971:
This is of course problematic; while any of the risk-free measures may theoretically be used to price a derivative, it is likely that each of them will give a different price. In theory, however, only one of these risk-free measures would be compatible with the market prices of volatility-dependent
967:
In the Heston model, we still have one asset (volatility is not considered to be directly observable or tradeable in the market) but we now have two Wiener processes - the first in the
Stochastic Differential Equation (SDE) for the stock price and the second in the SDE for the variance of the stock
1099:
of the objective function with respect to the model parameters. This was usually computed with a finite difference approximation although it is less accurate, less efficient and less elegant than an analytical gradient because an insightful expression of the latter became available only when a new
1036:
An explicit solution of the Heston price equation in terms of the volatility was developed by
Kouritzin. This can be combined with known weak solutions for the volatility equation and Girsanov's theorem to produce explicit weak solutions of the Heston model. Such solutions are useful for efficient
867:
Now consider each of the underlying assets as providing a constraint on the set of equivalent measures, as its expected discount process must be equal to a constant (namely, its initial value). By adding one asset at a time, we may consider each additional constraint as reducing the dimension of
756:
A risk-neutral measure, also known as an equivalent martingale measure, is one which is equivalent to the real-world measure, and which is arbitrage-free: under such a measure, the discounted price of each of the underlying assets is a martingale. See
921:, we have one asset and one Wiener process. The dimension of the set of equivalent martingale measures is zero; hence it can be shown that there is a single value for the drift, and thus a single risk-neutral measure, under which the discounted asset
764:
In the Black-Scholes and Heston frameworks (where filtrations are generated from a linearly independent set of Wiener processes alone), any equivalent measure can be described in a very loose sense by adding a drift to each of the Wiener
444:
297:
984:). Hence we could add a volatility-dependent asset; by doing so, we add an additional constraint, and thus choose a single risk-free measure which is compatible with the market. This measure may be used for pricing.
165:
724:
1026:
An expression of the characteristic function of the Heston model that is both numerically continuous and easily differentiable with respect to the parameters was introduced by Cui et al.
1840:
497:
199:
2375:
962:
1383:
Christoffersen, P.; Heston, S.; Jacobs, K. (2009). "The shape and term structure of the index option smirk: Why multifactor stochastic volatility models work so well".
753:
To price a derivative whose payoff is a function of one or more underlying assets, we evaluate the expected value of its discounted payoff under a risk-neutral measure.
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by one dimension. Hence we can see that in the general situation described above, the dimension of the set of equivalent martingale measures is
2582:
1069:. Sometimes the model is also calibrated to the variance swap term-structure as in Guillaume and Schoutens. Yet another approach is to include
68:
2872:
2862:
2385:
2772:
2736:
768:
By selecting certain values for the drifts described above, we may obtain an equivalent measure which fulfills the arbitrage-free condition.
2689:
3040:
2777:
1887:
1788:
1016:
A derivation of closed-form option prices for the double Heston model was given by
Christoffersen et al. and by Gauthier and Possamai.
1084:
Under the Heston model, the price of vanilla options is given analytically, but requires a numerical method to compute the integral.
2842:
2420:
2390:
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2677:
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1812:
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1954:
1870:
1781:
2832:
2468:
1505:
Cui, Y.; Del BaĂąo Rollin, S.; Germano, G. (2017). "Full and fast calibration of the Heston stochastic volatility model".
1100:
representation of the characteristic function was introduced by Cui et al. in 2017 . Another possibility is to resort to
2827:
1169:(1993). "A closed-form solution for options with stochastic volatility with applications to bond and currency options".
2430:
2014:
1959:
1875:
1045:
There are few known parameterisations of the volatility surface based on the Heston model (Schonbusher, SVI and gSVI).
968:
price. Here, the dimension of the set of equivalent martingale measures is one; there is no unique risk-free measure.
2762:
2752:
2395:
2365:
749:; this is explained in further depth in the above article. For our purposes, it is sufficient to note the following:
43:
model: such a model assumes that the volatility of the asset is not constant, nor even deterministic, but follows a
3076:
2767:
1932:
1830:
2478:
2054:
1835:
1586:
Kouritzin, M. (2018). "Explicit Heston solutions and stochastic approximation for path-dependent option pricing".
816:, the space of possible drifts. Consider the set of equivalent martingale measures to be isomorphic to a manifold
687:
3071:
2847:
2648:
2562:
2547:
1937:
1105:
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minimizing the squared difference between the prices observed in the market and those calculated from the model.
