Fast strong approximation Monte Carlo schemes for stochastic volatility models

Quantitative Finance

Volumn 6, Issue 6, 2006, Pages 513-536

  Kahl, Christian ^a,b   Jäckel, Peter ^a  

^a Quantitative Analytics Group   (United Kingdom)

^b UNIVERSITY OF WUPPERTAL   (Germany)

Author keywords

Hyperbolic Ornstein Uhlenbeck process; Hyperbolic volatility; Stochastic numerical integration; Stochastic volatility models; Strong approximation error

Indexed keywords

EID: 33751534524     PISSN: 14697688     EISSN: 14697696     Source Type: Journal    
DOI: 10.1080/14697680600841108     Document Type: Review

References (49)

1. Abe, K.E.S., Strong Taylor schemes for stochastic volatility. Working Paper, 2004 (www.maths.ox.ac.uk/schmitz/project2.htm).
2. Andersen, L. and Andreasen, J., Volatility skews and extensions of the Libor market model. Appl. Math. Financ., 2000, 7(1), 1-32.
3. Andersen, L. and Brotherton-Ratcliffe, R., Extended Libor market models with stochastic volatility. Working Paper, 2001 (Gen Re Securities).
4. Andersen, L. and Piterbarg, V., Moment explosions in stochastic volatilty models. Technical Report, 2004 (Bank of America) (ssrn.com/abstract=559481).
5. Beckers, S., The constant elasticity of variance model and its implications for option pricing. J. Financ., 1980, XXXV(3), 661-673.
6. Black, F. and Scholes, M., The pricing of options and corporate liabilities. J. Polit. Econ., 1973, 81, 637-654.
7. Brennan, M.J. and Schwartz, E.S., Analyzing convertible bonds. J. Fincnc. Quant. Anal., 1980, 15, 907-929.
8. Broadie, M. and Kaya, Ö., Exact simulation of stochastic volatility and other affine jump diffusion processes. Working Paper, 2004 (Columbia University, New York) (www.orie.cornell.edu/~aberndt/FEseminar/papers04/ exact_sim_200409.pdf).
9. Chan, C.K., Karolyi, G.A., Longstaff, F.A. and Sanders, A.B., An empirical comparison of alternate models of the short-term interest rate. J. Financ., 1992, 47(3), 1209-1227 (www.cob.ohio-state.edu/~sanders/ckls.pdt).
10. Cox, J.C., Notes on option pricing I; constant elasticity of variance diffusions. Working Paper, 1975 (Stanford University).
11. Cox, J.C., Ingersoll, J.E. and Ross, S.A., A theory of the term structure of interest rates. Econometrica, 1985, 53, 385-408.
12. Cox, J.C. and Ross, S.A., The valuation of options for alternative stochastic processes. J. Financ. Econ., 1976, 3, 145-166.
13. Doob, J.L., The Brownian movement and stochastic equations. Ann. Math., 1942, 43, 351-369.
14. Doss, H., Liens entre équations différentielles stochastiques ordinaires. Ann. Inst. Henrie Poincaré Probab. Stat., 1977, 13, 99-125.
15. Euler, L., Institutiones Calculi Integralis, Volume 1. St Peterslarg, 1913.
16. Reprinted in Opera Omnia, Series I, volume 11, Birkhäuser, 1913.
17. Gaines, J.G. and Lyons, T.J., Random generation of stochastic area integrals. SIAM J. Appl. Math., 1994, 54(4), 1132-1146.
18. Geman, H. and Yor, M., Bessel processes, Asian options, and perpetuities. Math. Financ., 1993, 3, 349-375.
19. Glasserman, P., Monte Carlo Methods in Financial Engineering, 2003 (Springer: Berlin).
20. Heston, S.L., A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ: Stud., 1993, 6, 327-343.
21. Hogan, M. and Weintraub, K., The lognormal interest rate model and eurodollar futures. Discussion Paper, 1993 (Citibank, New York).
22. Hull, J. and White, A., An analysis of the bias in option pricing caused by a stochastic volatility. Adv. Futures Options Res., 1988, 3, 27-61.
23. Jäckel, P., Monte Carlo Methods in Finance, 2002 (John Wiley and Sons: New York).
24. Kahl, C., Positive numerical integration of stochastic differential equations. Master's thesis, Bergische Universität Wuppertal, 2004 (www.math.uni-wuppertal.de/~kahl/publications/DT.pdf).
