1070:
511:
94:. In these models the underlying volatility does not feature any new randomness but it isn't a constant either. In local volatility models the volatility is a non-trivial function of the underlying asset, without any extra randomness. According to this definition, models like constant elasticity of variance would be local volatility models, although they are sometimes classified as stochastic volatility models. The classification can be a little ambiguous in some cases.
1065:{\displaystyle {\begin{aligned}{\widehat {\sigma }}^{2}&=\left({\frac {1}{n}}\sum _{i=1}^{n}{\frac {(\ln S_{t_{i}}-\ln S_{t_{i-1}})^{2}}{t_{i}-t_{i-1}}}\right)-{\frac {1}{n}}{\frac {(\ln S_{t_{n}}-\ln S_{t_{0}})^{2}}{t_{n}-t_{0}}}\\&={\frac {1}{n}}\sum _{i=1}^{n}(t_{i}-t_{i-1})\left({\frac {\ln {\frac {S_{t_{i}}}{S_{t_{i-1}}}}}{t_{i}-t_{i-1}}}-{\frac {\ln {\frac {S_{t_{n}}}{S_{t_{0}}}}}{t_{n}-t_{0}}}\right)^{2};\end{aligned}}}
4662:
2694:) model is another popular model for estimating stochastic volatility. It assumes that the randomness of the variance process varies with the variance, as opposed to the square root of the variance as in the Heston model. The standard GARCH(1,1) model has the following form for the continuous variance differential:
78:
model. In particular, models based on Black-Scholes assume that the underlying volatility is constant over the life of the derivative, and unaffected by the changes in the price level of the underlying security. However, these models cannot explain long-observed features of the implied volatility
3242:
An alternative to calibration is statistical estimation, thereby accounting for parameter uncertainty. Many frequentist and
Bayesian methods have been proposed and implemented, typically for a subset of the abovementioned models. The following list contains extension packages for the open source
3281:
Many numerical methods have been developed over time and have solved pricing financial assets such as options with stochastic volatility models. A recent developed application is the local stochastic volatility model. This local stochastic volatility model gives better results in pricing new
2796:
The GARCH model has been extended via numerous variants, including the NGARCH, TGARCH, IGARCH, LGARCH, EGARCH, GJR-GARCH, Power GARCH, Component GARCH, etc. Strictly, however, the conditional volatilities from GARCH models are not stochastic since at time
3117:
Once a particular SV model is chosen, it must be calibrated against existing market data. Calibration is the process of identifying the set of model parameters that are most likely given the observed data. One popular technique is to use
2972:
1774:
97:
The early history of stochastic volatility has multiple roots (i.e. stochastic process, option pricing and econometrics), it is reviewed in
Chapter 1 of Neil Shephard (2005) "Stochastic Volatility," Oxford University Press.
2791:
419:
1487:
1394:
2191:
1161:
1672:
The popular Heston model is a commonly used SV model, in which the randomness of the variance process varies as the square root of variance. In this case, the differential equation for variance takes the form:
3247:
that have been specifically designed for heteroskedasticity estimation. The first three cater for GARCH-type models with deterministic volatilities; the fourth deals with stochastic volatility estimation.
195:
3178:
2389:
2455:
516:
2673:
2977:
However the meaning of the parameters is different from Heston model. In this model, both mean reverting and volatility of variance parameters are stochastic quantities given by
3075:, at any reasonable timescale. This led to adopting a fractional stochastic volatility (FSV) model, leading to an overall Rough FSV (RFSV) where "rough" is to highlight that
2628:
2601:
1523:
3005:
2251:
Some argue that because the CEV model does not incorporate its own stochastic process for volatility, it is not truly a stochastic volatility model. Instead, they call it a
1556:
87:
and expiry. By assuming that the volatility of the underlying price is a stochastic process rather than a constant, it becomes possible to model derivatives more accurately.
2847:
3035:
2246:
2220:
3216:
2512:
3352:
1184:
447:
268:
3107:
1296:
1242:
299:
3237:
3045:
Using estimation of volatility from high frequency data, smoothness of the volatility process has been questioned. It has been found that log-volatility behaves as a
2311:
2051:
2031:
1817:
1797:
1215:
503:
475:
247:
1980:
1950:
1897:
1867:
1636:
1606:
219:
3073:
2566:
2539:
2485:
2077:
2004:
1656:
1269:
2855:
1837:
1576:
1920:
4752:
2291:
1679:
4205:
3780:
2700:
323:
4545:
4235:
1402:
3383:
2092:
3746:
1304:
4699:
2107:
3239:
to try to minimize these errors. Once the calibration has been performed, it is standard practice to re-calibrate the model periodically.
1082:
4367:
1191:
3255:: ARFIMA, in-mean, external regressors and various GARCH flavors, with methods for fit, forecast, simulation, inference and plotting.
4103:
3500:
4605:
3773:
3740:
3527:
Fabienne Comte and Eric
Renault (1998). Long memory in continuous-time stochastic volatility models. Math. Finance, 8(4), 291–323
314:
116:
3125:
2319:
106:
Starting from a constant volatility approach, assume that the derivative's underlying asset price follows a standard model for
2397:
67:
such as the price level of the underlying security, the tendency of volatility to revert to some long-run mean value, and the
4819:
4540:
4185:
3693:
3518:
Jim
Gatheral, Thibault Jaisson and Mathieu Rosenbaum (2018). Volatility is rough. Quantitative Finance 18(6), Pages 933-949
4809:
3317:
2633:
4580:
4118:
3972:
3766:
4199:
3553:
4814:
4377:
3642:"Ancillarity-Sufficiency Interweaving Strategy (ASIS) for Boosting MCMC Estimation of Stochastic Volatility Models"
3337:
3332:
3046:
2809:
The 3/2 model is similar to the Heston model, but assumes that the randomness of the variance process varies with
4481:
4292:
107:
4692:
4600:
4595:
3274:
3244:
4250:
4195:
3302:
3109:. The RFSV model is consistent with time series data, allowing for improved forecasts of realized volatility.
1187:
4550:
4220:
4210:
4078:
3919:
3851:
3488:
3218:, compute the residual errors when applying the historic price data to the resulting model, and then adjust
4499:
4347:
4332:
4297:
4240:
3595:
2293:(related to any asset e.g. an index, interest rate, bond, currency or equity) under stochastic volatility
4747:
4732:
4560:
4411:
4327:
3904:
3701:
3347:
3327:
2264:
4225:
2606:
2571:
1495:
4768:
4708:
4514:
4471:
4461:
4451:
4172:
4113:
4048:
4002:
3997:
3871:
3831:
3798:
3342:
3322:
2980:
1528:
56:
45:
41:
22:
4230:
2812:
90:
A middle ground between the bare Black-Scholes model and stochastic volatility models is covered by
4685:
4519:
4307:
4053:
3459:"A Continuous Time GARCH Process Driven by a LĂ©vy Process: Stationarity and Second Order Behaviour"
3261:: Part of the Rmetrics environment for teaching "Financial Engineering and Computational Finance".
3010:
2225:
2199:
2101:
model describes the relationship between volatility and price, introducing stochastic volatility:
4722:
4570:
4555:
4524:
4509:
4476:
4342:
4133:
4098:
3861:
3826:
3789:
3674:
3656:
3610:
3576:
3426:
3408:
3193:
3119:
2490:
425:
37:
3264:
75:
2083:
Some parametrisation of the volatility surface, such as 'SVI', are based on the Heston model.
