystyle \Gamma }
7806:{\displaystyle \Theta }
7489:{\displaystyle \Delta }
5703:{\displaystyle \varpi }
4657:{\displaystyle \Gamma }
3925:{\displaystyle \Theta }
3645:{\displaystyle \Delta }
2433:
2322:
1765:{\displaystyle \Gamma }
1474:{\displaystyle \Omega }
1326:{\displaystyle \tau \,}
1293:{\displaystyle \tau \,}
1202:{\displaystyle \Theta }
1101:{\displaystyle \kappa }
923:
564:{\displaystyle \Delta }
464:risk-free interest rate
376:{\displaystyle \Gamma }
339:{\displaystyle \Theta }
258:{\displaystyle \Gamma }
236:{\displaystyle \Delta }
209:{\displaystyle \Theta }
157:{\displaystyle \Delta }
126:{\displaystyle \sigma }
12418:Sovereign wealth funds
12190:High-frequency trading
12039:Fixed income arbitrage
11618:Forward Rate Agreement
10979:Cite journal requires
10857:"Derivatives – Greeks"
10563:Hull, John C. (1993).
10133:, various measures of
10096:
10089:
10022:
9750:
9582:
9505:
9422:
9346:
9267:
9228:
9161:
8930:
8789:
8657:
8599:
8455:
8401:
8373:
8273:
8176:
8152:
7977:
7807:
7782:
7669:
7641:
7581:
7527:
7490:
7462:
7368:
7277:
7234:
6722:
6642:
6585:
6519:
6368:
6179:
6015:
5821:
5704:
5681:
5503:
5418:
5277:
5108:
4940:
4803:
4658:
4630:
4597:
4573:
4511:
4447:
4422:
4356:
4295:
4271:
4096:
3926:
3902:
3789:
3761:
3701:
3646:
3618:
3514:
3415:
3378:
3354:
3318:
3303:Annual dividend yield
3294:
3270:
3246:
3211:
3184:
3086:
2980:
2862:
2726:the gamma of the gamma
2715:
2607:
2524:
2421:
2277:
2146:
2018:
1893:, also referred to as
1871:
1853:
1766:
1723:
1657:
1636:
1600:
1552:
1475:
1448:
1406:
1360:
1336:
1327:
1294:
1257:
1203:
1150:
1126:
1102:
1045:. Presumably the name
1039:
1015:
983:
910:
824: therefore:
655:
609:
589:
565:
377:
340:
300:
259:
237:
210:
186:
158:
127:
12260:Structured securities
12076:Distressed securities
12048:Statistical arbitrage
12034:Equity market neutral
12029:Convertible arbitrage
11846:Credit default option
11190:Employee stock option
10756:(inactive 2024-03-19)
10611:John Wiley & Sons
10538:Steiner, Bob (2013).
10425:Chriss, Neil (1996).
10105:financial instruments
10090:
10023:
9751:
9583:
9506:
9423:
9347:
9268:
9238:
9229:
9162:
8931:
8790:
8658:
8600:
8456:
8402:
8374:
8274:
8177:
8175:{\displaystyle \rho }
8153:
7978:
7808:
7783:
7670:
7642:
7582:
7528:
7491:
7463:
7369:
7278:
7235:
6723:
6643:
6586:
6520:
6369:
6180:
6016:
5822:
5705:
5682:
5504:
5419:
5278:
5109:
4941:
4804:
4659:
4631:
4598:
4574:
4512:
4448:
4423:
4357:
4296:
4294:{\displaystyle \rho }
4272:
4097:
3927:
3903:
3790:
3762:
3702:
3647:
3619:
3515:
3416:
3379:
3355:
3319:
3295:
3271:
3247:
3212:
3210:{\displaystyle \Phi }
3185:
3087:
2981:
2863:
2716:
2608:
2525:
2422:
2312: < 0 or
2278:
2147:
2019:
1869:
1854:
1767:
1724:
1658:
1656:{\displaystyle \psi }
1637:
1601:
1553:
1476:
1449:
1407:
1361:
1359:{\displaystyle \rho }
1328:
1295:
1258:
1214:): the "time decay."
1204:
1151:
1127:
1125:{\displaystyle \tau }
1103:
1040:
1016:
984:
911:
656:
610:
590:
566:
378:
341:
301:
260:
238:
211:
187:
159:
142:Value (V)
128:
64:financial instruments
48:derivative instrument
12513:Mathematical finance
12378:Financial endowments
12323:Fundamental analysis
12071:Shareholder activism
12053:Volatility arbitrage
11800:Inflation derivative
11785:Commodity derivative
11757:Single-stock futures
11747:Normal backwardation
11737:Interest rate future
11578:Conditional variance
11084:Derivative (finance)
10705:www.traderbrains.com
10680:www.traderbrains.com
10594:Omega – Investopedia
10216:convexity in finance
10035:
9762:
9594:
9516:
9433:
9357:
9278:
9244:
9174:
8948:
8809:
8668:
8614:
8466:
8411:
8391:
8284:
8188:
8166:
7988:
7819:
7797:
7681:
7655:
7592:
7538:
7500:
7480:
7379:
7289:
7267:
6740:
6658:
6596:
6539:
6388:
6195:
6031:
5841:
5715:
5694:
5519:
5434:
5297:
5119:
4956:
4819:
4670:
4648:
4609:
4587:
4522:
4460:
4437:
4367:
4307:
4285:
4107:
3938:
3916:
3801:
3775:
3712:
3658:
3636:
3525:
3427:
3405:
3367:
3331:
3307:
3283:
3259:
3235:
3201:
3174:
3012:
2901:
2758:
2643:
2560:
2452:
2345:
2319: > 0.)
2201:
2045:
1916:
1783:
1756:
1684:
1647:
1642:(also known as psi,
1626:
1565:
1500:
1465:
1438:
1373:
1350:
1316:
1283:
1221:
1193:
1140:
1116:
1092:
1067:by analogy with how
1053:looked like a Latin
1025:
1005:
946:
779:
739:risk-neutral measure
622:
599:
579:
555:
367:
330:
286:
249:
227:
200:
172:
148:
117:
32:mathematical finance
12492:List of hedge funds
12482:Hedge fund managers
12398:Insurance companies
12383:Fund of hedge funds
12291:Black–Scholes model
12205:Proprietary trading
12180:Algorithmic trading
12147:Fund of hedge funds
11952:Business portal
11805:Property derivative
10748:(May 2003): 49–57,
10343:Vanna–Volga pricing
10235:effective convexity
10221:For a bond with an
7188:
7112:
6509:
6491:
3317:{\displaystyle q\,}
3293:{\displaystyle r\,}
3269:{\displaystyle K\,}
3245:{\displaystyle S\,}
3168:Black–Scholes model
3150:Black–Scholes model
1736:Second-order Greeks
1273:the passage of time
453:Black–Scholes model
44:partial derivatives
12348:Technical analysis
11810:Weather derivative
11795:Freight derivative
11777:Exotic derivatives
11697:Commodities future
11384:Intermarket spread
11147:Synthetic position
11075:Derivatives market
11020:2007-09-28 at the
10807:(July 2003): 43–57
10231:effective duration
10107:are listed below.
10085:
10018:
9746:
9578:
9501:
9418:
9342:
9263:
9224:
9157:
9155:
8926:
8785:
8653:
8595:
8451:
8397:
8369:
8269:
8172:
8148:
7973:
7803:
7778:
7665:
7637:
7577:
7523:
7486:
7458:
7364:
7273:
7230:
7228:
7174:
7095:
6718:
6638:
6581:
6515:
6495:
6477:
6364:
6175:
6011:
5817:
5700:
5677:
5499:
5414:
5273:
5104:
4936:
4799:
4654:
4626:
4593:
4569:
4507:
4443:
4418:
4352:
4291:
4267:
4092:
3922:
3898:
3785:
3757:
3697:
3642:
3614:
3510:
3411:
3374:
3350:
3314:
3290:
3266:
3242:
3207:
3180:
3082:
2976:
2858:
2711:
2624:Third-order Greeks
2603:
2520:
2417:
2288:implied volatility
2273:
2142:
2014:
1872:
1849:
1762:
1719:
1653:
1632:
1596:
1548:
1471:
1444:
1402:
1356:
1323:
1290:
1253:
1199:
1146:
1122:
1098:
1035:
1011:
979:
906:
651:
605:
585:
561:
537:First-order Greeks
451:The Greeks in the
373:
336:
296:
255:
233:
206:
182:
154:
123:
72:risk sensitivities
12523:Options (finance)
12500:
12499:
12356:
12355:
12159:
12158:
12126:Long/short equity
12102:Convergence trade
12086:Special situation
11960:
11959:
11861:Equity derivative
11851:Credit derivative
11819:Other derivatives
11790:Energy derivative
11752:Perpetual futures
11633:Overnight indexed
11583:Constant maturity
11544:
11543:
11491:Finite difference
11424:Protective option
10411:978-0-13-636002-5
10227:yield to maturity
10211:second derivative
10197:Key rate duration
10170:modified duration
9955:
9932:
9874:
9827:
9797:
9720:
9685:
9667:
9654:
9605:
9564:
9527:
9489:
9406:
9393:
9340:
9324:
9311:
9261:
9151:
9125:
9122:
9097:
9037:
9034:
9009:
8939:
8938:
8923:
8884:
8855:
8777:
8774:
8740:
8724:
8651:
8592:
8589:
8523:
8520:
8449:
8268:
8051:
8048:
7882:
7879:
7775:
7727:
7363:
7276:{\displaystyle V}
7205:
7172:
7171:
7129:
7093:
7092:
7043:
7025:
7024:
6987:
6961:
6958:
6928:
6851:
6848:
6818:
6731:
6730:
6715:
6712:
6415:
6346:
6343:
6331:
6261:
6258:
6172:
6131:
6128:
5997:
5994:
5966:
5939:
5936:
5908:
5905:
5801:
5783:
5780:
5669:
5626:
5623:
5568:
5496:
5411:
5372:
5343:
5270:
5267:
5249:
5101:
5098:
5080:
4928:
4925:
4891:
4875:
4796:
4793:
4727:
4724:
4623:
4170:
4167:
4001:
3998:
3895:
3847:
3414:{\displaystyle V}
3327:Time to maturity
3115:Correlation delta
3080:
3043:
3032:
3018:
2974:
2930:
2907:
2856:
2812:
2801:
2787:
2764:
2709:
2672:
2649:
2601:
2518:
2481:
2458:
2415:
2378:
2351:
2271:
2234:
2207:
2140:
2100:
2077:
2051:
2009:
1972:
1945:
1922:
1847:
1810:
1717:
1594:
1546:
1533:
1400:
1251:
1212:Option time value
1063:was derived from
1014:{\textstyle \nu }
977:
895:
878:
867:
853:
836:
825:
808:
791:
649:
608:{\displaystyle S}
588:{\displaystyle V}
509:, and the use of
433:
432:
427:Schwarz's theorem
416:
415:
102:Option parameter
86:Use of the Greeks
16:(Redirected from
12530:
12518:Financial ratios
12490:
12489:
12480:
12479:
12470:
12469:
12460:
12459:
12403:Investment banks
12250:Foreign exchange
12220:
12219:
12167:
12007:
12006:
11987:
11980:
11973:
11964:
11963:
11950:
11949:
11722:Forwards pricing
11496:Garman–Kohlhagen
11097:
11096:
11068:
11061:
11054:
11045:
11044:
10989:
10988:
10982:
10977:
10975:
10967:
10957:
10948:
10939:
10938:
10936:
10934:
10924:
10918:
10917:
10915:
10913:
10906:Correlation Risk
10901:
10895:
10894:
10892:
10890:
10880:
10871:
10870:
10868:
10867:
10853:
10847:
10842:
10836:
10831:
10822:
10817:Pierino Ursone.
10815:
10809:
10808:
10805:Wilmott Magazine
10796:
10783:
10778:
10772:
10771:
10765:
10757:
10746:Wilmott Magazine
10743:
10734:
10715:
10714:
10712:
10711:
10696:
10690:
10689:
10687:
10686:
10671:
10665:
10664:
10654:
10634:
10625:
10624:
10602:
10596:
10591:
10585:
10584:
10569:(2nd ed.).
10560:
10554:
10553:
10535:
10529:
10528:
10526:
10524:
10513:
10507:
10506:
10484:
10451:
10450:
10422:
10416:
10415:
10400:(3rd ed.).
