Options & Derivatives

Black-Scholes model for European option pricing and the put-call parity relationship.

Black-Scholes Call Option Price

European call option. S₀ = spot price, K = strike, r = risk-free rate, T = time to maturity, N(·) = standard normal CDF.

C = S_0 \cdot N(d_1) - K \cdot e^{-rT} \cdot N(d_2)

Black-Scholes d₁ Parameter

σ = volatility. Then d₂ = d₁ − σ√T̅.

d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2) \cdot T}{\sigma \sqrt{T}}

Put-Call Parity

Relationship between European call and put prices with the same strike and expiry.

C + K \cdot e^{-rT} = P + S_0

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