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2567:
1989:
2069:
2044:
2787:
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1905:
1011:
A derivation of closed-form option prices for the time-dependent Heston model was presented by
Benhamou et al.
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2972:
2663:
2445:
2184:
2049:
1860:
452:
2267:
1709:
Damghani, Babak
Mahdavi; Kos, Andrew (2013). "De-arbitraging with a weak smile: Application to skew risk".
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32:
1323:
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2257:
2147:
1021:
An extension of the Heston model with stochastic interest rates was given by
Grzelak and Oosterlee.
918:
1272:
2987:
2952:
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2837:
2607:
2602:
2425:
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1336:
1194:
1186:
1059:
28:
2668:
1227:
Albrecher, H.; Mayer, P.; Schoutens, W.; Tistaert, J. (January 2007), "The little Heston trap",
654:
If the parameters obey the following condition (known as the Feller condition) then the process
1252:
3007:
2812:
2463:
2220:
2137:
2106:
1999:
1979:
1969:
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1820:
1648:
1644:
1428:
1424:
1388:
1384:
1348:
1344:
994:
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512:
439:{\displaystyle d\nu _{t}=\kappa (\theta -\nu _{t})\,dt+\xi {\sqrt {\nu _{t}}}\,dW_{t}^{\nu },}
308:
302:
2673:
2410:
599:
541:
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2019:
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24:
1683:
Le Floc'h, Fabien (2018). "An adaptive Filon quadrature for stochastic volatility models".
1643:
Guillaume, Florence; Schoutens, Wim (2013). "Heston model: The variance swap calibration".
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1031:
The use of the model in a local stochastic volatility context was given by Van Der Weijst.
627:
292:{\displaystyle d{\sqrt {\nu _{t}}}=-\theta {\sqrt {\nu _{t}}}\,dt+\delta \,dW_{t}^{\nu }.}
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2004:
1984:
1949:
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871:
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819:
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775:
644:, the volatility of the volatility, or 'vol of vol', which determines the variance of ν
591:
500:
44:
1423:
Gauthier, P.; Possamai, D. (2009). "Efficient simulation of the double Heston model".
3060:
2899:
2440:
2277:
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2230:
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1994:
1910:
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1732:
1617:
1055:
981:
1536:
1340:
1198:
1104:. For example, the tangent mode of algorithmic differentiation may be applied using
1006:
A discussion of the implementation of the Heston model was given by Kahl and Jäckel.
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2919:
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2009:
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summarized the various quadratures applied and proposed an efficient adaptive
36:
2506:
1042:
High precision reference prices are available in a blog post by Alan Lewis.
1696:
1281:
1182:
1332:
1317:
Benhamou, E.; Gobet, E.; Miri, M. (2009). "Time dependent Heston model".
160:{\displaystyle dS_{t}=\mu S_{t}\,dt+{\sqrt {\nu _{t}}}S_{t}\,dW_{t}^{S},}
2333:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
1773:
1724:
1190:
1480:
1741:
1742:"CLOSED FORM SOLUTION FOR HESTON PDE BY GEOMETRICALTRANSFORMATIONS"
1600:
1519:
1143:
558:, the long variance, or long-run average variance of the price; as
1632:
https://financepress.com/2019/02/15/heston-model-reference-prices/
812:
Wiener processes. The set of equivalent measures is isomorphic to
332:, the instantaneous variance, is given by a Feller square-root or
62:, the price of the asset, is determined by a stochastic process,
840:; initially, consider the situation where we have no assets and
1777:
1552:"Numerical solutions for the stochastic local volatility model"
1096:
1054:
The calibration of the Heston model is often formulated as a
2313:
Autoregressive conditional heteroskedasticity (ARCH) model
1841:
Independent and identically distributed random variables
1588:
International
Journal of Theoretical and Applied Finance
1144:
MATLAB code for implementation by Kahl, Jäckel and Lord
2318:
Autoregressive integrated moving average (ARIMA) model
927:
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345:
311:
213:
176:
71:
1465:"On the Heston model with stochastic interest rates"
1125:(another name for the equivalent martingale measure)
792:
underlying assets and a linearly independent set of
745:
A fundamental concept in derivatives pricing is the
503:(i.e., continuous random walks) with correlation Ď.
