Pricing a European Call by Hand With Black–Scholes

With a non-dividend-paying stock at $105$, a strike of $100$, six months remaining, a continuously compounded rate of $4\%$, and annual volatility of $25\%$, the European call is worth about $11.17$ per share under Black–Scholes. Convert inputs to years and decimals, calculate $d_1$ and $d_2$, look up two normal probabilities, and subtract the discounted strike term from the stock term.

Let $N$ denote the standard normal cumulative distribution function. The option value is $C=S N(d_1)-Ke^{-r\tau}N(d_2)$. Under the risk-neutral lognormal model, $N(d_2)$ is the chance of exercise. The $N(d_1)$ term appears after integrating the stock-weighted portion of the payoff, which shifts the normal argument by $\sigma\sqrt{\tau}$. Look up the lower-tail area in the normal table. Some printed tables report the area between zero and the argument, in which case add $0.5$ for a positive argument.

The stock term is about $71.7681$ and the strike term about $60.5955$. Their difference is $11.1725$. For a quick no-arbitrage check, the price exceeds the lower bound $105-98.0199\approx6.98$ and is below the spot price of $105$. A contract representing 100 shares would have a model premium of roughly $1{,}117.25$ before rounding to the market's quote increment.

Keep annual volatility in decimal form and enter the maturity in years. The $\sigma\sqrt{\tau}$ term is the standard deviation of the log return over the remaining time. When you reach the last line, multiply $N(d_1)$ by today's spot and $N(d_2)$ by the discounted strike, then take the difference. As a check, the price must exceed the positive forward intrinsic value $S-Ke^{-r\tau}$ in this example and stay below the spot. Using the raw strike in the second term would ignore when it is paid.

Compare your hand calculation with the options pricer using the same maturity, spot, strike, rate and volatility.