On Numerical Pricing of Put-call parities for Asian Options driven by New Time-Fractional Black-Scholes Evolution Equation

Abstract

The objective of this paper is twofold. Firstly, to derive time-fractional evolution equation modeling the premium of Asian option with arithmetic and geometric in setting when the strike price is fixed and floating. To do so we will assume that underlying security satisfies the time-fractional stochastic differential equation and then we run a no-arbitrage argument and fractional version of Ito's formula to arrive at fractional evolution equation modeling our Asian option. Secondly, to formulate a time-fractional Black-Scholes evolution equation satisfied by the difference of premiums of Asian put and call options i.e. parity of put and call option, subject to a terminal condition. We have also proved the convergence results for the algorithm employed to compute the solution.