Chan, Terence Pricing contingent claims on stocks driven by Lévy processes. (English) Zbl 1054.91033 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) Cited in 98 Documents 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 PDF BibTeX XML Cite \textit{T. Chan}, Ann. Appl. Probab. 9, No. 2, 504--528 (1999; Zbl 1054.91033) Full Text: DOI OpenURL 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.