997:, as well as a corresponding restriction on the "drift coefficients" of the forward rates. These, in turn, are functions of the volatility(s) of the forward rates. A "simple" discretized expression for the drift then allows for forward rates to be expressed in a binomial lattice. For these forward rate-based models, dependent on volatility assumptions, the lattice might not recombine. (This means that an "up-move" followed by a "down-move" will not give the same result as a "down-move" followed by an "up-move".) In this case, the Lattice is sometimes referred to as a "bush", and the number of nodes grows exponentially as a function of number of time-steps. A recombining binomial tree methodology is also available for the Libor Market Model.
1223:(CBs) the approach of Tsiveriotis and Fernandes (1998) is to divide the value of the bond at each node into an "equity" component, arising from situations where the CB will be converted, and a "debt" component, arising from situations where CB is redeemed. Correspondingly, twin trees are constructed where discounting is at the risk free and credit risk adjusted rate respectively, with the sum being the value of the CB. There are other methods, which similarly combine an equity-type tree with a short-rate tree. An alternate approach, originally published by
20:
31:
1208:. To accommodate this in the lattice framework, the OIS rate and the relevant LIBOR rate are jointly modeled in a three-dimensional tree, constructed such that LIBOR swap rates are matched. With the zeroeth step thus accomplished, the valuation will proceed largely as previously, using steps 1 and onwards, but here with cashflows based on the LIBOR "dimension", and discounting using the corresponding nodes from the OIS "dimension".
3354:
856:. This approach is useful when the underlying's behavior departs (markedly) from normality. A related use is to calibrate the tree to the volatility smile (or surface), by a "judicious choice" of parameter values—priced here, options with differing strikes will return differing implied volatilities. For pricing American options, an
764:, sensitivity to time, is likewise estimated given the option price at the first node in the tree and the option price for the same spot in a later time step. (Second time step for trinomial, third for binomial. Depending on method, if the "down factor" is not the inverse of the "up factor", this method will not be precise.) For
823:
As regards the construction, for an R-IBT the first step is to recover the "Implied Ending Risk-Neutral
Probabilities" of spot prices. Then by the assumption that all paths which lead to the same ending node have the same risk-neutral probability, a "path probability" is attached to each ending node.
791:
can be constructed. Here, the tree is solved such that it successfully reproduces selected (all) market prices, across various strikes and expirations. These trees thus "ensure that all
European standard options (with strikes and maturities coinciding with the tree nodes) will have theoretical values
732:
over the time-step, and option valuation is then based on the value of the share at the up-, down- and middle-nodes in the later time-step. As for the binomial, a similar (although smaller) range of methods exist. The trinomial model is considered to produce more accurate results than the binomial
958:
In these cases the valuation is largely as above, but requires an additional, zeroeth, step of constructing an interest rate tree, on which the price of the underlying is then based. The next step also differs: the underlying price here is built via "backward induction" i.e. flows backwards from
832:
corresponding to each node in the tree, such that these are consistent with observed option prices (i.e. with the volatility surface). Thereafter the up-, down- and middle-probabilities are found for each node such that: these sum to 1; spot prices adjacent time-step-wise evolve risk neutrally,
1115:
in the construction relative to equity implied trees: for interest rates, the volatility is known for each time-step, and the node-values (i.e. interest rates) must be solved for specified risk neutral probabilities; for equity, on the other hand, a single volatility cannot be specified per
2078:
1153:
on the bond price and / or the option price there before stepping-backwards one time-step. (And noting that these options are not mutually exclusive, and so a bond may have several options embedded; hybrid securities are treated below.) For other,
959:
maturity, accumulating the present value of scheduled cash flows at each node, as opposed to flowing forwards from valuation date as above. The final step, option valuation, then proceeds as standard. See top for graphic, and aside for description.
772:, sensitivity to input volatility, the measurement is indirect, as the value must be calculated a second time on a new lattice built with these inputs slightly altered - and the sensitivity here is likewise returned via finite difference. See also
792:
which match their market prices". Using the calibrated lattice one can then price options with strike / maturity combinations not quoted in the market, such that these prices are consistent with observed volatility patterns. There exist both
657:
are often modeled using a lattice framework, though with modified assumptions. In each of these cases, a third step is to determine whether the option is to be exercised or held, and to then apply this value at the node in question. Some
1982:
1119:
Once calibrated, the interest rate lattice is then used in the valuation of various of the fixed income instruments and derivatives. The approach for bond options is described aside—note that this approach addresses the problem of
824:
Thereafter "it's as simple as One-Two-Three", and a three step backwards recursion allows for the node probabilities to be recovered for each time step. Option valuation then proceeds as standard, with these substituted for
1040:
observed market prices. The tree is then built as a function of these parameters. In the latter case, the calibration is directly on the lattice: the fit is to both the current term structure of interest rates (i.e. the
542:
2085:
1859:
1269:
The calculation of "Greeks" for interest rate derivatives proceeds as for equity. There is however an additional requirement, particularly for hybrid securities: that is, to estimate sensitivities related to
1132:
for bonds in step 1, and swaptions for bond options in step 2. For caps (and floors) step 1 and 2 are combined: at each node the value is based on the relevant nodes at the later step, plus, for any caplet
717:. Further enhancements are designed to achieve stability relative to Black-Scholes as the number of time-steps changes. More recent models, in fact, are designed around direct convergence to Black-Scholes.
837:; state prices similarly "grow" at the risk free rate. (The solution here is iterative per time step as opposed to simultaneous.) As for R-IBTs, option valuation is then by standard backward recursion.
860:-generated ending distribution may be combined with an R-IBT. This approach is limited as to the set of skewness and kurtosis pairs for which valid distributions are available. The more recent
337:
1990:
693:(CRR) in 1979; see diagram for formulae. Over 20 other methods have been developed, with each "derived under a variety of assumptions" as regards the development of the underlying's price.
1080:
The volatility structure—i.e. vertical node-spacing—here reflects the volatility of rates during the quarter, or other period, corresponding to the lattice time-step. (Some analysts use "
909:(or $ 1), plus coupon (in cents) if relevant; if the bond-date and tree-date do not coincide, these are then discounted to the start of the time-step using the node-specific short-rate;
1246:
on the firm: where the value of the firm is less than the value of the outstanding debt shareholders would choose not to repay the firm's debt; they would choose to repay—and not to
1785:
Grant, Dwight M.; Vora, Gautam (26 February 2009). "Implementing No-Arbitrage Term
Structure of Interest Rate Models in Discrete Time When Interest Rates Are Normally Distributed".
812:(DKC; superseding the DK-IBT). The former is easier built, but is consistent with one maturity only; the latter will be consistent with, but at the same time requires, known (or
261:
1177:
by building a corresponding DKC tree (based on every second time-step in the CRR tree: as DKC is trinomial whereas CRR is binomial) and then using this for option valuation.
