Option Pricing under Feller-Lévy models.

This repository contains the C++ codes for my work as a research assistant at the Seminar for Applied Mathematics at ETH Zürich.
The topic of my project was: Option Pricing under Feller-Lévy models.
The codes contain the Finite Element Method implementation for option pricing.

Overview

The arbitrage-free price of financial products with payoff at time $t \in [0,T]$ , is given by $V(t,x) = \mathbb{E}[e^{-rT}g(X_T) | X_t=x]$ .

The stock price process follows $dX_t = b(X_{t^-})dt + a(X_{t^-})^{1/\alpha}d L_t^{\alpha}, \ \ \ X_0=x,$ , where $(L_t^{\alpha})_{t\ge0}$ is an $\alpha$ -stable Lévy process, and hence the stock price is a jump process.

Using the Feynman-Kac theorem, the fractional partial differential equation governing the price of the option is given by

where the risk free interest rate, and the payoff.

Finally, the finite element method is applied to the above equation to solve for the option price process. However, a special numerical treatment is required for the discretization of the fractional laplace operator $(-\Delta)^{\alpha/2}$ given by $(-\Delta)^{-(1-\beta)}= \frac{2\sin(\pi(1-\beta))}{\pi} \int_{-\infty}^{\infty} \text{e}^{2(1-\beta) x} (\mathcal{I}-\text{e}^{2x}\Delta)^{-1}dx$ , and that is taken care of in this project.