Section 25.1

Payoffs and Strategies

Payoffs and P&L of option strategies (spreads, straddles, covered calls), breakevens, expected payoffs under simple distributions.

Toolkit

Static option payoffs are piecewise-linear in the underlying price $S_T$ at expiry; build any strategy’s payoff by summing the payoffs of its legs, with long call $(S_T-K)^+$ , long put $(K-S_T)^+$ , short positions negated, and P&L equal to payoff minus net premium paid (plus stock gain/loss if shares are held). Key facts:

Covered call = long stock + short call: payoff capped at $K$ once the stock is called away; profit $=\min(S_T,K)-S_0+c$ .
Spreads (bull/bear call or put) have payoff bounded by the strike width, with breakeven where payoff equals net premium.
Straddles/strangles sum call and put payoffs; breakevens lie symmetrically around the strikes shifted by total premium.
Fair/expected value of an option is $\mathbb{E}[(X-K)^+]=\int_K^\infty (x-K)\,f(x)\,dx = \mathbb{P}(X>K)\,\mathbb{E}[X-K\mid X>K]$ , computed directly from the terminal distribution of $X$ .

Problems