213:
1586:
934:
at earlier nodes, value is a function of the expected value of the option at the nodes in the later time step, discounted at the short-rate of the current node; where
603:
For equity and commodities the application is as follows. The first step is to trace the evolution of the option's key underlying variable(s), starting with today's
391:
364:
1227:(1994), does not decouple the components, rather, discounting is at a conversion-probability-weighted risk-free and risky interest rate within a single tree. See
1769:
760:, being sensitivities of option value w.r.t. price, are approximated given differences between option prices - with their related spot - in the same time step.
171:
151:
3191:
1116:
time-step, i.e. we have a "smile", and the tree is built by solving for the probabilities corresponding to specified values of the underlying at each node.
1036:
to the model. In the former case, the approach is to "calibrate" the model parameters, such that bond prices produced by the model, in its continuous form,
3103:
1670:
618:
with constant volatility is usually assumed. The next step is to value the option recursively: stepping backwards from the final time-step, where we have
2337:
111:
24:
1871:
1453:
737:, as the number of steps increases, the results rapidly converge, and the binomial model is then preferred due to its simpler implementation. For
411:
1169:(YTM), employs modified equity-lattice methods. Here the analyst builds a CRR tree of YTM, applying a constant volatility assumption, and then
875:, multinomial lattices can be built, although the number of nodes increases exponentially with the number of underlyers. As an alternative,
2897:
2472:
2012:
1345:
Cox, J. C., Ross, S. A., & Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of financial
Economics, 7(3), 229–263.
894:
0. Construct an interest-rate tree, which, as described in the text, will be consistent with the current term structure of interest rates.
1656:
1418:
1490:
1137:) maturing in the time-step, the difference between its reference-rate and the short-rate at the node (and reflecting the corresponding
701:, and hence produce the "same" option price as Black-Scholes: to achieve this, these will variously seek to agree with the underlying's
2902:
1603:
1363:
919:
of nodes in the later time step, plus coupon payments during the current time step, similarly discounted to the start of the time-step.
1700:
3237:
1962:
2927:
2431:
1638:
107:
2441:
2349:
2327:
2304:
2278:
2254:
2231:
1134:
983:
401:
at each final node of the tree—i.e. at expiration of the option—the option value is simply its intrinsic, or exercise, value;
1914:
1104:
733:
model when fewer time steps are modelled, and is therefore used when computational speed or resources may be an issue. For
3059:
1934:
1749:
1158:, similar adjustments are made to steps 1 and onward. For the "Greeks", largely as for equity, see under next section.
2416:
2397:
2375:
2206:
2183:
1298:
are introduced. Here, similar to rho and vega above, the interest rate tree is rebuilt for an upward and then downward
986:. As for equity, trinomial trees may also be employed for these models; this is usually the case for Hull-White trees.
923:
2. Construct a corresponding bond-option tree, where the option on the bond is valued largely as for an equity option:
622:
at each node; and applying risk neutral valuation at each earlier node, where option value is the probability-weighted
269:
1948:
1155:
1125:
627:
2795:
1831:
1766:
1521:
94:
lattices are additionally useful in that they address many of the issues encountered with continuous models, such as
78:
is required at "all" times (any time) before and including maturity. A continuous model, on the other hand, such as
3399:
3297:
2465:
1009:
2047:
1081:
1046:
110:
fail to account for optimal decisions to terminate the derivative by early exercise, though methods now exist for
1290:
do not consider how changes in interest rates will alter the cash flows due to option exercise. To address this,
650:
2145:
1021:
3394:
3232:
2877:
1181:
1074:
749:
678:
671:
3389:
1193:
46:(black vs red): the short rate is the top value; the development of the bond value shows pull-to-par clearly
1013:
597:
1108:
631:
3272:
2810:
2664:
2458:
1879:
1299:
1232:
3384:
3379:
3128:
3069:
2891:
1762:
1450:
879:, for example, can be priced using an "approximating distribution" via an Edgeworth (or Johnson) tree.
654:
645:, or to hold the option, may be modeled at each discrete time/price combination; this is also true for
218:
2409:
Valuation in a World of CVA, DVA, and FVA: A Tutorial on Debt
Securities and Interest Rate Derivatives
1170:
3404:
3173:
2984:
2133:
1254:)—otherwise. Lattice models have been developed for equity analysis here, particularly as relates to
955:
741:
the trinomial model (or adaptations) is sometimes more stable and accurate, regardless of step-size.
619:
176:
91:
2109:
1900:
1620:
3292:
3287:
1926:
1054:
857:
686:
576: + 1 is captured in a branch. This process is iterated until every possible path between
2062:
3242:
2942:
2887:
2770:
2611:
2543:
1967:
1821:
1430:
1058:
865:
710:
698:
612:
79:
2265:
2019:
1000:
As regards the short-rate models, these are, in turn, further categorized: these will be either
3039:
3024:
2989:
2932:
1930:
1205:
1070:
801:
667:
43:
1684:
1173:
at each node; prices here are thus pulling-to-par. The second step is to then incorporate any
3252:
3019:
2917:
2596:
1986:
1255:
1228:
1189:
793:
2079:"Valuation Considerations Related to Complex Financial Instruments for Investment Companies"
1537:
1358:
1149:
a third step would be required: at each node in the time-step incorporate the effect of the
714:
3206:
3163:
3153:
3143:
3138:
2864:
2805:
2740:
2694:
2689:
2563:
2523:
2490:
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2198:
1434:
1414:
1001:
994:
967:
369:
342:
87:
59:
1733:
8:
3211:
2999:
2922:
2745:
2013:"A Tree Model for Pricing Convertible Bonds with Equity, Interest Rate, and Default Risk"
1704:
1295:
1258:. Relatedly, as regards corporate debt pricing, the relationship between equity holders'
1185:
1062:
761:
757:
753:
694:
548:
value is the greater of this and the exercise value given the corresponding equity value.
2146:
A Binomial
Lattice Method for Pricing Corporate Debt and Modeling Chapter 11 Proceedings
1621:"Wiley: Advanced Modelling in Finance using Excel and VBA - Mary Jackson, Mike Staunton"
1323:
769:
3262:
3247:
3216:
3201:
3168:
3034:
2825:
2790:
2553:
2518:
2481:
1802:
1291:
1251:
1174:
1138:
1096:
979:
938:
value is the greater of this and the exercise value given the corresponding bond value.
784:
765:
642:
608:
556:
In general the approach is to divide time between now and the option's expiration into
156:
136:
75:
1567:
3267:
3257:
3196:
3183:
3158:
3044:
2830:
2626:
2437:
2427:
2412:
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2371:
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2300:
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2250:
2227:
2202:
2179:
1827:
1806:
1664:
1336:
Hull, J. C. (2006). Options, futures, and other derivatives. Pearson
Education India.