25. Kahl, C. and Jäckel, P., Not-so-complex logarithms in the Heston model. Wilmott, 2005, September, 14-103.
26. Kahl, C., and Schurz, H., Balanced Milstein methods for ordinary SDEs. Technical Report, 2005 (Department of Mathematics, Southern Illinois University).
27. Karatzas, I. and Shreve, S.E., Brownian Motion and Stochastic Calculus, 1991 (Springer: Berlin).
28. Karlin, S. and Taylor, M., A Second Course in Stochastic Processes, 1981 (Academic Press: New York).
29. Kloeden, P.E. and Platen, E., Numerical Solution of Stochastic Differential Equations, 1992, 1995, 1999 (Springer: Berlin).
30. Lévy, P., Wiener's random function, and other Laplacian random functions, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles, 1951, pp. 171-187.
31. Marsaglia, G., Anantharayanan, K. and Paul, N.J., Improvements on fast methods for generating normal random variables. Informat. Process. Lett., 1976, 5, 27-30.
32. Marsaglia, G., MacLaren, M.D. and Bray, T.A., A fast procedure for generating normal random variables. Commun. ACM, 1964, 5, 27-30.
33. Maruyama, G., Continuous Markov processes and stochastic equations. Rend. Circ. Mat. Palermo, 1955, 4, 48-90.
34. Matsumoto, M. and Nishimura, T., Mersenne Twister: a 623-dimensionally equidistributed uniform pseudorandom number generator. ACM Trans. Model. Comput. Simulat., 1998, 8(1), 3-30.
35. Milstein, G.N., Approximate integration of stochastic differential equations. Theory Probab. Appl., 1974, 19, 557-562.
36. Milstein, G.N., Platen, E. and Schurz, H., Balanced implicit methods for stiff stochastic systems. SIAM, 1998, 38(3), 1010-1019.
37. Newton, N.J., Asymptotically efficient Runge-Kutta methods for a class of Ito and Stratonovich equations. SIAM J. Appl. Math., 1991, 51, 542-567.
38. Newton, N.J., Variance reduction for simulated diffusions. SIAM J. Appl. Math., 1994, 54, 1780-1805.
39. Pardoux, E. and Talay, M., Discretization and simulation of stochastic differential equations. Acta Appl. Math., 1985, 3, 23-47.
40. Sandmann, K. and Sondermann, D., On the stability of lognormal interest rate models and the pricing of Eurodollar futures. Discussion Paper, 1994 (Department of Statistics, Faculty of Economics, SFB 303, Universität Bonn) (ftp://ftp.wipol.uni-bonn.de/pub/RePEc/bon/bonsfb/bonsfb263.pdf).
41. Sandmann, K. and Sondermann, D., A note on the stability of lognormal interest rate models and the pricing of eurodollar futures. Math. Financ., 1997a, 7(2), 119.
42. Sandmann, K. and Sondermann, D., Log-normal interest rate models: stability and methodology. Discussion Paper, 1997b (Department of Statistics, Faculty of Economics, SFB 303, Universität Bonn) (ftp://ftp.wipol.uni-bonn. de/pub/RePEc/bon/bonsfb/bonsfb398.pdf).
43. Schöbel, R. and Zhu, J., Stochastic volatility with an Ornstein Uhlenbeck process: an extension. Eur. Finance Rev., 1999, 3, 23-46 (ssrn.com/abstract=100831).
44. Schurz, H., Numerical regularization for SDE's: construction of nonnegative solutions. Dyn. Systems Appl., 1996, 5, 323-352.
45. Scott, L., Option pricing when the variance changes randomly: theory, estimation and an application. J. Financ. Quant. Anal., 1987, 22, 419-438.
46. Stein, E.M. and Stein, J.C., Stock price distribution with stochastic volatility: an analytic approach. Rev. Financ. Stud., 1991, 4, 727-752.
47. Uhlenbeck, G.E. and Ornstein, L.S., On the theory of Brownian motion. Phys. Rev., 1930, 36, 823-841.
48. Wiggins, J., Option values under stochastic volatility: theory and empirical estimates. J. Financ. Econ., 1987, 19, 351-372.
49. Wong, E. and Zakai, M., On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Stat., 1965, 36, 1560-1564.