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On the martingale property of stochastic exponentials

Part of: Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Bernard Wong  and
C. C. Heyde
Bernard Wong*
Affiliation:
University of New South Wales and Australian National University
C. C. Heyde*
Affiliation:
Australian National University and Columbia University
*
∗ Postal address: School of Actuarial Studies, University of New South Wales, Sydney, NSW 2052, Australia. Email address: [email protected]
∗∗ Postal address: Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
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Abstract

We present a necessary and sufficient condition for a stochastic exponential to be a true martingale. It is proved that the criteria for the true martingale property are related to whether a related process explodes. An alternative and interesting interpretation of this result is that the stochastic exponential is a true martingale if and only if under a ‘candidate measure’ the integrand process is square integrable over time. Applications of our theorem to problems arising in mathematical finance are also given.

Keywords

Explosionmartingale propertyequivalent local martingale measure

MSC classification

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004 , pp. 654 - 664
Copyright
Copyright © Applied Probability Trust 2004 

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References

Anderson, L., and Andreasen, J. (2000). Volatility skews and extensions of the LIBOR market model. Appl. Math. Finance 7, 1–32.10.1080/135048600450275Google Scholar
Cox, J., and Ross, S. (1976). The valuation of options for alternative stochastic processes. J. Financial Econom. 3, 145–166.10.1016/0304-405X(76)90023-4Google Scholar
Delbaen, F., and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 464–520.10.1007/BF01450498Google Scholar
Delbaen, F., and Shirakawa, H. (2002a). A note on option pricing under the constant elasticity of variance model. Asia-Pacific Financial Markets 9, 83–157.Google Scholar
Delbaen, F., and Shirakawa, H. (2002b). No arbitrage condition for positive diffusion price processes. Asia-Pacific Financial Markets 9, 159–168.10.1023/A:1024173029378Google Scholar
Geman, H., El Karoui, N., and Rochet, J. (1995). Changes of numeraire, changes of probability measure and option pricing. J. Appl. Prob. 32, 443–458.10.2307/3215299Google Scholar
Heath, D., and Platen, E. (2002). Consistent pricing and hedging for modified constant elasticity of variance model. Quantitative Finance 2, 459–467.10.1080/14697688.2002.0000013Google Scholar
Heath, D., Platen, E., and Schweizer, M. (2001). A comparison of two quadratic approaches to hedging in incomplete markets. Math. Finance 11, 385–413.10.1111/1467-9965.00122Google Scholar
Hobson, D., and Rogers, C. (1998). Complete markets with stochastic volatility. Math. Finance 8, 27–48.10.1111/1467-9965.00043Google Scholar
Johnson, G, and Helms, L. L. (1963). Class D supermartingales. Bull. Amer. Math. Soc. 69, 59–62.10.1090/S0002-9904-1963-10857-5Google Scholar
Kadota, T., and Shepp, L. (1970). Conditions for absolute continuity between a certain pair of probability measures. Z. Wahrscheinlichkeitsth. 16, 250–260.10.1007/BF00534599Google Scholar
Kallianpur, H. (1980). Stochastic Filtering Theory. Springer, New York.10.1007/978-1-4757-6592-2Google Scholar
Karatzas, I., and Shreve, S. (1991). Brownian Motion and Stochastic Calculus. Springer, New York.Google Scholar
Kazamaki, N. (1994). Continuous Exponential Martingales and BMO (Lecture Notes Math. 1579). Springer, New York.Google Scholar
Levental, S., and Skorohod, A. V. (1995). A necessary and sufficient condition for absence of arbitrage with tame portfolios. Ann. Appl. Prob. 5, 906–925.10.1214/aoap/1177004599Google Scholar
Liptser, R. S., and Shiryayev, A. N. (1977). Statistics of Random Processes, Vol. 1. Springer, New York. Russian original: Nauka, Moscow, 1974.10.1007/978-1-4757-1665-8Google Scholar
Lucic, V. (2003). Forward start options in stochastic volatility models. Willmott Magazine, September 2003.10.1002/wilm.42820030518Google Scholar
McKean, H. (1969). Stochastic Integrals. Academic Press, New York.Google Scholar
Revuz, D., and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, New York.10.1007/978-3-662-06400-9Google Scholar
Rogers, L. C. G., and Williams, D. (2000). Diffusions, Markov Processes and Martingales, Vol. 2, 2nd edn. Cambridge University Press.Google Scholar
Rydberg, T. (1997). A note on the existence of equivalent martingale measures in a Markovian setting. Finance Stoch. 1, 251–257.10.1007/s007800050024Google Scholar
Sin, C. (1998). Complications with stochastic volatility models. Adv. Appl. Prob. 30, 256–268.10.1239/aap/1035228003Google Scholar
Stein, E., and Stein, J. (1991). Stock price distributions with stochastic volatility: an analytic approach. Rev. Financial Studies 4, 727–752.10.1093/rfs/4.4.727Google Scholar
Wong, B., and Heyde, C. C. (2003). Change of measure for stochastic volatility models. Working Paper, University of New South Wales.Google Scholar