1303:
1279:
1259:
1220:
1219:, incorporating both equity- and bond-like features are also valued using trees. For
1166:
1092:
405:
1983:"Description of Tree Model for the Valuation of a Convertible Bond with Credit Risk"
3148:
3087:
3082:
3064:
2994:
2760:
2755:
2727:
2679:
2558:
2498:
2385:
2129:
2058:
1794:
1688:
1239:
1050:
1049:. Here, calibration means that the interest-rate-tree reproduces the prices of the
1017:
990:
971:
963:
947:
853:
817:
780:
745:
1380:
3358:
3328:
3323:
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2785:
2780:
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2684:
2601:
2581:
2359:
1773:
1503:
1457:
1367:
1275:
1216:
1201:
1150:
797:
776:, the estimated time to exercise, which is typically calculated using a lattice.
690:
663:
646:
638:
615:
568: + 1 such that every possible change in the state of the world between
103:
83:
71:
35:
19:
1360:
A Synthesis of
Binomial Option Pricing Models for Lognormally Distributed Assets
1161:
An alternative approach to modeling (American) bond options, particularly those
3333:
3318:
3118:
3029:
2979:
2956:
2937:
2765:
2707:
2674:
2669:
2649:
2573:
2367:
2073:
1287:
1129:
916:
834:
805:
734:
721:
702:
659:
173:) to the current price, such that in the next period the price will either be
3373:
3313:
3282:
3123:
3049:
3009:
3004:
2840:
2712:
2659:
2654:
2636:
2533:
2513:
2288:
2223:
2043:
1716:
1573:
1398:
Efficient procedures for valuing
European and American path-dependent options
1283:
1224:
1142:
1088:
for the time-step; to be market-consistent, analysts generally prefer to use
1005:
951:
913:
876:
738:
623:
99:
63:
16:
Method for evaluating stock options that divides time into discrete intervals
2217:
1798:
1469:
537:{\displaystyle C_{t-\Delta t,i}=e^{-r\Delta t}(pC_{t,i+1}+(1-p)C_{t,i-1})\,}
3133:
2907:
2835:
2815:
2775:
2644:
2616:
2606:
2548:
1847:
1397:
1162:
1146:
975:
935:
813:
725:
545:
67:
1192:", whereas previously it was off a single, "self discounting", curve; see
674:
would be preferred. (Although, tree-based methods have been developed. )
30:
3014:
2882:
2853:
2849:
2800:
2591:
2586:
2319:
1734:
Efficient
Calibration of Trinomial Trees for One-Factor Short Rate Models
1243:
1121:
1042:
1025:
829:
809:
682:
95:
39:
816:) prices at all time-steps and nodes. (DKC is effectively a discretized
637:
As stated above, the lattice approach is particularly useful in valuing
3338:
2974:
2969:
2735:
2621:
1550:
1263:
1066:
1065:(such as Ho-Lee), calibration may be performed analytically, while for
1057:; note the parallel to the implied trees for equity above, and compare
906:
898:
872:
706:
604:
1032:
from the model, while for arbitrage-free models the yield curve is an
2528:
2450:
1750:
Pricing Interest Rate-dependent Financial Claims with Option Features
1247:
928:
1504:"Option Pricing & Stock Price Probability Calculators - Hoadley"
1470:
On the Relation Between Binomial and Trinomial Option Pricing Models
1111:.) Given this functional link to volatility, note now the resultant
3098:
2820:
2717:
2538:
2390:
Stochastic Calculus for Finance I: The Binomial Asset Pricing Model
1100:
1037:
849:
845:
2425:
51:
1848:
The Libor Market Model: A Recombining Binomial Tree Methodology
3353:
750:
where the sensitivities are calculated using finite differences
697:, as the number of time-steps increases, these converge to the
1451:
The Convergence of Binomial Trees for Pricing the American Put
2267:
Tree methods in finance, Encyclopedia of Quantitative Finance
1197:
1128:. For swaptions the logic is almost identical, substituting
773:
897:
1. Construct a corresponding tree of bond-prices, where the
2240:
2215:
2192:
1053:—and any other interest-rate sensitive securities—used in
868:, as this is capable of accommodating all possible pairs.
133:
Either forward-construct, applying an up or down factor (
681:; the standard ("canonical") method is that proposed by
1540:. Goldman Sachs, Quantitative Strategies Research Notes
1024:). This distinction: for equilibrium-based models the
901:
bond is valued at each node by "backwards induction":
600:
until a fair value of the option today is calculated.
962:
The initial lattice is built by discretizing either a
626:
of the up- and down-nodes in the later time-step. See
588:
is mapped. Probabilities are then estimated for every
2176:
Valuation of Interest-Sensitive Financial Instruments
1171:
calculates the bond price as a function of this yield
414:
372:
345:
272:
221:
179:
159:
139:
2293:
Valuation of fixed income securities and derivatives
1872:"Calibrating the Ornstein-Uhlenbeck (Vasicek) model"
1585:
Jim Clark, Les Clewlow and Chris Strickland (2008).
564:, the model has a finite number of outcomes at time
266:
or given that the tree is recombining, directly via
1536:Emanuel Derman, Iraj Kani, and Neil Chriss (1996).
117:
2010:
1846:S. Derrick, D. Stapleton and R. Stapleton (2005).
1381:Simple, Fast and Flexible Pricing of Asian Options
536:
385:
358:
331:
255:
207:
165:
145:
2336:
1587:Calibrating trees to the market prices of options
332:{\displaystyle S_{n}=S_{0}\times u^{N_{u}-N_{d}}}
3371:
2342:Applied Derivatives: Options, Futures, and Swaps
1669:: CS1 maint: bot: original URL status unknown (
1266:proceedings has also been modelled via lattice.
2406:
2358:
2173:
1752:, Ch 11. in Rendleman (2002), per Bibliography.
1654:
1538:Implied Trinomial Trees of the Volatility Smile
1306:given the corresponding changes in bond value.
1194:Interest rate swap § Valuation and pricing
1126:Black–Scholes model § Valuing bond options
1073:; see for example, the boxed-description under
544:, p being the probability of an up move; where
2195:Valuation of Interest Rate Swaps and Swaptions
2150:Journal of Financial and Quantitative Analysis
1745:
1743:
1741:
1124:experienced under closed form approaches; see
632:Rational pricing § Risk neutral valuation
98:. The method is also used for valuing certain
2466:
2384:
2316:The Complete Guide to Option Pricing Formulas
2287:
1659:. Archived from the original on 22 June 2007.
