Understand short straddle and strangle

5.4.7 · Stock-Market › Options Strategies

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First principles se derivation

Hum payoff ko individual short legs ke payoffs ko expiry par sum karke banate hain. Maano $S_T$ = expiry par stock price.

Step 1 — Short call payoff. Jab tum ek call sell karte ho, expiry par tumhe buyer ko $\max (S_T-K,0)$ dena hota hai aur tumne premium $C$ rakha hua hai. $\text{Short Call}=C-\max (S_T-K,0)$ Ye step kyun? Call buyer sirf tab exercise karta hai jab $S_T>K$; usse neeche, option worthless hai aur tum poora $C$ rakhte ho.

Step 2 — Short put payoff. $\text{Short Put}=P-\max (K-S_T,0)$ Ye step kyun? Put buyer sirf tab exercise karta hai jab $S_T<K$ (woh tumhe $K$ par market se upar sell karta hai).

Step 3 — Dono ko add karo (Straddle, same strike $K$). $\Pi =(C+P)-\max (S_T-K,0)-\max (K-S_T,0)$ Kyunki ek waqt mein sirf ek hi max-term nonzero hoti hai, yeh simplify hota hai: $\boxed{\Pi =(C+P)-|S_T-K|}$ Ye step kyun? $\max (S_T-K,0)+\max (K-S_T,0)=|S_T-K|$ — exactly ek side "in the money" hoti hai. Yeh sabse clean form hai: profit = premium minus kitni door price $K$ se bhatki.

Step 4 — Breakevens (Straddle). $\Pi =0$ set karo: $|S_T-K|=C+P⟹\boxed{S_T=K\pm (C+P)}$ Ye step kyun? Tum break even karte ho jab move se loss exactly tumhara collected premium kha jaata hai.

Step 4b — Loss formula mein symmetric hai, lekin magnitude mein nahi. Upside: $S_T$ bina limit ke badh sakti hai, isliye $\Pi =(C+P)-(S_T-K)$ → $-\infty$. Unlimited. Downside: $S_T$ zero se neeche nahi ja sakti, isliye worst case $S_T=0$ hai: ${\Pi }_{\min }^{\text{down}}=(C+P)-K⟹\boxed{\text{max downside loss}=K-(C+P)}$ Ye step kyun? Stock price zero par floored hoti hai, jo short put ki kitni takleef de sakti hai isko cap karti hai.

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

• $K_p$ aur $K_c$ ke beech: dono max terms 0 hain → $\Pi =C+P$ (max profit, flat).
• Breakevens: $\boxed{S_T=K_p-(C+P)}$ aur $\boxed{S_T=K_c+(C+P)}$.
• Wahi asymmetry: upside loss unlimited; downside $K_p-(C+P)$ par capped ($S_T=0$ par).

Worked Examples

Common Mistakes

Flashcards

Connections

• Long Straddle and Strangle — mirror image: buyers jo BIG move par bet lagaate hain.
• Implied Volatility — tum inhe tab sell karte ho jab IV high ho (overpriced fear).
• Option Greeks - Vega — ye positions short vega hain.
• Option Greeks - Theta — short options time decay se gain karte hain (theta positive).
• Iron Condor and Iron Butterfly — losses cap karne ke liye wings kharid ke "defined-risk" versions.
• Put-Call Parity — yeh underpin karta hai ki $C$ aur $P$ har strike par kaise relate karte hain.

Concept Map