Stochastic volatility for Lévy processes

Mathematical Finance

Volumn 13, Issue 3, 2003, Pages 345-382

  Carr, Peter ^a   Geman, Hélyette ^b   Madan, Dilip B ^c   Yor, Marc ^d  

^a NEW YORK UNIVERSITY   (United States)

^b UNIVERSITÉ PARIS DAUPHINE   (France)

^c UNIVERSITY OF MARYLAND   (United States)

^d SORBONNE UNIVERSITÉ   (France)

Author keywords

L vy marginal; Leverage; Martingale marginal; OU equation; Static arbitrage; Stochastic exponential; Variance gamma

Indexed keywords

EID: 0038742720     PISSN: 09601627     EISSN: None     Source Type: Journal    
DOI: 10.1111/1467-9965.00020     Document Type: Article

References (56)

1. ANÉ, T., and H. GEMAN (2000): Order Flow, Transaction Clock and Normality of Asset Returns, J. Finance 55, 2259-2284.
2. BARNDORFF-NIELSEN, O. E. (1998): Processes of Normal Inverse Gaussian type, Finance Stoch. 2, 41-68.
3. BARNDORFF-NIELSEN, O. E., E. NICCOLATO, and N. SHEPARD (2001): Some Recent Developments in Stochastic Volatility Modeling, Quant. Finance, forthcoming.
4. BARNDORFF-NIELSEN, O. E., and N. SHEPHARD (2001): Non-Gaussian Ornstein-Uhlenbeck-Based Models and Some of Their Uses in Financial Economics (with discussion), J. Royal Stat. Soc. B, 63, 167-241.
5. BATES, D. (1996): Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutschemark Options, Rev. Financial Stud. 9, 69-108.
6. BATES, D. (2000): Post-'87 Crash Fears in S&P 500 Futures Options, J. Econometrics 94, 181-238.
7. BLACK, F., and M. SCHOLES (1973): The Pricing of Options and Corporate Liabilities, J. Polit. Econ. 81, 637-654.
8. BOYARCHENKO, S., and S. LEVENDORSKII (2000): Option Pricing for Truncated Lévy Processes, Inter. J. Theor. Appl. Fin. 3, 549-552.
9. BREEDEN, D. T., and R. H. LITZENBERGER (1978): Prices of State Contingent Claims Implicit in Option Prices, J. Business 51, 621-652.
10. CARR, P., H. GEMAN, D. MADAN, and M. YOR (2002): The Fine Structure of Asset Returns: An Empirical Investigation, J. Business 75, 2, 305-332.
11. CARR, P., and D. MADAN (1998): Option Valuation Using the Fast Fourier Transform, J. Computat. Finance 2, 61-73.
12. CARR, P., and L. WU (2002): Time-changed Lévy Processes and Option Pricing, J. Financial Economics, forthcoming.
13. CLARK, P. (1973): A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices, Econometrica 41, 135-156.
14. COX, J., J. INGERSOLL, and S. ROSS (1985): A Theory of the Term Structure of Interest rates, Econometrica 53, 385-408.
15. DUAN, J. (1995): The Garch Option Pricing Model, Math. Finance 5, 1-32.
16. DUFFIE, D., J. PAN, and K. SINGLETON (2000): Transform Analysis and Asset Pricing for Affine Jump-Diffusions, Econometrica 68, 1343-1376.
17. EBERLEIN, E., U. KELLER, and K. PRAUSE (1998): New Insights into Smile, Mispricing and Value at Risk, J. Business 71, 371-406.
18. ENGLE, R. (1982): Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation, Econometrica 50, 987-1008.
19. FITZSIMMONS, P. (1987): A Converse to a Theorem of P. Lévy, Ann. Probab. 15, 1515-1523.
20. GEMAN, H., D. MADAN, and M. YOR (2001): Time Changes for Lévy Processes, Math. Finance 11, 79-96.
21. GEMAN, H., D. MADAN, and M. YOR (2002): Stochastic Volatility, Jumps and Hidden Time Changes, Finance Stoch. 6, 63-90.
22. GETOOR, R. K. (1979): The Brownian Escape Process, Ann. Probab. 1, 864-867.
23. GRAVERSEN, S. E., and A. N. SHIRYAEV (2000): An Extension of P. Lévy's Distributional Properties to the Case of Brownian Motion with Drift, Bernoulli 6, 615-620.
24. HAGAN, P., and D. WOODWARD, (2002): Using Lévy Distributions to Price and Manage Option Risk. Working paper, Bear Sterns, New York.
25. HESTON, S. (1993): A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, Rev. Financial Stud. 6, 327-343.
26. HULL, J., and A. WHITE (1987): The Pricing of Options on Assets with Stochastic Volatilities, J. Finance 42, 281-300.
27. JUREK, Z. J., and W. VERVAAT (1983): An Integral Representation for Self-Decomposable Banach Space Valued Random Variables, Z. Wahrsch. verw. Gebiete 62, 247-262.