1109:Interest rate cap § Implied Volatilities
598:probabilities flow backwards through the tree
1963:"Valuing Convertible Bonds with Credit Risk"
1767:The Heath-Jarrow-Morton Term Structure Model
1522:Calculating the Greeks in the Binomial Model
882:
828:. For DKC, the first step is to recover the
628:Binomial options pricing model § Method
397:2. Construct the corresponding option tree:
2216:Gerald Buetow & James Sochacki (2001).
1738:
1419:African Institute for Mathematical Sciences
728:in 1986. Here, the share price may remain
2473:
2459:
2313:
2219:Term-Structure Models Using Binomial Trees
2110:"Not Found - Business Valuation Resources"
1819:
748:can be estimated directly on the lattice,
666:, are also easily modeled here; for other
2263:
1784:
1569:The Volatility Smile and Its Implied Tree
1204:in question, while discounting is at the
1196:. Here, payoffs are set as a function of
905:at its final nodes, bond value is simply
533:
129:1. Construct the tree of equity-prices:
2134:Option Pricing Applications in Valuation
2048:Valuing Convertible Bonds as Derivatives
2018:. Journal of Derivatives. Archived from
1901:"embedded option, thefreedictionary.com"
1353:
1351:
1322:Staff, Investopedia (17 November 2010).
779:When it is important to incorporate the
82:, would only allow for the valuation of
70:, a typical example would be pricing an
29:
18:
3298:Power reverse dual-currency note (PRDC)
3238:Constant proportion portfolio insurance
2178:(1st ed.). John Wiley & Sons.
1445:
1443:
560:discrete periods. At the specific time
3372:
2480:
2241:Les Clewlow; Chris Strickland (1998).
1532:
1530:
1274:in interest rates. For a bond with an
946:Lattices are commonly used in valuing
927:at option maturity, value is based on
596: + 1 path. The outcomes and
108:Monte Carlo methods for option pricing
2454:
2193:Gerald Buetow; Frank Fabozzi (2000).
1980:
1721:Mathematica in Education and Research
1593:, August 2008. (Archived, 2015-06-30)
1566:Emanuel Derman and Iraj Kani (1994).
1491:Pricing Options Using Trinomial Trees
1370:. Journal of Applied Finance, Vol. 18
1348:
1321:
1156:more exotic interest rate derivatives
768:, sensitivity to interest rates, and
3233:Collateralized debt obligation (CDO)
2144:Mark Broadie and Ozgur Kaya (2007).
1655:Mark Rubinstein (January 15, 1995).
1440:
1211:
1141:and notional-value exchanged). For
2222:. The Research Foundation of AIMR (
1820:Rubinstein, Mark (1 January 1999).
1527:
634:for logic and formulae derivation.
611:is consistent with its volatility;
126:Tree-based equity option valuation:
13:
1961:Tsiveriotis and Fernandes (1998).
1732:M. Leippold and Z. Wiener (2003).
1715:S. Benninga and Z. Wiener. (1998).
720:A variant on the Binomial, is the
677:The simplest lattice model is the
454:
426:
23:Binomial Lattice for equity, with
14:
3416:
2344:(1st ed.). Wiley-Blackwell.
1685:Pricing Basket Options with Smile
1396:John Hull and Alan White. (1993)
1300:parallel shift in the yield curve
1229:Convertible bond § Valuation
891:Tree-based bond option valuation:
866:Johnson "family" of distributions
256:{\displaystyle S_{down}=S\cdot d}
3352:
1936:Multi-Curve Modeling Using Trees
1385:Journal of Computational Finance
1084:", i.e. of the rates applicable
1069:models the calibration is via a
931:for all nodes in that time-step;
912:at each earlier node, it is the
118:Equity and commodity derivatives
2167:
2161:See Fabozzi under Bibliography.
2155:
2138:
2123:
2102:
2067:
2052:
2037:
2004:
1974:
1955:
1941:
1920:
1907:
1893:
1864:
1852:
1840:
1813:
1778:
1755:
1726:
1709:
1693:
1677:
1648:
1631:
1613:
1596:
1579:
1560:
1543:
1514:
1496:
1479:
1462:
844:allow for an analyst-specified
208:{\displaystyle S_{up}=S\cdot u}
3060:Year-on-year inflation-indexed
2243:Implementing Derivative Models
1717:Binomial Term Structure Models
1429:Prof. Markus K. Brunnermeier.
1423:
1407:
1390:
1373:
1339:
1330:
1315:
995:martingale probability measure
679:binomial options pricing model
641:, where the choice whether to
530:
505:
493:
462:
366:is the number of up ticks and
58:is a technique applied to the
1:
3070:Zero-coupon inflation-indexed
1701:Term Structure Lattice Models
1415:Pricing in the Binomial Model
1309:
1915:American Bond Option Pricing
1645:, Volume 11, Pages 1165-1176
1175:term structure of volatility
1055:constructing the yield curve
993:implies that there exists a
886:
630:for more detail, as well as
393:is the number of down ticks.
121:
7:
3273:Foreign exchange derivative
2665:Callable bull/bear contract
1949:"Pricing Convertible Bonds"
1787:The Journal of Fixed Income
1233:Contingent convertible bond
852:in spot price returns; see
86:, where exercise is on the
10:
3421:
2433:Binomial Models in Finance
1763:Louisiana State University
1637:Jean-Guy Simonato (2011).
1476:, Winter 2000, 8 (2) 47-50
1474:The Journal of Derivatives
1431:Multi-period Model Options
1357:Chance, Don M. March 2008
1182:2007–2008 financial crisis
1103:-prices of each component
978:-based model, such as the
3347:
3306:
3225:
3182:
3174:Stock market index future
3078:
2955:
2863:
2726:
2635:
2572:
2506:
2497:
2488:
2364:Rubinstein On Derivatives
2063:Valuing Firms in Distress
1860:Interest Rate Derivatives
1836:– via Google Books.
1823:Rubinstein on Derivatives
1045:), and the corresponding
991:condition of no arbitrage
956:interest rate derivatives
883:Interest rate derivatives
643:exercise the option early
92:interest rate derivatives
74:, where a decision as to
3293:Mortgage-backed security
3288:Interest rate derivative
3263:Equity-linked note (ELN)
3248:Credit-linked note (CLN)
2426:John van der Hoek &
2407:Donald J. Smith (2017).
2174:David F. Babbel (1996).
2011:D. R. Chambers, Qin Lu.
1858:Dr. Graeme West (2010).
1683:Isabel Ehrlich (2012).
1604:Edgeworth Binomial Trees
1602:Mark Rubinstein (1998).
1549:Mark Rubinstein (1994).
1468:Mark Rubinstein (2000).
1379:Timothy Klassen. (2001)
1188:is (generally) under a "
1061:. For models assuming a
842:Edgeworth binomial trees
406:value is via expectation
60:valuation of derivatives
3400:Trees (data structures)
3243:Contract for difference
2544:Risk-free interest rate
2264:Rama Cont, ed. (2010).