10391:
10382:
10381:
10359:
10099:Related measures
10094:
10092:
10091:
10086:
10081:
10080:
10056:
10055:
10027:
10025:
10024:
10019:
10011:
10010:
9998:
9997:
9975:
9974:
9956:
9948:
9943:
9942:
9933:
9925:
9917:
9916:
9899:
9898:
9897:
9896:
9891:
9890:
9889:
9875:
9867:
9862:
9861:
9856:
9846:
9845:
9828:
9820:
9810:
9809:
9808:
9807:
9798:
9790:
9782:
9781:
9755:
9753:
9752:
9747:
9745:
9744:
9743:
9742:
9737:
9736:
9735:
9721:
9713:
9708:
9707:
9702:
9701:
9700:
9686:
9678:
9668:
9660:
9655:
9653:
9649:
9648:
9632:
9628:
9627:
9611:
9606:
9598:
9587:
9585:
9584:
9579:
9577:
9576:
9575:
9574:
9565:
9557:
9546:
9545:
9528:
9520:
9510:
9508:
9507:
9502:
9500:
9499:
9490:
9482:
9471:
9470:
9452:
9427:
9425:
9424:
9419:
9417:
9416:
9407:
9399:
9394:
9386:
9369:
9368:
9351:
9349:
9348:
9343:
9341:
9336:
9335:
9334:
9325:
9317:
9312:
9304:
9295:
9290:
9289:
9272:
9270:
9269:
9264:
9262:
9257:
9233:
9231:
9230:
9225:
9220:
9219:
9195:
9194:
9166:
9164:
9163:
9158:
9156:
9152:
9147:
9139:
9138:
9126:
9124:
9123:
9118:
9112:
9108:
9107:
9098:
9090:
9079:
9061:
9052:
9051:
9038:
9036:
9035:
9030:
9024:
9020:
9019:
9010:
9002:
8991:
8973:
8964:
8963:
8935:
8933:
8932:
8927:
8924:
8919:
8918:
8917:
8908:
8907:
8897:
8895:
8894:
8885:
8880:
8879:
8878:
8869:
8868:
8858:
8856:
8851:
8846:
8845:
8830:
8829:
8794:
8792:
8791:
8786:
8783:
8779:
8778:
8776:
8775:
8770:
8764:
8763:
8754:
8741:
8736:
8731:
8725:
8720:
8719:
8710:
8705:
8704:
8689:
8688:
8662:
8660:
8659:
8654:
8652:
8650:
8636:
8632:
8631:
8621:
8604:
8602:
8601:
8596:
8593:
8591:
8590:
8585:
8580:
8579:
8569:
8565:
8564:
8548:
8546:
8545:
8524:
8522:
8521:
8516:
8507:
8503:
8502:
8486:
8484:
8483:
8460:
8458:
8457:
8452:
8450:
8448:
8447:
8446:
8433:
8429:
8428:
8418:
8406:
8404:
8403:
8398:
8378:
8376:
8375:
8370:
8361:
8360:
8333:
8332:
8308:
8307:
8278:
8276:
8275:
8270:
8266:
8259:
8258:
8234:
8233:
8212:
8211:
8181:
8179:
8178:
8173:
8157:
8155:
8154:
8149:
8143:
8142:
8124:
8123:
8096:
8095:
8077:
8076:
8052:
8050:
8049:
8044:
8038:
8031:
8030:
8015:
8014:
7995:
7982:
7980:
7979:
7974:
7968:
7967:
7952:
7951:
7924:
7923:
7908:
7907:
7883:
7881:
7880:
7875:
7869:
7862:
7861:
7846:
7845:
7826:
7812:
7810:
7809:
7804:
7787:
7785:
7784:
7779:
7776:
7771:
7766:
7765:
7750:
7749:
7728:
7723:
7718:
7717:
7702:
7701:
7674:
7672:
7671:
7666:
7664:
7663:
7646:
7644:
7643:
7638:
7632:
7631:
7613:
7612:
7586:
7584:
7583:
7578:
7572:
7571:
7556:
7555:
7532:
7530:
7529:
7524:
7516:
7495:
7493:
7492:
7487:
7467:
7465:
7464:
7459:
7450:
7449:
7422:
7421:
7397:
7396:
7373:
7371:
7370:
7365:
7361:
7354:
7353:
7329:
7328:
7307:
7306:
7282:
7280:
7279:
7274:
7250:
7249:
7239:
7237:
7236:
7231:
7229:
7218:
7217:
7216:
7215:
7206:
7198:
7187:
7182:
7173:
7164:
7160:
7142:
7141:
7140:
7139:
7130:
7122:
7111:
7106:
7094:
7085:
7081:
7056:
7055:
7054:
7053:
7044:
7036:
7026:
7017:
7013:
6988:
6983:
6975:
6974:
6962:
6960:
6959:
6954:
6948:
6944:
6940:
6939:
6938:
6929:
6921:
6893:
6875:
6866:
6865:
6852:
6850:
6849:
6844:
6838:
6834:
6830:
6829:
6828:
6819:
6811:
6783:
6765:
6756:
6755:
6727:
6725:
6724:
6719:
6716:
6714:
6713:
6708:
6699:
6695:
6694:
6678:
6676:
6675:
6647:
6645:
6644:
6639:
6633:
6632:
6614:
6613:
6590:
6588:
6587:
6582:
6576:
6575:
6560:
6559:
6524:
6522:
6521:
6516:
6514:
6510:
6508:
6503:
6490:
6485:
6470:
6469:
6460:
6459:
6441:
6440:
6431:
6430:
6416:
6414:
6413:
6404:
6403:
6402:
6392:
6373:
6371:
6370:
6365:
6362:
6358:
6357:
6356:
6347:
6345:
6344:
6339:
6333:
6332:
6327:
6322:
6321:
6287:
6262:
6260:
6259:
6254:
6239:
6235:
6234:
6218:
6216:
6215:
6184:
6182:
6181:
6176:
6173:
6168:
6161:
6160:
6151:
6150:
6140:
6132:
6130:
6129:
6124:
6122:
6121:
6108:
6107:
6103:
6096:
6095:
6086:
6085:
6068:
6067:
6051:
6049:
6048:
6020:
6018:
6017:
6012:
6009:
6005:
5998:
5996:
5995:
5990:
5984:
5983:
5974:
5967:
5959:
5951:
5947:
5940:
5938:
5937:
5932:
5926:
5925:
5916:
5909:
5907:
5906:
5901:
5896:
5895:
5885:
5881:
5880:
5864:
5862:
5861:
5826:
5824:
5823:
5818:
5812:
5811:
5806:
5802:
5794:
5784:
5782:
5781:
5776:
5771:
5770:
5757:
5752:
5751:
5736:
5735:
5709:
5707:
5706:
5701:
5686:
5684:
5683:
5678:
5675:
5671:
5670:
5668:
5660:
5659:
5658:
5649:
5648:
5632:
5627:
5625:
5624:
5619:
5613:
5612:
5611:
5602:
5598:
5582:
5569:
5564:
5559:
5558:
5543:
5542:
5508:
5506:
5505:
5500:
5497:
5492:
5491:
5482:
5477:
5476:
5461:
5460:
5423:
5421:
5420:
5415:
5412:
5407:
5406:
5405:
5396:
5395:
5385:
5383:
5382:
5373:
5368:
5367:
5366:
5357:
5356:
5346:
5344:
5339:
5334:
5333:
5318:
5317:
5282:
5280:
5279:
5274:
5271:
5269:
5268:
5263:
5251:
5250:
5245:
5240:
5239:
5205:
5200:
5199:
5184:
5183:
5162:
5161:
5143:
5142:
5113:
5111:
5110:
5105:
5102:
5100:
5099:
5094:
5082:
5081:
5076:
5071:
5070:
5036:
5031:
5030:
5015:
5014:
4993:
4992:
4977:
4976:
4945:
4943:
4942:
4937:
4934:
4930:
4929:
4927:
4926:
4921:
4915:
4914:
4905:
4892:
4887:
4882:
4876:
4871:
4870:
4861:
4856:
4855:
4840:
4839:
4808:
4806:
4805:
4800:
4797:
4795:
4794:
4789:
4784:
4783:
4773:
4769:
4768:
4752:
4750:
4749:
4728:
4726:
4725:
4720:
4711:
4707:
4706:
4690:
4688:
4687:
4663:
4661:
4660:
4655:
4635:
4633:
4632:
4627:
4624:
4616:
4602:
4600:
4599:
4594:
4578:
4576:
4575:
4570:
4565:
4564:
4546:
4545:
4516:
4514:
4513:
4508:
4503:
4502:
4487:
4486:
4452:
4450:
4449:
4444:
4427:
4425:
4424:
4419:
4413:
4412:
4394:
4393:
4361:
4359:
4358:
4353:
4347:
4346:
4331:
4330:
4300:
4298:
4297:
4292:
4276:
4274:
4273:
4268:
4262:
4261:
4243:
4242:
4215:
4214:
4196:
4195:
4171:
4169:
4168:
4163:
4157:
4150:
4149:
4130:
4128:
4127:
4101:
4099:
4098:
4093:
4087:
4086:
4071:
4070:
4043:
4042:
4027:
4026:
4002:
4000:
3999:
3994:
3988:
3981:
3980:
3961:
3959:
3958:
3931:
3929:
3928:
3923:
3907:
3905:
3904:
3899:
3896:
3891:
3886:
3885:
3870:
3869:
3848:
3843:
3838:
3837:
3822:
3821:
3794:
3792:
3791:
3786:
3784:
3783:
3766:
3764:
3763:
3758:
3752:
3751:
3733:
3732:
3706:
3704:
3703:
3698:
3692:
3691:
3676:
3675:
3651:
3649:
3648:
3643:
3623:
3621:
3620:
3615:
3609:
3608:
3590:
3589:
3565:
3564:
3543:
3542:
3519:
3517:
3516:
3511:
3505:
3504:
3486:
3485:
3464:
3463:
3448:
3447:
3420:
3418:
3417:
3412:
3388:
3387:
3383:
3381:
3380:
3375:
3359:
3357:
3356:
3351:
3323:
3321:
3320:
3315:
3299:
3297:
3296:
3291:
3275:
3273:
3272:
3267:
3251:
3249:
3248:
3243:
3216:
3214:
3213:
3208:
3189:
3187:
3186:
3181:
3156:European options
3091:
3089:
3088:
3083:
3081:
3079:
3078:
3077:
3064:
3060:
3059:
3049:
3044:
3042:
3034:
3033:
3030:
3024:
3019:
3016:
2985:
2983:
2982:
2977:
2975:
2973:
2965:
2964:
2951:
2947:
2946:
2936:
2931:
2929:
2921:
2913:
2908:
2905:
2867:
2865:
2864:
2859:
2857:
2855:
2847:
2846:
2833:
2829:
2828:
2818:
2813:
2811:
2803:
2802:
2799:
2793:
2788:
2786:
2778:
2770:
2765:
2762:
2720:
2718:
2717:
2712:
2710:
2708:
2707:
2706:
2693:
2689:
2688:
2678:
2673:
2671:
2663:
2655:
2650:
2647:
2612:
2610:
2609:
2604:
2602:
2600:
2599:
2598:
2585:
2581:
2580:
2570:
2529:
2527:
2526:
2521:
2519:
2517:
2502:
2498:
2497:
2487:
2482:
2480:
2472:
2464:
2459:
2456:
2426:
2424:
2423:
2418:
2416:
2414:
2399:
2395:
2394:
2384:
2379:
2377:
2369:
2368:
2367:
2357:
2352:
2349:
2282:
2280:
2279:
2274:
2272:
2270:
2269:
2268:
2255:
2251:
2250:
2240:
2235:
2233:
2225:
2224:
2223:
2213:
2208:
2205:
2151:
2149:
2148:
2143:
2141:
2139:
2124:
2120:
2119:
2109:
2101:
2099:
2091:
2083:
2078:
2076:
2068:
2060:
2052:
2049:
2023:
2021:
2020:
2015:
2010:
2008:
1993:
1989:
1988:
1978:
1973:
1971:
1963:
1962:
1961:
1951:
1946:
1944:
1936:
1928:
1923:
1920:
1858:
1856:
1855:
1850:
1848:
1846:
1845:
1844:
1831:
1827:
1826:
1816:
1811:
1809:
1801:
1793:
1771:
1769:
1768:
1763:
1728:
1726:
1725:
1720:
1718:
1716:
1708:
1700:
1662:
1660:
1659:
1654:
1641:
1639:
1638:
1633:
1605:
1603:
1602:
1597:
1595:
1587:
1557:
1555:
1554:
1549:
1547:
1539:
1534:
1532:
1524:
1516:
1480:
1478:
1477:
1472:
1453:
1451:
1450:
1445:
1411:
1409:
1408:
1403:
1401:
1399:
1391:
1383:
1365:
1363:
1362:
1357:
1332:
1330:
1329:
1324:
1299:
1297:
1296:
1291:
1262:
1260:
1259:
1254:
1252:
1250:
1242:
1234:
1208:
1206:
1205:
1200:
1161:percentage point
1155:
1153:
1152:
1147:
1131:
1129:
1128:
1123:
1107:
1105:
1104:
1099:
1044:
1042:
1041:
1036:
1034:
1033:
1020:
1018:
1017:
1012:
988:
986:
985:
980:
978:
976:
968:
960:
955:
954:
915:
913:
912:
907:
896:
893:
879:
876:
868:
865:
854:
851:
837:
834:
826:
823:
809:
806:
792:
789:
743:out-of-the-money
660:
658:
657:
652:
650:
648:
640:
632:
614:
612:
611:
606:
594:
592:
591:
586:
573:first derivative
570:
568:
567:
562:
531:particle physics
382:
380:
379:
374:
345:
343:
342:
337:
305:
303:
302:
297:
295:
294:
264:
262:
261:
256:
242:
240:
239:
234:
215:
213:
212:
207:
191:
189:
188:
183:
181:
180:
163:
161:
160:
155:
132:
130:
129:
124:
94:
90:
89:
80:hedge parameters
21:
12538:
12537:
12533:
12532:
12531:
12529:
12528:
12527:
12503:
12502:
12501:
12496:
12448:
12434:Fund governance
12422:
12352:
12276:Absolute return
12264:
12215:
12209:
12200:Program trading
12195:Prime brokerage
12168:
12155:
12135:
12131:Trend following
12116:Dedicated short
12090:
12057:
12014:
12002:
11996:
11991:
11961:
11956:
11944:
11936:
11922:Great Recession
11917:Government debt
11895:
11871:Fund derivative
11814:
11771:
11732:Futures pricing
11707:Dividend future
11702:Currency future
11685:
11667:
11540:
11516:Put–call parity
11452:
11439:Vertical spread
11374:Diagonal spread
11344:Calendar spread
11315:
11224:
11161:
11086:
11077:
11072:
11022:Wayback Machine
10998:
10993:
10992:
10980:
10978:
10969:
10968:
10955:
10949:
10942:
10932:
10930:
10926:
10925:
10921:
10911:
10909:
10903:
10902:
10898:
10888:
10886:
10882:
10881:
10874:
10865:
10863:
10855:
10854:
10850:
10843:
10839:
10832:
10825:
10816:
10812:
10797:
10786:
10779:
10775:
10759:
10758:
10741:
10735:
10718:
10709:
10707:
10697:
10693:
10684:
10682:
10672:
10668:
10635:
10628:
10621:
10603:
10599:
10592:
10588:
10581:
10561:
10557:
10550:
10536:
10532:
10522:
10520:
10514:
10510:
10503:
10485:
10454:
10447:
10423:
10419:
10412:
10392:
10385:
10378:
10370:. p. 263.
