Basic credit risk analysis with Levy processes and numerical FFT method

Abstract

Levy processes application is becoming a hot topic in financial modeling and empirical calibration on recent decades. Due to the infinite divisibility, independent and stationary increments properties, Levy processes match the market price dynamics intuitively.

In this thesis, some properties of Levy processes which outbreak the restricts of classic continuous Black-Scholes model with jumps are explored. Moreover, the explicit sensitivities for the bond price according to the log-normal distributed compound Poisson processes are deduced strictly. Meanwhile, the analytic illustrations are provided.

To find the inherent Levy processes evidences of the market, the S&P 500 index option prices are studied since those are the easiest and representative data source. Besides the classic Black-Scholes model, the Heston model is considered since its stochastic volatility embedding. Then non-iid models which violate the assumption of identically and independently distributed jumps are checked for next. Furthermore, Levy processes are discussed for the Partial Integro-Differential Equation (PIDE) question and numerical estimation by applying Fast Fourier Transform (FFT) algorithm. By investigating the typical three Levy processes: General Hyperbolic model (GH), Normal Inverse Gaussian model (NIG), Carr-Geman-Madan-Yor model (CGMY), numerical signs about the parameters sensitivities show up. The empirical indications and comparisons which reveal the more stable prediction of Levy processes are observed as well.