Expanding Forward Starting Options

Abstract

Using the properties of the Affine and Quadratic models, we derive the price of a forward starting call option in a jump-diffusion model. Conditioning that price at the determination time of the strike we use a general pricing approximation technique for call options in a jump-diffusion model to calculate the price at that time. We then show that the resulting price no longer depends on the stock price but only depends on the instantaneous variance process and the time to maturity. Using this property, we apply Ito's lemma to each of the expanded terms of the call price and compute their expected value.

Keywords: Jump-Diffusion, Forward Start Option, Price Expansion, Malliavin Calculus