Super-Diffusive Noise Source in Asset Dynamics

Max-Olivier Hongler

Ecole Polytechnique Fédérale de Lausanne (EPFL), IMT/STI/LPM, Lausanne, Switzerland.

DOI: 10.4236/jmf.2013.31004                

Abstract

Given an asset with value S_t, we revisit the Black and Scholes dynamics  when the driving noise ξ_t is a non-Gaussian super-diffusive stochastic process with variance of the type . This super-diffusive quadratic variance behavior, synthesizes a ballistic component which would occur in strongly fluctuating environments. When , the assets can, with high probability, be driven towards the bankruptcy . This extra dynamic feature significantly affects the management of an optimal portfolio. In this context, we focus on basic decisions like: 1) determine the optimal level to sell the asset; 2) determine how to balance a portfolio which incorporates such a high volatility asset; and 3) when facing incertitudes on the asset’s growth rate μ, construct an optimal adaptive portfolio control. In all mentioned cases and despite the presence of this highly non-Gaussian noise source, we are able to deliver simple exact and fully explicit optimal control rules.

 

Keywords

Black-Scholes Dynamics; Non-Gaussian Volatility; Optimal Stopping; Adaptive Optimal Control; Exact Solutions

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	O. E. Barndorff-Nielsen and N. Shepard, “Non-Gaussian Ornstein-Uhlenbeck-Based Models and Some of Their Uses in Financial Economics,” Journal of the Royal Statistical Society, Vol. 63, No. 2, 2001, pp. 167-241.
[2]	R. C. Dalang and M.-O. Hongler, “The Right Time to Sell a Stock Whose Price Is Driven by Markovian Noise,” Annals of Applied Probability, Vol. 14, No. 4, 2004, pp. 2176-2201. doi:10.1214/105051604000000747
[3]	M.-O. Hongler, R. Filliger and P. Blanchard, “Soluble Models for a Dynamics Driven by Super-Diffuisve Noise,” Physica A: Statistical Mechanics and Its Applications, Vol. 370, No. 2, 2006, pp. 301-315.
[4]	M.-O. Hongler, R. Filliger, P. Blanchard and J. Rodriguez, “On Stochastic Processes Driven by Ballistic Noise Sources,” In: A. Adhikari, M. R. Adhikari and Y. P. Chaubey, Eds., Contemporary Topics in Mathematics and Statistics with Applications, Asian Books Private Ltd., New Delhi, 2003.
[5]	I. Benjamini and S. Lee, “Conditional Diffusions Which Are Brownian Bridges,” Journal of Theoretical Probability, Vol. 10, No. 3, 1997, pp. 733-736. doi:10.1023/A:1022657828923
[6]	L. C. G. Rogers and J. W. Pitman, “Markov Functions,” Annals of Probability, Vol. 9, No. 4, 1981, pp. 573-582. doi:10.1214/aop/1176994363
[7]	B. Oksendal, “Stochastic Differential Equations—An Introduction with Applications,” Springer, 1998.
[8]	I. Karatzas, “Adaptive Control of a Diffusion to a Goal and a Parabolic Monge-Ampère Equation,” The Asian Journal of Mathematics, Vol. 1, No. 2, 1997, pp. 295313.
[9]	M. Kulldorff, “Optimal Control of Favorable Games with a Time Limit,” SIAM Journal on Control and Optimization, Vol. 31, No. 1, 1993, pp. 52-69. doi:10.1137/0331005
[10]	R. Filliger and M.-O. Hongler, “Explicit Gittins’ Indices for a Class of Super-Diffusive Processes,” Journal of Applied Probability, Vol. 44, No. 2, 2007, pp. 554-559. doi:10.1239/jap/1183667421