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

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.

### 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)
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### 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
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