1968:Journal of Fixed Income
1799:10.3905/jfi.1999.319247
1493:. University of Warwick
1413:Ronnie Becker. (N.D.).
1302:and these measures are
1059:Bootstrapping (finance)
802:implied trinomial trees
699:Log-normal distribution
649:. For similar reasons,
66:model is required. For
3025:Forward Rate Agreement
1639:Johnson binomial trees
1608:Journal of Derivatives
1551:Implied Binomial Trees
1449:Mark s. Joshi (2008).
1402:Journal of Derivatives
1304:calculated numerically
1282:based calculations of
1075:Black–Derman–Toy model
1071:root-finding algorithm
862:Johnson binomial trees
794:implied binomial trees
715:as measured discretely
668:Path-Dependent Options
655:employee stock options
538:
387:
360:
333:
257:
209:
167:
147:
88:option's maturity date
47:
27:
3395:Models of computation
3253:Credit default option
2597:Employee stock option
1987:University of Waikato
1699:Martin Haugh (2010).
1387:, 4 (3) 89-124 (2001)
1324:"Lattice-Based Model"
1252:exercise their option
1190:multi-curve framework
539:
388:
386:{\displaystyle N_{d}}
361:
359:{\displaystyle N_{u}}
334:
258:
210:
168:
148:
33:
22:
3390:Mathematical finance
3207:Inflation derivative
3192:Commodity derivative
3164:Single-stock futures
3154:Normal backwardation
3144:Interest rate future
2985:Conditional variance
2491:Derivative (finance)
2411:. World Scientific.
1913:Riskworx (c. 2000).
1643:Quantitative Finance
1435:Princeton University
1047:volatility structure
412:
370:
343:
270:
219:
177:
157:
137:
112:solving this problem
3359:Business portal
3212:Property derivative
2314:Espen Haug (2006).
2117:www.bvresources.com
1705:Columbia University
1520:Don Chance. (2010)
1242:can be viewed as a
1082:realized volatility
1063:normal distribution
840:As an alternative,
713:at each time-step,
102:, where because of
3217:Weather derivative
3202:Freight derivative
3184:Exotic derivatives
3104:Commodities future
2791:Intermarket spread
2554:Synthetic position
2482:Derivatives market
1772:2015-09-23 at the
1761:Prof. Don Chance,
1555:Journal of Finance
1456:2015-07-02 at the
1366:2016-03-04 at the
1292:effective duration
1139:day count fraction
1097:implied volatility
980:LIBOR market model
800:IBTs (R-IBT), and
534:
404:at earlier nodes,
383:
356:
329:
253:
205:
163:
143:
48:
28:
3385:Short-rate models
3380:Options (finance)
3367:
3366:
3268:Equity derivative
3258:Credit derivative
3226:Other derivatives
3197:Energy derivative
3159:Perpetual futures
3040:Overnight indexed
2990:Constant maturity
2951:
2950:
2898:Finite difference
2831:Protective option
2443:978-0-387-25898-0
2428:Robert J. Elliott
2351:978-0-631-21590-5
2338:Richard Rendleman
2329:978-0-07-138997-6
2306:978-1-883249-25-0
2280:978-0-470-05756-8
2256:978-0-471-96651-7
2233:978-0-943205-53-3
1657:"Rainbow Options"
1572:. Research Note,
1280:yield to maturity
1260:limited liability
1221:convertible bonds
1217:Hybrid securities
1212:Hybrid securities
1167:yield to maturity
1093:interest rate cap
1051:zero-coupon bonds
1002:equilibrium-based
944:
943:
607:, such that this
554:
553:
166:{\displaystyle d}
146:{\displaystyle u}
3412:
3405:Financial models
3357:
3356:
3129:Forwards pricing
2903:Garman–Kohlhagen
2504:
2503:
2475:
2468:
2461:
2452:
2451:
2447:
2422:
2403:
2381:
2366:(1st ed.).
2355:
2333:
2310:
2295:(3rd ed.).
2284:
2272:
2260:
2237:
2212:
2189:
2162:
2159:
2153:
2152:, Vol. 42, No. 2
2142:
2136:
2130:Aswath Damodaran
2127:
2121:
2120:
2114:
2106:
2100:
2099:
2097:
2096:
2090:
2084:. Archived from
2083:
2071:
2065:
2059:Aswath Damodaran
2056:
2050:
2041:
2035:
2034:
2032:
2030:
2024:
2017:
2008:
2002:
2001:
1999:
1998:
1989:. Archived from
1978:
1972:
1959:
1953:
1952:
1945:
1939:
1924:
1918:
1911:
1905:
1904:
1897:
1891:
1890:
1888:
1887:
1878:. Archived from
1868:
1862:
1856:
1850:
1844:
1838:
1837:
1817:
1811:
1810:
1782:
1776:
1759:
1753:
1747:
1736:
1730:
1724:
1713:
1707:
1697:
1691:
1689:Imperial College
1681:
1675:
1674:
1668:
1660:
1652:
1646:
1635:
1629:
1628:
1617:
1611:
1600:
1594:
1583:
1577:
1564:
1558:
1547:
1541:
1534:
1525:
1518:
1512:
1511:
1500:
1494:
1483:
1477:
1466:
1460:
1447:
1438:
1427:
1421:
1411:
1405:
1394:
1388:
1377:
1371:
1355:
1346:
1343:
1337:
1334:
1328:
1327:
1319:
1256:distressed firms
1238:More generally,
1200:specific to the
1095:prices, and the