1168:
431:
252:
4575:
4565:
4504:
4491:
4466:
4352:
4138:
3934:
3496:
3444:
3379:
3273:: Efficient algorithms for fully Bayesian estimation of stochastic volatility (SV) models via
3270:
3078:
1274:
1220:
273:
3538:
3285:
There are also alternate statistical estimation libraries in other languages such as Python:
3252:
3221:
2967:{\displaystyle d\nu _{t}=\nu _{t}(\omega -\theta \nu _{t})\,dt+\xi \nu _{t}^{3/2}\,dB_{t}.\,}
2296:
2036:
2016:
1802:
1782:
1200:
480:
452:
224:
4737:
4727:
4456:
4446:
4436:
4395:
4390:
4372:
4302:
4068:
4063:
4035:
3987:
3866:
3806:
3666:
3620:
3568:
3470:
3418:
3312:
3258:
2679:
2252:
1955:
1925:
1872:
1842:
1611:
1581:
203:
91:
80:
52:
3052:
2544:
2517:
2463:
2273:
model (Stochastic Alpha, Beta, Rho), introduced by Hagan et al. describes a single forward
2062:
1989:
1641:
1247:
4778:
4666:
4636:
4631:
4585:
4421:
4416:
4362:
4272:
4180:
4153:
4093:
4088:
4058:
4007:
3992:
3909:
3889:
3714:
1822:
1561:
2568:
are two correlated Wiener processes (i.e. Brownian motions) with correlation coefficient
1902:
1769:{\displaystyle d\nu _{t}=\theta (\omega -\nu _{t})\,dt+\xi {\sqrt {\nu _{t}}}\,dB_{t}\,}
4641:
4626:
4426:
4337:
4287:
4264:
4245:
4073:
4015:
3982:
3977:
3957:
3881:
2276:
2196:
Conceptually, in some markets volatility rises when prices rise (e.g. commodities), so
2009:
In other words, the Heston SV model assumes that the variance is a random process that
1076:
302:
64:
60:
2678:
The main feature of the SABR model is to be able to reproduce the smile effect of the
4803:
4621:
4590:
4431:
4357:
4317:
4312:
4148:
4020:
3967:
3962:
3944:
3841:
3821:
3292:
Includes
Bayesian and classical inference support for GARCH and beta-t-EGARCH models.
3678:
3580:
3430:
4441:
4215:
4143:
4123:
4083:
3952:
3924:
3914:
3856:
3307:
3181:
2801:
the volatility is completely pre-determined (deterministic) given previous values.
1667:
84:
83:
and skew, which indicate that implied volatility does tend to vary with respect to
3734:
3185:
3422:
3373:
4322:
4190:
4161:
4157:
4108:
3899:
3894:
3596:"Dealing with Stochastic Volatility in Time Series Using the R Package stochvol"
1983:
3670:
4646:
4282:
4277:
4043:
3929:
2786:{\displaystyle d\nu _{t}=\theta (\omega -\nu _{t})\,dt+\xi \nu _{t}\,dB_{t}\,}
414:{\displaystyle S_{t}=S_{0}e^{(\mu -{\frac {1}{2}}\sigma ^{2})t+\sigma W_{t}}.}
74:
Stochastic volatility models are one approach to resolve a shortcoming of the
48:
3474:
1271:. This variance function is also modeled as Brownian motion, and the form of
3836:
2222:. In other markets, volatility tends to rise as prices fall, modelled with
3267:: Bayesian estimation of the GARCH(1,1) model with Student's t innovations.
55:. The name derives from the models' treatment of the underlying security's
4661:
3625:
4773:
4406:
4128:
4025:
3846:
3572:
3457:
Kluppelberg, Claudia; Lindner, Alexander; Maller, Ross (September 2004).
3399:
J Gatheral, A Jacquier (2014). "Arbitrage-free SVI volatility surfaces".
310:
68:
33:
3641:
3554:"Bayesian Estimation of the GARCH(1,1) Model with Student-t Innovations"
3458:
1482:{\displaystyle d\nu _{t}=\alpha _{\nu ,t}\,dt+\beta _{\nu ,t}\,dB_{t}\,}
3122:(MLE). For instance, in the Heston model, the set of model parameters
1389:{\displaystyle dS_{t}=\mu S_{t}\,dt+{\sqrt {\nu _{t}}}S_{t}\,dW_{t}\,}
3758:
2186:{\displaystyle dS_{t}=\mu S_{t}\,dt+\sigma S_{t}^{\,\gamma }\,dW_{t}}
1819:
is the rate at which the variance reverts toward its long-term mean,
3495:(3rd ed.). Cambridge: Cambridge University Press. p. 461.
1156:{\displaystyle \operatorname {E} \left={\frac {n-1}{n}}\sigma ^{2}.}
3661:
3615:
1197:
For a stochastic volatility model, replace the constant volatility
1186:
is the starting point for non-stochastic volatility models such as
221:
is the constant drift (i.e. expected return) of the security price
3752:
3413:
2691:
2056:
exhibits a volatility proportional to the square root of its level
3289:
306:
4681:
3762:
3694:"Numerical Solutions for the Stochastic Local Volatility Model"
2690:
The
Generalized Autoregressive Conditional Heteroskedasticity (
2059:
and whose source of randomness is correlated (with correlation
4788:
4783:
3180:
can be estimated applying an MLE algorithm such as the Powell
40:
is itself randomly distributed. They are used in the field of
4677:
3741:
A closed-form solution for options with stochastic volatility
3753:
Accelerating the
Calibration of Stochastic Volatility Models
190:{\displaystyle dS_{t}=\mu S_{t}\,dt+\sigma S_{t}\,dW_{t}\,}
3173:{\displaystyle \Psi _{0}=\{\omega ,\theta ,\xi ,\rho \}\,}
2079:) with the randomness of the underlying's price processes.
2384:{\displaystyle dF_{t}=\sigma _{t}F_{t}^{\beta }\,dW_{t},}
3187:
to observations of historic underlying security prices.
2450:{\displaystyle d\sigma _{t}=\alpha \sigma _{t}\,dZ_{t},}
2013:
exhibits a tendency to revert towards a long-term mean
2514:
are the current forward price and volatility, whereas
3640:
Kastner, Gregor; FrĂĽhwirth-Schnatter, Sylvia (2014).
3224:
3196:
3128:
3081:
3055:
3013:
2983:
2858:
2815:
2703:
2636:
2609:
2574:
2547:
2520:
2493:
2466:
2400:
2322:
2299:
2279:
2228:
2202:
2110:
2065:
2039:
2019:
1992:
1958:
1928:
1905:
1875:
1845:
1825:
1805:
1785:
1682:
1644:
1614:
1608:
is another standard gaussian that is correlated with
1584:
1564:
1531:
1498:
1405:
1307:
1277:
1250:
1223:
1203:
1171:
1085:
514:
483:
455:
434:
326:
276:
255:
227:
206:
119:
3443:
PS Hagan, D Kumar, A Lesniewski, DE Woodward (2002)
4761:
4715:
4614:
4533:
4490:
4386:
4263:
4171:
4034:
3943:
3880:
3814:
3805:
3231:
3210:
3172:
3101:
3067:
3029:
2999:
2966:
2841:
2785:
2668:{\displaystyle 0\leq \beta \leq 1,\;\alpha \geq 0}
2667:
2622:
2595:
2560:
2533:
2506:
2479:
2449:
2383:
2305:
2285:
2240:
2214:
2185:
2071:
2045:
2025:
1998:
1974:
1944:
1914:
1891:
1861:
1831:
1811:
1791:
1768:
1650:
1630:
1600:
1570:
1550:
1517:
1481:
1388:
1290:
1263:
1236:
1209:
1178:
1155:
1064:
497:
469:
441:
413:
293:
262:
241:
213:
189:
3353:Volatility, uncertainty, complexity and ambiguity
1298:depends on the particular SV model under study.
