Understand ratio spreads

WHY does this strategy exist?

• WHAT you believe: stock drifts up a little, lands near the short strike, then goes no further.
• WHY it feels smart: you get paid to be bullish in a narrow zone.
• HOW you pay for it: with the risk of being wrong in the strong direction.

Building it from first principles

Let's derive the payoff of the canonical 1×2 call ratio spread with strikes $K_1 < K_2$.

Ingredients (all per share, expiry $T$):

• Long 1 call at $K_1$, premium $c_1$ paid.
• Short 2 calls at $K_2$, premium $c_2$ each received.

Net premium (positive = credit received): $P = 2c_2 - c_1$

Why this step? Every option's cost is a cash flow now; we net them. Selling two brings in $2c_2$, buying one costs $c_1$.

Payoff of a call at expiry for underlying $S$: $\text{long call}(K) = \max(S-K,0), \qquad \text{short call}(K) = -\max(S-K,0)$

Combine — total value at expiry: $V(S) = \underbrace{\max(S-K_1,0)}_{\text{long}} - 2\underbrace{\max(S-K_2,0)}_{\text{two shorts}}$

Total profit = value at expiry + net premium: $\boxed{\;\Pi(S) = \max(S-K_1,0) - 2\max(S-K_2,0) + P\;}$

Why add $P$? $P$ is the cash you already banked (or paid). Profit = what the position is worth at expiry PLUS what you pocketed setting it up.

Reading the profit in three zones

Why these pieces? Each $\max$ "switches on" at its strike. Below $K_1$ nothing is intrinsic → just keep $P$. Between strikes only the long is in-the-money → rises 1-for-1. Above $K_2$ two shorts kick in, so slope becomes $1-2=-1$ → you lose money as $S$ climbs.

Key points (derive them, don't memorize):

• Max profit occurs exactly at $S=K_2$ (the peak of the tent): $\Pi_{\max} = (K_2-K_1)+P$ Why? Long call fully in-money, shorts still worthless.

• Upper breakeven — set the $S>K_2$ piece to 0: $-S+(2K_2-K_1)+P = 0 \;\Rightarrow\; S_{BE}^{up} = 2K_2 - K_1 + P$ Why? Above here the naked short overwhelms the profit.

• Lower behavior: if $P>0$ (credit), you keep $P$ even if the stock collapses — there is no downside loss. If $P<0$ (debit), your loss below $K_1$ is just $|P|$.

• Uncapped loss as $S\to\infty$: slope $-1$, so $\Pi\to-\infty$. This is the naked-short danger.

Worked examples

Common mistakes

Active recall

Concept Map

Hinglish (regional understanding)