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Complications with stochastic volatility models

Part of: Stochastic analysis

Published online by Cambridge University Press:  01 July 2016

Carlos A. Sin
Carlos A. Sin*
Affiliation:
UBS
*
∗ Postal address: UBS-Limited, 100 Liverpool Street, London EC2M 2RH, UK.
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Abstract

We show a class of stock price models with stochastic volatility for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what is usually assumed in the finance literature. We also show the existence of equivalent martingale measures, and provide one explicit example.

Keywords

Strictly local martingaleschange of probability measureoption pricing

MSC classification

Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.)

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 1 , March 1998 , pp. 256 - 268
Copyright
Copyright © Applied Probability Trust 1998 

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References

Ansel, J.-P. and Stricker, C. (1993). Unicité et existence de la loi minimale. In Séminaire de Probabilités XXVII (Lecture Notes in Mathematics 1557). Springer, Berlin, pp. 22–29.Google Scholar
Dalang, R., Morton, A. and Willinger, W. (1990). Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stoch. Stoch. Rep. 29, 185–201.CrossRefGoogle Scholar
Delbaen, F. (1992). Representing martingale measures when asset prices are continuous and bounded. Math. Finance 2, 107–130.Google Scholar
Delbaen, F. and Schachermayer, W. (1994a). Arbitrage and free lunch with bounded risk for unbounded continuous processes. Math. Finance 4, 343–348.CrossRefGoogle Scholar
Delbaen, F. and Schachermayer, W. (1994b). A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520.CrossRefGoogle Scholar
Delbaen, F. and Schachermayer, W. (1995). The no-arbitrage property under a change of numéraire. Stoch. Stoch. Rep. 53, 213–226.Google Scholar
Dupire, B. (1992). Arbitrage pricing with stochastic volatility. Preprint.Google Scholar
Eisenberg, L. and Jarrow, R. (1994). Option pricing with random volatilities in complete markets. Rev. Quant. Finance and Accounting 4, 5–17.Google Scholar
Elliot, R. (1982). Stochastic Calculus and Applications. Springer, New York.Google Scholar
Harrison, M. and Kreps, D. (1979). Martingales and arbitrage in multiperiod security markets. J. Econ. Theory 20, 381–408.Google Scholar
Harrison, M. and Pliska, S. (1981). Martingales and stochastic integrals in the theory of continuous trading. Stoch. Proc. Appl. 11, 215–260.CrossRefGoogle Scholar
Heath, D. and Sin, C. (1996). A note on local martingale measures and the fundamental theorem of asset pricing. Preprint, Cornell University.Google Scholar
Heston, S. (1993). A closed form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327–343.CrossRefGoogle Scholar
Hofmann, N., Platen, E. and Schweizer, M. (1992). Option pricing under incompleteness and stochastic volatility. Math. Finance 2, 153–187.CrossRefGoogle Scholar
Hull, J. and White, A. (1987). The pricing of options on assets with stochastic volatility. J. Finance 42, 281–300.Google Scholar
Hull, J. and White, A. (1988). An analysis of the bias in option pricing caused by a stochastic volatility. Adv. Futures Options Res. 3, 29–61.Google Scholar
Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.Google Scholar
Johnson, H. and Shanno, D. (1987). Option pricing when the variance is changing. J. Financial Quant. Anal. 22, 143–151.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. (1988). Brownian Motion and Stochastic Calculus. Springer, New York.CrossRefGoogle Scholar
Lakner, P. (1993). Martingale measures for right-continuous processes which are bounded below. Math. Finance 3, 43–53.CrossRefGoogle Scholar
Schachermayer, W. (1993). A counterexample to several problems in the theory of asset pricing. Math. Finance 3, 217–229.CrossRefGoogle Scholar
Schachermayer, W. (1994). Martingale measures for discrete-time processes with infinite horizon. Math. Finance 4, 25–55.CrossRefGoogle Scholar
Scott, L. (1987). Option pricing when the variance changes randomly: theory, estimation and an application. J. Financial Quant. Anal. 22, 419–438.CrossRefGoogle Scholar
Scott, L. (1992). Stock market volatility and the pricing of index options. Technical Report, University of Georgia.Google Scholar
Sin, C. (1996). Strictly local martingales and hedge ratios on stochastic volatility models. , Cornell University.Google Scholar
Stein, E. and Stein, J. (1991). Stock price distributions with stochastic volatility: an analytic approach. Rev. Financial Studies 4, 727–752.CrossRefGoogle Scholar
Stroock, D.W. and Varadhan, S. (1979). Multidimensional Diffusion Processes. Springer, Berlin.Google Scholar
Wiggins, J. (1987). Option values under stochastic volatility: theory and empirical estimates. J Financial Econ. 19, 351–372.Google Scholar