10360:
10356:
10351:
10313:Alpha (finance)
10309:
10293:
10287:
10254:
10248:
10225:, the standard
10223:embedded option
10172:, which is the
10123:
10115:Main articles:
10113:
10101:
10076:
10072:
10051:
10047:
10036:
10033:
10032:
10006:
10002:
9970:
9966:
9947:
9938:
9934:
9924:
9912:
9908:
9907:
9903:
9892:
9885:
9881:
9880:
9879:
9866:
9857:
9841:
9837:
9833:
9832:
9819:
9818:
9814:
9803:
9799:
9789:
9777:
9773:
9772:
9768:
9763:
9760:
9759:
9738:
9731:
9727:
9726:
9725:
9712:
9703:
9696:
9692:
9691:
9690:
9677:
9676:
9672:
9659:
9644:
9640:
9633:
9623:
9619:
9612:
9610:
9597:
9595:
9592:
9591:
9570:
9566:
9556:
9541:
9537:
9536:
9532:
9519:
9517:
9514:
9513:
9495:
9491:
9481:
9466:
9462:
9448:
9434:
9431:
9430:
9412:
9408:
9398:
9385:
9364:
9360:
9358:
9355:
9354:
9330:
9326:
9316:
9303:
9296:
9294:
9285:
9281:
9279:
9276:
9275:
9256:
9245:
9242:
9241:
9215:
9211:
9190:
9186:
9175:
9172:
9171:
9154:
9153:
9146:
9134:
9130:
9117:
9113:
9103:
9099:
9089:
9075:
9062:
9060:
9053:
9047:
9043:
9040:
9039:
9029:
9025:
9015:
9011:
9001:
8987:
8974:
8972:
8965:
8959:
8955:
8951:
8949:
8946:
8945:
8913:
8909:
8903:
8899:
8898:
8896:
8890:
8889:
8874:
8870:
8864:
8860:
8859:
8857:
8850:
8841:
8837:
8819:
8815:
8810:
8807:
8806:
8769:
8765:
8759:
8755:
8753:
8746:
8742:
8732:
8730:
8715:
8711:
8709:
8700:
8696:
8678:
8674:
8669:
8666:
8665:
8637:
8627:
8623:
8622:
8620:
8615:
8612:
8611:
8584:
8575:
8571:
8570:
8560:
8556:
8549:
8547:
8535:
8531:
8515:
8508:
8498:
8494:
8487:
8485:
8473:
8469:
8467:
8464:
8463:
8442:
8438:
8434:
8424:
8420:
8419:
8417:
8412:
8409:
8408:
8392:
8389:
8388:
8356:
8352:
8328:
8324:
8297:
8293:
8285:
8282:
8281:
8254:
8250:
8229:
8225:
8201:
8197:
8189:
8186:
8185:
8167:
8164:
8163:
8138:
8134:
8113:
8109:
8091:
8087:
8066:
8062:
8043:
8039:
8026:
8022:
8004:
8000:
7996:
7994:
7989:
7986:
7985:
7963:
7959:
7941:
7937:
7919:
7915:
7897:
7893:
7874:
7870:
7857:
7853:
7835:
7831:
7827:
7825:
7820:
7817:
7816:
7798:
7795:
7794:
7770:
7761:
7757:
7739:
7735:
7722:
7713:
7709:
7691:
7687:
7682:
7679:
7678:
7659:
7658:
7656:
7653:
7652:
7627:
7623:
7602:
7598:
7593:
7590:
7589:
7567:
7563:
7545:
7541:
7539:
7536:
7535:
7512:
7501:
7498:
7497:
7481:
7478:
7477:
7445:
7441:
7417:
7413:
7386:
7382:
7380:
7377:
7376:
7349:
7345:
7324:
7320:
7296:
7292:
7290:
7287:
7286:
7268:
7265:
7264:
7227:
7226:
7211:
7207:
7197:
7193:
7189:
7183:
7178:
7159:
7135:
7131:
7121:
7117:
7113:
7107:
7099:
7080:
7073:
7058:
7057:
7049:
7045:
7035:
7031:
7027:
7012:
7005:
6990:
6989:
6982:
6970:
6966:
6953:
6949:
6934:
6930:
6920:
6907:
6903:
6889:
6876:
6874:
6867:
6861:
6857:
6854:
6853:
6843:
6839:
6824:
6820:
6810:
6797:
6793:
6779:
6766:
6764:
6757:
6751:
6747:
6743:
6741:
6738:
6737:
6707:
6700:
6690:
6686:
6679:
6677:
6665:
6661:
6659:
6656:
6655:
6628:
6624:
6603:
6599:
6597:
6594:
6593:
6571:
6567:
6549:
6545:
6540:
6537:
6536:
6504:
6499:
6486:
6481:
6465:
6461:
6455:
6451:
6436:
6432:
6426:
6422:
6421:
6417:
6409:
6405:
6398:
6397:
6393:
6391:
6389:
6386:
6385:
6352:
6348:
6338:
6334:
6326:
6317:
6313:
6288:
6286:
6267:
6263:
6253:
6240:
6230:
6226:
6219:
6217:
6205:
6201:
6196:
6193:
6192:
6156:
6152:
6146:
6142:
6141:
6139:
6123:
6117:
6113:
6109:
6091:
6087:
6081:
6077:
6076:
6072:
6063:
6059:
6052:
6050:
6038:
6034:
6032:
6029:
6028:
5989:
5985:
5979:
5975:
5973:
5972:
5968:
5958:
5931:
5927:
5921:
5917:
5915:
5914:
5910:
5900:
5891:
5887:
5886:
5876:
5872:
5865:
5863:
5851:
5847:
5842:
5839:
5838:
5807:
5793:
5789:
5788:
5775:
5766:
5762:
5761:
5756:
5747:
5743:
5725:
5721:
5716:
5713:
5712:
5695:
5692:
5691:
5661:
5654:
5650:
5644:
5640:
5633:
5631:
5618:
5614:
5607:
5603:
5588:
5584:
5583:
5581:
5574:
5570:
5563:
5554:
5550:
5532:
5528:
5520:
5517:
5516:
5487:
5483:
5481:
5472:
5468:
5450:
5446:
5435:
5432:
5431:
5401:
5397:
5391:
5387:
5386:
5384:
5378:
5377:
5362:
5358:
5352:
5348:
5347:
5345:
5338:
5329:
5325:
5307:
5303:
5298:
5295:
5294:
5262:
5252:
5244:
5235:
5231:
5206:
5204:
5195:
5191:
5173:
5169:
5157:
5153:
5132:
5128:
5120:
5117:
5116:
5093:
5083:
5075:
5066:
5062:
5037:
5035:
5026:
5022:
5004:
5000:
4988:
4984:
4966:
4962:
4957:
4954:
4953:
4920:
4916:
4910:
4906:
4904:
4897:
4893:
4883:
4881:
4866:
4862:
4860:
4851:
4847:
4829:
4825:
4820:
4817:
4816:
4788:
4779:
4775:
4774:
4764:
4760:
4753:
4751:
4739:
4735:
4719:
4712:
4702:
4698:
4691:
4689:
4677:
4673:
4671:
4668:
4667:
4649:
4646:
4645:
4615:
4610:
4607:
4606:
4588:
4585:
4584:
4560:
4556:
4535:
4531:
4523:
4520:
4519:
4498:
4494:
4476:
4472:
4461:
4458:
4457:
4438:
4435:
4434:
4408:
4404:
4383:
4379:
4368:
4365:
4364:
4342:
4338:
4320:
4316:
4308:
4305:
4304:
4286:
4283:
4282:
4257:
4253:
4232:
4228:
4210:
4206:
4185:
4181:
4162:
4158:
4145:
4141:
4131:
4129:
4117:
4113:
4108:
4105:
4104:
4082:
4078:
4060:
4056:
4038:
4034:
4016:
4012:
3993:
3989:
3976:
3972:
3962:
3960:
3948:
3944:
3939:
3936:
3935:
3917:
3914:
3913:
3890:
3881:
3877:
3859:
3855:
3842:
3833:
3829:
3811:
3807:
3802:
3799:
3798:
3779:
3778:
3776:
3773:
3772:
3747:
3743:
3722:
3718:
3713:
3710:
3709:
3687:
3683:
3665:
3661:
3659:
3656:
3655:
3637:
3634:
3633:
3604:
3600:
3579:
3575:
3560:
3556:
3532:
3528:
3526:
3523:
3522:
3500:
3496:
3475:
3471:
3459:
3455:
3437:
3433:
3428:
3425:
3424:
3406:
3403:
3402:
3368:
3365:
3364:
3332:
3329:
3328:
3308:
3305:
3304:
3284:
3281:
3280:
3279:Risk-free rate
3260:
3257:
3256:
3236:
3233:
3232:
3219:standard normal
3202:
3199:
3198:
3192:standard normal
3175:
3172:
3171:
3152:
3146:
3105:
3073:
3069:
3065:
3055:
3051:
3050:
3048:
3035:
3029:
3025:
3023:
3015:
3013:
3010:
3009:
3000:
2960:
2956:
2952:
2942:
2938:
2937:
2935:
2922:
2914:
2912:
2904:
2902:
2899:
2898:
2881:
2842:
2838:
2834:
2824:
2820:
2819:
2817:
2804:
2798:
2794:
2792:
2779:
2771:
2769:
2761:
2759:
2756:
2755:
2746:
2702:
2698:
2694:
2684:
2680:
2679:
2677:
2664:
2656:
2654:
2646:
2644:
2641:
2640:
2631:
2626:
2594:
2590:
2586:
2576:
2572:
2571:
2569:
2561:
2558:
2557:
2547:
2503:
2493:
2489:
2488:
2486:
2473:
2465:
2463:
2455:
2453:
2450:
2449:
2436:
2400:
2390:
2386:
2385:
2383:
2370:
2363:
2362:
2358:
2356:
2348:
2346:
2343:
2342:
2325:
2318:
2311:
2304:
2297:
2264:
2260:
2256:
2246:
2242:
2241:
2239:
2226:
2219:
2218:
2214:
2212:
2204:
2202:
2199:
2198:
2173:
2125:
2115:
2111:
2110:
2108:
2092:
2084:
2082:
2069:
2061:
2059:
2048:
2046:
2043:
2042:
2029:
1994:
1984:
1980:
1979:
1977:
1964:
1957:
1956:
1952:
1950:
1937:
1929:
1927:
1919:
1917:
1914:
1913:
1888:
1840:
1836:
1832:
1822:
1818:
1817:
1815:
1802:
1794:
1792:
1784:
1781:
1780:
1757:
1754:
1753:
1745:
1738:
1709:
1701:
1699:
1685:
1682:
1681:
1648:
1645:
1644:
1627:
1624:
1623:
1615:
1586:
1566:
1563:
1562:
1538:
1525:
1517:
1515:
1501:
1498:
1497:
1466:
1463:
1462:
1439:
1436:
1435:
1426:
1392:
1384:
1382:
1374:
1371:
1370:
1351:
1348:
1347:
1339:
1317:
1314:
1313:
1284:
1281:
1280:
1243:
1235:
1233:
1222:
1219:
1218:
1194:
1191:
1190:
1181:
1173:Volatility risk
1169:option straddle
1141:
1138:
1137:
1117:
1114:
1113:
1093:
1090:
1089:
1029:
1028:
1026:
1023:
1022:
1006:
1003:
1002:
969:
961:
959:
950:
949:
947:
944:
943:
926:
892:
875:
866: and
864:
850:
833:
822:
805:
788:
780:
777:
776:
766:put–call parity
761:
735:Brownian motion
719:
713:
679:put–call parity
667:
641:
633:
631:
623:
620:
619:
600:
597:
596:
580:
577:
576:
556:
553:
552:
544:
539:
499:
437:risk management
424:
422:
368:
365:
364:
331:
328:
327:
290:
289:
287:
284:
283:
250:
247:
246:
228:
225:
224:
201:
198:
197:
176:
175:
173:
170:
169:
149:
146:
145:
136:
118:
115:
114:
113:
108:
98:
88:
28:
23:
22:
18:Delta (finance)
15:
12:
11:
5:
12536:
12526:
12525:
12520:
12515:
12498:
12497:
12495:
12494:
12484:
12474:
12464:
12453:
12450:
12449:
12447:
12446:
12441:
12436:
12430:
12428:
12424:
12423:
12421:
12420:
12415:
12410:
12408:Merchant banks
12405:
12400:
12395:
12390:
12385:
12380:
12375:
12373:Family offices
12370:
12364:
12362:
12358:
12357:
12354:
12353:
12351:
12350:
12345:
12340:
12335:
12333:Securitization
12330:
12325:
12320:
12302:
12288:
12283:
12278:
12272:
12270:
12266:
12265:
12263:
12262:
12257:
12252:
12247:
12242:
12237:
12232:
12226:
12224:
12217:
12211:
12210:
12208:
12207:
12202:
12197:
12192:
12187:
12182:
12176:
12174:
12170:
12169:
12162:
12160:
12157:
12156:
12154:
12153:
12143:
12141:
12137:
12136:
12134:
12133:
12128:
12123:
12118:
12113:
12104:
12098:
12096:
12092:
12091:
12089:
12088:
12083:
12081:Risk arbitrage
12078:
12073:
12067:
12065:
12059:
12058:
12056:
12055:
12050:
12045:
12036:
12031:
12026:
12020:
12018:
12016:relative value
12004:
11998:
11997:
11990:
11989:
11982:
11975:
11967:
11958:
11957:
11955:
11954:
11941:
11938:
11937:
11935:
11934:
11929:
11927:Municipal debt
11924:
11919:
11914:
11912:Corporate debt
11909:
11903:
11901:
11897:
11896:
11894:
11893:
11888:
11883:
11878:
11873:
11868:
11863:
11858:
11853:
11848:
11843:
11838:
11833:
11828:
11822:
11820:
11816:
11815:
11813:
11812:
11807:
11802:
11797:
11792:
11787:
11781:
11779:
11773:
11772:
11770:
11769:
11764:
11759:
11754:
11749:
11744:
11739:
11734:
11729:
11724:
11719:
11714:
11712:Forward market
11709:
11704:
11699:
11694:
11688:
11686:
11684:
11683:
11678:
11672:
11669:
11668:
11666:
11665:
11660:
11655:
11650:
11645:
11640:
11635:
11630:
11625:
11620:
11615:
11610:
11605:
11600:
11595:
11593:Credit default
11590:
11585:
11580:
11575:
11570:
11565:
11560:
11554:
11552:
11546:
11545:
11542:
11541:
11539:
11538:
11533:
11528:
11523:
11518:
11513:
11508:
11503:
11498:
11493:
11488:
11478:
11473:
11468:
11462:
11460:
11454:
11453:
11451:
11450:
11436:
11431:
11426:
11421:
11416:
11411:
11406:
11401:
11396:
11391:
11389:Iron butterfly
11386:
11381:
11376:
11371:
11366:
11361:
11359:Covered option
11356:
11351:
11346:
11341:
11336:
11331:
11325:
11323:
11317:
11316:
11314:
11313:
11308:
11303:
11298:
11297:Mountain range
11295:
11290:
11285:
11280:
11275:
11270:
11265:
11260:
11255:
11250:
11245:
11240:
11234:
11232:
11226:
11225:
11223:
11222:
11217:
11212:
11207:
11202:
11197:
11192:
11187:
11182:
11177:
11171:
11169:
11163:
11162:
11160:
11159:
11154:
11149:
11144:
11139:
11134:
11129:
11124:
11119:
11114:
11109:
11103:
11101:
11094:
11088:
11087:
11082:
11079:
11078:
11071:
11070:
11063:
11056:
11048:
11042:
11041:
11029:
11028:
11011:
10997:
10996:External links
10994:
10991:
10990:
10981:|journal=
10940:
10919:
10896:
10872:
10848:
10837:
10823:
10810:
10784:
10773:
10716:
10691:
10666:
10626:
10619:
10597:
10586:
10579:
10555:
10548:
10530:
10508:
10501:
10452:
10445:
10417:
10410:
10383:
10376:
10353:
10352:
10350:
10347:
10346:
10345:
10340:
10335:
10330:
10325:
10320:
10318:Beta (finance)
10315:
10308:
10305:
10289:Main article:
10286:
10283:
10273:, such as the
10252:Beta (finance)
10250:Main article:
10247:
10244:
10207:interest rates
10203:Bond convexity
10153:. (Related is
10121:Bond convexity
10112:
10109:
10100:
10097:
10084:
10079:
10075:
10071:
10068:
10065:
10062:
10059:
10054:
10050:
10046:
10043:
10040:
10029:
10028:
10017:
10014:
10009:
10005:
10001:
9996:
9993:
9990:
9987:
9984:
9981:
9978:
9973:
9969:
9965:
9962:
9959:
9954:
9951:
9946:
9941:
9937:
9931:
9928:
9923:
9920:
9915:
9911:
9906:
9902:
9895:
9888:
9884:
9878:
9873:
9870:
9865:
9860:
9855:
9852:
9849:
9844:
9840:
9836:
9831:
9826:
9823:
9817:
9813:
9806:
9802:
9796:
9793:
9788:
9785:
9780:
9776:
9771:
9767:
9741:
9734:
9730:
9724:
9719:
9716:
9711:
9706:
9699:
9695:
9689:
9684:
9681:
9675:
9671:
9666:
9663:
9658:
9652:
9647:
9643:
9639:
9636:
9631:
9626:
9622:
9618:
9615:
9609:
9604:
9601:
9590:Then we have:
9573:
9569:
9563:
9560:
9555:
9552:
9549:
9544:
9540:
9535:
9531:
9526:
9523:
9498:
9494:
9488:
9485:
9480:
9477:
9474:
9469:
9465:
9461:
9458:
9455:
9451:
9447:
9444:
9441:
9438:
9415:
9411:
9405:
9402:
9397:
9392:
9389:
9384:
9381:
9378:
9375:
9372:
9367:
9363:
9339:
9333:
9329:
9323:
9320:
9315:
9310:
9307:
9302:
9299:
9293:
9288:
9284:
9260:
9255:
9252:
9249:
9223:
9218:
9214:
9210:
9207:
9204:
9201:
9198:
9193:
9189:
9185:
9182:
9179:
9168:
9167:
9150:
9145:
9142:
9137:
9133:
9129:
9121:
9116:
9111:
9106:
9102:
9096:
9093:
9088:
9085:
9082:
9078:
9074:
9071:
9068:
9065:
9059:
9056:
9054:
9050:
9046:
9042:
9041:
9033:
9028:
9023:
9018:
9014:
9008:
9005:
9000:
8997:
8994:
8990:
8986:
8983:
8980:
8977:
8971:
8968:
8966:
8962:
8958:
8954:
8953:
8937:
8936:
8922:
8916:
8912:
8906:
8902:
8893:
8888:
8883:
8877:
8873:
8867:
8863:
8854:
8849:
8844:
8840:
8836:
8833:
8828:
8825:
8822:
8818:
8814:
8804:
8800:
8799:
8796:
8795:
8782:
8773:
8768:
8762:
8758:
8752:
8749:
8745:
8739:
8735:
8729:
8723:
8718:
8714:
8708:
8703:
8699:
8695:
8692:
8687:
8684:
8681:
8677:
8673:
8663:
8649:
8646:
8643:
8640:
8635:
8630:
8626:
8619:
8607:
8606:
8588:
8583:
8578:
8574:
8568:
8563:
8559:
8555:
8552:
8544:
8541:
8538:
8534:
8530:
8527:
8519:
8514:
8511:
8506:
8501:
8497:
8493:
8490:
8482:
8479:
8476:
8472:
8461:
8445:
8441:
8437:
8432:
8427:
8423:
8416:
8396:
8384:
8383:
8380:
8379:
8367:
8364:
8359:
8355:
8351:
8348:
8345:
8342:
8339:
8336:
8331:
8327:
8323:
8320:
8317:
8314:
8311:
8306:
8303:
8300:
8296:
8292:
8289:
8279:
8265:
8262:
8257:
8253:
8249:
8246:
8243:
8240:
8237:
8232:
8228:
8224:
8221:
8218:
8215:
8210:
8207:
8204:
8200:
8196:
8193:
8183:
8171:
8159:
8158:
8146:
8141:
8137:
8133:
8130:
8127:
8122:
8119:
8116:
8112:
8108:
8105:
8102:
8099:
8094:
8090:
8086:
8083:
8080:
8075:
8072:
8069:
8065:
8061:
8058:
8055:
8047:
8042:
8037:
8034:
8029:
8025:
8021:
8018:
8013:
8010:
8007:
8003:
7999:
7993:
7983:
7971:
7966:
7962:
7958:
7955:
7950:
7947:
7944:
7940:
7936:
7933:
7930:
7927:
7922:
7918:
7914:
7911:
7906:
7903:
7900:
7896:
7892:
7889:
7886:
7878:
7873:
7868:
7865:
7860:
7856:
7852:
7849:
7844:
7841:
7838:
7834:
7830:
7824:
7814:
7802:
7790:
7789:
7774:
7769:
7764:
7760:
7756:
7753:
7748:
7745:
7742:
7738:
7734:
7731:
7726:
7721:
7716:
7712:
7708:
7705:
7700:
7697:
7694:
7690:
7686:
7676:
7662:
7648:
7647:
7635:
7630:
7626:
7622:
7619:
7616:
7611:
7608:
7605:
7601:
7597:
7587:
7575:
7570:
7566:
7562:
7559:
7554:
7551:
7548:
7544:
7533:
7522:
7519:
7515:
7511:
7508:
7505:
7485:
7473:
7472:
7469:
7468:
7456:
7453:
7448:
7444:
7440:
7437:
7434:
7431:
7428:
7425:
7420:
7416:
7412:
7409:
7406:
7403:
7400:
7395:
7392:
7389:
7385:
7374:
7360:
7357:
7352:
7348:
7344:
7341:
7338:
7335:
7332:
7327:
7323:
7319:
7316:
7313:
7310:
7305:
7302:
7299:
7295:
7284:
7272:
7260:
7259:
7256:
7253:
7241:
7240:
7225:
7222:
7214:
7210:
7204:
7201:
7196:
7192:
7186:
7181:
7177:
7170:
7167:
7163:
7158:
7155:
7152:
7149:
7146:
7138:
7134:
7128:
7125:
7120:
7116:
7110:
7105:
7102:
7098:
7091:
7088:
7084:
7079:
7076:
7074:
7072:
7069:
7066:
7063:
7060:
7059:
7052:
7048:
7042:
7039:
7034:
7030:
7023:
7020:
7016:
7011:
7008:
7006:
7004:
7001:
6998:
6995:
6992:
6991:
6986:
6981:
6978:
6973:
6969:
6965:
6957:
6952:
6947:
6943:
6937:
6933:
6927:
6924:
6919:
6916:
6913:
6910:
6906:
6902:
6899:
6896:
6892:
6888:
6885:
6882:
6879:
6873:
6870:
6868:
6864:
6860:
6856:
6855:
6847:
6842:
6837:
6833:
6827:
6823:
6817:
6814:
6809:
6806:
6803:
6800:
6796:
6792:
6789:
6786:
6782:
6778:
6775:
6772:
6769:
6763:
6760:
6758:
6754:
6750:
6746:
6745:
6729:
6728:
6711:
6706:
6703:
6698:
6693:
6689:
6685:
6682:
6674:
6671:
6668:
6664:
6653:
6649:
6648:
6636:
6631:
6627:
6623:
6620:
6617:
6612:
6609:
6606:
6602:
6591:
6579:
6574:
6570:
6566:
6563:
6558:
6555:
6552:
6548:
6544:
6534:
6530:
6529:
6526:
6525:
6513:
6507:
6502:
6498:
6494:
6489:
6484:
6480:
6476:
6473:
6468:
6464:
6458:
6454:
6450:
6447:
6444:
6439:
6435:
6429:
6425:
6420:
6412:
6408:
6401:
6396:
6383:
6379:
6378:
6375:
6374:
6361:
6355:
6351:
6342:
6337:
6330:
6325:
6320:
6316:
6312:
6309:
6306:
6303:
6300:
6297:
6294:
6291:
6285:
6282:
6279:
6276:
6273:
6270:
6266:
6257:
6252:
6249:
6246:
6243:
6238:
6233:
6229:
6225:
6222:
6214:
6211:
6208:
6204:
6200:
6190:
6186:
6185:
6171:
6167:
6164:
6159:
6155:
6149:
6145:
6138:
6135:
6127:
6120:
6116:
6112:
6106:
6102:
6099:
6094:
6090:
6084:
6080:
6075:
6071:
6066:
6062:
6058:
6055:
6047:
6044:
6041:
6037:
6026:
6022:
6021:
6008:
6004:
6001:
5993:
5988:
5982:
5978:
5971:
5965:
5962:
5957:
5954:
5950:
5946:
5943:
5935:
5930:
5924:
5920:
5913:
5904:
5899:
5894:
5890:
5884:
5879:
5875:
5871:
5868:
5860:
5857:
5854:
5850:
5846:
5836:
5832:
5831:
5828:
5827:
5815:
5810:
5805:
5800:
5797:
5792:
5787:
5779:
5774:
5769:
5765:
5760:
5755:
5750:
5746:
5742:
5739:
5734:
5731:
5728:
5724:
5720:
5710:
5699:
5688:
5687:
5674:
5667:
5664:
5657:
5653:
5647:
5643:
5639:
5636:
5630:
5622:
5617:
5610:
5606:
5601:
5597:
5594:
5591:
5587:
5580:
5577:
5573:
5567:
5562:
5557:
5553:
5549:
5546:
5541:
5538:
5535:
5531:
5527:
5524:
5514:
5510:
5509:
5495:
5490:
5486:
5480:
5475:
5471:
5467:
5464:
5459:
5456:
5453:
5449:
5445:
5442:
5439:
5429:
5425:
5424:
5410:
5404:
5400:
5394:
5390:
5381:
5376:
5371:
5365:
5361:
5355:
5351:
5342:
5337:
5332:
5328:
5324:
5321:
5316:
5313:
5310:
5306:
5302:
5292:
5288:
5287:
5284:
5283:
5266:
5261:
5258:
5255:
5248:
5243:
5238:
5234:
5230:
5227:
5224:
5221:
5218:
5215:
5212:
5209:
5203:
5198:
5194:
5190:
5187:
5182:
5179:
5176:
5172:
5168:
5165:
5160:
5156:
5152:
5149:
5146:
5141:
5138:
5135:
5131:
5127:
5124:
5114:
5097:
5092:
5089:
5086:
5079:
5074:
5069:
5065:
5061:
5058:
5055:
5052:
5049:
5046:
5043:
5040:
5034:
5029:
5025:
5021:
5018:
5013:
5010:
5007:
5003:
4999:
4996:
4991:
4987:
4983:
4980:
4975:
4972:
4969:
4965:
4961:
4951:
4947:
4946:
4933:
4924:
4919:
4913:
4909:
4903:
4900:
4896:
4890:
4886:
4880:
4874:
4869:
4865:
4859:
4854:
4850:
4846:
4843:
4838:
4835:
4832:
4828:
4824:
4814:
4810:
4809:
4792:
4787:
4782:
4778:
4772:
4767:
4763:
4759:
4756:
4748:
4745:
4742:
4738:
4734:
4731:
4723:
4718:
4715:
4710:
4705:
4701:
4697:
4694:
4686:
4683:
4680:
4676:
4665:
4653:
4641:
4640:
4637:
4636:
4622:
4619:
4614:
4604:
4592:
4580:
4579:
4568:
4563:
4559:
4555:
4552:
4549:
4544:
4541:
4538:
4534:
4530:
4527:
4517:
4506:
4501:
4497:
4493:
4490:
4485:
4482:
4479:
4475:
4471:
4468:
4465:
4455:
4442:
4429:
4428:
4416:
4411:
4407:
4403:
4400:
4397:
4392:
4389:
4386:
4382:
4378:
4375:
4372:
4362:
4350:
4345:
4341:
4337:
4334:
4329:
4326:
4323:
4319:
4315:
4312:
4302:
4290:
4278:
4277:
4265:
4260:
4256:
4252:
4249:
4246:
4241:
4238:
4235:
4231:
4227:
4224:
4221:
4218:
4213:
4209:
4205:
4202:
4199:
4194:
4191:
4188:
4184:
4180:
4177:
4174:
4166:
4161:
4156:
4153:
4148:
4144:
4140:
4137:
4134:
4126:
4123:
4120:
4116:
4112:
4102:
4090:
4085:
4081:
4077:
4074:
4069:
4066:
4063:
4059:
4055:
4052:
4049:
4046:
4041:
4037:
4033:
4030:
4025:
4022:
4019:
4015:
4011:
4008:
4005:
3997:
3992:
3987:
3984:
3979:
3975:
3971:
3968:
3965:
3957:
3954:
3951:
3947:
3943:
3933:
3921:
3909:
3908:
3894:
3889:
3884:
3880:
3876:
3873:
3868:
3865:
3862:
3858:
3854:
3851:
3846:
3841:
3836:
3832:
3828:
3825:
3820:
3817:
3814:
3810:
3806:
3796:
3782:
3768:
3767:
3755:
3750:
3746:
3742:
3739:
3736:
3731:
3728:
3725:
3721:
3717:
3707:
3695:
3690:
3686:
3682:
3679:
3674:
3671:
3668:
3664:
3653:
3641:
3629:
3628:
3625:
3624:
3612:
3607:
3603:
3599:
3596:
3593:
3588:
3585:
3582:
3578:
3574:
3571:
3568:
3563:
3559:
3555:
3552:
3549:
3546:
3541:
3538:
3535:
3531:
3520:
3508:
3503:
3499:
3495:
3492:
3489:
3484:
3481:
3478:
3474:
3470:
3467:
3462:
3458:
3454:
3451:
3446:
3443:
3440:
3436:
3432:
3422:
3410:
3398:
3397:
3394:
3391:
3386:
3385:
3372:
3361:
3348:
3345:
3342:
3339:
3336:
3325:
3312:
3301:
3288:
3277:
3264:
3253:
3240:
3206:
3179:
3154:The Greeks of
3145:
3142:
3104:
3101:
3093:
3092:
3076:
3072:
3068:
3063:
3058:
3054:
3047:
3041:
3038:
3028:
3022:
2999:
2996:
2987:
2986:
2972:
2969:
2963:
2959:
2955:
2950:
2945:
2941:
2934:
2928:
2925:
2920:
2917:
2911:
2880:
2877:
2869:
2868:
2854:
2851:
2845:
2841:
2837:
2832:
2827:
2823:
2816:
2810:
2807:
2797:
2791:
2785:
2782:
2777:
2774:
2768:
2745:
2742:
2722:
2721:
2705:
2701:
2697:
2692:
2687:
2683:
2676:
2670:
2667:
2662:
2659:
2653:
2630:
2627:
2625:
2622:
2614:
2613:
2597:
2593:
2589:
2584:
2579:
2575:
2568:
2565:
2546:
2540:
2531:
2530:
2516:
2513:
2509:
2506:
2501:
2496:
2492:
2485:
2479:
2476:
2471:
2468:
2462:
2435:
2432:
2428:
2427:
2413:
2410:
2406:
2403:
2398:
2393:
2389:
2382:
2376:
2373:
2366:
2361:
2355:
2324:
2321:
2316:
2309:
2302:
2295:
2284:
2283:
2267:
2263:
2259:
2254:
2249:
2245:
2238:
2232:
2229:
2222:
2217:
2211:
2184:vega convexity
2172:
2169:
2153:
2152:
2138:
2135:
2131:
2128:
2123:
2118:
2114:
2107:
2104:
2098:
2095:
2090:
2087:
2081:
2075:
2072:
2067:
2064:
2058:
2055:
2028:
2025:
2013:
2007:
2004:
2000:
1997:
1992:
1987:
1983:
1976:
1970:
1967:
1960:
1955:
1949:
1943:
1940:
1935:
1932:
1926:
1887:
1884:
1863:short options.
