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A class of complete benchmark models with intensity-based jumps

Part of: Mathematical economics Stochastic processes Applications

Published online by Cambridge University Press:  14 July 2016

Eckhard Platen
Eckhard Platen*
Affiliation:
University of Technology, Sydney
*
∗ Postal address: School of Finance and Economics and Department of Mathematical Sciences, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia. Email address: eckhard.platen@uts.edu.au
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Abstract

This paper proposes a class of complete financial market models, the benchmark models, with security price processes that exhibit intensity-based jumps. The benchmark or reference unit is chosen to be the growth-optimal portfolio. Primary security account prices, when expressed in units of the benchmark, turn out to be local martingales. In the proposed framework an equivalent risk-neutral measure need not exist. Benchmarked fair derivative prices are obtained as conditional expectations of future benchmarked prices under the real-world probability measure. This concept of fair pricing generalizes the classical risk-neutral approach and the actuarial present-value pricing methodology.

Keywords

Benchmark modeljump diffusionsgrowth-optimal portfolioevent risk premiumfair pricingactuarial pricinginsurance

MSC classification

Primary: 91B24: Price theory and market structure
Secondary: 60G30: Continuity and singularity of induced measures 62P20: Applications to economics
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 1 , March 2004 , pp. 19 - 34
Copyright
Copyright © Applied Probability Trust 2004 

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References

Ansel, J. P., and Stricker, C. (1994). Couverture des actifs contingents. Ann. Inst. H. Poincaré Prob. Statist. 30, 303–315.Google Scholar
Bajeux-Besnainou, I., and Portait, R. (1997). The numéraire portfolio: a new perspective on financial theory. Europ. J. Finance 3, 291–309.Google Scholar
Bardhan, I., and Chao, X. (1993). Pricing options on securities with discontinuous returns. Stoch. Process. Appl. 48, 123–137.Google Scholar
Becherer, D. (2001). The numéraire portfolio for unbounded semimartingales. Finance Stoch. 5, 327–341.Google Scholar
Bühlmann, H. (1992). Stochastic discounting. Insurance Math. Econom. 11, 113–127.Google Scholar
Davis, M. H. A. (1997). Option pricing in incomplete markets. In Mathematics of Derivative Securities, eds Dempster, M. A. H. and Pliska, S. R., Cambridge University Press, pp. 227–254.Google Scholar
Delbaen, F., and Schachermayer, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 215–250.Google Scholar
Föllmer, H., and Schweizer, M. (1991). Hedging of contingent claims under incomplete information. In Applied Stochastic Analysis (Stoch. Monogr. 5), eds Davis, M. and Elliott, R., Gordon and Breach, London, pp. 389–414.Google Scholar
Geman, S., El Karoui, N., and Rochet, J. C. (1995). Changes of numéraire, changes of probability measures and pricing of options. J. Appl. Prob. 32, 443–458.Google Scholar
Gerber, H. U. (1990). Life Insurance Mathematics. Springer, Berlin.Google Scholar
Goll, T., and Kallsen, J. (2003). A complete explicit solution to the log-optimal portfolio problem. Ann. Appl. Prob. 13, 774–799.CrossRefGoogle Scholar
Heath, D., and Platen, E. (2002). Consistent pricing and hedging for a modified constant elasticity of variance model. Quant. Finance 2, 459–467.Google Scholar
Heath, D., and Platen, E. (2002). Perfect hedging of index derivatives under a minimal market model. Internat. J. Theory Appl. Finance 5, 757–774.CrossRefGoogle Scholar
Heath, D., and Platen, E. (2002). Pricing and hedging of index derivatives under an alternative asset price model with endogenous stochastic volatility. In Recent Developments in Mathematical Finance, ed. Yong, J., World Scientific, River Edge, NJ, pp. 117–126.Google Scholar
Jacod, J., Méléard, S., and Protter, P. (2000). Explicit form and robustness of martingale representations. Ann. Prob. 28, 1747–1780.CrossRefGoogle Scholar
Karatzas, I., and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus (Graduate Texts Math. 113), 2nd edn. Springer, New York.Google Scholar
Karatzas, I., and Shreve, S. E. (1998). Methods of Mathematical Finance (Appl. Math. 39). Springer, New York.Google Scholar
Kelly, J. R. (1956). A new interpretation of information rate. Bell Syst. Tech. J. 35, 917–926.Google Scholar
Korn, R. (2001). Value preserving strategies and a general framework for local approaches to optimal portfolios. Math. Finance 10, 227–241.Google Scholar
Korn, R. and Schäl, M. (1999). On value preserving and growth-optimal portfolios. Math. Meth. Operat. Res. 50, 189–218.Google Scholar
Kramkov, D. O., and Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Prob. 9, 904–950.Google Scholar
Long, J. B. (1990). The numéraire portfolio. J. Financial Econom. 26, 29–69.Google Scholar
Merton, R. C. (1973). An intertemporal capital asset pricing model. Econometrica 41, 867–888.Google Scholar
Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. J. Financial Econom. 2, 125–144.Google Scholar
Platen, E. (2002). Arbitrage in continuous complete markets. Adv. Appl. Prob. 34, 540–558.Google Scholar
Protter, P. (1990). Stochastic Integration and Differential Equations (Appl. Math. 21). Springer, Berlin.Google Scholar