972:Black Derman Toy
964:short-rate model
887:
854:Edgeworth series
818:local volatility
781:volatility smile
647:Bermudan options
639:American options
543:
541:
540:
535:
529:
528:
489:
488:
461:
460:
439:
438:
392:
390:
389:
384:
382:
381:
365:
363:
362:
357:
355:
354:
338:
336:
335:
330:
328:
327:
326:
325:
313:
312:
295:
294:
282:
281:
262:
260:
259:
254:
240:
239:
214:
212:
211:
206:
192:
191:
172:
170:
169:
164:
152:
150:
149:
144:
122:
84:European options
3420:
3419:
3415:
3414:
3413:
3411:
3410:
3409:
3370:
3369:
3368:
3363:
3351:
3343:
3329:Great Recession
3324:Government debt
3302:
3278:Fund derivative
3221:
3178:
3139:Futures pricing
3114:Dividend future
3109:Currency future
3092:
3074:
2947:
2923:Put–call parity
2859:
2846:Vertical spread
2781:Diagonal spread
2751:Calendar spread
2722:
2631:
2568:
2493:
2484:
2479:
2444:
2419:
2400:
2378:
2360:Mark Rubinstein
2352:
2330:
2307:
2281:
2270:
2257:
2234:
2209:
2186:
2170:
2165:
2160:
2156:
2143:
2139:
2128:
2124:
2112:
2108:
2107:
2103:
2094:
2092:
2088:
2081:
2077:
2072:
2068:
2057:
2053:
2042:
2038:
2028:
2026:
2022:
2015:
2009:
2005:
1996:
1994:
1979:
1975:
1960:
1956:
1947:
1946:
1942:
1925:
1921:
1912:
1908:
1899:
1898:
1894:
1885:
1883:
1870:
1869:
1865:
1857:
1853:
1845:
1841:
1834:
1818:
1814:
1783:
1779:
1774:Wayback Machine
1760:
1756:
1748:
1739:
1731:
1727:
1714:
1710:
1698:
1694:
1682:
1678:
1662:
1661:
1653:
1649:
1636:
1632:
1619:
1618:
1614:
1601:
1597:
1584:
1580:
1565:
1561:
1548:
1544:
1535:
1528:
1519:
1515:
1508:www.hoadley.net
1502:
1501:
1497:
1484:
1480:
1467:
1463:
1458:Wayback Machine
1448:
1441:
1428:
1424:
1412:
1408:
1395:
1391:
1378:
1374:
1368:Wayback Machine
1356:
1349:
1344:
1340:
1335:
1331:
1320:
1316:
1312:
1278:, the standard
1276:embedded option
1272:overall changes
1214:
1151:embedded option
989:Under HJM, the
885:
744:Various of the
735:vanilla options
724:, developed by
703:central moments
664:barrier options
616:Brownian motion
512:
508:
472:
468:
447:
443:
419:
415:
413:
410:
409:
377:
373:
371:
368:
367:
350:
346:
344:
341:
340:
321:
317:
308:
304:
303:
299:
290:
286:
277:
273:
271:
268:
267:
226:
222:
220:
217:
216:
184:
180:
178:
175:
174:
158:
155:
154:
138:
135:
134:
120:
106:in the payoff,
104:path dependence
76:option exercise
72:American option
17:
12:
11:
5:
3418:
3408:
3407:
3402:
3397:
3392:
3387:
3382:
3365:
3364:
3362:
3361:
3348:
3345:
3344:
3342:
3341:
3336:
3334:Municipal debt
3331:
3326:
3321:
3319:Corporate debt
3316:
3310:
3308:
3304:
3303:
3301:
3300:
3295:
3290:
3285:
3280:
3275:
3270:
3265:
3260:
3255:
3250:
3245:
3240:
3235:
3229:
3227:
3223:
3222:
3220:
3219:
3214:
3209:
3204:
3199:
3194:
3188:
3186:
3180:
3179:
3177:
3176:
3171:
3166:
3161:
3156:
3151:
3146:
3141:
3136:
3131:
3126:
3121:
3119:Forward market
3116:
3111:
3106:
3101:
3095:
3093:
3091:
3090:
3085:
3079:
3076:
3075:
3073:
3072:
3067:
3062:
3057:
3052:
3047:
3042:
3037:
3032:
3027:
3022:
3017:
3012:
3007:
3002:
3000:Credit default
2997:
2992:
2987:
2982:
2977:
2972:
2967:
2961:
2959:
2953:
2952:
2949:
2948:
2946:
2945:
2940:
2935:
2930:
2925:
2920:
2915:
2910:
2905:
2900:
2895:
2885:
2880:
2875:
2869:
2867:
2861:
2860:
2858:
2857:
2843:
2838:
2833:
2828:
2823:
2818:
2813:
2808:
2803:
2798:
2796:Iron butterfly
2793:
2788:
2783:
2778:
2773:
2768:
2766:Covered option
2763:
2758:
2753:
2748:
2743:
2738:
2732:
2730:
2724:
2723:
2721:
2720:
2715:
2710:
2705:
2704:Mountain range
2702:
2697:
2692:
2687:
2682:
2677:
2672:
2667:
2662:
2657:
2652:
2647:
2641:
2639:
2633:
2632:
2630:
2629:
2624:
2619:
2614:
2609:
2604:
2599:
2594:
2589:
2584:
2578:
2576:
2570:
2569:
2567:
2566:
2561:
2556:
2551:
2546:
2541:
2536:
2531:
2526:
2521:
2516:
2510:
2508:
2501:
2495:
2494:
2489:
2486:
2485:
2478:
2477:
2470:
2463:
2455:
2449:
2448:
2442:
2423:
2418:978-9813222748
2417:
2404:
2399:978-0387249681
2398:
2382:
2377:978-1899332533
2376:
2356:
2350:
2334:
2328:
2311:
2305:
2285:
2279:
2261:
2255:
2245:. New Jersey:
2238:
2232:
2213:
2208:978-1883249892
2207:
2190:
2185:978-1883249151
2184:
2169:
2166:
2164:
2163:
2154:
2137:
2122:
2101:
2074:Grant Thornton
2066:
2051:
2036:
2003:
1973:
1954:
1940:
1919:
1917:, riskworx.com
1906:
1892:
1863:
1851:
1839:
1832:
1826:. Risk Books.
1812:
1777:
1754:
1737:
1725:
1708:
1692:
1676:
1647:
1630:
1612:
1610:, Spring 1998.
