Pricing and Hedging in Affine Models with Possibility of Default

Abstract

We propose a general framework for the simultaneous modeling of equity, government bonds, corporate bonds, and derivatives. Uncertainty is generated by a general affine Markov process. The setting allows for stochastic volatility, jumps, the possibility of default, and correlation between different assets. We show how to calculate discounted complex moments by solving a coupled system of generalized Riccati equations. This yields an efficient method to compute prices of power payoffs. European calls and puts as well as binaries and asset-or-nothing options can be priced with the fast Fourier transform methods of Carr and Madan [J. Comput. Finance, 2 (1999), pp. 61–73] and Lee [J. Comput. Finance, 7 (2005), pp. 51–86]. Other European payoffs can be approximated with a linear combination of government bonds, power payoffs, and vanilla options. We show the results to be superior to using only government bonds and power payoffs or government bonds and vanilla options. We also give conditions for European continent claims in our framework to be replicable if enough financial instruments are liquidly tradable and study dynamic hedging strategies. As an example we discuss a Heston-type stochastic volatility model with possibility of default and stochastic interest rates.