3735:Stochastic Volatility and Mean-variance Analysis
71:of the volatility process itself, among others.
1839:is the volatility of the variance process, and
3552:Ardia, David; Hoogerheide, Lennart F. (2010).
3375:The Volatility Surface: A Practitioner's Guide
4693:
3774:
3190:In this case, you start with an estimate for
8:
3166:
3142:
2849:. The form of the variance differential is:
4700:
4686:
4678:
3811:
3781:
3767:
3759:
3649:Computational Statistics and Data Analysis
2655:
2616:
1165:This basic model with constant volatility
3660:
3624:
3614:
3412:
3282:financial assets such as forex options.
3228:
3223:
3207:
3201:
3195:
3169:
3133:
3127:
3091:
3080:
3054:
3021:
3012:
2991:
2982:
2963:
2954:
2946:
2936:
2932:
2927:
2910:
2901:
2879:
2866:
2857:
2829:
2825:
2820:
2814:
2782:
2776:
2768:
2762:
2745:
2736:
2711:
2702:
2635:
2608:
2573:
2552:
2546:
2525:
2519:
2498:
2492:
2471:
2465:
2438:
2430:
2424:
2408:
2399:
2372:
2364:
2358:
2353:
2343:
2330:
2321:
2298:
2278:
2227:
2201:
2177:
2169:
2163:
2162:
2157:
2140:
2134:
2118:
2109:
2064:
2038:
2018:
1991:
1966:
1957:
1936:
1927:
1904:
1883:
1874:
1853:
1844:
1824:
1804:
1784:
1765:
1759:
1751:
1743:
1737:
1724:
1715:
1690:
1681:
1643:
1622:
1613:
1592:
1583:
1563:
1536:
1530:
1503:
1497:
1478:
1472:
1464:
1452:
1438:
1426:
1413:
1404:
1385:
1379:
1371:
1365:
1353:
1347:
1337:
1331:
1315:
1306:
1282:
1276:
1255:
1249:
1228:
1222:
1202:
1175:
1170:
1144:
1122:
1109:
1098:
1097:
1084:
1049:
1035:
1022:
1006:
1001:
989:
984:
978:
969:
951:
938:
916:
911:
899:
894:
888:
879:
858:
845:
832:
821:
807:
788:
775:
763:
751:
746:
725:
720:
704:
694:
671:
658:
646:
628:
623:
602:
597:
581:
575:
564:
550:
532:
521:
520:
515:
513:
494:
488:
482:
466:
460:
454:
438:
433:
400:
378:
364:
354:
344:
331:
325:
290:
284:
275:
259:
254:
238:
232:
226:
210:
205:
186:
180:
172:
166:
149:
143:
127:
118:
4606:Power reverse dual-currency note (PRDC)
4546:Constant proportion portfolio insurance
3537:Ghalanos, Alexios (20 September 2023).
3364:
3710:
3699:
3737:, Hyungsok Ahn, Paul Wilmott, (2006).
3514:
3512:
3493:Introductory Econometrics for Finance
2093:Constant elasticity of variance model
7:
4541:Collateralized debt obligation (CDO)
428:to estimate the constant volatility
3539:"rugarch: Univariate GARCH models"
3372:Jim Gatheral (18 September 2006).
3225:
3198:
3130:
1086:
16:When variance is a random variable
14:
1982:are correlated with the constant
1638:with constant correlation factor
4660:
2623:{\displaystyle \beta ,\;\alpha }
2596:{\displaystyle -1<\rho <1}
1899:, a gaussian with zero mean and
1799:is the mean long-term variance,
1518:{\displaystyle \alpha _{\nu ,t}}
315:stochastic differential equation
313:. The explicit solution of this
270:is the constant volatility, and
3603:Journal of Statistical Software
3000:{\displaystyle \theta \nu _{t}}
1551:{\displaystyle \beta _{\nu ,t}}
4368:Year-on-year inflation-indexed
2907:
2885:
2842:{\displaystyle \nu _{t}^{3/2}}
2742:
2723:
1721:
1702:
870:
838:
760:
707:
643:
584:
384:
355:
32:models are those in which the
1:
4378:Zero-coupon inflation-indexed
3692:van der Weijst, Roel (2017).
3318:Markov switching multifractal
3120:maximum likelihood estimation
3049:with Hurst exponent of order
3423:10.1080/14697688.2013.819986
3030:{\displaystyle \xi \nu _{t}}
2241:{\displaystyle \gamma <1}
2215:{\displaystyle \gamma >1}
1244:that models the variance of
426:maximum likelihood estimator
4581:Foreign exchange derivative
3973:Callable bull/bear contract
3749:, Alireza Javaheri, (2005).
3747:Inside Volatility Arbitrage
3211:{\displaystyle \Psi _{0}\,}
2507:{\displaystyle \sigma _{0}}
4836:
3671:10.1016/j.csda.2013.01.002
3333:Stochastic volatility jump
3113:Calibration and estimation
3047:fractional Brownian motion
2603:. The constant parameters
2262:
2090:
1665:
20:
4655:
4482:Stock market index future
3796:
1192:Cox–Ross–Rubinstein model
1179:{\displaystyle \sigma \,}
442:{\displaystyle \sigma \,}
263:{\displaystyle \sigma \,}
108:geometric Brownian motion
4601:Mortgage-backed security
4596:Interest rate derivative
4571:Equity-linked note (ELN)
4556:Credit-linked note (CLN)
3594:Kastner, Gregor (2016).
3275:Markov chain Monte Carlo
3102:{\displaystyle H<1/2}
1291:{\displaystyle \nu _{t}}
1237:{\displaystyle \nu _{t}}
294:{\displaystyle dW_{t}\,}
4551:Contract for difference
3852:Risk-free interest rate
3755:, Kilin, Fiodar (2006).
