Option Prices Expansions and Applications

Abstract

Daniel Bloch presents a general pricing approximation technique for European call option prices in a jump-diffusion model with stochastic interest rates. The author considers the dynamics of the logarithm of the forward price, under the forward measure, in the class of multifactor Affine and Quadratic models without jumps, and devise the dynamics of its associated variance swap. Bloch expresses the expected future average volatility as a function of that variance swap, and uses classical Ito's calculus to expand prices around the Black formula. The author then adds jumps to the dynamics of the log-forward price and condition the expectation of the call price with respect to the number of jumps. Applying a change of measure, option prices decompose into a weighted sum of approximated multifactor Affine and Quadratic models. Bloch uses these prices decompositions to define an analytical formula that approximate the implied volatility surface in the class of Affine and Quadratic models with jumps. At last, the author compares a few prices approximations against Monte Carlo simulations for a range of maturities covering one year, and provide accuracy measures. Bloch then uses these prices as benchmark to measure the accuracy of the implied volatility surface expansion. The author obtains very low MAE and RMSE, both on the prices and IVS, demonstrating that the series expansions are very precise for short and long maturities. Applications include: fast prices and Greeks estimation for European options, initial values for numerical computation of IVS, and control variate for complex pricing models in Monte Carlo and Machine Learning.