Fast pricing of path-dependent interest rate options with jumps in continuous and discrete time and stochastic volatility - Silva, Allan Jonathan da (2021)
The COS method was introduced in (FANG; OOSTERLEE, 2008) and then was applied to price a variety of stock options. The work of the thesis adapts the Fourier-cosine series (COS) method to price path-dependent interest rate options, particularly the IDI Option, with a broad range of affine jump-diffusion models. The use of models which does not pursue analytical solution to its corresponding characteristic function, such as interest rate process with stochastic volatility, correlation and jumps is, inter alia, the focus of the thesis. We calculate the option prices of such sophisticated models by numerically solving the characteristic function which values enter in the terms of the cosine series. Enormous precision and computational speed are the qualities of the pricing and hedging estimates here obtained. We also introduce a novel stochastically correlated two-factor affine diffusion process under the CIR format, for which we obtained the closed-form expression of the associated conditional characteristic function and the moment generating function and, in turn, the prices of a bond and the IDI option. Finally, we introduce models with jumps with predetermined times to occur and a discrete and arbitrary set of possible jump sizes. The jump size distribution we introduce - a modified version of the Skellam distribution, departs from the Gaussian framework to a much more realistic format, and we directly apply it to obtain closed form expressions to the price of the zero-Coupon bonds, the IDI options and the Monetary Policy Committee (COPOM) option. We also extend this format to include random time jumps and stochastic volatility. As far as the author is aware, this is the first time that Skellam distributions are used in finance.