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Option Pricing in Markets with Unknown Stochastic Dynamics


  • Hanno Gottschalk
  • Elpida Nizami
  • Marius Schubert


We consider arbitrage free valuation of European options in Black-Scholes and Merton markets, where the general structure of the market is known, however the specific parameters are not known. In order to reflect this subjective uncertainty of a market participant, we follow a Bayesian approach to option pricing. Here we use historic discrete or continuous observations of the market to set up posterior distributions for the future market. Given a subjective physical measure for the market dynamics, we derive the existence of arbitrage free pricing rules by constructing subjective option pricing measures. The non-uniqueness of such measures can be proven using the freedom of choice of prior distributions. The subjective market measure thus turns out to model an incomplete market. In addition, for the Black-Scholes market we prove that in the high frequency limit (or the long time limit) of observations, Bayesian option prices converge to the standard BS-Option price with the true volatility. In contrast to this, in the Merton market with normally distributed jumps Bayesian prices do not converge to standard Merton prices with the true parameters, as only a finite number of jump events can be observed in finite time. However, we prove that this convergence holds true in the limit of long observation times.

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  • Hanno Gottschalk & Elpida Nizami & Marius Schubert, 2016. "Option Pricing in Markets with Unknown Stochastic Dynamics," Papers 1602.04848,, revised Jan 2017.
  • Handle: RePEc:arx:papers:1602.04848

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    References listed on IDEAS

    1. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters,in: Theory Of Valuation, chapter 8, pages 229-288 World Scientific Publishing Co. Pte. Ltd..
    2. Rombouts, Jeroen V.K. & Stentoft, Lars, 2014. "Bayesian option pricing using mixed normal heteroskedasticity models," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 588-605.
    3. Darsinos, T. & Satchell, S.E., 2001. "Bayesian Analysis of the Black-Scholes Option Price," Cambridge Working Papers in Economics 0102, Faculty of Economics, University of Cambridge.
    4. Marco Frittelli, 2000. "The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets," Mathematical Finance, Wiley Blackwell, vol. 10(1), pages 39-52.
    5. Catherine S. Forbes & Gael M. Martin & Jill Wright, 2003. "Bayesian Estimation of a Stochastic Volatility Model Using Option and Spot Prices: Application of a Bivariate Kalman Filter," Monash Econometrics and Business Statistics Working Papers 17/03, Monash University, Department of Econometrics and Business Statistics.
    6. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    7. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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