Black Scholes Calculator

The Black-Scholes pricing model relies on determining intermediate variables $d_1$ and $d_2$ to compute the cumulative normal distribution functions of the underlying asset. The primary equations are defined as:

$d_1 = \frac{\ln(S / K) + (r - q + \frac{\sigma^2}{2})t}{\sigma\sqrt{t}}$
$d_2 = d_1 - \sigma\sqrt{t}$

Where $S$ is the stock price, $K$ is the strike price, $r$ is the risk-free interest rate, $q$ is the dividend yield, $\sigma$ is the volatility, and $t$ is the time to maturity. Once $d_1$ and $d_2$ are established, the call option price ($C$) and put option price ($P$) are derived by discounting the expected terminal values against their respective cumulative normal probabilities ($N$).

Consider a European call and put option on a non-dividend-paying stock ($q = 0$) with the following market inputs: Stock Price ($S$) = $100, Strike Price ($K$) = $100, Risk-Free Rate ($r$) = 5% (0.05), Volatility ($\sigma$) = 20% (0.20), and Time to Maturity ($t$) = 1 year (1.0).

1. Calculate $d_1$: $\ln(100/100) + (0.05 + 0.20^2/2)(1) / (0.20 \times \sqrt{1}) = (0 + 0.07) / 0.20 = 0.35$.
2. Calculate $d_2$: $0.35 - 0.20 \times \sqrt{1} = 0.15$.
3. Determine cumulative normal values: $N(d_1) = N(0.35) \approx 0.6368$ and $N(d_2) = N(0.15) \approx 0.5596$.
4. Compute Call Price: $100 \times 0.6368 - 100 \times e^{-0.05(1)} \times 0.5596 \approx $10.45.

Always input annualized figures for both the risk-free rate and volatility to ensure mathematical consistency. Remember that the standard Black-Scholes model evaluates European options, meaning they cannot be exercised prior to the expiration date. Finally, recognize that implied volatility changes dynamically in live markets, requiring regular model updates.