Random field HJM model : a unified framework for fixed income, equity and credit markets

Abstract

The HJM framework was originally introduced for the modelling of the dynamics of the instantaneous forward rates in the fixed income market. This approach was later extended to the case with uncertainties described by a Gaussian random field, which is termed as the random field HJM model. In this thesis, this approach is extended and applied to both equity and credit markets. In each case, it is shown that how to yield a consistent condition such that the market is arbitrage-free or the market observables are martingales. In the first part, this approach is applied to the equity market. A unified framework is proposed for the joint modelling of an index and its local volatility surface. This model enables index and volatility derivatives to be priced consistently and reproduces empirical features of the equity market, including the strong negative correlation between the changes of SPX and VIX, any observed term structure of variance swap rates, and the volatility skews and term structures observed in SPX and VIX options. Starting from an initial collection of index options and their associated local volatility surface, it is shown in the framework of the HJM philosophy that how to construct the arbitrage-free evolution of this local volatility surface from a Gaussian random field. The relationship between the local volatility model and the variance rate model is also presented. In the end, it is shown that how to implement this model numerically by the Monte Carlo simulation and the finite difference method, and how to simultaneously calibrate to the prices of SPX options and VIX options across strikes and maturities.

In the second part, this approach is applied to the credit market. Contagion credit and counterparty credit risks are two of the central topics in the credit risk management after the sub-prime crisis. Starting from a Gaussian random field, it is shown in the framework of the HJM philosophy that how to construct the dynamics of the conditional joint density of the default times given the risk-free market information via the change-of-measure technique. It is a top-down approach in the analysis of the contagion credit risk with multiple default events. In the same framework, the dynamics of the forward risk-free rate can also be described and a general pricing formula for credit products can be derived. Furthermore, this contagion model is applied to the pricing of the credit valuation adjustment by taking into account the unilateral or bilateral counterparty credit risk. The influence of the default of the counterparty on the underlying reference entity is considered so as to avoid the wrong-way risk in particular. Finally, the bilateral credit valuation adjustment for credit default swaps is illustrated through a numerical study.