Two-dimensional Haar wavelet based approximation technique to study the sensitivities of the price of an option

Abstract

In the present work, a two-dimensional Haar wavelet method is proposed to study the sensitivities of the price of an option. The method is appropriate to study these sensitivities as it explicitly gives the values of all the derivatives of the solution. A Black–Scholes model for European style options is considered to analyze the physical and numerical aspects of the put and the call option Greeks. We use the concept of coordinate transformation to make the Black–Scholes equation dimensionless and to resolve the obstacle in approximating the Greeks having non-smoothness at the strike price. The infinite spatial domain is truncated into the finite domain to avoid large truncation errors. Through rigorous analysis, the method is shown first-order accurate in the L2−norm. The numerical computations performed to approximate the option price and various Greeks, like delta, theta, gamma, and so forth, confirm the theoretical results in L2−norm. The relative errors and the maximum absolute errors are also presented. The motivational work of option Greeks analysis may leave a significant impact on financial institutes; it helps them to manage the risk by setting the portfolio's new value and to estimate the probability of losing money.