3232:{\displaystyle \Psi \,}
3041:Rough volatility models
2306:{\displaystyle \sigma }
2046:{\displaystyle \theta }
2026:{\displaystyle \omega }
1812:{\displaystyle \theta }
1792:{\displaystyle \omega }
1210:{\displaystyle \sigma }
498:{\displaystyle t_{i}\,}
470:{\displaystyle S_{t}\,}
449:for given stock prices
242:{\displaystyle S_{t}\,}
92:local volatility models
4333:Forward Rate Agreement
3709:Cite journal requires
3475:10.1239/jap/1091543413
3233:
3212:
3174:
3103:
3069:
3031:
3001:
2968:
2843:
2787:
2669:
2624:
2597:
2562:
2535:
2508:
2481:
2451:
2385:
2307:
2287:
2242:
2216:
2187:
2073:
2047:
2027:
2000:
1976:
1975:{\displaystyle dB_{t}}
1946:
1945:{\displaystyle dW_{t}}
1916:
1893:
1892:{\displaystyle dW_{t}}
1863:
1862:{\displaystyle dB_{t}}
1833:
1813:
1793:
1770:
1652:
1632:
1631:{\displaystyle dW_{t}}
1602:
1601:{\displaystyle dB_{t}}
1572:
1558:are some functions of
1552:
1519:
1483:
1390:
1292:
1265:
1238:
1211:
1180:
1157:
1066:
837:
580:
499:
471:
443:
415:
295:
264:
243:
215:
214:{\displaystyle \mu \,}
191:
4820:Derivatives (finance)
4748:Jump-diffusion models
4743:Stochastic volatility
4733:Volatility clustering
4561:Credit default option
3905:Employee stock option
3626:10.18637/jss.v069.i05
3348:Volatility clustering
3328:SABR volatility model
3243:statistical software
3234:
3213:
3175:
3104:
3070:
3068:{\displaystyle H=0.1}
3032:
3002:
2969:
2844:
2788:
2670:
2625:
2598:
2563:
2561:{\displaystyle Z_{t}}
2536:
2534:{\displaystyle W_{t}}
2509:
2482:
2480:{\displaystyle F_{0}}
2452:
2386:
2308:
2288:
2265:SABR volatility model
2259:SABR volatility model
2243:
2217:
2188:
2074:
2072:{\displaystyle \rho }
2048:
2028:
2001:
1999:{\displaystyle \rho }
1977:
1947:
1917:
1894:
1864:
1834:
1814:
1794:
1771:
1653:
1651:{\displaystyle \rho }
1633:
1603:
1573:
1553:
1520:
1484:
1391:
1293:
1266:
1264:{\displaystyle S_{t}}
1239:
1212:
1181:
1158:
1067:
817:
560:
500:
472:
444:
416:
296:
265:
244:
216:
192:
30:stochastic volatility
4810:Mathematical finance
4769:Volatility arbitrage
4716:Modelling volatility
4515:Inflation derivative
4500:Commodity derivative
4472:Single-stock futures
4462:Normal backwardation
4452:Interest rate future
4293:Conditional variance
3799:Derivative (finance)
3743:, SL Heston, (1993).
3573:10.32614/RJ-2010-014
3401:Quantitative Finance
3323:Risk-neutral measure
3222:
3194:
3126:
3079:
3053:
3011:
2981:
2856:
2813:
2701:
2634:
2607:
2572:
2545:
2518:
2491:
2464:
2398:
2320:
2297:
2277:
2226:
2200:
2108:
2063:
2037:
2017:
1990:
1956:
1926:
1903:
1873:
1843:
1832:{\displaystyle \xi }
1823:
1803:
1783:
1680:
1642:
1612:
1582:
1571:{\displaystyle \nu }
1562:
1529:
1496:
1403:
1305:
1275:
1248:
1221:
1201:
1169:
1083:
512:
481:
453:
432:
324:
274:
253:
225:
204:
117:
42:mathematical finance
23:Volatility (finance)
4667:Business portal
4520:Property derivative
3445:Managing smile risk
3303:Black–Scholes model
2945:
2838:
2460:The initial values
2363:
2168:
1922:variance. However,
1188:Black–Scholes model
477:at different times
4762:Trading volatility
4723:Implied volatility
4525:Weather derivative
4510:Freight derivative
4492:Exotic derivatives
4412:Commodities future
4099:Intermarket spread
3862:Synthetic position
3790:Derivatives market
3447:, Wilmott, 84-108.
3229:
3208:
3170:
3099:
3065:
3027:
2997:
2964:
2923:
2839:
2816:
2783:
2665:
2620:
2593:
2558:
2531:
2504:
2477:
2447:
2381:
2349:
2303:
2283:
2238:
2212:
2183:
2153:
2069:
2043:
2023:
1996:
1972:
1942:
1915:{\displaystyle dt}
1912:
1889:
1859:
1829:
1809:
1789:
1766:
1648:
1628:
1598:
1568:
1548:
1515:
1479:
1386:
1288:
1261:
1234:
1207:
1176:
1153:
1062:
1060:
495:
467:
439:
411:
291:
260:
239:
211:
187:
38:stochastic process
4815:Options (finance)
4797:
4796:
4675:
4674:
4576:Equity derivative
4566:Credit derivative
4534:Other derivatives
4505:Energy derivative
4467:Perpetual futures
4348:Overnight indexed
4298:Constant maturity
4259:
4258:
4206:Finite difference
4139:Protective option
3385:978-0-470-06825-0
2286:{\displaystyle F}
1749:
1359:
1138:
1106:
1042:
1014:
964:
930:
815:
795:
702:
684:
558:
529:
372:
309:and unit rate of
4827:
4738:Local volatility
4728:Volatility smile
4702:
4695:
4688:
4679:
4665:
4664:
4437:Forwards pricing
4211:Garman–Kohlhagen
3812:
3783:
3776:
3769:
3760:
3719:
3718:
3712:
3707:
3705:
3697:
3689:
3683:
3682:
3664:
3646:
3637:
3631:
3630:
3628:
3618:
3600:
3591:
3585:
3584:
3558:
3549:
3543:
3542:
3534:
3528:
3525:
3519:
3516:
3507:
3506:
3485:
3479:
3478:
3454:
3448:
3441:
3435:
3434:
3416:
3396:
3390:
3389:
3369:
3313:Local volatility
3238:
3236:
3235:
3230:
3217:
3215:
3214:
3209:
3206:
3205:
3179:
3177:
3176:
3171:
3138:
3137:
3108:
3106:
3105:
3100:
3095:
3074:
3072:
3071:
3066:
3036:
3034:
3033:
3028:
3026:
3025:
3006:
3004:
3003:
2998:
2996:
2995:
2973:
2971:
2970:
2965:
2959:
2958:
2944:
2940:
2931:
2906:
2905:
2884:
2883:
2871:
2870:
2848:
2846:
2845:
2840:
2837:
2833:
2824:
2792:
2790:
2789:
2784:
2781:
2780:
2767:
2766:
2741:
2740:
2716:
2715:
2680:volatility smile
2674:
2672:
2671:
2666:
2629:
2627:
2626:
2621:
2602:
2600:
2599:
2594:
2567:
2565:
2564:
2559:
2557:
2556:
2540:
2538:
2537:
2532:
2530:
2529:
2513:
2511:
2510:
2505:
2503:
2502:
2486:
2484:
2483:
2478:
2476:
2475:
2456:
2454:
2453:
2448:
2443:
2442:
2429:
2428:
2413:
2412:
2390:
2388:
2387:
2382:
2377:
2376:
2362:
2357:
2348:
2347:
2335:
2334:
2312:
2310:
2309:
2304:
2292:
2290:
2289:
2284:
2253:local volatility
2247:
2245:
2244:
2239:
2221:
2219:
2218:
2213:
2192:
2190:
2189:
2184:
2182:
2181:
2167:
2161:
2139:
2138:
2123:
2122:
2078:
2076:
2075:
2070:
2052:
2050:
2049:
2044:
2032:
2030:
2029:
2024:
2005:
2003:
2002:
1997:
1981:
1979:
1978:
1973:
1971:
1970:
1951:
1949:
1948:
1943:
1941:
1940:
1921:
1919:
1918:
1913:
1898:
1896:
