9.7 Option strategies: payoffs, Greeks and exact conditions

Two exact conditions apply to every position

• Unhedged, held to expiry: the position makes money iff its payoff exceeds the initial cost compounded: $\Pi(S_T)>V_0e^{rT}$. Only the terminal distribution matters.
• Delta-hedged at implied vol $\sigma_i$: by (9.11) summed over legs, the P&L is $\tfrac12\int e^{-rt}\big(\sum_j q_j\Gamma_j\big)S^2(\sigma_t^2-\sigma_i^2)dt$ plus the vega P&L from changes in implied vol, $\sum_jq_j\mathcal V_j\,d\sigma_{i,j}$. The position makes money iff realised variance, weighted by the net dollar gamma along the path, exceeds implied (for net long gamma), plus whatever implied vol moves contribute.

Under $\mathbb Q$ every position has zero expected excess return. A positive expectation needs $\mathbb P$ to differ from $\mathbb Q$: a directional view, or a variance premium.

9.7.1Vertical spreads#

Bull call spread $K_1<K_2$: long call $K_1$, short call $K_2$. Payoff $\min\{(S_T-K_1)^+,K_2-K_1\}$. Max profit $K_2-K_1-De^{rT}$, max loss $De^{rT}$ (debit $D$), breakeven $K_1+De^{rT}$.

Greeks: delta $>0$; gamma positive near $K_1$, negative near $K_2$; vega likewise changes sign. The spread is long vol below the midpoint and short vol above it.

Example: 25,000/25,300 call spread, 7 days. Debit ₹7,651 per lot (117.7 points). Max profit $300-117.8=182.2$ points (₹11,840 per lot). Delta +16.3 units per lot (0.25 of a lot), gamma +0.0090, vega +14,088 (₹141 per vol point).

Exact condition held to expiry: $S_T>K_1+De^{rT}=25{,}117.8$. Bear put spreads mirror this.

9.7.2Straddles and strangles#

Long straddle at $K$: long call and put. Payoff $|S_T-K|$. Long gamma, long vega, short theta.

• Held to expiry, profit iff $|S_T-K|>(C+P)e^{rT}$. The 7-day ATM straddle costs 359.6 points, so Nifty must finish more than about 1.44% from 25,000.
• Delta-hedged: profit iff the dollar-gamma-weighted realised variance exceeds implied. The two conditions differ. A straddle can expire worthless at the strike after a week of violent two-way moves, and a hedged holder would have made money; an unhedged one loses the premium.

Short strangle $K_p<K_c$: short put $K_p$, short call $K_c$. Example 24,500/25,500, 7 days: credit 61.02 points (₹3,967 per lot). Per lot: delta −2.35 units, gamma −0.0622, vega −96,937 (−₹969 per vol point), theta +331,836 per year (≈ +₹909 per calendar day). Breakevens 24,438.9 and 25,561.1 (short strikes ∓ compounded credit). Max profit is the credit; the loss is unbounded on both sides.

Exact condition held to expiry: $K_p-Ce^{rT}<S_T<K_c+Ce^{rT}$. Delta-hedged: implied must exceed gamma-weighted realised.

9.7.3Iron condor#

A short strangle with protective wings: long put $K_1$, short put $K_2$, short call $K_3$, long call $K_4$, $K_1<K_2\le K_3<K_4$. Max profit = credit; max loss = wing width $-$ credit (for equal wings); breakevens $K_2-Ce^{rT}$ and $K_3+Ce^{rT}$.

Example 24,300/24,500/25,500/25,700: credit 37.37 points (₹2,429 per lot). Max loss $(200-37.42)\times65=\text{₹}10{,}568$ per lot. Breakevens 24,462.6 and 25,537.4. Per lot: gamma −0.0276, vega −43,060 (−₹431 per vol point), theta ≈ +₹403 per day.

Under the pricing measure the probability of finishing between the breakevens is 76.7%, and the probability of the maximum loss is 12.1% (Problem 10). A 77% win rate is what you pay for; it is not an edge. The wings cut tail loss and margin, at the cost of giving up part of the variance premium.

9.7.4Calendar spreads#

Long call calendar at $K$: short near-expiry call, long far-expiry call. Example: short 7-day, long 35-day 25,000 calls: debit 281.9 points (₹18,325 per lot). Per lot: vega +108,558 (₹1,086 per vol point), gamma −0.0320, theta +₹457 per day.

The calendar is long vega (the far option has more vega) and short gamma (the near option has more gamma). It makes money if spot stays near $K$ through the near expiry, so the short leg decays, or if far-dated implied vol rises. The exact condition at the near expiry $T_1$: $C^{far}(S_{T_1},\sigma_i^{far}(T_1))-(S_{T_1}-K)^+>De^{rT_1}$. The far leg's value at $T_1$ depends on its implied vol then, so a calendar is a bet on the term structure as well as on realised vol. SEBI removed the calendar-spread margin benefit on expiry day (effective 1 Feb 2025; 9.10), so on a Tuesday the short near leg is margined on its own.

9.7.5Ratio spreads#

1×2 call ratio spread: long one call $K_1$, short two calls $K_2>K_1$. Example 25,000/25,300: debit 41.2 points per unit. Breakevens 25,041.3 and 25,558.7. Max profit at $S_T=K_2$: $300-41.2e^{rT}\approx258.7$ points. Above the upper breakeven the loss grows one-for-one with no limit. Per lot: delta −1.71, gamma −0.0394, vega −61,362: short vol.

It makes money if the market rises moderately and stops, which is a view on the distribution's shape, not just direction. The backspread (short one, long two) is the mirror image: long vol, profits from a large move up.

9.7.6Covered calls#

Long the underlying, short a call $K$: payoff $\min(S_T,K)$, which by put–call parity equals a short put plus a bond paying $K$. The covered call is a short-vol, negatively skewed position. It underperforms the underlying iff $S_T>K+Ce^{rT}$, and it is not "safer" than holding stock except by the premium amount. Index buy-write products earn a variance premium plus equity beta; they lag the index in strong rallies.

9.7.7Summary table#

Structure	Delta	Gamma	Vega	Theta	Makes money (held to expiry)	Makes money (delta-hedged)
Bull call spread	+	±	±	±	$S_T>K_1+De^{rT}$	net-gamma-weighted $\sigma_r^2-\sigma_i^2$, sign depends on spot
Long straddle	≈0	+	+	−	$\lvert S_T-K\rvert>(C+P)e^{rT}$	gamma-weighted $\sigma_r^2>\sigma_i^2$
Short strangle	≈0	−	−	+	$K_p-Ce^{rT}<S_T<K_c+Ce^{rT}$	gamma-weighted $\sigma_r^2<\sigma_i^2$
Short iron condor	≈0	−	−	+	$K_2-Ce^{rT}<S_T<K_3+Ce^{rT}$	as short strangle, tails capped
Long calendar	≈0	−	+	+	spot near $K$ at $T_1$, far IV up	realised low near $T_1$, far IV up
1×2 call ratio	±	− near $K_2$	−	+	$S_T$ between breakevens	short vol around $K_2$
Covered call	+	−	−	+	vs. stock: $S_T<K+Ce^{rT}$	short vol, long beta