1860:
1859:
1843:
1839:
1835:
1830:
1825:
1821:
1814:
1808:
1805:
1800:
1797:
1791:
1788:
1761:
1744:
1741:
1737:
1734:
1730:
1729:
1715:
1712:
1707:
1704:
1698:
1695:
1692:
1689:
1652:
1631:
1614:
1611:
1593:
1590:
1585:
1582:
1579:
1576:
1573:
1570:
1561:It holds that
1559:
1558:
1545:
1542:
1537:
1531:
1528:
1523:
1520:
1514:
1511:
1508:
1505:
1470:
1443:
1425:
1422:
1413:
1412:
1398:
1395:
1390:
1387:
1381:
1378:
1355:
1338:
1335:
1321:
1288:
1264:
1263:
1249:
1246:
1241:
1238:
1232:
1229:
1226:
1198:
1180:
1177:
1145:
1121:
1097:
1032:
1021:), written as
1010:
990:
989:
975:
972:
967:
964:
958:
953:
925:
922:
917:
916:
905:
902:
899:
891:
888:
885:
882:
874:
871:
863:
860:
857:
849:
846:
843:
840:
832:
829:
821:
818:
815:
812:
804:
801:
798:
795:
787:
784:
760:
757:
715:Main article:
712:
709:
666:
663:
662:
661:
647:
644:
639:
636:
630:
627:
604:
584:
560:
543:
540:
538:
535:
498:
495:
431:
430:
418:
417:
414:
413:
411:
406:
404:
400:
399:
394:
389:
384:
372:
360:
359:
357:
352:
347:
335:
323:
322:
317:
312:
307:
293:
279:
278:
273:
268:
254:
244:
232:
220:
219:
205:
195:
179:
167:
153:
143:
139:
138:
133:
122:
110:
104:
103:
100:
87:
84:
26:
9:
6:
4:
3:
2:
12535:
12524:
12521:
12519:
12516:
12514:
12511:
12510:
12508:
12493:
12485:
12483:
12475:
12473:
12465:
12463:
12455:
12454:
12451:
12445:
12442:
12440:
12437:
12435:
12432:
12431:
12429:
12425:
12419:
12416:
12414:
12413:Pension funds
12411:
12409:
12406:
12404:
12401:
12399:
12396:
12394:
12391:
12389:
12386:
12384:
12381:
12379:
12376:
12374:
12371:
12369:
12368:Vulture funds
12366:
12365:
12363:
12359:
12349:
12346:
12344:
12341:
12339:
12336:
12334:
12331:
12329:
12326:
12324:
12321:
12318:
12314:
12310:
12306:
12303:
12300:
12299:delta neutral
12296:
12292:
12289:
12287:
12284:
12282:
12279:
12277:
12274:
12273:
12271:
12267:
12261:
12258:
12256:
12255:Money markets
12253:
12251:
12248:
12246:
12243:
12241:
12238:
12236:
12233:
12231:
12228:
12227:
12225:
12221:
12218:
12212:
12206:
12203:
12201:
12198:
12196:
12193:
12191:
12188:
12186:
12183:
12181:
12178:
12177:
12175:
12171:
12166:
12152:
12151:Multi-manager
12148:
12145:
12144:
12142:
12138:
12132:
12129:
12127:
12124:
12122:
12119:
12117:
12114:
12112:
12108:
12105:
12103:
12100:
12099:
12097:
12093:
12087:
12084:
12082:
12079:
12077:
12074:
12072:
12069:
12068:
12066:
12064:
12060:
12054:
12051:
12049:
12046:
12044:
12040:
12037:
12035:
12032:
12030:
12027:
12025:
12022:
12021:
12019:
12017:
12012:
12008:
12005:
11999:
11995:
11988:
11983:
11981:
11976:
11974:
11969:
11968:
11965:
11953:
11948:
11943:
11942:
11939:
11933:
11930:
11928:
11925:
11923:
11920:
11918:
11915:
11913:
11910:
11908:
11907:Consumer debt
11905:
11904:
11902:
11900:Market issues
11898:
11892:
11889:
11887:
11884:
11882:
11879:
11877:
11876:Fund of funds
11874:
11872:
11869:
11867:
11864:
11862:
11859:
11857:
11854:
11852:
11849:
11847:
11844:
11842:
11839:
11837:
11834:
11832:
11829:
11827:
11824:
11823:
11821:
11817:
11811:
11808:
11806:
11803:
11801:
11798:
11796:
11793:
11791:
11788:
11786:
11783:
11782:
11780:
11778:
11774:
11768:
11765:
11763:
11760:
11758:
11755:
11753:
11750:
11748:
11745:
11743:
11740:
11738:
11735:
11733:
11730:
11728:
11725:
11723:
11720:
11718:
11717:Forward price
11715:
11713:
11710:
11708:
11705:
11703:
11700:
11698:
11695:
11693:
11690:
11689:
11687:
11682:
11679:
11677:
11674:
11673:
11670:
11664:
11661:
11659:
11656:
11654:
11651:
11649:
11646:
11644:
11641:
11639:
11636:
11634:
11631:
11629:
11628:Interest rate
11626:
11624:
11621:
11619:
11616:
11614:
11611:
11609:
11606:
11604:
11601:
11599:
11596:
11594:
11591:
11589:
11586:
11584:
11581:
11579:
11576:
11574:
11571:
11569:
11566:
11564:
11561:
11559:
11556:
11555:
11553:
11551:
11547:
11537:
11534:
11532:
11529:
11527:
11524:
11522:
11521:MC Simulation
11519:
11517:
11514:
11512:
11509:
11507:
11504:
11502:
11499:
11497:
11494:
11492:
11489:
11486:
11482:
11481:Black–Scholes
11479:
11477:
11474:
11472:
11469:
11467:
11464:
11463:
11461:
11459:
11455:
11448:
11444:
11440:
11437:
11435:
11434:Risk reversal
11432:
11430:
11427:
11425:
11422:
11420:
11417:
11415:
11412:
11410:
11407:
11405:
11402:
11400:
11397:
11395:
11392:
11390:
11387:
11385:
11382:
11380:
11377:
11375:
11372:
11370:
11367:
11365:
11364:Credit spread
11362:
11360:
11357:
11355:
11352:
11350:
11347:
11345:
11342:
11340:
11337:
11335:
11332:
11330:
11327:
11326:
11324:
11322:
11318:
11312:
11309:
11307:
11304:
11302:
11299:
11296:
11294:
11291:
11289:
11288:Interest rate
11286:
11284:
11283:Forward start
11281:
11279:
11276:
11274:
11271:
11269:
11266:
11264:
11261:
11259:
11256:
11254:
11251:
11249:
11246:
11244:
11241:
11239:
11236:
11235:
11233:
11231:
11227:
11221:
11218:
11216:
11213:
11211:
11210:Option styles
11208:
11206:
11203:
11201:
11198:
11196:
11193:
11191:
11188:
11186:
11183:
11181:
11178:
11176:
11173:
11172:
11170:
11168:
11164:
11158:
11155:
11153:
11150:
11148:
11145:
11143:
11140:
11138:
11135:
11133:
11130:
11128:
11127:Open interest
11125:
11123:
11120:
11118:
11115:
11113:
11110:
11108:
11107:Delta neutral
11105:
11104:
11102:
11098:
11095:
11093:
11089:
11085:
11080:
11076:
11069:
11064:
11062:
11057:
11055:
11050:
11049:
11046:
11039:
11036:
11035:
11034:
11033:
11027:
11023:
11019:
11016:
11012:
11009:
11005:
11004:
11003:
11002:
10986:
10973:
10965:
10964:10.18452/3572
10961:
10954:
10947:
10945:
10929:
10923:
10908:
10907:
10900:
10885:
10879:
10877:
10862:
10858:
10852:
10846:
10841:
10835:
10830:
10828:
10820:
10814:
10806:
10802:
10795:
10793:
10791:
10789:
10782:
10777:
10769:
10763:
10755:
10751:
10747:
10740:
10733:
10731:
10729:
10727:
10725:
10723:
10721:
10706:
10702:
10695:
10681:
10677:
10670:
10662:
10658:
10653:
10648:
10644:
10640:
10633:
10631:
10622:
10620:9780470689684
10616:
10612:
10608:
10601:
10595:
10590:
10582:
10580:9780136390145
10576:
10572:
10571:Prentice-Hall
10568:
10567:
10559:
10551:
10549:9780273750604
10545:
10541:
10534:
10519:
10512:
10504:
10502:9780071389976
10498:
10494:
10490:
10483:
10481:
10479:
10477:
10475:
10473:
10471:
10469:
10467:
10465:
10463:
10461:
10459:
10457:
10448:
10446:9780786310258
10442:
10438:
10434:
10430:
10429:
10421:
10413:
10407:
10403:
10399:
10398:
10390:
10388:
10379:
10377:9780071453158
10373:
10369:
10365:
10358:
10354:
10344:
10341:
10339:
10336:
10334:
10331:
10329:
10326:
10324:
10323:Delta neutral
10321:
10319:
10316:
10314:
10311:
10310:
10304:
10302:
10298:
10292:
10282:
10278:
10276:
10272:
10267:
10263:
10259:
10253:
10243:
10241:
10236:
10232:
10228:
10224:
10219:
10217:
10212:
10208:
10204:
10200:
10198:
10194:
10190:
10188:
10183:
10179:
10175:
10171:
10167:
10164:Analogous to
10162:
10160:
10159:credit spread
10156:
10152:
10148:
10144:
10140:
10136:
10135:bond duration
10132:
10128:
10122:
10118:
10117:Bond duration
10108:
10106:
10095:
10077:
10073:
10066:
10063:
10060:
10052:
10048:
10041:
10038:
10015:
10012:
10007:
10003:
9999:
9991:
9988:
9979:
9976:
9971:
9967:
9963:
9957:
9952:
9949:
9944:
9939:
9935:
9929:
9926:
9921:
9918:
9913:
9909:
9904:
9900:
9893:
9886:
9882:
9876:
9871:
9868:
9863:
9858:
9850:
9847:
9842:
9838:
9829:
9824:
9821:
9815:
9811:
9804:
9800:
9794:
9791:
9786:
9783:
9778:
9774:
9769:
9765:
9758:
9757:
9756:
9739:
9732:
9728:
9722:
9717:
9714:
9709:
9704:
9697:
9693:
9687:
9682:
9679:
9673:
9669:
9664:
9661:
9656:
9645:
9641:
9634:
9624:
9620:
9613:
9607:
9602:
9599:
9588:
9571:
9567:
9561:
9558:
9553:
9550:
9547:
9542:
9538:
9533:
9529:
9524:
9521:
9511:
9496:
9492:
9486:
9483:
9478:
9475:
9472:
9467:
9463:
9459:
9453:
9449:
9445:
9439:
9436:
9428:
9413:
9409:
9403:
9400:
9395:
9390:
9387:
9382:
9379:
9376:
9373:
9370:
9365:
9361:
9352:
9337:
9331:
9327:
9321:
9318:
9313:
9308:
9305:
9300:
9297:
9291:
9286:
9282:
9273:
9258:
9253:
9250:
9247:
9237:
9236:Micro proof:
9234:
9216:
9212:
9205:
9202:
9199:
9191:
9187:
9180:
9177:
9148:
9143:
9140:
9135:
9131:
9127:
9119:
9114:
9109:
9104:
9100:
9094:
9091:
9086:
9080:
9076:
9072:
9066:
9063:
9057:
9055:
9048:
9044:
9031:
9026:
9021:
9016:
9012:
9006:
9003:
8998:
8992:
8988:
8984:
8978:
8975:
8969:
8967:
8960:
8956:
8944:
8943:
8942:
8920:
8914:
8910:
8904:
8900:
8886:
8881:
8875:
8871:
8865:
8861:
8852:
8842:
8838:
8831:
8826:
8823:
8820:
8816:
8812:
8802:
8801:
8797:
8780:
8771:
8766:
8760:
8756:
8750:
8747:
8743:
8737:
8727:
8721:
8716:
8712:
8701:
8697:
8690:
8685:
8682:
8679:
8675:
8671:
8647:
8641:
8633:
8628:
8617:
8609:
8608:
8586:
8581:
8576:
8572:
8561:
8557:
8550:
8542:
8539:
8536:
8532:
8528:
8525:
8517:
8512:
8509:
8499:
8495:
8488:
8480:
8477:
8474:
8470:
8443:
8439:
8430:
8425:
8414:
8386:
8385:
8381:
8357:
8353:
8349:
8340:
8337:
8329:
8325:
8321:
8312:
8304:
8301:
8298:
8294:
8290:
8287:
8280:
8255:
8251:
8241:
8238:
8230:
8226:
8216:
8208:
8205:
8202:
8198:
8194:
8191:
8184:
8169:
8161:
8160:
8139:
8135:
8131:
8120:
8117:
8114:
8110:
8106:
8103:
8100:
8092:
8088:
8084:
8073:
8070:
8067:
8063:
8059:
8056:
8053:
8045:
8040:
8035:
8027:
8023:
8016:
8011:
8008:
8005:
8001:
7997:
7991:
7984:
7964:
7960:
7948:
7945:
7942:
7938:
7934:
7931:
7928:
7920:
7916:
7904:
7901:
7898:
7894:
7890:
7887:
7884:
7876:
7871:
7866:
7858:
7854:
7847:
7842:
7839:
7836:
7832:
7828:
7822:
7815:
7792:
7791:
7772:
7762:
7758:
7751:
7746:
7743:
7740:
7736:
7732:
7729:
7724:
7714:
7710:
7703:
7698:
7695:
7692:
7688:
7684:
7650:
7649:
7628:
7624:
7620:
7609:
7606:
7603:
7599:
7595:
7588:
7568:
7564:
7552:
7549:
7546:
7542:
7534:
7520:
7513:
7509:
7503:
7475:
7474:
7470:
7446:
7442:
7438:
7429:
7426:
7418:
7414:
7410:
7401:
7393:
7390:
7387:
7383:
7375:
7350:
7346:
7336:
7333:
7325:
7321:
7311:
7303:
7300:
7297:
7293:
7285:
7270:
7262:
7261:
7257:
7254:
7252:
7251:
7248:
7246:
7223:
7220:
7212:
7208:
7202:
7199:
7194:
7190:
7179:
7175:
7168:
7165:
7161:
7156:
7153:
7150:
7147:
7144:
7136:
7132:
7126:
7123:
7118:
7114:
7108:
7100:
7096:
7089:
7086:
7082:
7077:
7075:
7067:
7050:
7046:
7040:
7037:
7032:
7028:
7021:
7018:
7014:
7009:
7007:
6999:
6993:
6984:
6979:
6976:
6971:
6967:
6963:
6955:
6950:
6945:
6941:
6935:
6931:
6925:
6922:
6917:
6914:
6911:
6908:
6904:
6900:
6894:
6890:
6886:
6880:
6877:
6871:
6869:
6862:
6858:
6845:
6840:
6835:
6831:
6825:
6821:
6815:
6812:
6807:
6804:
6801:
6798:
6794:
6790:
6784:
6780:
6776:
6770:
6767:
6761:
6759:
6752:
6748:
6736:
6735:
6734:
6709:
6704:
6701:
6691:
6687:
6680:
6672:
6669:
6666:
6662:
6651:
6650:
6629:
6625:
6621:
6610:
6607:
6604:
6600:
6592:
6572:
6568:
6556:
6553:
6550:
6546:
6542:
6535:
6532:
6531:
6527:
6511:
6505:
6500:
6496:
6492:
6487:
6482:
6478:
6474:
6466:
6462:
6456:
6452:
6448:
6445:
6437:
6433:
6427:
6423:
6418:
6410:
6406:
6394:
6381:
6380:
6376:
6359:
6353:
6349:
6340:
6335:
6328:
6323:
6318:
6314:
6310:
6307:
6301:
6298:
6295:
6289:
6283:
6280:
6277:
6274:
6271:
6268:
6264:
6255:
6250:
6247:
6244:
6241:
6231:
6227:
6220:
6212:
6209:
6206:
6202:
6198:
6188:
6187:
6169:
6165:
6162:
6157:
6153:
6147:
6143:
6133:
6125:
6118:
6114:
6110:
6104:
6100:
6097:
6092:
6088:
6082:
6078:
6073:
6064:
6060:
6053:
6045:
6042:
6039:
6035:
6024:
6023:
6006:
6002:
5999:
5991:
5986:
5980:
5976:
5969:
5963:
5955:
5952:
5948:
5944:
5941:
5933:
5928:
5922:
5918:
5911:
5902:
5897:
5892:
5888:
5877:
5873:
5866:
5858:
5855:
5852:
5848:
5844:
5834:
5833:
5829:
5808:
5803:
5798:
5795:
5790:
5785:
5777:
5772:
5767:
5763:
5758:
5748:
5744:
5737:
5732:
5729:
5726:
5722:
5718:
5697:
5690:
5689:
5672:
5665:
5662:
5655:
5651:
5645:
5641:
5637:
5634:
5628:
5620:
5615:
5608:
5604:
5599:
5595:
5592:
5589:
5585:
5578:
5575:
5571:
5565:
5555:
5551:
5544:
5539:
5536:
5533:
5529:
5525:
5522:
5512:
5511:
5493:
5488:
5484:
5473:
5469:
5462:
5457:
5454:
5451:
5447:
5443:
5440:
5437:
5427:
5426:
5408:
5402:
5398:
5392:
5388:
5374:
5369:
5363:
5359:
5353:
5349:
5340:
5330:
5326:
5319:
5314:
5311:
5308:
5304:
5300:
5290:
5289:
5285:
5264:
5259:
5256:
5253:
5246:
5241:
5236:
5232:
5228:
5225:
5219:
5216:
5213:
5207:
5196:
5192:
5185:
5180:
5177:
5174:
5170:
5166:
5158:
5154:
5150:
5139:
5136:
5133:
5129:
5125:
5122:
5115:
5095:
5090:
5087:
5084:
5077:
5072:
5067:
5063:
5059:
5056:
5050:
5047:
5044:
5038:
5027:
5023:
5016:
5011:
5008:
5005:
5001:
4997:
4989:
4985:
4973:
4970:
4967:
4963:
4959:
4952:
4949:
4948:
4931:
4922:
4917:
4911:
4907:
4901:
4898:
4894:
4888:
4878:
4872:
4867:
4863:
4852:
4848:
4841:
4836:
4833:
4830:
4826:
4822:
4812:
4811:
4790:
4785:
4780:
4776:
4765:
4761:
4754:
4746:
4743:
4740:
4736:
4732:
4729:
4721:
4716:
4713:
4703:
4699:
4692:
4684:
4681:
4678:
4674:
4643:
4642:
4638:
4620:
4617:
4590:
4582:
4581:
4561:
4557:
4553:
4542:
4539:
4536:
4532:
4528:
4525:
4518:
4499:
4495:
4483:
4480:
4477:
4473:
4469:
4466:
4463:
4456:
4454:
4440:
4431:
4430:
4409:
4405:
4401:
4390:
4387:
4384:
4380:
4376:
4373:
4370:
4363:
4343:
4339:
4327:
4324:
4321:
4317:
4313:
4310:
4303:
4288:
4280:
4279:
4258:
4254:
4250:
4239:
4236:
4233:
4229:
4225:
4222:
4219:
4211:
4207:
4203:
4192:
4189:
4186:
4182:
4178:
4175:
4172:
4164:
4159:
4154:
4146:
4142:
4135:
4132:
4124:
4121:
4118:
4114:
4110:
4103:
4083:
4079:
4067:
4064:
4061:
4057:
4053:
4050:
4047:
4039:
4035:
4023:
4020:
4017:
4013:
4009:
4006:
4003:
3995:
3990:
3985:
3977:
3973:
3966:
3963:
3955:
3952:
3949:
3945:
3941:
3934:
3911:
3910:
3892:
3882:
3878:
3871:
3866:
3863:
3860:
3856:
3852:
3849:
3844:
3834:
3830:
3823:
3818:
3815:
3812:
3808:
3804:
3770:
3769:
3748:
3744:
3740:
3729:
3726:
3723:
3719:
3715:
3708:
3688:
3684:
3672:
3669:
3666:
3662:
3654:
3631:
3630:
3626:
3605:
3601:
3597:
3586:
3583:
3580:
3576:
3572:
3569:
3561:
3557:
3553:
3544:
3539:
3536:
3533:
3529:
3521:
3501:
3497:
3487:
3482:
3479:
3476:
3472:
3468:
3460:
3456:
3444:
3441:
3438:
3434:
3430:
3423:
3408:
3400:
3399:
3395:
3392:
3390:
3389:
3370:
3362:
3346:
3343:
3340:
3337:
3334:
3326:
3310:
3302:
3286:
3278:
3262:
3255:Strike price
3254:
3238:
3230:
3229:
3228:
3227:For a given:
3225:
3223:
3220:
3196:
3193:
3190:(phi) is the
3177:
3169:
3165:
3161:
3157:
3151:
3141:
3139:
3135:
3132:
3128:
3126:
3122:
3120:
3116:
3112:
3110:
3100:
3098:
3074:
3070:
3061:
3056:
3045:
3039:
3020:
3008:
3007:
3006:
3004:
2995:
2991:
2970:
2961:
2957:
2948:
2943:
2932:
2926:
2909:
2897:
2896:
2895:
2893:
2889:
2885:
2876:
2874:
2852:
2843:
2839:
2830:
2825:
2814:
2808:
2789:
2783:
2766:
2754:
2753:
2752:
2750:
2741:
2739:
2738:delta-hedging
2735:
2731:
2727:
2703:
2699:
2690:
2685:
2674:
2668:
2651:
2639:
2638:
2637:
2635:
2621:
2619:
2595:
2591:
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2577:
2566:
2563:
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2552:
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2537:
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2494:
2483:
2477:
2469:
2460:
2448:
2447:
2446:
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2431:
2411:
2404:
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2391:
2380:
2374:
2353:
2341:
2340:
2339:
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2333:
2329:
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2308:
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2294:
2289:
2265:
2261:
2252:
2247:
2236:
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2196:
2195:
2193:
2189:
2185:
2181:
2177:
2168:
2164:
2162:
2158:
2136:
2129:
2121:
2116:
2105:
2102:
2096:
2079:
2073:
2056:
2053:
2041:
2040:
2039:
2037:
2033:
2024:
2011:
2005:
1998:
1990:
1985:
1974:
1968:
1947:
1941:
1924:
1910:
1908:
1904:
1900:
1896:
1892:
1883:
1879:
1877:
1868:
1864:
1841:
1837:
1828:
1823:
1812:
1806:
1789:
1779:
1778:
1777:
1775:
1751:
1750:
1740:
1733:
1713:
1705:
1696:
1693:
1690:
1687:
1680:
1679:
1678:
1676:
1671:
1667:
1663:
1650:
1629:
1621:
1620:
1610:
1607:
1591:
1588:
1583:
1577:
1571:
1568:
1543:
1540:
1535:
1529:
1521:
1512:
1506:
1503:
1496:
1495:
1494:
1492:
1488:
1484:
1460:
1459:
1454:
1441:
1432:
1431:
1421:
1417:
1396:
1388:
1379:
1376:
1369:
1368:
1367:
1353:
1345:
1344:
1334:
1319:
1310:
1307:
1301:
1286:
1278:
1274:
1270:
1247:
1239:
1230:
1227:
1217:
1216:
1215:
1213:
1209:
1187:
1186:
1176:
1174:
1170:
1164:
1162:
1157:
1135:
1132:) or capital
1119:
1111:
1095:
1087:
1086:
1080:
1078:
1074:
1070:
1066:
1062:
1058:
1057:
1052:
1048:
1008:
1000:
999:
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940:
938:
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930:
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900:
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819:
816:
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774:
773:
769:
767:
756:
754:
749:
744:
740:
736:
732:
728:
724:
718:
708:
705:
704:selling short
700:
694:
690:
688:
684:
680:
676:
672:
665:Practical use
645:
637:
628:
618:
617:
616:
602:
582:
575:of the value
574:
550:
549:
534:
532:
528:
524:
520:
516:
512:
508:
504:
494:
490:
488:
485:; as well as
484:
480:
476:
472:
467:
465:
461:
458:
454:
449:
447:
446:delta hedging
442:
438:
428:
420:
419:
412:
410:
407:
405:
402:
401:
398:
395:
393:
390:
388:
385:
362:
361:
358:
356:
353:
351:
348:
325:
324:
321:
318:
316:
313:
311:
308:
281:
280:
277:
274:
272:
269:
267:
245:
222:
221:
218:
196:
194:
168:
166:
144:
141:
140:
134:
120:
111:
106:
105:
95:
92:
91:
83:
81:
77:
76:risk measures
73:
69:
68:Greek letters
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
12294:
12245:Fixed income
12121:Global macro
12063:Event-driven
11727:Forward rate
11638:Total return
11526:Real options
11429:Ratio spread
11409:Naked option
11369:Debit spread
11200:Fixed income
11151:
11142:Strike price
11032:Online tools
11031:
11030:
11000:
10999:
10972:cite journal
10931:. Retrieved
10922:
10910:. Retrieved
10905:
10899:
10887:. Retrieved
10864:. Retrieved
10860:
10851:
10840:
10818:
10813:
10804:
10776:
10745:
10708:. Retrieved
10704:
10694:
10683:. Retrieved
10679:
10669:
10642:
10638:
10606:
10600:
10589:
10565:
10558:
10539:
10533:
10521:. Retrieved
10516:Suma, John.
10511:
10488:
10427:
10420:
10396:
10363:
10357:
10294:
10279:
10257:
10255:
10220:
10201:
10192:
10186:
10177:
10173:
10163:
10124:
10102:
10030:
9589:
9512:
9429:
9353:
9274:
9239:
9235:
9169:
8940:
7263:fair value (
7242:
6732:
4433:
3401:fair value (
3231:Stock price
3226:
3166:) under the
3153:
3137:
3136:
3130:
3129:
3124:
3123:
3118:
3114:
3113:
3106:
3096:
3094:
3002:
3001:
2992:
2988:
2891:
2887:
2883:
2882:
2872:
2870:
2748:
2747:
2733:
2729:
2725:
2723:
2633:
2632:
2620:strategies.
2615:
2548:
2543:
2532:
2442:
2438:
2437:
2429:
2335:
2331:
2327:
2326:
2313:
2306:
2299:
2292:
2285:
2187:
2183:
2179:
2175:
2174:
2165:
2156:
2154:
2035:
2031:
2030:
1911:
1906:
1898:
1894:
1890:
1889:
1880:
1873:
1861:
1747:
1746:
1739:
1731:
1643:
1617:
1616:
1608:
1560:
1482:
1456:
1434:
1428:
1427:
1418:
1414:
1341:
1340:
1311:
1305:
1302:
1276:
1272:
1268:
1265:
1189:
1183:
1182:
1165:
1158:
1112:(as is tau (
1109:
1083:
1081:
1076:
1072:
1068:
1064:
1060:
1055:
1050:
1046:
997:
992:
991:
928:
927:
918:
770:
762:
748:At-the-money
731:in-the-money
726:
720:
695:
691:
686:
682:
668:
546:
545:
522:
500:
491:
468:
450:
434:
363:Gamma (
326:Theta (
223:Delta (
79:
75:
71:
35:
29:
12472:Hedge funds
12235:Derivatives
12230:Commodities
12185:Day trading
12095:Directional
11994:Hedge funds
11658:Zero Coupon
11588:Correlation
11536:Vanna–Volga
11394:Iron condor
11180:Bond option
10275:S&P 500
10189:-elasticity
10143:basis point
10125:In trading
7245:Black model
6652:dual gamma
6533:dual delta
3363:Volatility
3138:Cross volga
3131:Cross vanna
3125:Cross gamma
3109:underlyings
2892:DgammaDtime
2888:gamma decay
2730:DgammaDspot
2441:(sometimes
2157:DdeltaDtime
2036:delta decay
1082:The symbol
699:delta-hedge
441:sensitivity
282:Vega (
97:Underlying
50:such as an
12507:Categories
12427:Governance
12001:Investment
11932:Tax policy
11648:Volatility
11558:Amortising
11399:Jelly roll
11334:Box spread
11329:Backspread
11321:Strategies
11157:Volatility
11152:the Greeks
11117:Expiration
10889:24 January
10866:2020-12-21
10710:2014-03-07
10685:2014-03-07
10652:2206.08753
10435:. p.
10349:References
10182:elasticity
10174:percentage
10166:the lambda
10129:and other
7243:Under the
3148:See also:
3097:DvommaDvol
2873:DgammaDvol
2336:DvegaDtime
2332:vega decay
2192:volatility
1907:DdeltaDvol
1903:spot price
1899:DdeltaDvol
1895:DvegaDspot
1878:of value.
1774:derivative
1677:products.
