Pricing contingent claims on stocks driven by Lévy processes. (English) Zbl 1054.91033

Chan, Terence

Ann. Appl. Probab. 9, No. 2, 504-528 (1999).

This paper studies martingale measures in a model where the asset price \(S\) is given as the stochastic exponential of a Lévy process \(Y\) satisfying an exponential moment condition and having jumps bounded from below; more precisely, \(S = {\mathcal E}\left( \int \sigma(s)\,dY_s + \int b(s)\,ds \right)\) for deterministic continuous functions \(\sigma,b\). The author determines the minimal equivalent martingale measure, a multiplicative variant of this, and shows that the minimal entropy martingale measure is given by a generalized Esscher transform. (This result is different from the one in H. U. Gerber and E. S. W. Shiu [Trans. Soc. Actuar. 69, 99–191 (1994)] because the latter paper considers a model with \(S = \exp( \sigma Y + b t)\).) Numerical examples show that prices computed under these measures differ very substantially.

Reviewer: Martin Schweizer (Zürich)

MSC:

91B28	Finance etc. (MSC2000)
60G35	Signal detection and filtering (aspects of stochastic processes)
60J27	Continuous-time Markov processes on discrete state spaces
60J75	Jump processes (MSC2010)

Keywords:

martingale measures; Lévy processes; incomplete markets; option pricing; minimal entropy; minimal martingale measure

References:

[1]	Aase, K. K. (1988). Contingent claims valuation when the security price is a combination of an It o process and a random point process. Stochastic Process. Appl. 28 185-220. · Zbl 0645.90009
[2]	Bardhan I. and Chao, X. (1993). Pricing options on securities with discontinuous returns. Stochastic Process. Appl. 48 123-137. · Zbl 0791.60050
[3]	Davis, M. H. A. (1994). A general option pricing formula. Preprint, Imperial College, London.
[4]	Elliott, R. J., Hunter, W. C., Kopp, P. E. and Madan, D. B. (1995). Pricing via multiplicative price decomposition. J. Financial Engineering 4 247-262.
[5]	Esscher, F. (1932). On the probability function in the collective theory of risk. Skandinavisk Aktuarietidskrift 15 175-195. · Zbl 0004.36101
[6]	F öllmer, H. and Schweizer, M. (1991). Hedging of contingent claims under incomplete information. In Applied Stochastic Analy sis (M. H. A. Davis and R. J. Elliott, eds.) 389-414. Gordon and Breach, New York. · Zbl 0738.90007
[7]	Gerber, H. U. and Shiu, E. S. W. (1994). Option pricing by Esscher transforms (with discussion). Trans. Soc. Actuaries 46 99-191.
[8]	Harrison, J. M. and Pliska, S. R. (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Process. Appl. 11 215-260. · Zbl 0482.60097
[9]	Harrison, J. M. and Pliska, S. R. (1983). A stochastic calculus model of continuous trading: Complete markets. Stochastic Process. Appl. 15 313-316. · Zbl 0511.60094
[10]	Jacod, J. and Shiry aev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, New York.
[11]	Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York. · Zbl 0734.60060
[12]	Liptser, R. Sh. and Shiry ayev, A. N. (1989). Theory of Martingales. Kluwer, Dordrecht. · Zbl 0728.60048
[13]	Protter, P. (1990). Stochastic Integration and Differential Equations. Springer, New York. · Zbl 0694.60047