2940:
2745:
2707:
2616:
2530:
2487:
2454:
2346:
2303:
2213:
2130:
1886:
1811:
1253:"Option valuation using the fast Fourier transform"
956:
906:
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852:
828:
804:
784:
718:
673:
636:
608:
582:
550:
528:
491:
438:
324:
291:
193:
159:
2200:Stochastic chains with memory of variable length
1297:"Not-so-complex logarithms in the Heston model"
1789:
8:
719:{\displaystyle 2\kappa \theta >\xi ^{2}.}
772:Consider a general situation where we have
2328:Autoregressiveâmoving-average (ARMA) model
1796:
1782:
1774:
1500:
1498:
562:tends to infinity, the expected value of ν
1599:
1518:
1322:
1271:
1236:
1077:as well, in order to capture the forward
948:
932:
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101:
95:
79:
70:
1507:European Journal of Operational Research
1463:Grzelak, L.A.; Oosterlee, C.W. (2011).
1155:
2634:Doob's martingale convergence theorems
1762:
1751:
1665:
1654:
1568:
1557:
1445:
1434:
1405:
1394:
1365:
1354:
492:{\displaystyle W_{t}^{S},W_{t}^{\nu }}
2386:Constant elasticity of variance (CEV)
2376:ChanâKarolyiâLongstaffâSanders (CKLS)
1469:SIAM Journal on Financial Mathematics
7:
1214:Paul Wilmott on Quantitative Finance
1161:
1159:
55:The basic Heston model assumes that
31:that describes the evolution of the
2873:Skorokhod's representation theorem
2654:Law of large numbers (weak/strong)
1065:The prices are typically those of
194:{\displaystyle {\sqrt {\nu _{t}}}}
14:
2843:Martingale representation theorem
1095:Calibration usually requires the
2888:Stochastic differential equation
2778:Doob's optional stopping theorem
2773:DoobâMeyer decomposition theorem
1685:Journal of Computational Finance
1260:Journal of Computational Finance
957:{\displaystyle e^{-\rho t}S_{t}}
2758:Convergence of random variables
2644:FisherâTippettâGnedenko theorem
1134:Martingale (probability theory)
506:The model has five parameters:
2356:Binomial options pricing model
384:
365:
1:
2823:Kolmogorov continuity theorem
2659:Law of the iterated logarithm
1550:van der Weijst, Roel (2017).
1295:Kahl, C.; Jäckel, P. (2005).
1108:in a straightforward manner.
590:, the correlation of the two
2828:Kolmogorov extension theorem
2507:Generalized queueing network
2015:Interacting particle systems
1251:Carr, P.; Madan, D. (1999).
1001:was shown by Carr and Madan.
1960:Continuous-time random walk
1216:(2nd ed.), p. 861
1171:Review of Financial Studies
3098:
2968:Extreme value theory (EVT)
2768:Doob decomposition theorem
2060:OrnsteinâUhlenbeck process
1831:Chinese restaurant process
1529:10.1016/j.ejor.2017.05.018
203:Ornstein-Uhlenbeck process
3036:
2848:Optional stopping theorem
2649:Large deviation principle
2401:HeathâJarrowâMorton (HJM)
2338:Moving-average (MA) model
2323:Autoregressive (AR) model
2148:Hidden Markov model (HMM)
2082:SchrammâLoewner evolution
1610:10.1142/S0219024918500061
1102:automatic differentiation
2763:DolĂŠans-Dade exponential
2593:Progressively measurable
2391:CoxâIngersollâRoss (CIR)
1740:Mario, Dell'Era (2014).
740:for the complete article
674:{\displaystyle \nu _{t}}
529:{\displaystyle \nu _{0}}
325:{\displaystyle \nu _{t}}
2983:Mathematical statistics
2973:Large deviations theory
2803:Infinitesimal generator
2664:Maximal ergodic theorem
2583:Piecewise-deterministic
2185:Random dynamical system
2050:Markov additive process
976:(for example, European
609:{\displaystyle \kappa }
551:{\displaystyle \theta }
536:, the initial variance.
2818:KarhunenâLoève theorem
2753:CameronâMartin formula
2717:BurkholderâDavisâGundy
2112:Variance gamma process
1761:Cite journal requires
1664:Cite journal requires
1567:Cite journal requires
1444:Cite journal requires
1404:Cite journal requires
1364:Cite journal requires
980:, or more explicitly,
964:will be a martingale.
