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Options in markets with unknown dynamics. (English) Zbl 1431.91393

Summary: 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. This is a statistical consequence of the self-similarity of the Brownian motion, as the information from an arbitrarily short time span equals information from eternal observation. In contrast to this, in the – non self similar – 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.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
60G51 Processes with independent increments; Lévy processes
62P05 Applications of statistics to actuarial sciences and financial mathematics
60G44 Martingales with continuous parameter
91B24 Microeconomic theory (price theory and economic markets)
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References:

[1] D. Applebaum. Levy processes and stochastic calculus. Cambridge University Press, Cambridge, 2004.; · Zbl 1073.60002
[2] F. Black and M. Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 3:637-654, 1973.; · Zbl 1092.91524
[3] T. Choi and R. V. Ramamoorthi. Remarks on consistency of posterior distributions. IMS Collections, 3:170-186, 2008.;
[4] R. Cont and P. Tankov. Financial modelling with jump processes. Chapman and Hall/CRC, London, 2004.; · Zbl 1052.91043
[5] T. Darsinos and S. Satchell. Bayesian analysis of the black-scholes option price, forecasting expected returns in the financial markets. IMS Collections, 3:170-186, 2007.;
[6] F. Delbaen andW. Schachermayer. A general version of the fundamental theorem of asset pricing. Math. Annalen, 300:463- 520, 1994.; · Zbl 0865.90014
[7] D. B. Flynn, S. D. Grose, G. M. Martin, and Vance L. Martin. Pricing australian and s& p200 options: A bayesian approach based on generalized distributional forms. Aust. N. Z. J. Stat., 47 (1):101-117, 2005.; · Zbl 1108.62109
[8] C. S. Forbes, G. M. Martin, and J. Wright. Bayesian estimation of a stochastic volatility model using option and spot prices: application of a bivariate kalman filter. Cambridge Working Papers in Economics 17/03, Monash University, Department of Econometrics and Business Statistics, January 2003.;
[9] S. J. Frame and C. A. Ramezani. Bayesian estimation of asymmetric jump-diffusion processes. Annals of Financial Economics, 09(03):1450008, 2014.;
[10] H. Gzyl, E. ter Horst, and S. W. Malone. Towards a bayesian framework for option pricing. Technical Report cs/0610053, 2006.;
[11] P. R. Halmos. Measure theory. Springer Verlag, Heidelberg NY, 1950.; · Zbl 0040.16802
[12] S.W. Ho, A. Lee, and A.Marsden. Use of bayesian estimates to determine the volatility parameter. Journ. of Risk and Financial Management, 3:74-96, 2011.;
[13] S. Iacus. Option pricing and estimation of financial models with R. Wiley & Sons, New York, 2011.;
[14] E. Jacquier and N. Polson. Bayesian econometrics in finance. Journal of Finance, 58:1269, 2003.;
[15] R. Kaila. The integrated volatility implied by option prices: A Bayesian approach. Phd thesis, Helsinki University of Technology Institute of Mathematics, 2008.;
[16] R. Merton. Theory of rational option pricing. Bell J. of Economics, 3:141-183, 1973.; · Zbl 1257.91043
[17] R. Merton. Option pricing when the underlying stock returns are discontinuous. Bell J. of Economics, 4:125-144, 1976.; · Zbl 1131.91344
[18] S. T. Rachev, J. S. J. Hasu, B. S. Baghasheva, and F. J. Fabozzi. Bayesian Methods in Finance. Wiley & Sons, New York, 2008.;
[19] J. V. K. Rombouts and L. Stentoft. Bayesian option pricing using mixed normal heteroskledasticity models. Computational Statistics and Data Analysis, 76:588-605, 2014.; · Zbl 1506.62157
[20] K.-I. Sato. L/evy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge, UK, 1999.; · Zbl 0973.60001
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