1595:
1578:
1559:
1542:
1526:
1513:
1495:
1478:
1461:
1439:
1422:
1406:
1389:
1372:
1347:
1338:
1329:
1313:
1311:
1308:
1262:and potential
1213:
1210:
1014:arbitrage-free
942:
941:
940:
939:
932:
921:
920:
917:expected value
910:
884:
881:
877:Basket options
835:dividend yield
833:incorporating
739:exotic options
722:Trinomial tree
660:exotic options
620:exercise value
552:
551:
550:
549:
532:
527:
524:
521:
518:
515:
511:
507:
504:
501:
498:
495:
492:
487:
484:
481:
478:
475:
471:
467:
464:
459:
456:
453:
450:
446:
442:
437:
434:
431:
428:
425:
422:
418:
402:
395:
394:
380:
376:
353:
349:
324:
320:
316:
311:
307:
302:
298:
293:
289:
285:
280:
276:
264:
252:
249:
246:
243:
238:
235:
232:
229:
225:
204:
201:
198:
195:
190:
187:
183:
162:
142:
119:
116:
100:exotic options
68:equity options
42:returning the
15:
9:
6:
4:
3:
2:
3417:
3406:
3403:
3401:
3398:
3396:
3393:
3391:
3388:
3386:
3383:
3381:
3378:
3377:
3375:
3360:
3355:
3350:
3349:
3346:
3340:
3337:
3335:
3332:
3330:
3327:
3325:
3322:
3320:
3317:
3315:
3314:Consumer debt
3312:
3311:
3309:
3307:Market issues
3305:
3299:
3296:
3294:
3291:
3289:
3286:
3284:
3283:Fund of funds
3281:
3279:
3276:
3274:
3271:
3269:
3266:
3264:
3261:
3259:
3256:
3254:
3251:
3249:
3246:
3244:
3241:
3239:
3236:
3234:
3231:
3230:
3228:
3224:
3218:
3215:
3213:
3210:
3208:
3205:
3203:
3200:
3198:
3195:
3193:
3190:
3189:
3187:
3185:
3181:
3175:
3172:
3170:
3167:
3165:
3162:
3160:
3157:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3135:
3132:
3130:
3127:
3125:
3124:Forward price
3122:
3120:
3117:
3115:
3112:
3110:
3107:
3105:
3102:
3100:
3097:
3096:
3094:
3089:
3086:
3084:
3081:
3080:
3077:
3071:
3068:
3066:
3063:
3061:
3058:
3056:
3053:
3051:
3048:
3046:
3043:
3041:
3038:
3036:
3035:Interest rate
3033:
3031:
3028:
3026:
3023:
3021:
3018:
3016:
3013:
3011:
3008:
3006:
3003:
3001:
2998:
2996:
2993:
2991:
2988:
2986:
2983:
2981:
2978:
2976:
2973:
2971:
2968:
2966:
2963:
2962:
2960:
2958:
2954:
2944:
2941:
2939:
2936:
2934:
2931:
2929:
2928:MC Simulation
2926:
2924:
2921:
2919:
2916:
2914:
2911:
2909:
2906:
2904:
2901:
2899:
2896:
2893:
2889:
2888:Black–Scholes
2886:
2884:
2881:
2879:
2876:
2874:
2871:
2870:
2868:
2866:
2862:
2855:
2851:
2847:
2844:
2842:
2841:Risk reversal
2839:
2837:
2834:
2832:
2829:
2827:
2824:
2822:
2819:
2817:
2814:
2812:
2809:
2807:
2804:
2802:
2799:
2797:
2794:
2792:
2789:
2787:
2784:
2782:
2779:
2777:
2774:
2772:
2771:Credit spread
2769:
2767:
2764:
2762:
2759:
2757:
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2725:
2719:
2716:
2714:
2711:
2709:
2706:
2703:
2701:
2698:
2696:
2695:Interest rate
2693:
2691:
2690:Forward start
2688:
2686:
2683:
2681:
2678:
2676:
2673:
2671:
2668:
2666:
2663:
2661:
2658:
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2648:
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2640:
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2623:
2620:
2618:
2617:Option styles
2615:
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2610:
2608:
2605:
2603:
2600:
2598:
2595:
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2585:
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2560:
2557:
2555:
2552:
2550:
2547:
2545:
2542:
2540:
2537:
2535:
2534:Open interest
2532:
2530:
2527:
2525:
2522:
2520:
2517:
2515:
2514:Delta neutral
2512:
2511:
2509:
2505:
2502:
2500:
2496:
2492:
2487:
2483:
2476:
2471:
2469:
2464:
2462:
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2435:
2434:
2429:
2424:
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2405:
2401:
2395:
2391:
2387:
2386:Steven Shreve
2383:
2379:
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2369:
2365:
2361:
2357:
2353:
2347:
2343:
2339:
2335:
2331:
2325:
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2317:
2312:
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2302:
2298:
2294:
2290:
2289:Frank Fabozzi
2286:
2282:
2276:
2269:
2268:
2262:
2258:
2252:
2248:
2244:
2239:
2235:
2229:
2225:
2224:CFA Institute
2221:
2220:
2214:
2210:
2204:
2200:
2196:
2191:
2187:
2181:
2177:
2172:
2171:
2158:
2151:
2147:
2141:
2135:
2131:
2126:
2118:
2111:
2105:
2091:on 2015-07-09
2087:
2080:
2075:
2070:
2064:
2060:
2055:
2049:
2045:
2044:Goldman Sachs
2040:
2025:on 2016-04-21
2021:
2014:
2007:
1993:on 2012-03-21
1992:
1988:
1984:
1977:
1970:
1969:
1964:
1958:
1950:
1944:
1938:
1937:
1932:
1928:
1923:
1916:
1910:
1902:
1896:
1882:on 2015-06-19
1881:
1877:
1876:www.sitmo.com
1873:
1867:
1861:
1855:
1849:
1843:
1835:
1833:9781899332533
1829:
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1800:
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1751:
1746:
1744:
1742:
1735:
1729:
1723:. Vol.7 No.3
1722:
1718:
1712:
1706:
1702:
1696:
1690:
1686:
1680:
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1658:
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1626:
1622:
1616:
1609:
1605:
1599:
1592:
1588:
1582:
1575:
1574:Goldman Sachs
1571:
1570:
1563:
1557:. July, 1994.
1556:
1552:
1546:
1539:
1533:
1531:
1523:
1517:
1509:
1505:
1499:
1492:
1488:
1482:
1475:
1471:
1465:
1459:
1455:
1452:
1446:
1444:
1436:
1432:
1426:
1420:
1416:
1410:
1404:, Fall, 21-31
1403:
1399:
1393:
1386:
1382:
1376:
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1354:
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1245:
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1234:
1230:
1226:
1225:Goldman Sachs
1222:
1218:
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1207:
1203:
1199:
1195:
1191:
1187:
1183:
1178:
1176:
1172:
1168:
1164:
1159:
1157:
1152:
1148:
1147:putable bonds
1144:
1140:
1136:
1131:
1127:
1123:
1117:
1114:
1110:
1106:
1102:
1098:
1094:
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915:
911:
908:
904:
903:
902:
900:
895:
892:
889:
888:
880:
878:
874:
871:For multiple
869:
867:
863:
859:
855:
851:
847:
843:
838:
836:
831:
827:
821:
819:
815:
811:
807:
803:
799:
795:
790:
789:implied trees
786:
782:
777:
775:
771:
767:
763:
759:
755:
751:
747:
742:
740:
736:
731:
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723:
718:
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692:
688:
684:
680:
675:
673:
669:
665:
661:
656:
652:
648:
644:
640:
635:
633:
629:
625:
624:present value
621:
617:
614:
610:
606:
601:
599:
595:
591:
587:
583:
579:
575:
571:
567:
563:
559:
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519:
516:
513:
509:
502:
499:
496:
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485:
482:
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476:
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457:
451:
448:
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440:
435:
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429:
423:
420:
416:
407:
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400:
399:
398:
378:
374:
351:
347:
322:
318:
314:
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305:
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296:
291:
287:
283:
278:
274:
265:
250:
247:
244:
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236:
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227:
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199:
196:
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188:
185:
181:
160:
140:
132:
131:
130:
127:
124:
123:
115:
113:
109:
105:
101:
97:
93:
89:
85:
81:
80:Black–Scholes
77:
73:
69:
65:
64:discrete time
61:
57:
56:lattice model
53:
45:
41:
37:
34:Tree for an (
32:
26:
21:
3134:Forward rate
3045:Total return
2933:Real options
2912:
2836:Ratio spread
2816:Naked option
2776:Debit spread
2607:Fixed income
2549:Strike price
2436:. Springer.