958:
908:
882:
854:
830:
806:
786:
720:
681:is strictly positive
675:
638:
610:
584:
552:
530:
493:
440:
326:
293:
195:
161:
3067:Derivatives (finance)
2948:Actuarial mathematics
2910:Uniform integrability
2905:Stratonovich integral
2833:LĂŠvyâProkhorov metric
2737:MarcinkiewiczâZygmund
2624:Central limit theorem
2226:Gaussian random field
2055:McKeanâVlasov process
1975:Dyson Brownian motion
1836:GaltonâWatson process
1697:10.21314/JCF.2018.356
1282:10.21314/JCF.1999.043
1139:SABR volatility model
1118:Stochastic volatility
1071:forward start options
1056:least squares problem
959:
909:
883:
855:
831:
807:
787:
721:
676:
639:
616:, the rate at which ν
611:
585:
583:{\displaystyle \rho }
553:
531:
494:
441:
327:
294:
196:
170:where the volatility
162:
41:stochastic volatility
3082:Mathematical finance
3023:Time series analysis
2978:Mathematical finance
2863:Reflection principle
2190:Regenerative process
1990:FlemingâViot process
1805:Stochastic processes
1333:10.2139/ssrn.1367955
1212:Wilmott, P. (2006),
1123:Risk-neutral measure
925:
892:
872:
844:
820:
796:
776:
747:risk-neutral measure
738:Risk-neutral measure
730:Risk-neutral measure
688:
658:
637:{\displaystyle \xi }
628:
600:
574:
542:
513:
453:
343:
309:
211:
174:
69:
3018:Stochastic analysis
2858:Quadratic variation
2853:Prokhorov's theorem
2788:FeynmanâKac formula
2258:Markov random field
1906:Birthâdeath process
1183:10.1093/rfs/6.2.327
919:Black-Scholes model
907:{\displaystyle m-n}
488:
470:
432:
285:
153:
2988:Probability theory
2868:Skorokhod integral
2838:Malliavin calculus
2421:Korn-Kreer-Lenssen
2305:Time series models
2268:PitmanâYor process
1725:10.1002/wilm.10201
1129:Girsanov's theorem
1060:objective function
954:
904:
878:
850:
826:
802:
782:
759:Girsanov's theorem
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51:Basic Heston model
29:mathematical model
3077:Options (finance)
3054:
3053:
3008:Signal processing
2727:Doob's upcrossing
2722:Doob's martingale
2686:EngelbertâSchmidt
2629:Donsker's theorem
2563:Feller-continuous
2431:RendlemanâBartter
2221:Dirichlet process
2138:Branching process
2107:Telegraph process
2000:Geometric process
1980:Empirical process
1970:Diffusion process
1826:Branching process
1821:Bernoulli process
1481:10.1137/090756119
1167:Heston, Steven L.
995:Fourier transform
881:{\displaystyle M}
860:is isomorphic to
853:{\displaystyle M}
829:{\displaystyle M}
805:{\displaystyle m}
785:{\displaystyle n}
412:
252:
229:
189:
123:
3089:
3072:Financial models
3028:Machine learning
2915:Usual hypotheses
2798:Girsanov theorem
2783:Dynkin's formula
2548:Continuous paths
2456:Actuarial models
2396:GarmanâKohlhagen
2366:BlackâKarasinski
2361:BlackâDermanâToy
2348:Financial models
2214:Fields and other
2143:Gaussian process
2092:Sigma-martingale
1896:Additive process
1798:
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1314:
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1307:
1304:Wilmott Magazine
1301:
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1248:
1242:
1241:
1240:
1229:Wilmott Magazine
1224:
1218:
1217:
1209:
1203:
1202:
1163:
1090:Filon quadrature
963:
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592:Wiener processes
589:
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527:
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501:Wiener processes
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413:
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331:
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305:then shows that
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25:Steven L. Heston
19:In finance, the
16:Model in finance
3097:
3096:
3092:
3091:
3090:
3088:
3087:
3086:
3057:
3056:
3055:
3050:
3032:
2993:Queueing theory
2936:
2878:Skorokhod space
2741:
2732:KunitaâWatanabe
2703:
2669:Sanov's theorem
2639:Ergodic theorem
2612:
2608:Time-reversible
2526:
2489:Queueing models
2483:
2479:SparreâAnderson
2469:CramĂŠrâLundberg
2450:
2436:SABR volatility
2342:
2299:
2251:Boolean network
2209:
2195:Renewal process
2126:
2075:Non-homogeneous
2065:Poisson process
1955:Contact process
1918:Brownian motion
1888:Continuous time
1882:
1876:Maximal entropy
1807:
1802:
1760:
1750:
1739:
1708:
1705:
1704:
1682:
1681:
1677:
1663:
1653:
1642:
1641:
1637:
1629:
1625:
1585:
1584:
1580:
1566:
1556:
1549:
1548:
1544:
1504:
1503:
1496:
1462:
1461:
1457:
1443:
1433:
1422:
1421:
1417:
1403:
1393:
1382:
1381:
1377:
1363:
1353:
1324:10.1.1.657.6271
1316:
1315:
1311:
1299:
1294:
1293:
1289:
1255:
1250:
1249:
1245:
1238:10.1.1.170.9335
1226:
1225:
1221:
1211:
1210:
1206:
1165:
1164:
1157:
1152:
1114:
1075:barrier options
1067:vanilla options
1052:
993:The use of the
990:
944:
928:
923:
922:
890:
889:
870:
869:
842:
841:
818:
817:
794:
793:
774:
773:
732:
703:
686:
685:
661:
656:
655:
649:
626:
625:
621:
598:
597:
572:
571:
567:
540:
539:
516:
511:
510:
451:
450:
402:
374:
349:
341:
340:
312:
307:
306:
242:
219:
209:
208:
179:
172:
171:
125:
113:
91:
75:
67:
66:
60:
53:
39:asset. It is a
17:
12:
11:
5:
3095:
3093:
3085:
3084:
3079:
3074:
3069:
3059:
3058:
3052:
3051:
3049:
3048:
3043:
3041:List of topics
3037:
3034:
3033:
3031:
3030:
3025:
3020:
3015:
3010:
3005:
3000:
2998:Renewal theory
2995:
2990:
2985:
2980:
2975:
2970:
2965:
2963:Ergodic theory
2960:
2955:
2953:Control theory
2950:
2944:
2942:
2938:
2937:
2935:
2934:
2933:
2932:
2927:
2917:
2912:
2907:
2902:
2897:
2896:
2895:
2885:
2883:Snell envelope
2880:
2875:
2870:
2865:
2860:
2855:
2850:
2845:
2840:
2835:
2830:
2825:
2820:
2815:
2810:
2805:
2800:
2795:
2790:
2785:
2780:
2775:
2770:
2765:
2760:
2755:
2749:
2747:
2743:
2742:
2740:
2739:
2734:
2729:
2724:
2719:
2713:
2711:
2705:
2704:
2702:
2701:
2682:BorelâCantelli
2671:
2666:
2661:
2656:
2651:
2646:
2641:
2636:
2631:
2626:
2620:
2618:
2617:Limit theorems
2614:
2613:
2611:
2610:
2605:
2600:
2595:
2590:
2585:
2580:
2575:
2570:
2565:
2560:
2555:
2550:
2545:
2540:
2534:
2532:
2528:
2527:
2525:
2524:
2519:
2514:
2509:
2504:
2499:
2493:
2491:
2485:
2484:
2482:
2481:
2476:
2471:
2466:
2460:
2458:
2452:
2451:
2449:
2448:
2443:
2438:
2433:
2428:
2423:
2418:
2413:
2408:
2403:
2398:
2393:
2388:
2383:
2378:
2373:
2368:
2363:
2358:
2352:
2350:
2344:
2343:
2341:
2340:
2335:
2330:
2325:
2320:
2315:
2309:
2307:
2301:
2300:
2298:
2297:
2292:
2287:
2286:
2285:
2280:
2270:
2265:
2260:
2255:
2254:
2253:
2248:
2238:
2236:Hopfield model
2233:
2228:
2223:
2217:
2215:
2211:
2210:
2208:
2207:
2202:
2197:
2192:
2187:
2182:
2181:
2180:
2175:
2170:
2165:
2155:
2153:Markov process
2150:
2145:
2140:
2134:
2132:
2128:
2127:
2125:
2124:
2122:Wiener sausage
2119:
2117:Wiener process
2114:
2109:
2104:
2099:
2097:Stable process
2094:
2089:
2087:Semimartingale
2084:
2079:
2078:
2077:
2072:
2062:
2057:
2052:
2047:
2042:
2037:
2032:
2030:Jump diffusion
2027:
2022:
2017:
2012:
2007:
2005:Hawkes process
2002:
1997:
1992:
1987:
1985:Feller process
1982:
1977:
1972:
1967:
1962:
1957:
1952:
1950:Cauchy process
1947:
1946:
1945:
1940:
1935:
1930:
1925:
1915:
1914:
1913:
1903:
1901:Bessel process
1898:
1892:
1890:
1884:
1883:
1881:
1880:
1879:
1878:
1873:
1868:
1863:
1853:
1848:
1843:
1838:
1833:
1828:
1823:
1817:
1815:
1809:
1808:
1803:
1801:
1800:
1793:
1786:
1778:
1772:
1771:
1763:|journal=
1737:
1703:
1702:
1675:
1666:|journal=
1635:
1623:
1578:
1569:|journal=
1542:
1513:(2): 625â638.
