Computational Accuracy of Crank-Nicolson Method for European Put Option Pricing Under Varying Volatility

Option pricing models are central to financial risk management and derivatives trading. The Black-Scholes-Merton (BSM) model provides closed-form solutions but relies on constant parameters, while numerical methods such as the Crank-Nicolson (CN) finite difference scheme offer flexibility for complex payoffs. This study investigates the computational accuracy of the Crank-Nicolson method in pricing European put options relative to the Black-Scholes exact values under varying volatility levels. Using $T=1$ year, $K=100$, $r=0.2$, and two initial stock prices $S_0=40$ and $S_0=50$, put prices were computed for volatilities ranging from $0.25$ to $0.95$. Also, figure results were obtained to validate the performances of BS and CN respectively. Four statistical error metrics: MAE, RMSE, MAPE, and Max Error were employed to quantify deviation. Present value variations across initial stock prices were also analyzed. Furthermore, Non-Central F Analysis was conducted to test the statistical significance of differences in mean put prices across stock price levels. Results show that CN closely approximates BSM at low to moderate volatilities $V \leq 0.55$ with $\mathrm{MAPE}<5%$, but accuracy deteriorates significantly at high volatility $V>0.65$, with Max Error of $7.19$. Present value differences decrease as $S_0$ increases, and the effect of $S_0$ on put prices is statistically significant with $F_{\mathrm{CAL}}=62.97$, $p<0.0001$. The study concludes that while CN is efficient for stable markets, grid refinement is necessary for high volatility regimes.