Understand short straddle and strangle

What are we building?

Derivation from first principles

We build the payoff by summing the payoffs of the individual short legs at expiry. Let $S_T$ = stock price at expiry.

Step 1 — A short call payoff. When you sell a call, at expiry you owe the buyer $\max(S_T - K, 0)$ and you kept the premium $C$. $\text{Short Call} = C - \max(S_T - K, 0)$ Why this step? A call buyer only exercises when $S_T > K$; below that, the option is worthless and you keep all of $C$.

Step 2 — A short put payoff. $\text{Short Put} = P - \max(K - S_T, 0)$ Why this step? A put buyer exercises only when $S_T < K$ (they sell to you at $K$ above market).

Step 3 — Add them (Straddle, same strike $K$). $\Pi = (C + P) - \max(S_T-K,0) - \max(K-S_T,0)$ Since only one of the two max-terms is nonzero at once, this simplifies to: $\boxed{\Pi = (C+P) - |S_T - K|}$ Why this step? $\max(S_T-K,0)+\max(K-S_T,0) = |S_T-K|$ — exactly one side is "in the money." This is the cleanest form: profit = premium minus how far the price wandered from $K$.

Step 4 — Breakevens (Straddle). Set $\Pi = 0$: $|S_T - K| = C+P \implies \boxed{S_T = K \pm (C+P)}$ Why this step? You break even when the loss from the move exactly eats your collected premium.

Step 4b — Loss is symmetric only in formula, not in magnitude. Upside: $S_T$ can rise without limit, so $\Pi = (C+P)-(S_T-K)$ → $-\infty$. Unlimited. Downside: $S_T$ cannot go below $0$, so the worst case is $S_T = 0$: $\Pi_{\min}^{\text{down}} = (C+P) - K \implies \boxed{\text{max downside loss} = K - (C+P)}$ Why this step? A stock price is floored at zero, which caps how far the short put can hurt you.

Step 5 — Strangle version. With $K_p < K_c$: $\Pi = (C+P) - \max(S_T-K_c,0) - \max(K_p-S_T,0)$

• Between $K_p$ and $K_c$: both max terms are 0 → $\Pi = C+P$ (max profit, flat).
• Breakevens: $\boxed{S_T = K_p - (C+P)}$ and $\boxed{S_T = K_c + (C+P)}$.
• Same asymmetry: upside loss unlimited; downside capped at $K_p - (C+P)$ (at $S_T=0$).

Worked Examples

Common Mistakes

Flashcards

Connections

• Long Straddle and Strangle — the mirror image: buyers betting on a BIG move.
• Implied Volatility — you sell these when IV is high (overpriced fear).
• Option Greeks - Vega — these positions are short vega.
• Option Greeks - Theta — short options gain from time decay (theta positive).
• Iron Condor and Iron Butterfly — the "defined-risk" versions with wings bought to cap losses.
• Put-Call Parity — underpins how $C$ and $P$ relate at each strike.

Concept Map

Hinglish (regional understanding)