1494:
1455:
1446:|journal=
1415:
1406:|journal=
1375:
1366:|journal=
1309:
1287:
1243:
1219:
1204:
1177:(2): 327â343.
1154:
1153:
1151:
1148:
1147:
1146:
1141:
1136:
1131:
1126:
1120:
1113:
1110:
1051:
1048:
1047:
1046:
1043:
1039:
1038:
1033:
1032:
1028:
1027:
1023:
1022:
1018:
1017:
1013:
1012:
1008:
1007:
1003:
1002:
989:
988:Implementation
986:
982:variance swaps
951:
947:
941:
938:
935:
931:
903:
900:
897:
877:
849:
825:
801:
781:
770:
769:
766:
762:
754:
743:
742:
731:
728:
727:
726:
715:
710:
706:
702:
699:
696:
693:
668:
664:
652:
651:
645:
633:
623:
617:
605:
595:
579:
569:
563:
547:
537:
523:
519:
486:
481:
477:
473:
468:
463:
459:
447:
446:
435:
430:
425:
421:
417:
409:
405:
399:
396:
393:
390:
386:
381:
377:
373:
370:
367:
364:
361:
356:
352:
348:
319:
315:
300:
299:
288:
283:
278:
274:
270:
266:
263:
260:
257:
249:
245:
239:
236:
233:
226:
222:
216:
186:
182:
168:
167:
156:
151:
146:
142:
138:
132:
128:
120:
116:
110:
107:
104:
98:
94:
90:
87:
82:
78:
74:
58:
52:
49:
45:random process
23:, named after
15:
13:
10:
9:
6:
4:
3:
2:
3094:
3083:
3080:
3078:
3075:
3073:
3070:
3068:
3065:
3064:
3062:
3047:
3044:
3042:
3039:
3038:
3035:
3029:
3026:
3024:
3021:
3019:
3016:
3014:
3011:
3009:
3006:
3004:
3001:
2999:
2996:
2994:
2991:
2989:
2986:
2984:
2981:
2979:
2976:
2974:
2971:
2969:
2966:
2964:
2961:
2959:
2956:
2954:
2951:
2949:
2946:
2945:
2943:
2939:
2931:
2928:
2926:
2923:
2922:
2921:
2918:
2916:
2913:
2911:
2908:
2906:
2903:
2901:
2900:Stopping time
2898:
2894:
2891:
2890:
2889:
2886:
2884:
2881:
2879:
2876:
2874:
2871:
2869:
2866:
2864:
2861:
2859:
2856:
2854:
2851:
2849:
2846:
2844:
2841:
2839:
2836:
2834:
2831:
2829:
2826:
2824:
2821:
2819:
2816:
2814:
2811:
2809:
2806:
2804:
2801:
2799:
2796:
2794:
2791:
2789:
2786:
2784:
2781:
2779:
2776:
2774:
2771:
2769:
2766:
2764:
2761:
2759:
2756:
2754:
2751:
2750:
2748:
2744:
2738:
2735:
2733:
2730:
2728:
2725:
2723:
2720:
2718:
2715:
2714:
2712:
2710:
2706:
2699:
2695:
2691:
2690:HewittâSavage
2687:
2683:
2679:
2675:
2674:Zeroâone laws
2672:
2670:
2667:
2665:
2662:
2660:
2657:
2655:
2652:
2650:
2647:
2645:
2642:
2640:
2637:
2635:
2632:
2630:
2627:
2625:
2622:
2621:
2619:
2615:
2609:
2606:
2604:
2601:
2599:
2596:
2594:
2591:
2589:
2586:
2584:
2581:
2579:
2576:
2574:
2571:
2569:
2566:
2564:
2561:
2559:
2556:
2554:
2551:
2549:
2546:
2544:
2541:
2539:
2536:
2535:
2533:
2529:
2523:
2520:
2518:
2515:
2513:
2510:
2508:
2505:
2503:
2500:
2498:
2495:
2494:
2492:
2490:
2486:
2480:
2477:
2475:
2472:
2470:
2467:
2465:
2462:
2461:
2459:
2457:
2453:
2447:
2444:
2442:
2439:
2437:
2434:
2432:
2429:
2427:
2424:
2422:
2419:
2417:
2414:
2412:
2409:
2407:
2404:
2402:
2399:
2397:
2394:
2392:
2389:
2387:
2384:
2382:
2379:
2377:
2374:
2372:
2371:BlackâScholes
2369:
2367:
2364:
2362:
2359:
2357:
2354:
2353:
2351:
2349:
2345:
2339:
2336:
2334:
2331:
2329:
2326:
2324:
2321:
2319:
2316:
2314:
2311:
2310:
2308:
2306:
2302:
2296:
2293:
2291:
2288:
2284:
2281:
2279:
2276:
2275:
2274:
2273:Point process
2271:
2269:
2266:
2264:
2261:
2259:
2256:
2252:
2249:
2247:
2244:
2243:
2242:
2239:
2237:
2234:
2232:
2231:Gibbs measure
2229:
2227:
2224:
2222:
2219:
2218:
2216:
2212:
2206:
2203:
2201:
2198:
2196:
2193:
2191:
2188:
2186:
2183:
2179:
2176:
2174:
2171:
2169:
2166:
2164:
2161:
2160:
2159:
2156:
2154:
2151:
2149:
2146:
2144:
2141:
2139:
2136:
2135:
2133:
2129:
2123:
2120:
2118:
2115:
2113:
2110:
2108:
2105:
2103:
2100:
2098:
2095:
2093:
2090:
2088:
2085:
2083:
2080:
2076:
2073:
2071:
2068:
2067:
2066:
2063:
2061:
2058:
2056:
2053:
2051:
2048:
2046:
2043:
2041:
2038:
2036:
2033:
2031:
2028:
2026:
2023:
2021:
2020:ItĂ´ diffusion
2018:
2016:
2013:
2011:
2008:
2006:
2003:
2001:
1998:
1996:
1995:Gamma process
1993:
1991:
1988:
1986:
1983:
1981:
1978:
1976:
1973:
1971:
1968:
1966:
1963:
1961:
1958:
1956:
1953:
1951:
1948:
1944:
1941:
1939:
1936:
1934:
1931:
1929:
1926:
1924:
1921:
1920:
1919:
1916:
1912:
1909:
1908:
1907:
1904:
1902:
1899:
1897:
1894:
1893:
1891:
1889:
1885:
1877:
1874:
1872:
1869:
1867:
1866:Self-avoiding
1864:
1862:
1859:
1858:
1857:
1854:
1852:
1851:Moran process
1849:
1847:
1844:
1842:
1839:
1837:
1834:
1832:
1829:
1827:
1824:
1822:
1819:
1818:
1816:
1814:
1813:Discrete time
1810:
1806:
1799:
1794:
1792:
1787:
1785:
1780:
1779:
1776:
1768:
1755:
1748:(6): 793â807.
1747:
1743:
1738:
1734:
1730:
1726:
1722:
1718:
1714:
1713:
1707:
1706:
1698:
1694:
1690:
1686:
1679:
1676:
1671:
1658:
1650:
1646:
1639:
1636:
1633:
1627:
1624:
1619:
1615:
1611:
1607:
1602:
1597:
1593:
1589:
1582:
1579:
1574:
1561:
1553:
1546:
1543:
1538:
1534:
1530:
1526:
1521:
1516:
1512:
1508:
1501:
1499:
1495:
1490:
1486:
1482:
1478:
1474:
1470:
1466:
1459:
1456:
1451:
1438:
1430:
1426:
1419:
1416:
1411:
1398:
1390:
1386:
1379:
1376:
1371:
1358:
1350:
1346:
1342:
1338:
1334:
1330:
1325:
1320:
1313:
1310:
1305:
1298:
1291:
1288:
1283:
1279:
1274:
1273:10.1.1.6.9994
1269:
1265:
1261:
1254:
1247:
1244:
1239:
1234:
1230:
1223:
1220:
1215:
1208:
1205:
1200:
1196:
1192:
1188:
1184:
1180:
1176:
1172:
1168:
1162:
1160:
1156:
1149:
1145:
1142:
1140:
1137:
1135:
1132:
1130:
1127:
1124:
1121:
1119:
1116:
1115:
1111:
1109:
1107:
1103:
1098:
1093:
1091:
1087:
1082:
1080:
1076:
1072:
1068:
1063:
1061:
1057:
1049:
1044:
1041:
1040:
1035:
1034:
1030:
1029:
1025:
1024:
1020:
1019:
1015:
1014:
1010:
1009:
1005:
1004:
1000:
999:value options
996:
992:
991:
987:
985:
983:
979:
975:
969:
965:
949:
945:
939:
936:
933:
929:
920:
915:
901:
898:
895:
875:
865:
863:
847:
839:
823:
815:
799:
779:
767:
763:
760:
755:
752:
751:
750:
748:
741:
739:
734:
733:
729:
713:
708:
704:
700:
697:
694:
691:
684:
683:
682:
666:
662:
648:
631:
624:
622:reverts to θ.