1895:
1890:
1888:
1887:
1868:
1866:
1865:
1860:
1858:
1857:
1838:
1836:
1835:
1830:
1818:
1816:
1815:
1810:
1798:
1796:
1795:
1790:
1775:
1773:
1772:
1767:
1764:
1763:
1750:
1748:
1747:
1738:
1720:
1719:
1695:
1694:
1657:
1655:
1654:
1649:
1637:
1635:
1634:
1629:
1627:
1626:
1607:
1605:
1604:
1599:
1597:
1596:
1577:
1575:
1574:
1569:
1557:
1555:
1554:
1549:
1547:
1546:
1524:
1522:
1521:
1516:
1514:
1513:
1488:
1486:
1485:
1480:
1477:
1476:
1463:
1462:
1437:
1436:
1418:
1417:
1395:
1393:
1392:
1387:
1384:
1383:
1370:
1369:
1360:
1358:
1357:
1348:
1336:
1335:
1320:
1319:
1297:
1295:
1294:
1289:
1287:
1286:
1270:
1268:
1267:
1262:
1260:
1259:
1243:
1241:
1240:
1235:
1233:
1232:
1217:with a function
1216:
1214:
1213:
1208:
1185:
1183:
1182:
1177:
1162:
1160:
1159:
1154:
1149:
1148:
1139:
1134:
1123:
1118:
1114:
1113:
1108:
1107:
1099:
1071:
1069:
1068:
1063:
1061:
1054:
1053:
1048:
1044:
1043:
1041:
1040:
1039:
1027:
1026:
1016:
1015:
1013:
1012:
1011:
1010:
996:
995:
994:
993:
979:
970:
965:
963:
962:
961:
943:
942:
932:
931:
929:
928:
927:
926:
906:
905:
904:
903:
889:
880:
869:
868:
850:
849:
836:
831:
816:
808:
800:
796:
794:
793:
792:
780:
779:
769:
768:
767:
758:
757:
756:
755:
732:
731:
730:
729:
705:
703:
695:
690:
686:
685:
683:
682:
681:
663:
662:
652:
651:
650:
641:
640:
639:
638:
609:
608:
607:
606:
582:
579:
574:
559:
551:
537:
536:
531:
530:
522:
504:
502:
501:
496:
493:
492:
476:
474:
473:
468:
465:
464:
448:
446:
445:
440:
420:
418:
417:
412:
407:
406:
405:
404:
383:
382:
373:
365:
349:
348:
336:
335:
300:
298:
297:
292:
289:
288:
269:
267:
266:
261:
248:
246:
245:
240:
237:
236:
220:
218:
217:
212:
196:
194:
193:
188:
185:
184:
171:
170:
148:
147:
132:
131:
81:volatility smile
79:surface such as
4835:
4834:
4830:
4829:
4828:
4826:
4825:
4824:
4800:
4799:
4798:
4793:
4779:Volatility swap
4757:
4711:
4706:
4676:
4671:
4659:
4651:
4637:Great Recession
4632:Government debt
4610:
4586:Fund derivative
4529:
4486:
4447:Futures pricing
4422:Dividend future
4417:Currency future
4400:
4382:
4255:
4231:Put–call parity
4167:
4154:Vertical spread
4089:Diagonal spread
4059:Calendar spread
4030:
3939:
3876:
3801:
3792:
3787:
3731:
3725:
3723:
3722:
3708:
3698:
3691:
3690:
3686:
3644:
3639:
3638:
3634:
3598:
3593:
3592:
3588:
3556:
3551:
3550:
3546:
3536:
3535:
3531:
3526:
3522:
3517:
3510:
3503:
3487:
3486:
3482:
3463:J. Appl. Probab
3456:
3455:
3451:
3442:
3438:
3398:
3397:
3393:
3386:
3371:
3370:
3366:
3361:
3299:
3277:(MCMC) methods.
3220:
3219:
3197:
3192:
3191:
3129:
3124:
3123:
3115:
3077:
3076:
3051:
3050:
3043:
3017:
3009:
3008:
2987:
2979:
2978:
2950:
2897:
2875:
2862:
2854:
2853:
2811:
2810:
2807:
2772:
2758:
2732:
2707:
2699:
2698:
2688:
2632:
2631:
2605:
2604:
2570:
2569:
2548:
2543:
2542:
2521:
2516:
2515:
2494:
2489:
2488:
2467:
2462:
2461:
2434:
2420:
2404:
2396:
2395:
2368:
2339:
2326:
2318:
2317:
2295:
2294:
2275:
2274:
2267:
2261:
2224:
2223:
2198:
2197:
2173:
2130:
2114:
2106:
2105:
2095:
2089:
2061:
2060:
2035:
2034:
2015:
2014:
1988:
1987:
1962:
1954:
1953:
1932:
1924:
1923:
1901:
1900:
1879:
1871:
1870:
1849:
1841:
1840:
1821:
1820:
1801:
1800:
1781:
1780:
1755:
1739:
1711:
1686:
1678:
1677:
1670:
1664:
1640:
1639:
1618:
1610:
1609:
1588:
1580:
1579:
1560:
1559:
1532:
1527:
1526:
1499:
1494:
1493:
1468:
1448:
1422:
1409:
1401:
1400:
1375:
1361:
1349:
1327:
1311:
1303:
1302:
1278:
1273:
1272:
1251:
1246:
1245:
1224:
1219:
1218:
1199:
1198:
1167:
1166:
1140:
1124:
1096:
1092:
1081:
1080:
1059:
1058:
1031:
1018:
1017:
1002:
997:
985:
980:
971:
947:
934:
933:
912:
907:
895:
890:
881:
878:
874:
873:
854:
841:
798:
797:
784:
771:
770:
759:
747:
742:
721:
716:
706:
667:
654:
653:
642:
624:
619:
598:
593:
583:
549:
545:
538:
519:
510:
509:
484:
479:
478:
456:
451:
450:
430:
429:
396:
374:
350:
340:
327:
322:
321:
280:
272:
271:
251:
250:
228:
223:
222:
202:
201:
176:
162:
139:
123:
115:
114:
104:
65:state variables
28:In statistics,
26:
17:
12:
11:
5:
4833:
4831:
4823:
4822:
4817:
4812:
4802:
4801:
4795:
4794:
4792:
4791:
4786:
4781:
4776:
4771:
4765:
4763:
4759:
4758:
4756:
4755:
4753:ARCH and GARCH
4750:
4745:
4740:
4735:
4730:
4725:
4719:
4717:
4713:
4712:
4707:
4705:
4704:
4697:
4690:
4682:
4673:
4672:
4670:
4669:
4656:
4653:
4652:
4650:
4649:
4644:
4642:Municipal debt
4639:
4634:
4629:
4627:Corporate debt
4624:
4618:
4616:
4612:
4611:
4609:
4608:
4603:
4598:
4593:
4588:
4583:
4578:
4573:
4568:
4563:
4558:
4553:
4548:
4543:
4537:
4535:
4531:
4530:
4528:
4527:
4522:
4517:
4512:
4507:
4502:
4496:
4494:
4488:
4487:
4485:
4484:
4479:
4474:
4469:
4464:
4459:
4454:
4449:
4444:
4439:
4434:
4429:
4427:Forward market
4424:
4419:
4414:
4409:
4403:
4401:
4399:
4398:
4393:
4387:
4384:
4383:
4381:
4380:
4375:
4370:
4365:
4360:
4355:
4350:
4345:
4340:
4335:
4330:
4325:
4320:
4315:
4310:
4308:Credit default
4305:
4300:
4295:
4290:
4285:
4280:
4275:
4269:
4267:
4261:
4260:
4257:
4256:
4254:
4253:
4248:
4243:
4238:
4233:
4228:
4223:
4218:
4213:
4208:
4203:
4193:
4188:
4183:
4177:
4175:
4169:
4168:
4166:
4165:
4151:
4146:
4141:
4136:
4131:
4126:
4121:
4116:
4111:
4106:
4104:Iron butterfly
4101:
4096:
4091:
4086:
4081:
4076:
4074:Covered option
4071:
4066:
4061:
4056:
4051:
4046:
4040:
4038:
4032:
4031:
4029:
4028:
4023:
4018:
4013:
4012:Mountain range
4010:
4005:
4000:
3995:
3990:
3985:
3980:
3975:
3970:
3965:
3960:
3955:
3949:
3947:
3941:
3940:
3938:
3937:
3932:
3927:
3922:
3917:
3912:
3907:
3902:
3897:
3892:
3886:
3884:
3878:
3877:
3875:
3874:
3869:
3864:
3859:
3854:
3849:
3844:
3839:
3834:
3829:
3824:
3818:
3816:
3809:
3803:
3802:
3797:
3794:
3793:
3788:
3786:
3785:
3778:
3771:
3763:
3757:
3756:
3750:
3744:
3738:
3730:
3727:
3721:
3720:
3711:|journal=
3684:
3632:
3586:
3544:
3529:
3520:
3508:
3501:
3480:
3469:(3): 601–622.