2432:
2408:
2392:. Springer.
2389:
2363:
2341:
2318:. New York:
2315:
2292:
2266:
2242:
2218:
2194:
2175:
2168:Bibliography
2157:
2149:
2140:
2125:
2116:
2104:
2093:. Retrieved
2086:the original
2069:
2054:
2039:
2027:. Retrieved
2020:the original
2006:
1995:. Retrieved
1991:the original
1976:
1966:
1957:
1943:
1935:
1922:
1909:
1895:
1884:. Retrieved
1880:the original
1875:
1866:
1854:
1842:
1822:
1815:
1793:(4): 85–98.
1790:
1786:
1780:
1757:
1728:
1720:
1711:
1695:
1679:
1650:
1642:
1633:
1625:eu.wiley.com
1624:
1615:
1607:
1598:
1590:
1581:
1568:
1562:
1554:
1545:
1516:
1507:
1498:
1486:
1485:Zaboronski
1481:
1473:
1464:
1425:
1409:
1401:
1392:
1384:
1375:
1359:
1341:
1332:
1317:
1271:
1268:
1237:
1215:
1186:swap pricing
1179:
1160:
1118:
1112:
1089:
1086:historically
1085:
1079:
1033:
1029:
999:
988:
976:forward rate
961:
954:, and other
948:bond options
945:
936:non-European
922:
896:
893:
890:
870:
861:
841:
839:
830:state prices
825:
822:
814:interpolated
788:
778:
743:
729:
726:Phelim Boyle
719:
695:In the limit
676:
651:real options
636:
602:
593:
589:
585:
581:
577:
573:
569:
565:
561:
557:
555:
546:non-European
396:
128:
125:
55:
49:
25:CRR formulae
3065:Zero Coupon
2995:Correlation
2943:Vanna–Volga
2801:Iron condor
2587:Bond option
2320:McGraw-Hill
1981:Kurt Hess.
1591:Energy Risk
1244:call option
1122:pull to par
1043:yield curve
1026:yield curve
711:log-moments
707:raw moments
96:pull to par
40:bond option
3374:Categories
3339:Tax policy
3055:Volatility
2965:Amortising
2806:Jelly roll
2741:Box spread
2736:Backspread
2728:Strategies
2564:Volatility
2559:the Greeks
2524:Expiration
2368:Risk Books
2297:John Wiley
2199:John Wiley
2095:2015-07-08
1997:2015-06-12
1931:Alan White
1886:2015-06-19
1687:. Thesis,
1310:References
1296:-convexity
1264:Chapter 11
1180:Since the
1113:difference
1067:log-normal
1022:subsequent
968:Hull–White
966:, such as
914:discounted
907:face value
899:underlying
873:underlyers
798:Rubinstein
691:Rubinstein
672:simulation
662:, such as
613:log-normal
605:spot price
62:, where a
3030:Inflation
2980:Commodity
2938:Trinomial
2873:Bachelier
2865:Valuation
2746:Butterfly
2680:Commodore
2529:Moneyness
2273:. Wiley.
1927:John Hull
1807:153599970
1288:convexity
1248:liquidate
1143:callable-
952:swaptions
929:moneyness
858:Edgeworth
730:unchanged
709:and / or
523:−
500:−
455:Δ
449:−
427:Δ
424:−
315:−
297:×
248:⋅
200:⋅
3169:Slippage
3099:Contango
3083:Forwards
3050:Variance
3010:Dividend
3005:Currency
2918:Margrabe
2913:Lattices
2892:equation
2878:Binomial
2826:Strangle
2821:Straddle
2718:Swaption
2700:Lookback
2685:Compound
2627:Warrants
2602:European
2582:American
2574:Vanillas
2539:Pin risk
2519:Exercise
2430:(2006).
2388:(2004).
2362:(2000).
2340:(2002).
2291:(1998).
2076:(2013).
2061:(2002).
2046:(1994).
1933:(2015).
1770:Archived
1665:cite web
1489:(2010).
1454:Archived
1364:Archived
1284:duration
1206:OIS rate
1135:floorlet
1101:Black-76
1099:for the
1038:best fit
864:use the
850:kurtosis
820:model.)
804:, often
796:, often
580:= 0 and
339:, where
36:embedded
3088:Futures
2708:Rainbow
2675:Cliquet
2670:Chooser
2650:Barrier
2637:Exotics
2499:Options
1090:current
1006:Vasicek
974:, or a
785:surface
609:process
52:finance
3149:Margin
3015:Equity
2908:Heston
2811:Ladder
2761:Condor
2756:Collar
2713:Spread
2660:Binary
2655:Basket
2440:
2415:
2396:
2374:
2348:
2326:
2303:
2277:
2253:
2230:
2205:
2182:
2029:31 May
1830:
1805:
1250:(i.e.
1240:equity
1163:struck
1107:; see
1105:caplet
1030:output
1028:is an
1018:Ho–Lee
810:Chriss
808:-Kani-
806:Derman
746:Greeks
90:. For
3020:Forex
2975:Basis
2970:Asset
2957:Swaps
2883:Black
2786:Fence
2645:Asian
2507:Terms
2271:(PDF)
2247:Wiley
2113:(PDF)
2089:(PDF)
2082:(PDF)
2023:(PDF)
2016:(PDF)
1803:S2CID
1487:et al
1202:tenor
1198:LIBOR
1130:swaps
1034:input
1012:) or
783:, or
774:Fugit
762:Theta
758:gamma
754:Delta
2854:Bull
2850:Bear
2592:Call
2438:ISBN
2413:ISBN
2394:ISBN
2372:ISBN
2346:ISBN
2324:ISBN
2301:ISBN
2275:ISBN
2251:ISBN
2228:ISBN
2203:ISBN
2180:ISBN
2031:2007
1929:and
1828:ISBN
1671:link
1294:and
1286:and
1145:and
1020:and
1008:and
848:and
846:skew
770:vega
756:and
689:and
687:Ross
653:and
572:and
54:, a
2622:Put
2226:).
1795:doi
1165:on
1010:CIR
984:HJM
982:or
970:or
766:rho
683:Cox
592:to
215:or
153:or
50:In
44:OAS
3376::
2852:,
2612:FX
2370:.
2322:.
2299:.
2249:.
2201:.
2197:.
2148:,
2132:.
2115:.
1985:.
1965:,
1874:.
1801:.
1789:.
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2000:.
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