620:
603:
596:
593:
577:
570:
566:
561:
545:
538:
521:
517:
509:
508:
507:
504:
502:
484:
479:
475:
471:
466:
461:
457:
433:
428:
423:
419:
415:
407:
403:
397:
394:
391:
388:
379:
375:
371:
368:
362:
359:
354:
350:
346:
339:
338:
337:
335:
317:
313:
304:
286:
281:
276:
272:
268:
264:
261:
258:
255:
247:
243:
237:
234:
231:
224:
220:
214:
207:
206:
205:
204:
184:
180:
154:
149:
144:
140:
136:
130:
126:
118:
114:
108:
105:
102:
96:
92:
88:
85:
80:
76:
72:
65:
64:
63:
61:
50:
48:
46:
42:
38:
34:
30:
26:
22:
2958:Econometrics
2920:Wiener space
2808:ItĂ´ integral
2709:Inequalities
2598:Self-similar
2568:GaussâMarkov
2558:Exchangeable
2538:CĂ dlĂ g paths
2474:Risk process
2426:LIBOR market
2405:
2295:Random graph
2290:Random field
2102:Superprocess
2040:LĂŠvy process
2035:Jump process
2010:Hunt process
1846:Markov chain
1754:cite journal
1745:
1719:(1): 40â49.
1716:
1710:
1691:(3): 65â88.
1688:
1684:
1678:
1657:cite journal
1638:
1626:
1591:
1587:
1581:
1560:cite journal
1545:
1510:
1506:
1472:
1468:
1458:
1437:cite journal
1418:
1397:cite journal
1378:
1357:cite journal
1312:
1303:
1290:
1266:(4): 61â73.
1263:
1259:
1246:
1228:
1222:
1213:
1207:
1174:
1170:
1106:dual numbers
1094:
1083:
1064:
1053:
970:
966:
916:
866:
861:
837:
836:embedded in
813:
771:
744:
735:
653:
646:
618:
564:
559:
505:
448:
301:
169:
56:
54:
21:Heston model
20:
18:
3003:Ruin theory
2941:Disciplines
2813:ItĂ´'s lemma
2588:Predictable
2263:Percolation
2246:Potts model
2241:Ising model
2205:White noise
2163:Differences
2025:ItĂ´ process
1965:Cox process
1861:Loop-erased
1856:Random walk
1594:: 1850006.
1475:: 255â286.
1058:, with the
1050:Calibration
1037:simulation.
568:tends to θ.
334:CIR process
303:ItĂ´'s lemma
201:follows an
3061:Categories
3013:Statistics
2793:Filtration
2694:Kolmogorov
2678:Blumenthal
2603:Stationary
2543:Continuous
2531:Properties
2416:HullâWhite
2158:Martingale
2045:Local time
1933:Fractional
1911:pure birth
1601:1608.02028
1520:1511.08718
1150:References
765:processes.
37:underlying
33:volatility
2925:Classical
1938:Geometric
1928:Excursion
1733:154646708
1618:158891879
1319:CiteSeerX
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265:δ
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3046:Category
2930:Abstract
2464:BĂźhlmann
2070:Compound
1537:25667130
1341:12804395
1199:16091300
1112:See also
1097:gradient
2553:Ergodic
2441:VaĹĄĂÄek
2283:Poisson
1943:Meander
1712:Wilmott
1649:2255550
1489:9132119
1429:1434853
1389:1447362
1349:1367955
1191:2962057
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2893:Tanaka
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2446:Wilkie
2411:HoâLee
2406:Heston
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2517:M/M/1
2512:M/G/1
2502:Fluid
2168:Local
1729:S2CID
1614:S2CID
1596:arXiv
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1187:JSTOR
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1073:, or
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2173:Sub-
2131:Both
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