3449:
3436:
3391:
3384:
3363:
3362:
3360:
3357:
3356:
3355:
3350:
3345:
3340:
3335:
3330:
3325:
3320:
3315:
3310:
3305:
3298:
3295:
3294:
3293:
3279:
3278:
3268:
3262:
3256:
3227:
3204:
3200:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3147:
3144:
3141:
3136:
3132:
3114:
3111:
3098:
3094:
3090:
3087:
3084:
3064:
3061:
3058:
3042:
3039:
3037:respectively.
3024:
3020:
3016:
2994:
2990:
2986:
2975:
2974:
2962:
2957:
2953:
2949:
2943:
2939:
2935:
2930:
2926:
2922:
2919:
2916:
2913:
2909:
2904:
2900:
2896:
2893:
2890:
2887:
2882:
2878:
2874:
2869:
2865:
2861:
2836:
2832:
2828:
2823:
2819:
2806:
2803:
2794:
2793:
2779:
2775:
2771:
2765:
2761:
2757:
2754:
2751:
2748:
2744:
2739:
2735:
2731:
2728:
2725:
2722:
2719:
2714:
2710:
2706:
2687:
2684:
2664:
2661:
2658:
2654:
2651:
2648:
2645:
2642:
2639:
2630:are such that
2619:
2615:
2612:
2592:
2589:
2586:
2583:
2580:
2577:
2555:
2551:
2528:
2524:
2501:
2497:
2474:
2470:
2458:
2457:
2446:
2441:
2437:
2433:
2427:
2423:
2419:
2416:
2411:
2407:
2403:
2392:
2391:
2380:
2375:
2371:
2367:
2361:
2356:
2352:
2346:
2342:
2338:
2333:
2329:
2325:
2302:
2282:
2263:Main article:
2260:
2257:
2237:
2234:
2231:
2211:
2208:
2205:
2194:
2193:
2180:
2176:
2172:
2166:
2160:
2156:
2152:
2149:
2146:
2143:
2137:
2133:
2129:
2126:
2121:
2117:
2113:
2091:Main article:
2088:
2085:
2081:
2080:
2068:
2057:
2054:
2042:
2022:
1995:
1969:
1965:
1961:
1939:
1935:
1931:
1911:
1908:
1886:
1882:
1878:
1856:
1852:
1848:
1828:
1808:
1788:
1777:
1776:
1762:
1758:
1754:
1746:
1742:
1736:
1733:
1730:
1727:
1723:
1718:
1714:
1710:
1707:
1704:
1701:
1698:
1693:
1689:
1685:
1666:Main article:
1663:
1660:
1647:
1625:
1621:
1617:
1595:
1591:
1587:
1567:
1545:
1542:
1539:
1535:
1512:
1509:
1506:
1502:
1490:
1489:
1475:
1471:
1467:
1461:
1458:
1455:
1451:
1447:
1444:
1441:
1435:
1432:
1429:
1425:
1421:
1416:
1412:
1408:
1397:
1396:
1382:
1378:
1374:
1368:
1364:
1356:
1352:
1346:
1343:
1340:
1334:
1330:
1326:
1323:
1318:
1314:
1310:
1285:
1281:
1258:
1254:
1231:
1227:
1206:
1174:
1152:
1147:
1143:
1137:
1133:
1130:
1127:
1121:
1117:
1112:
1105:
1102:
1095:
1091:
1088:
1077:expected value
1073:
1072:
1057:
1052:
1047:
1038:
1034:
1030:
1025:
1021:
1009:
1005:
1000:
992:
988:
983:
977:
974:
968:
960:
957:
954:
950:
946:
941:
937:
925:
922:
919:
915:
910:
902:
898:
893:
887:
884:
877:
872:
867:
864:
861:
857:
853:
848:
844:
840:
835:
830:
827:
824:
820:
814:
811:
806:
803:
801:
799:
791:
787:
783:
778:
774:
766:
762:
754:
750:
745:
741:
738:
735:
728:
724:
719:
715:
712:
709:
701:
698:
693:
689:
680:
677:
674:
670:
666:
661:
657:
649:
645:
637:
634:
631:
627:
622:
618:
615:
612:
605:
601:
596:
592:
589:
586:
578:
573:
570:
567:
563:
557:
554:
548:
544:
541:
539:
535:
528:
525:
518:
517:
491:
487:
463:
459:
437:
422:
421:
410:
403:
399:
395:
392:
389:
386:
381:
377:
371:
368:
363:
360:
357:
353:
347:
343:
339:
334:
330:
303:Wiener process
301:is a standard
287:
283:
279:
258:
235:
231:
209:
198:
197:
183:
179:
175:
169:
165:
161:
158:
155:
152:
146:
142:
138:
135:
130:
126:
122:
103:
100:
63:, governed by
61:random process
15:
13:
10:
9:
6:
4:
3:
2:
4832:
4821:
4818:
4816:
4813:
4811:
4808:
4807:
4805:
4790:
4787:
4785:
4782:
4780:
4777:
4775:
4772:
4770:
4767:
4766:
4764:
4760:
4754:
4751:
4749:
4746:
4744:
4741:
4739:
4736:
4734:
4731:
4729:
4726:
4724:
4721:
4720:
4718:
4714:
4710:
4703:
4698:
4696:
4691:
4689:
4684:
4683:
4680:
4668:
4663:
4658:
4657:
4654:
4648:
4645:
4643:
4640:
4638:
4635:
4633:
4630:
4628:
4625:
4623:
4622:Consumer debt
4620:
4619:
4617:
4615:Market issues
4613:
4607:
4604:
4602:
4599:
4597:
4594:
4592:
4591:Fund of funds
4589:
4587:
4584:
4582:
4579:
4577:
4574:
4572:
4569:
4567:
4564:
4562:
4559:
4557:
4554:
4552:
4549:
4547:
4544:
4542:
4539:
4538:
4536:
4532:
4526:
4523:
4521:
4518:
4516:
4513:
4511:
4508:
4506:
4503:
4501:
4498:
4497:
4495:
4493:
4489:
4483:
4480:
4478:
4475:
4473:
4470:
4468:
4465:
4463:
4460:
4458:
4455:
4453:
4450:
4448:
4445:
4443:
4440:
4438:
4435:
4433:
4432:Forward price
4430:
4428:
4425:
4423:
4420:
4418:
4415:
4413:
4410:
4408:
4405:
4404:
4402:
4397:
4394:
4392:
4389:
4388:
4385:
4379:
4376:
4374:
4371:
4369:
4366:
4364:
4361:
4359:
4356:
4354:
4351:
4349:
4346:
4344:
4343:Interest rate
4341:
4339:
4336:
4334:
4331:
4329:
4326:
4324:
4321:
4319:
4316:
4314:
4311:
4309:
4306:
4304:
4301:
4299:
4296:
4294:
4291:
4289:
4286:
4284:
4281:
4279:
4276:
4274:
4271:
4270:
4268:
4266:
4262:
4252:
4249:
4247:
4244:
4242:
4239:
4237:
4236:MC Simulation
4234:
4232:
4229:
4227:
4224:
4222:
4219:
4217:
4214:
4212:
4209:
4207:
4204:
4201:
4197:
4196:Black–Scholes
4194:
4192:
4189:
4187:
4184:
4182:
4179:
4178:
4176:
4174:
4170:
4163:
4159:
4155:
4152:
4150:
4149:Risk reversal
4147:
4145:
4142:
4140:
4137:
4135:
4132:
4130:
4127:
4125:
4122:
4120:
4117:
4115:
4112:
4110:
4107:
4105:
4102:
4100:
4097:
4095:
4092:
4090:
4087:
4085:
4082:
4080:
4079:Credit spread
4077:
4075:
4072:
4070:
4067:
4065:
4062:
4060:
4057:
4055:
4052:
4050:
4047:
4045:
4042:
4041:
4039:
4037:
4033:
4027:
4024:
4022:
4019:
4017:
4014:
4011:
4009:
4006:
4004:
4003:Interest rate
4001:
3999:
3998:Forward start
3996:
3994:
3991:
3989:
3986:
3984:
3981:
3979:
3976:
3974:
3971:
3969:
3966:
3964:
3961:
3959:
3956:
3954:
3951:
3950:
3948:
3946:
3942:
3936:
3933:
3931:
3928:
3926:
3925:Option styles
3923:
3921:
3918:
3916:
3913:
3911:
3908:
3906:
3903:
3901:
3898:
3896:
3893:
3891:
3888:
3887:
3885:
3883:
3879:
3873:
3870:
3868:
3865:
3863:
3860:
3858:
3855:
3853:
3850:
3848:
3845:
3843:
3842:Open interest
3840:
3838:
3835:
3833:
3830:
3828:
3825:
3823:
3822:Delta neutral
3820:
3819:
3817:
3813:
3810:
3808:
3804:
3800:
3795:
3791:
3784:
3779:
3777:
3772:
3770:
3765:
3764:
3761:
3754:
3751:
3748:
3745:
3742:
3739:
3736:
3733:
3732:
3728:
3726:
3716:
3703:
3695:
3688:
3685:
3680:
3676:
3672:
3668:
3663:
3658:
3654:
3650:
3643:
3636:
3633:
3627:
3622:
3617:
3612:
3608:
3604:
3597:
3590:
3587:
3582:
3578:
3574:
3570:
3566:
3562:
3561:The R Journal
3555:
3548:
3545:
3540:
3533:
3530:
3524:
3521:
3515:
3513:
3509:
3504:
3502:9781107661455
3498:
3494:
3490:
3489:Brooks, Chris
3484:
3481:
3476:
3472:
3468:
3464:
3460:
3453:
3450:
3446:
3440:
3437:
3432:
3428:
3424:
3420:
3415:
3410:
3406:
3402:
3395:
3392:
3387:
3381:
3377:
3376:
3368:
3365:
3358:
3354:
3351:
3349:
3346:
3344:
3341:
3339:
3336:
3334:
3331:
3329:
3326:
3324:
3321:
3319:
3316:
3314:
3311:
3309:
3306:
3304:
3301:
3300:
3296:
3291:
3288:
3287:
3286:
3283:
3276:
3272:
3269:
3266:
3263:
3260:
3257:
3254:
3251:
3250:
3249:
3246:
3240:
3202:
3188:
3186:
3183:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3139:
3134:
3121:
3112:
3110:
3096:
3092:
3088:
3085:
3082:
3062:
3059:
3056:
3048:
3040:
3038:
3022:
3018:
3014:
2992:
2988:
2984:
2960:
2955:
2951:
2947:
2941:
2937:
2933:
2928:
2924:
2920:
2917:
2914:
2911:
2902:
2898:
2894:
2891:
2888:
2880:
2876:
2872:
2867:
2863:
2859:
2852:
2851:
2850:
2834:
2830:
2826:
2821:
2817:
2804:
2802:
2800:
2777:
2773:
2769:
2763:
2759:
2755:
2752:
2749:
2746:
2737:
2733:
2729:
2726:
2720:
2717:
2712:
2708:
2704:
2697:
2696:
2695:
2693:
2685:
2683:
2681:
2676:
2662:
2659:
2656:
2652:
2649:
2646:
2643:
2640:
2637:
2617:
2613:
2610:
2590:
2587:
2584:
2581:
2578:
2575:
2553:
2549:
2526:
2522:
2499:
2495:
2472:
2468:
2444:
2439:
2435:
2431:
2425:
2421:
2417:
2414:
2409:
2405:
2401:
2394:
2393:
2378:
2373:
2369:
2365:
2359:
2354:
2350:
2344:
2340:
2336:
2331:
2327:
2323:
2316:
2315:
2314:
2300:
2280:
2272:
2266:
2258:
2256:
2254:
2249:
2235:
2232:
2229:
2209:
2206:
2203:
2178:
2174:
2170:
2164:
2158:
2154:
2150:
2147:
2144:
2141:
2135:
2131:
2127:
2124:
2119:
2115:
2111:
2104:
2103:
2102:
2100:
2094:
2086:
2084:
2066:
2058:
2055:
2040:
2020:
2012:
2011:
2010:
2007:
1993:
1985:
1967:
1963:
1959:
1937:
1933:
1929:
1909:
1906:
1884:
1880:
1876:
1854:
1850:
1846:
1826:
1806:
1786:
1760:
1756:
1752:
1744:
1740:
1734:
1731:
1728:
1725:
1716:
1712:
1708:
1705:
1699:
1696:
1691:
1687:
1683:
1676:
1675:
1674:
1669:
1661:
1659:
1645:
1623:
1619:
1615:
1593:
1589:
1585:
1565:
1543:
1540:
1537:
1533:
1510:
1507:
1504:
1500:
1473:
1469:
1465:
1459:
1456:
1453:
1449:
1445:
1442:
1439:
1433:
1430:
1427:
1423:
1419:
1414:
1410:
1406:
1399:
1398:
1380:
1376:
1372:
1366:
1362:
1354:
1350:
1344:
1341:
1338:
1332:
1328:
1324:
1321:
1316:
1312:
1308:
1301:
1300:
1299:
1283:
1279:
1256:
1252:
1229:
1225:
1204:
1195:
1193:
1189:
1172:
1163:
1150:
1145:
1141:
1135:
1131:
1128:
1125:
1119:
1115:
1110:
1103:
1100:
1093:
1089:
1078:
1055:
1050:
1045:
1036:
1032:
1028:
1023:
1019:
1007:
1003:
998:
990:
986:
981:
975:
972:
966:
958:
955:
952:
948:
944:
939:
935:
923:
920:
917:
913:
908:
900:
896:
891:
885:
882:
875:
865:
862:
859:
855:
851:
846:
842:
833:
828:
825:
822:
818:
812:
809:
804:
802:
789:
785:
781:
776:
772:
764:
752:
748:
743:
739:
736:
733:
726:
722:
717:
713:
710:
699:
696:
691:
687:
678:
675:
672:
668:
664:
659:
655:
647:
635:
632:
629:
625:
620:
616:
613:
610:
603:
599:
594:
590:
587:
576:
571:
568:
565:
561:
555:
552:
546:
542:
540:
533:
526:
523:
508:
507:
506:
489:
485:
461:
457:
435:
427:
408:
401:
397:
393:
390:
387:
379:
375:
369:
366:
361:
358:
351:
345:
341:
337:
332:
328:
320:
319:
318:
316:
312:
308:
304:
285:
281:
277:
256:
233:
229:
207:
181:
177:
173:
167:
163:
159:
156:
153:
150:
144:
140:
136:
133:
128:
124:
120:
113:
112:
111:
109:
101:
99:
95:
93:
88:
86:
82:
77:
76:Black–Scholes
72:
70:
66:
62:
58:
54:
50:
47:
43:
39:
35:
31:
24:
19:
4742:
4442:Forward rate
4353:Total return
4241:Real options
4144:Ratio spread
4124:Naked option
4084:Debit spread
3915:Fixed income
3857:Strike price
3724:
3702:cite journal
3687:
3652:
3648:
3635:
3606:
3602:
3589:
3567:(2): 41–47.