1666:percentage
1487:percentage
1483:elasticity
1306:increasing
937:volatility
933:volatility
753:dual delta
135:Passage of
112:Volatility
107:Spot price
99:parameter
56:parameters
12361:Investors
12011:Arbitrage
11623:Inflation
11573:Commodity
11531:Trinomial
11466:Bachelier
11458:Valuation
11339:Butterfly
11273:Commodore
11122:Moneyness
10266:portfolio
10260:(β) of a
10147:the yield
10067:φ
10042:φ
9989:−
9977:−
9958:⋅
9922:−
9877:⋅
9864:−
9848:−
9830:⋅
9812:⋅
9787:−
9723:⋅
9710:−
9688:⋅
9670:⋅
9635:φ
9614:φ
9608:⋅
9554:−
9548:⋅
9479:−
9473:⋅
9440:
9383:
9371:⋅
9301:
9259:τ
9254:σ
9206:φ
9181:φ
9149:τ
9144:σ
9141:−
9120:τ
9115:σ
9110:τ
9101:σ
9087:−
9067:
9032:τ
9027:σ
9022:τ
9013:σ
8979:
8921:σ
8882:σ
8853:τ
8832:φ
8827:τ
8821:−
8772:τ
8767:σ
8751:−
8722:σ
8691:φ
8686:τ
8680:−
8672:−
8648:σ
8645:∂
8639:∂
8625:∂
8587:τ
8582:σ
8551:φ
8543:τ
8537:−
8518:τ
8513:σ
8489:φ
8481:τ
8475:−
8436:∂
8422:∂
8395:Γ
8350:−
8344:Φ
8338:−
8322:−
8316:Φ
8305:τ
8299:−
8291:τ
8288:−
8245:Φ
8239:−
8220:Φ
8209:τ
8203:−
8195:τ
8192:−
8170:ρ
8132:−
8126:Φ
8121:τ
8115:−
8101:−
8085:−
8079:Φ
8074:τ
8068:−
8046:τ
8036:σ
8017:φ
8012:τ
8006:−
7992:−
7954:Φ
7949:τ
7943:−
7910:Φ
7905:τ
7899:−
7885:−
7877:τ
7867:σ
7848:φ
7843:τ
7837:−
7823:−
7801:Θ
7773:τ
7752:φ
7747:τ
7741:−
7725:τ
7704:φ
7699:τ
7693:−
7621:−
7615:Φ
7610:τ
7604:−
7596:−
7558:Φ
7553:τ
7547:−
7518:∂
7507:∂
7484:Δ
7439:−
7433:Φ
7427:−
7411:−
7405:Φ
7394:τ
7388:−
7340:Φ
7334:−
7315:Φ
7304:τ
7298:−
7195:−
7185:∞
7176:∫
7169:π
7157:−
7119:−
7104:∞
7101:−
7097:∫
7090:π
7062:Φ
7033:−
7022:π
6994:φ
6985:τ
6980:σ
6977:−
6956:τ
6951:σ
6946:τ
6932:σ
6918:−
6912:−
6881:
6846:τ
6841:σ
6836:τ
6822:σ
6802:−
6771:
6710:τ
6705:σ
6681:φ
6673:τ
6667:−
6622:−
6616:Φ
6611:τ
6605:−
6562:Φ
6557:τ
6551:−
6543:−
6449:−
6407:σ
6395:−
6341:τ
6336:σ
6329:τ
6324:σ
6311:−
6308:τ
6299:−
6275:τ
6256:τ
6251:σ
6248:τ
6221:φ
6213:τ
6207:−
6199:−
6170:σ
6163:−
6137:Γ
6126:τ
6115:σ
6098:−
6054:φ
6046:τ
6040:−
5992:τ
5987:σ
5961:Γ
5956:−
5934:τ
5929:σ
5903:τ
5898:σ
5867:φ
5859:τ
5853:−
5845:−
5814:Γ
5778:τ
5773:σ
5738:φ
5733:τ
5727:−
5698:ϖ
5666:τ
5629:−
5621:τ
5616:σ
5593:−
5566:τ
5545:φ
5540:τ
5534:−
5523:−
5494:σ
5463:φ
5458:τ
5452:−
5444:τ
5438:−
5409:σ
5370:σ
5341:τ
5320:φ
5315:τ
5309:−
5265:τ
5260:σ
5257:τ
5247:τ
5242:σ
5229:−
5226:τ
5217:−
5186:φ
5181:τ
5175:−
5167:−
5151:−
5145:Φ
5140:τ
5134:−
5123:−
5096:τ
5091:σ
5088:τ
5078:τ
5073:σ
5060:−
5057:τ
5048:−
5017:φ
5012:τ
5006:−
4998:−
4979:Φ
4974:τ
4968:−
4923:τ
4918:σ
4902:−
4873:σ
4842:φ
4837:τ
4831:−
4823:−
4791:τ
4786:σ
4755:φ
4747:τ
4741:−
4722:τ
4717:σ
4693:φ
4685:τ
4679:−
4652:Γ
4613:Δ
4591:λ
4554:−
4548:Φ
4543:τ
4537:−
4529:τ
4489:Φ
4484:τ
4478:−
4470:τ
4464:−
4441:ϵ
4432:epsilon (
4402:−
4396:Φ
4391:τ
4385:−
4377:τ
4371:−
4333:Φ
4328:τ
4322:−
4314:τ
4289:ρ
4251:−
4245:Φ
4240:τ
4234:−
4220:−
4204:−
4198:Φ
4193:τ
4187:−
4165:τ
4155:σ
4136:φ
4125:τ
4119:−
4111:−
4073:Φ
4068:τ
4062:−
4029:Φ
4024:τ
4018:−
4004:−
3996:τ
3986:σ
3967:φ
3956:τ
3950:−
3942:−
3920:Θ
3893:τ
3872:φ
3867:τ
3861:−
3845:τ
3824:φ
3819:τ
3813:−
3741:−
3735:Φ
3730:τ
3724:−
3716:−
3678:Φ
3673:τ
3667:−
3640:Δ
3598:−
3592:Φ
3587:τ
3581:−
3570:−
3554:−
3548:Φ
3540:τ
3534:−
3491:Φ
3483:τ
3477:−
3469:−
3450:Φ
3445:τ
3439:−
3371:σ
3344:−
3335:τ
3205:Φ
3178:φ
3071:σ
3067:∂
3053:∂
3040:σ
3037:∂
3027:∂
2971:τ
2968:∂
2954:∂
2940:∂
2927:τ
2924:∂
2919:Γ
2916:∂
2853:σ
2850:∂
2836:∂
2822:∂
2806:∂
2796:∂
2784:σ
2781:∂
2776:Γ
2773:∂
2696:∂
2682:∂
2666:∂
2661:Γ
2658:∂
2618:butterfly
2588:∂
2574:∂
2564:ϖ
2512:∂
2508:σ
2505:∂
2491:∂
2478:σ
2475:∂
2470:ρ
2467:∂
2412:τ
2409:∂
2405:σ
2402:∂
2388:∂
2375:τ
2372:∂
2360:∂
2262:σ
2258:∂
2244:∂
2231:σ
2228:∂
2216:∂
2188:DvegaDvol
2134:∂
2130:τ
2127:∂
2113:∂
2106:−
2094:∂
2089:Θ
2086:∂
2074:τ
2071:∂
2066:Δ
2063:∂
2057:−
2006:σ
2003:∂
1996:∂
1982:∂
1966:∂
1954:∂
1942:σ
1939:∂
1934:Δ
1931:∂
1876:convexity
1834:∂
1820:∂
1804:∂
1799:Δ
1796:∂
1787:Γ
1760:Γ
1711:∂
1703:∂
1694:ψ
1688:ε
1651:ψ
1630:ε
1584:×
1581:Δ
1575:Ω
1569:λ
1536:×
1527:∂
1519:∂
1510:Ω
1504:λ
1469:Ω
1442:λ
1394:∂
1386:∂
1377:ρ
1354:ρ
1320:τ
1287:τ
1248:τ
1245:∂
1237:∂
1231:−
1225:Θ
1197:Θ
1144:Λ
1120:τ
1096:κ
1009:ν
974:σ
971:∂
963:∂
901:−
887:Δ
870:Δ
845:Δ
828:Δ
800:Δ
797:−
783:Δ
723:moneyness
717:Moneyness
643:∂
635:∂
626:Δ
559:Δ
457:financial
371:Γ
334:Θ
253:Γ
231:Δ
204:Θ
152:Δ
121:σ
60:portfolio
12003:strategy
11762:Slippage
11692:Contango
11676:Forwards
11643:Variance
11603:Dividend
11598:Currency
11511:Margrabe
11506:Lattices
11485:equation
11471:Binomial
11419:Strangle
11414:Straddle
11311:Swaption
11293:Lookback
11278:Compound
11220:Warrants
11195:European
11175:American
11167:Vanillas
11132:Pin risk
11112:Exercise
11018:Archived
10933:22 March
10912:22 March
10762:citation
10307:See also
10301:swaption
4583:lambda (
1670:dividend
1491:leverage
1277:decrease
1269:positive
493:Greeks.
306:)
243:)
40:calculus
12223:Markets
12214:Related
12173:Trading
11681:Futures
11301:Rainbow
11268:Cliquet
11263:Chooser
11243:Barrier
11230:Exotics
11092:Options
10661:4134885
10271:indices
10168:is the
8387:gamma (
7793:theta (
7476:delta (
6382:ultima
4644:gamma (
3912:theta (
3632:delta (
3217:is the
1619:Epsilon
1613:Epsilon
1485:is the
764:due to
737:in the
727:implied
12295:Greeks
12240:Equity
11742:Margin
11608:Equity
11501:Heston
11404:Ladder
11354:Condor
11349:Collar
11306:Spread
11253:Binary
11248:Basket
11001:Theory
10659:
10617:
10577:
10546:
10499:
10443:
10408:
10374:
10209:, the
8941:where
8803:vomma
8610:vanna
8267:
7651:vega (
7362:
6733:where
6189:color
6025:zomma
5835:speed
5291:vomma
4950:charm
4813:vanna
3771:vega (
3017:Ultima
3003:Ultima
2998:Ultima
1675:equity
1430:Lambda
1424:Lambda
1134:lambda
1075:, and
1059:, and
527:quarks
460:models
409:Ultima
403:Vomma
52:option
36:Greeks
34:, the
12338:Short
12328:Hedge
12309:alpha
12216:terms
12140:Other
11613:Forex
11568:Basis
11563:Asset
11550:Swaps
11476:Black
11379:Fence
11238:Asian
11100:Terms
10956:(PDF)
10742:(PDF)
10647:arXiv
10523:7 Jan
10297:fugit
10291:Fugit
10285:Fugit
10262:stock
10127:bonds
8162:rho (
7258:Puts
7255:Calls
5513:veta
5428:vera
4281:rho (
3396:Puts
3393:Calls
3160:calls
3031:vomma
2906:Color
2884:Color
2879:Color
2800:vanna
2763:Zomma
2749:Zomma
2744:Zomma
2734:Speed
2648:Speed
2634:Speed
2629:Speed
2536:Maple
2443:rhova
2206:Vomma
2186:, or
2180:volga
2176:Vomma
2171:Vomma
2161:theta
2050:Charm
2032:Charm
2027:Charm
1921:Vanna
1891:Vanna
1886:Vanna
1749:Gamma
1743:Gamma
1481:, or
1458:omega
1185:Theta
1179:Theta
1085:kappa
1077:theta
548:Delta
542:Delta
511:sigma
503:alpha
497:Names
487:gamma
479:theta
471:delta
397:Color
392:Zomma
387:Speed
350:Charm
315:Vomma
310:Vanna
276:Charm
271:Vanna
266:Gamma
217:Theta
165:Delta
137:time
12313:beta
12269:Misc
11447:Bull
11443:Bear
11185:Call
10985:help
10935:2018
10914:2018
10891:2017
10768:link
10657:SSRN
10643:June
10639:Risk
10615:ISBN
10575:ISBN
10544:ISBN
10525:2010
10497:ISBN
10441:ISBN
10406:ISBN
10372:ISBN
10295:The
10258:beta
10256:The
10246:Beta
10233:and
10193:unit
10187:semi
10178:unit
10155:CS01
10139:DV01
10119:and
9240:let
8605:(*)
7788:(*)
3197:and
3164:puts
3162:and
3119:cega
2457:Vera
2439:Vera
2434:Vera
2350:Veta
2328:Veta
2323:Veta
2298:and
1897:and
1110:vega
1069:beta
1061:vega
1047:vega
993:Vega
929:Vega
924:Vega
894:call
835:call
790:call
671:call
507:beta
505:and
481:and
475:vega
355:Veta
320:Veta
193:Vega
11215:Put
10960:doi
10750:doi
10437:308
10264:or
10161:.)
10031:So
2890:or
2728:or
2334:or
2034:or
1343:Rho
1337:Rho
1279:to
1073:eta
1065:vee
1056:vee
877:put
852:put
807:put
675:put
529:in
515:tau
483:rho
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62:of
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