The Linear-Rational Wishart Term Structure Model

Overview & Synopsis

The modeling of interest rates has undergone significant changes since the 2008 financial crisis, leading to the development of multi-curve models. This thesis presents a new multi-curve interest rate model, linear-rational, which is based on the affine Wishart process. In this framework, the pricing of zero-coupon bonds and swaps is efficiently calculated, and a semi-closed formula for swaptions is also derived. Additionally, the thesis proposes a fast and precise approximation of swaption prices. The model was calibrated using a 3-month sample of swaption data, resulting in conclusive and robust outcomes. The model is effective in handling the stochastic correlation between the OIS term structure and the EURIBOR-OIS spread term structure, thanks to the specific properties of the Wishart process. Analytical valuation formulas for constant maturity swaps (CMS) and their associated derivatives were obtained. This method enables an efficient estimation of CMS prices by utilizing a polynomial function of the model parameters, with accurate results confirmed and compared to a Monte Carlo approach. Additionally, this technique can be applied to other derivatives, including in-arrears swaps and caps. The model is extended with jumps, which helps reproduce the observed smile and skew in the market. The thesis utilizes the affine property of the Wishart process with jumps to derive an explicit formula that relates the volatility of the swaptions/caps/floors market to the model parameters. Different swaption hedge ratios (delta, vega) and sensitivities are derived and thoroughly analyzed.