Pricing Bermudan options under local Lévy models with default

Abstract

We consider a defaultable asset whose risk-neutral pricing dynamics are described by an exponential Lévy-type martingale. This class of models allows for a local volatility, local default intensity and a locally dependent Lévy measure. We present a pricing method for Bermudan options based on an analytical approximation of the characteristic function combined with the COS method. Due to a special form of the obtained characteristic function the price can be computed using a Fast Fourier Transform-based algorithm resulting in a fast and accurate calculation. The Greeks can be computed at almost no additional computational cost. Error bounds for the approximation of the characteristic function as well as for the total option price are given.

Computing Credit Valuation Adjustment for Bermudan Options With Wrong Way Risk

“…, Q, we approximate the optimal exercise log-stock values by a constant x j m,q , using the following expression: The COS method can also be employed when the ChF function can be approximated. Borovykh et al (2016) provide a pricing method for Bermudan options based on an analytic approximation of the ChF using the COS method under local Lévy models with default.…”

Bermudan Option Valuation Under State-Dependent Models

“…After some algebraic manipulations, see for details Borovykh et al (2016), we find that the approximation of order n is a function of the form…”

“…It is well-known that a product of a Hankel or Toeplitz matrix with a vector can be calculated using FFTs, see Borovykh et al (2016) for full details. Using the fact that an FFT can be computed with computational complexity O.N log 2 N/, we find that for a Bermudan option with M exercise dates the overall computational complexity is O..M 1/N log 2 N/.…”

“…One of the problems when considering such a state-dependent model is the fact that there is no explicit density function or characteristic function available. In order to still be able to price derivatives, we derive the characteristic function by means of an advanced Taylor expansion of the state-dependent coefficients, as first presented in Pagliarani et al (2013) for a simplified model and similar to the derivations in Borovykh et al (2016) for the local Lévy model. This Taylor expansion allows one to rewrite the fundamental solution of the related Cauchy problem in terms of solutions of simplified Cauchy problems, which we then solve in the Fourier space to obtain the approximated characteristic function.…”

On the data-driven COS method

“…Ruijter and Oosterlee [6] discretized the governing asset SDEs first and then worked with the ChF of the discrete asset process, within the framework of the COS method. Borovykh et al [7] used the Taylor expansion to derive a ChF for which they could even price Bermudan options highly efficiently. In this work, we extend the applicability of the COS method to the situation where only data (samples from an unknown distribution) are available.…”