3564:
3560:
3547:
3532:
3523:
3492:
3483:
3466:
3462:
3452:
3439:
3404:
3400:
3394:
3374:
3367:
3338:Subordinator
3308:Heston model
3284:
3280:
3241:
3189:
3182:Directed Set
3116:
3044:
2976:
2808:
2798:
2795:
2689:
2677:
2459:
2270:
2268:
2250:
2195:
2098:
2096:
2082:
2008:
1778:
1671:
1668:Heston model
1662:Heston model
1491:
1196:
1164:
1074:
423:
199:
105:
96:
89:
85:strike price
73:
44:to evaluate
29:
27:
18:
4373:Zero Coupon
4303:Correlation
4251:Vanna–Volga
4109:Iron condor
3895:Bond option
3655:: 408–423.
3609:(5): 1–30.
2686:GARCH model
1984:correlation
102:Basic model
4804:Categories
4709:Volatility
4647:Tax policy
4363:Volatility
4273:Amortising
4114:Jelly roll
4049:Box spread
4044:Backspread
4036:Strategies
3872:Volatility
3867:the Greeks
3832:Expiration
3662:1706.05280
3616:1906.12134
3359:References
3343:Volatility
3265:bayesGARCH
2033:at a rate
305:with zero
57:volatility
51:, such as
49:securities
46:derivative
4338:Inflation
4288:Commodity
4246:Trinomial
4181:Bachelier
4173:Valuation
4054:Butterfly
3988:Commodore
3837:Moneyness
3414:1204.0646
3407:: 59–71.
3378:. Wiley.
3226:Ψ
3199:Ψ
3164:ρ
3158:ξ
3152:θ
3146:ω
3131:Ψ
3019:ν
3015:ξ
2989:ν
2985:θ
2925:ν
2921:ξ
2899:ν
2895:θ
2892:−
2889:ω
2877:ν
2864:ν
2818:ν
2805:3/2 model
2760:ν
2756:ξ
2734:ν
2730:−
2727:ω
2721:θ
2709:ν
2660:≥
2657:α
2647:≤
2644:β
2641:≤
2618:α
2611:β
2585:ρ
2576:−
2496:σ
2422:σ
2418:α
2406:σ
2360:β
2341:σ
2301:σ
2230:γ
2204:γ
2165:γ
2151:σ
2128:μ
2087:CEV model
2067:ρ
2041:θ
2021:ω
1994:ρ
1869:is, like
1827:ξ
1807:θ
1787:ω
1741:ν
1735:ξ
1713:ν
1709:−
1706:ω
1700:θ
1688:ν
1646:ρ
1566:ν
1538:ν
1534:β
1505:ν
1501:α
1454:ν
1450:β
1428:ν
1424:α
1411:ν
1351:ν
1325:μ
1280:ν
1226:ν
1205:σ
1173:σ
1142:σ
1129:−
1104:^
1101:σ
1090:
1029:−
976:
967:−
956:−
945:−
921:−
886:
863:−
852:−
819:∑
782:−
740:
734:−
714:
692:−
676:−
665:−
633:−
617:
611:−
591:
562:∑
527:^
524:σ
436:σ
394:σ
376:σ
362:−
359:μ
257:σ
208:μ
160:σ
137:μ
21:See also
4774:Straddle
4477:Slippage
4407:Contango
4391:Forwards
4358:Variance
4318:Dividend
4313:Currency
4226:Margrabe
4221:Lattices
4200:equation
4186:Binomial
4134:Strangle
4129:Straddle
4026:Swaption
4008:Lookback
3993:Compound
3935:Warrants
3910:European
3890:American
3882:Vanillas
3847:Pin risk
3827:Exercise
3679:17019876
3581:17324384
3491:(2014).
3431:41434372
3297:See also
3271:stochvol
311:variance
69:variance
34:variance
4396:Futures
4016:Rainbow
3983:Cliquet
3978:Chooser
3958:Barrier
3945:Exotics
3807:Options
3729:Sources
3253:rugarch
3184:method
2255:model.
53:options
4457:Margin
4323:Equity
4216:Heston
4119:Ladder
4069:Condor
4064:Collar
4021:Spread
3968:Binary
3963:Basket
3677:
3579:
3499:
3429:
3382:
3290:PyFlux
3259:fGarch
1986:value
1779:where
1578:, and
1492:where
200:where
4328:Forex
4283:Basis
4278:Asset
4265:Swaps
4191:Black
4094:Fence
3953:Asian
3815:Terms
3675:S2CID
3657:arXiv
3645:(PDF)
3611:arXiv
3599:(PDF)
3577:S2CID
3557:(PDF)
3427:S2CID
3409:arXiv
2692:GARCH
59:as a
36:of a
4162:Bull
4158:Bear
3900:Call
3715:help
3497:ISBN
3380:ISBN
3086:<
3007:and
2588:<
2582:<
2541:and
2487:and
2271:SABR
2269:The
2233:<
2207:>
2097:The
1952:and
1525:and
1190:and
1075:its
424:The
307:mean
4789:VIX
4784:IVX
3930:Put
3667:doi
3621:doi
3569:doi
3471:doi
3419:doi
3063:0.1
2099:CEV
1079:is
505:is
317:is
4806::
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3704:}}
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