28. KELLERER, H. G. (1972): Markov-Komposition und eine Anwendung auf Martingale, Math. Annalan 198, 99-122.
29. KONIKOV, M., and D. MADAN (2002): Stochastic Volatility via Markov Chains, Rev. Derivat. Res. 5, 81-115.
30. KOPONEN, I. (1995): Analytic Approach to the Problem of Convergence of Truncated Lévy flights towards the Gaussian stochastic Process, Physical Rev. E 52, 1197-1199.
31. KOU, S., and H. WANG (2001): Option Pricing under a Double Exponential Jump Diffusion Model. Working paper, Columbia University.
32. LAMBERTON, D., and B. LAPEYRE (1996): Introduction to Stochastic Calculus Applied to Finance. New York: Chapman and Hall.
33. LAMPERTI, J. (1962): Semi-Stable Stochastic Processes, Trans. Am. Math. Soc. 104, 62-78.
34. MADAN, D., P. CARR, and E. CHANG (1998): The Variance Gamma Process and Option Pricing, European Finance Rev. 2, 79-105.
35. MADAN, D. B., and F. MILNE (1991): Option Pricing with VG Martingale Components, Math. Finance 1, 39-56.
36. MADAN, D. B., and E. SENETA (1990): The Variance Gamma (VG) Model for Share Market Returns, J. Business 63, 511-524.
37. MADAN, D. B., and M. YOR (2002): Making Markov Martingales Meet Marginals: with Explicit Constructions, Bernoulli 8, 509-536.
38. MANTEGNA, R. N., and H. E. STANLEY (1994): Stochastic Process with Ultraslow Convergence to a Gaussian: The Truncated Lévy Flight, Physics Rev. Lett. 73, 2946; also in M. F. Shlesinger, G. M. Zaslevsky, and U. Frisch, eds. (1995): Lévy Flights and Related Topics in Physics. Berlin: Springer.
39. MANTEGNA, R. N., and H. E. STANLEY (1994): Stochastic Process with Ultraslow Convergence to a Gaussian: The Truncated Lévy Flight, Physics Rev. Lett. 73, 2946; also in M. F. Shlesinger, G. M. Zaslevsky, and U. Frisch, eds. (1995): Lévy Flights and Related Topics in Physics. Berlin: Springer.
40. MERTON, R. C. (1973): Theory of Rational Option Pricing, Bell J. Econ. Mgemt. Sci. 4, 141-183.
41. MERTON, R. C. (1976): Option Pricing When Underlying Asset Returns Are Discontinuous, J. Financial Econ. 3, 125-144.
42. NICOLATO, E., and E. VENARDOS (2001): Option Pricing in Stochastic Volatility Models of the Ornstein-Uhlenbeck Type with a Leverage Effect. Working paper, Department of Mathematical Sciences, Aarhus University.
43. PITMAN, J., and M. YOR (1997): The Two Parameter Poisson-Dirichlet Laws, Ann. Probab. 25, 855-900.
44. PITMAN, J., and M. YOR (1982): A Decomposition of Bessel Bridges, Z. Wahrsch. verw. Gebeite 59, 425-457.
45. ROTHSCHILD, M., and J. STIGLITZ (1970): Increasing Risk I, A Definition, J. Econ. Theory 2, 225-243.
46. ROTHSCHILD, M., and J. STIGLITZ (1971): Increasing Risk II, Its Economic Consequences, J. Econ. Theory 3, 66-84.
47. ROTHSCHILD, M., and J. STIGLITZ (1972): Addendum to Increasing Risk I, A Definition, J. Econ. Theory 5, 306.
48. SATO, K. (1991): Self Similar Processes with Independent Increments, Prob. Theory & Rel. Fields 89, 285-300.
49. SATO, K. (1999): Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
50. SHIGA, T., and S. WATANABE (1973): Bessel Diffusions as a One-Parameter Family of Diffusion Processes, Z. Wahrsch. verw. Gebiete 27, 37-46.
51. SHIRYAEV, A. N., and A. CHERNY (1999): Some Distributional Properties of Brownian Motion with Drift, and an Extension of P. Lévy's Theorem, Theory Probab. Appl. 44, 412-418.
52. SIN, C. (2002): Modelling Yield Curve Volatility Skew Using Jumps and Stochastic Volatility. Working paper, UBS Warburg.
53. TAYLOR, H. M. (1975): A Stopped Brownian Motion Formula, Ann. Probab. 3, 234-246.
54. WILLIAMS, D. (1976): On a Stopped Brownian Motion Formula of H. M. Taylor, Sem. Probab. Strasbourg X. Lecture Notes in Mathematics 511, 235-239. Berlin: Springer.
55. YOR, M. (1992): Some Aspects of Brownian Motion, Part I: Some Special Functional. Lectures in Mathematics, ETH Zürich. Berlin: Birkhäuser Verlag.
56. YOR, M. (1992): Sur Certaines Fonctionnelles Exponentielles du Mouvement Brownien Réel, J. Appl. Probab. 29, 202-208.