Simulation of Asian options delta using a quadratic congruential pseudo-random numbers generator

Abstract

In this paper, we use the Monte Carlo simulation (MC) to calculate delta of an Asian option. Where delta measures the rate of change of option value (V) with respect to changes in the underlying asset's price (S): ∆ = ∂V ∂S. A. Moussi et al. Moreover, we use the quadratic congruential pseudorandom numbers generator, which have the following form: X n+1 = (aX 2 n + bX n + c) mod m, in order to evaluate its performance. We also compare results with two other chosen congruential generators.

Estimators of sensitivities of an Asian option: numerical analysis

“…The quadratic PRNG will be used in all simulations [3]. Hence, we can extract a reduced centered normal law variate N (0, 1) from X n , by Box-Muller method (see [9]).…”

“…The price is computed by using equation (3). Note that price and sensitivities of continuously sampled Asian options can be estimated by taking sufficiently large values of m. However one must account for the inherent discretization bias resulting from the approximation of continuous time processes through discrete sampling.…”

“…simulate sample paths of the value which one wants to estimate , 2. re-simulate N independent reproductions of this value , 3. take the average of all returns.…”

“…For generating random variables following standard normal distribution, we will use the quadratic Pseudo Random Numbers Generator (PRNG)(see Moussi el al. [3]). In what follow, we expose a method for increasing the efficiency of Monte Carlo simulation by reducing the variance of simulation estimates.…”