Options Pricing: Black-Scholes and the Greeks

Calls, puts and payoff

An option grants the right but not obligation to buy (call) or sell (put) an asset at a fixed strike price K on or before expiry. A European option is only exercisable at expiry, which is exactly what Black-Scholes prices. At expiry, call payoff = max(S_T − K, 0) and put payoff = max(K − S_T, 0) — the classic hockey-stick diagram, flat at −premium below the strike and rising linearly above it. Option value splits into intrinsic value (immediate exercise value) and time value, which decays as expiry approaches — theta decay.

The Black-Scholes formula

Fischer Black, Myron Scholes and Robert Merton (1973) derived option price by constructing a continuously-rebalanced delta hedge that eliminates all risk, under geometric Brownian motion and no-arbitrage. N(d₂) is the risk-neutral probability the call finishes in-the-money; N(d₁) is delta, the hedge ratio. Six inputs — spot price S, strike K, time to expiry T, volatility σ, risk-free rate r, and dividend yield q — fully determine the price. Scholes and Merton received the 1997 Nobel Memorial Prize in Economics for the formula; Fischer Black had died in 1995 and the prize is never awarded posthumously.

C = S·e^(−qT)·N(d₁) − K·e^(−rT)·N(d₂)
d₁ = [ln(S/K) + (r − q + σ²/2)·T] / (σ·√T)
d₂ = d₁ − σ·√T
Put via put-call parity: P = C − S·e^(−qT) + K·e^(−rT)

The Greeks

Delta (∂C/∂S, 0 to 1 for calls) is the hedge ratio — shares needed per option to stay delta-neutral. Gamma (∂²C/∂S²) is delta's rate of change, highest near-the-money close to expiry, and measures how often a hedge needs rebalancing. Theta (∂C/∂t) is time decay, almost always negative for buyers and accelerating in the final weeks. Vega (∂C/∂σ) measures sensitivity to volatility — long options are always long vega. Rho (∂C/∂r) measures interest-rate sensitivity, typically small except for long-dated LEAPS.

Delta hedging and common strategies

A delta hedge creates zero first-order sensitivity to the stock price: sell a call with delta 0.6, buy 0.6 shares per option, and the combined portfolio delta is zero. Because delta drifts as the stock moves, the hedge is rebalanced continuously in theory — that ongoing rebalancing cost is exactly the fair option premium. Large banks run "delta-neutral" books that are still exposed to gamma, vega and theta simultaneously. Common structures: a covered call (own stock + sell call) caps upside for premium income; a protective put acts as insurance against downside; a straddle (buy call + put at the same strike) profits from large moves in either direction; an iron condor profits from a range-bound market.

Frequently asked questions

What do delta and gamma actually measure?

Delta (∂C/∂S) is the rate of change of the option price with the stock price, ranging 0 to 1 for a call — it also tells you the hedge ratio, how many shares to hold per option for a delta-neutral position. Gamma (∂²C/∂S²) is the rate of change of delta itself, peaking near-the-money close to expiry, and measures how often a delta hedge needs rebalancing.

What is delta hedging?

Delta hedging builds a position with zero first-order sensitivity to the stock price: if you sell a call with delta 0.6, you buy 0.6 shares per option so the portfolio's combined delta is zero. Because delta drifts as the stock price moves, the hedge must be rebalanced continuously in theory, or discretely in practice — the cost of that rebalancing over time equals the fair option premium.

Who won the Nobel Prize for Black-Scholes?

Myron Scholes and Robert Merton received the 1997 Nobel Memorial Prize in Economics for the option-pricing formula. Fischer Black, the formula's co-author, had died in 1995, and the Nobel Prize is not awarded posthumously, so his name does not appear on the award despite being central to the original 1973 derivation.

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