💹 Option Pricing

Black-Scholes prices a European option from five inputs: current stock price S, strike price K, time to expiry T (in years), risk-free interest rate r, and volatility σ (the annualised standard deviation of returns). The formula computes two intermediate terms, d1 and d2, from these inputs and plugs them into the cumulative normal distribution to get the call price C = S·N(d1) − K·e^(−rT)·N(d2), with the put price following from put-call parity.

The Monte Carlo panel exists to demonstrate that the two very different approaches agree, which builds confidence in both. It simulates 50 random stock price paths using geometric Brownian motion, S(T) = S(0)·exp((r−σ²/2)T + σ·√T·Z), then averages the discounted payoff max(S(T)−K,0)·e^(−rT) across all paths. Because Monte Carlo works for options with no closed-form solution (like Asian or barrier options), showing it converge to the Black-Scholes price here confirms the simulation engine is implemented correctly.

Delta shows how much the option price moves per £1 change in the stock price and ranges from 0 to 1 for a call. Gamma is the rate of change of delta, highest for at-the-money options near expiry. Vega measures sensitivity to a 1-point change in volatility — options are always worth more when volatility rises. Theta is the daily time decay, typically negative, showing how much value the option loses purely from the passage of time. Rho measures sensitivity to the risk-free interest rate.

Put-call parity states that Call − Put should equal S − K·e^(−rT) for options with the same strike and expiry. The simulation displays "OK" whenever the discrepancy is below 0.01, which covers ordinary floating-point rounding in the normal-distribution approximation used for N(d1) and N(d2). A larger discrepancy would indicate a genuine arbitrage opportunity or a pricing bug, neither of which should occur with the closed-form Black-Scholes formula used here.

Try the Short expiry preset (T ≈ 0.08 years, about a month) against the LEAP preset (T = 2 years): the short-dated option has almost no time value left, so its price sits close to its intrinsic value and theta decay is steep and visible day to day. The 2-year LEAP retains much more time value, decays far more slowly, and is far more sensitive to changes in volatility (higher vega) since there is more time for the stock to move.