Accurate and Efficient Computations of the Greeks for Options Near Expiry Using the Black-Scholes Equations

We investigate the accurate computations for the Greeks using the numerical solutions of the Black-Scholes partial differential equation. In particular, we study the behaviors of the Greeks close to the maturity time and in the neighborhood around the strike price. The Black-Scholes equation is discretized using a nonuniform finite difference method. We propose a new adaptive time-stepping algorithm based on local truncation error. As a test problem for our numerical method, we consider a European cash-or-nothing call option. To show the effect of the adaptive stepping strategy, we calculate option price and its Greeks with various tolerances. Several numerical results confirm that the proposed method is fast, accurate, and practical in computing option price and the Greeks.