A numerical method for solving the underlying price problem driven by a fractional Levy process

Abstract

We consider European style options with risk-neutral parameters and time-fractional Levy diffusion equation of the exponential option pricing model in this paper. In a real market, volatility is a measure of the quantity of inflation in asset prices and changes. This makes it essential to accurately measure portfolio volatility, asset valuation, risk management, and monetary policy. We consider volatility as a function of time. Estimating volatility in the time-fractional Levy diffusion equation is an inverse problem. We use a numerical technique based on Chebyshev wavelets to estimate volatility and the price of European call and put options. To determine unknown values, the minimization of a least-squares function is used. Because the obtained corresponding system of linear equations is ill-posed, we use the Levenberg-Marquardt regularization technique. Finally, the proposed numerical algorithm has been used in a numerical example. The results demonstrate the accuracy and effectiveness of the methodology used.

Keywords

• european options
• time-fractional levy diffusion equation
• volatility
• chebyshev wavelets
• levenberg-marquardt regularization