Options & Hedging: Black-Scholes and the Greeks

An option costs a fraction of the underlying asset but can return multiples — or expire worthless. Black-Scholes derived the first closed-form price for European options in 1973, winning a Nobel Prize. The "Greeks" measure exactly how the price changes with market conditions, and delta hedging uses them to build theoretically riskless positions.

1. Call and Put Options

An option is a contract granting the right but not obligation to buy or sell an asset at a fixed price (strike price K) on or before an expiry date.

Call option: Right to buy the underlying at K. Profitable when asset price S > K at expiry.
Put option: Right to sell the underlying at K. Profitable when S < K at expiry.
American option: Can be exercised any time before expiry.
European option: Only exercisable at expiry. Simpler mathematically — Black-Scholes prices these exactly.

The price paid for the option contract is the premium. If the option expires worthless, the buyer loses the premium; the seller keeps it. The seller ("writer") of a call faces theoretically unlimited liability if the stock rises — they must deliver shares at K regardless of current price S.

2. Payoff Diagrams

At expiry, the payoff (before accounting for premium paid) is:

Call payoff = max(S_T - K, 0) Put payoff = max(K - S_T, 0) Profit (call buyer) = max(S_T - K, 0) - C (C = premium paid) Profit (put buyer) = max(K - S_T, 0) - P

The call buyer profits if S_T > K + C (stock must rise above strike plus the premium). The put buyer profits if S_T < K − P. Between these bounds, the option expires worthless and the premium is lost.

The payoff diagram is the hockey-stick shape: flat at −premium for S < K (call loses premium), rising linearly for S > K.

3. What Determines Option Price

Option value has two components:

Intrinsic value: max(S − K, 0) for calls. The immediate exercise value.
Time value (extrinsic): Premium above intrinsic. Reflects uncertainty — more time means more chance the stock moves favorably. Decays as expiry approaches (theta decay).

Six variables determine option price in the Black-Scholes model:

S — current stock price (↑S → call price ↑, put price ↓)
K — strike price (↑K → call ↓, put ↑)
T — time to expiry in years (↑T → both call and put ↑)
σ — volatility (annualized standard deviation of log-returns) (↑σ → both ↑)
r — risk-free interest rate (↑r → call ↑, put ↓)
q — dividend yield (↑q → call ↓, put ↑)

4. The Black-Scholes Formula

Fischer Black, Myron Scholes, and Robert Merton (1973) derived the option price by constructing a continuously-rebalanced delta hedge that eliminates all risk. Under geometric Brownian motion for stock price and no-arbitrage, the call price is:

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 N(x) = standard normal CDF (probability that Z ≤ x) Put price by put-call parity: P = C − S·e^{−qT} + K·e^{−rT}

Interpretation: N(d₂) is the risk-neutral probability that the call finishes in-the-money. N(d₁) is the delta — the hedge ratio. The formula decomposes the call value into the discounted expected stock receipt (S·N(d₁)) minus the discounted expected payment of the strike (K·e^{−rT}·N(d₂)).

Nobel Prize: Scholes and Merton received the 1997 Nobel Memorial Prize in Economics. Fischer Black died in 1995; the prize is not awarded posthumously. Merton's continuous-time hedging argument is mathematically deeper than the original derivation.

5. The Greeks

Delta

∂C/∂S. Rate of change of option price with stock price. Call: 0 to 1. Put: −1 to 0. At-the-money: ≈ 0.5. How many shares needed to delta-hedge.

Gamma

∂²C/∂S². Rate of change of delta. Highest near ATM, near expiry. Measures delta hedge rebalancing frequency needed. Long gamma = benefits from large moves.

Theta

∂C/∂t. Time decay. Almost always negative for buyers. ATM option loses value roughly ∝ √T as expiry approaches. Accelerates in final weeks.

Vega

∂C/∂σ. Sensitivity to volatility. Long options are always long vega (benefit from rising volatility). Maximum near ATM, longer expiry.

Rho

∂C/∂r. Sensitivity to interest rate. Typically small effect. Calls have positive rho (benefit from higher rates); puts negative. Important for LEAPS (long-dated options).

6. Delta Hedging

A delta hedge creates a position with zero first-order sensitivity to stock price changes. If you sell a call with Δ = 0.6 on 100 shares, you buy 60 shares of the underlying (or 0.6 shares per option):

Portfolio = -1 call + Δ shares \partialPortfolio/\partialS = -\partialC/\partialS + Δ = -Δ + Δ = 0

The hedge must be rebalanced continuously as Δ changes with S. This dynamic hedging is what makes Black-Scholes work: the hedging cost over time equals the fair option premium. In practice, traders rebalance discretely (daily or when Δ drifts beyond a threshold) — gamma measures the P&L from discrete rebalancing.

A "delta-neutral" book has zero Δ exposure but may still be exposed to gamma, vega, and theta. Large investment banks run complex optimization over all Greeks across millions of options positions.

7. Common Option Strategies

Covered call: Own stock + sell call. Limits upside; collects premium. Lowers cost basis.
Protective put: Own stock + buy put. Acts as insurance — limits downside to K − S₀ + P. Costs premium.
Straddle: Buy call and put at same K, T. Profits from large moves in either direction. Costs 2× premium (long volatility).
Strangle: Buy OTM call and OTM put. Cheaper than straddle; requires larger move to profit.
Iron condor: Sell strangle (short volatility) + buy wider strangle (cap losses). Profits from range-bound market. Net credit, non-directional.
Vertical spread (bull/bear): Buy one option + sell another at different K. Limits cost and max profit. Well-defined risk profile.