Stochastic volatility stock price-coefficient estimation and option pricing using a recombining tree. Sharp estimation of the almost-sure Lyapunov exponent for the Anderson model in continuous space

Ionut Florescu, Purdue University

Abstract

In the first part of the thesis we attempt to deal with the problem of estimating option pries when the volatility component of the price is stochastic. The model we use is: dSt = μStdt + σ(Yt)StdWt, where Yt is a mean-reverting type process. First, we show how to estimate the distribution of the volatility component, using an algorithm due to [1]. Second, using this distribution we are able to construct a tree model which converges to the solution of the given equation. In order to price options on the stock, we use the Monte Carlo method to compute the expected price using smaller, recombining trees. Finally, we use this method to compute the price of European Call Options on the SP500 index price in April. We use daily data and our method gives good results that are proximate to the reported bid-ask spread. The second part of the thesis studies the exponential behavior of the solution of the stochastic Anderson differential equation: u( t, x) = 1 + [special characters omitted]κΔxu(s, x) ds + [special characters omitted]W (ds, x) u ( s, x), when x the spatial parameter of the Gaussian field W is continuous i.e., x ∈ R. We give a partial existence result of the Lyapunov exponent defined as limt→∞ t −1 log u(t, x). Furthermore, we find upper and lower bounds for lim supt →∞ t−1 log u(t, x), respectively lim inft →∞ t−1 log u(t, x) as functions of the regularity constant κ. Our bounds are better than the previously known results or much easier to prove. When the absolute modulus of continuity of the process W is in the logarithmic scale our bounds are optimal.

Degree

Ph.D.

Advisors

Viens, Purdue University.

Subject Area

Statistics

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