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On martingale measures and pricing for continuous bond-stock market with stochastic bond

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  • Nikolai Dokuchaev

Abstract

This papers addresses the stock option pricing problem in a continuous time market model where there are two stochastic tradable assets, and one of them is selected as a num\'eraire. It is shown that the presence of arbitrarily small stochastic deviations in the evolution of the num\'eraire process causes significant changes in the market properties. In particular, an equivalent martingale measure is not unique for this market, and there are non-replicable claims. The martingale prices and the hedging error can vary significantly and take extreme values, for some extreme choices of the equivalent martingale measures. Some rational choices of the equivalent martingale measures are suggested and discussed, including implied measures calculated from observed bond prices. This allows to calculate the implied market price of risk process.

Suggested Citation

  • Nikolai Dokuchaev, 2011. "On martingale measures and pricing for continuous bond-stock market with stochastic bond," Papers 1108.0719, arXiv.org, revised Sep 2014.
    • Handle: RePEc:arx:papers:1108.0719
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    References listed on IDEAS

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    1. Kerry Back, 2010. "Martingale Pricing," Annual Review of Financial Economics, Annual Reviews, vol. 2(1), pages 235-250, December.
    2. Nikolai Dokuchaev, 2011. "Option Pricing Via Maximization Over Uncertainty And Correction Of Volatility Smile," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(04), pages 507-524.
    3. Cheng, Susan T., 1991. "On the feasibility of arbitrage-based option pricing when stochastic bond price processes are involved," Journal of Economic Theory, Elsevier, vol. 53(1), pages 185-198, February.
    4. Yong-Jin Kim & Naoto Kunitomo, 1999. "Pricing Options under Stochastic Interest Rates: A New Approach," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 6(1), pages 49-70, January.
    5. Benninga, Simon & Björk, Tomas & Wiener, Zvi, 2002. "On the Use of Numeraires in Option pricing," SSE/EFI Working Paper Series in Economics and Finance 484, Stockholm School of Economics.
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