Finance-Theory · Unit 12 · Video 5 · Interactive Practice

Call Spreads, Butterflies and the VIX

IKey Formulas

Formula	Name	What it says
$\text{Max}[0,\, S_T - 50] - \text{Max}[0,\, S_T - 60]$	50/60 call spread payoff	Zero below 50, rises one for one, capped at $60 - 50 = 10$
$C(50) - C(60) \lt C(50)$	Cost of the call spread	The premium received on the written 60 call lowers the cost
$\text{Max}[0,\, S_T - 50] - 2\,\text{Max}[0,\, S_T - 55] + \text{Max}[0,\, S_T - 60]$	50/55/60 butterfly payoff	A tent: peak 5 at $S_T = 55$ , never negative
$C_{\text{market}} = C(S_t, K, \tau, r, \sigma)$ , solved for $\sigma$	Implied volatility	Every input but $\sigma$ is observed; price rises with $\sigma$ , so one price gives one $\sigma$

Key Insight: Each option adds one kink at its strike, where the slope rises by one per option bought and falls by one per option written. Enough kinks at the right strikes approximate any payoff.

IISpreads, Leg by Leg

A spread's payoff is the sum of its legs, so it can be checked one stock price at a time.

Position S_T 55

IIISpanning: Kinks Can Follow Any Curve

Each option adds one kink at its strike; with enough strikes the straight pieces follow any payoff curve.

Strikes 4

IVImplied Volatility: Running the Model Backward

Option price rises with volatility, so each observed market price pins down exactly one implied volatility.

Market price 2.60

💡 The VIX applies this inversion to S&P 500 puts and calls across a wide range of strikes, reporting the market's 30-day expected volatility as an annualized standard deviation: it looks forward, while volatility computed from past returns looks back.

VQuiz Questions

Problem 1 · Call Spread Payoff

Given: You buy a call with strike $K = 40$ and write a call with strike $K = 50$ on the same stock and maturity. At maturity $S_T = 58$ . Find the payoff of the position.

✅ Correct! Above the upper strike the spread is capped at

50 - 40 = 10

❌ That is only the bought 40 call. You also owe

58 - 50 = 8

on the call you wrote.

❌ That is what you owe on the written call, not the payoff of the whole position.

❌ Check the sign of the written leg. A written call's payoff enters with a minus sign: you pay it, you do not receive it.

❌ Not quite. Evaluate each leg at

S_T = 58

and combine them.

Show solution

\text{Payoff} = \text{Max}[0,\, S_T - 40] - \text{Max}[0,\, S_T - 50]

= \text{Max}[0,\, 58 - 40] - \text{Max}[0,\, 58 - 50] = 18 - 8 = 10

Above 50 every extra dollar earned on the bought 40 call is paid away on the written 50 call, so the payoff stays at its maximum $50 - 40 = 10$ however high the stock goes.

Problem 2 · Cheaper, but Capped

Given: A call with strike 70 costs $5.40 and a call with strike 85 costs $1.90 (same stock and maturity). You buy the 70 call and write the 85 call. Net profit = payoff − cost today; ignore interest on the premiums.

Net profit of the call spread at $S_T = 80$ ?

Above which stock price does a bare 70 call earn a larger net profit than the spread?

✅ Correct! The spread costs $3.50 instead of $5.40; the price of that saving is every dollar of profit above 85, which only wins for the bare call once the stock passes 85 + 1.90 = 86.90.

❌ That is the payoff, not the profit. Shift the payoff down by the cost of the position.

❌ That subtracts the full 70-call premium. The $1.90 received for writing the 85 call lowers the cost to $3.50.

❌ The 85-call premium is received, not paid. You subtracted both premiums, 10 − 5.40 − 1.90. Writing an option brings cash in today.

❌ At 85 both positions have the same payoff (15), but the bare call cost $1.90 more, so it is still behind there.

❌ The gap to close is the 85-call premium ($1.90), not the spread's cost of $3.50.

❌ $75.40 is where the bare call breaks even (70 + 5.40), not where it overtakes the spread.

❌ Check the net profit. Payoff at 80 is 10; cost of the spread is 5.40 − 1.90.

❌ Check the crossover. Compare the spread's flat profit above 85 with the bare call's rising profit.

Show solution

Cost of the spread: pay $5.40, receive $1.90, so the net cost is $3.50.

At $S_T = 80$ : payoff $= \text{Max}[0,\, 80 - 70] - \text{Max}[0,\, 80 - 85] = 10 - 0 = 10$ , so net profit is 10 − 3.50 = $6.50. The bare 70 call would earn only 10 − 5.40 = $4.60 here.

Crossover: above 85 the spread's profit is flat at 15 − 3.50 = 11.50. The bare call's profit is $S_T - 70 - 5.4$ , which exceeds 11.50 when $S_T > 86.9$ , i.e. above $86.90. That is 85 plus the premium collected on the 85 call.

Problem 3 · A Butterfly from Its Kinks

Given: A payoff built from calls on one stock is 0 below 40, rises with slope 1 to 5 at $S_T = 45$ , falls with slope $-1$ to 0 at $S_T = 50$ , and is 0 above 50. It is bought for a premium of $1.20.

What position sits at the strike 45?

Payoff at $S_T = 48$ ?

Largest possible net profit?

✅ Correct! Buy a 40 call, write two 45 calls, buy a 50 call: the tent pays in a calm market, and since its payoff is never negative the most it can lose is the $1.20 premium.

❌ Writing one call only flattens the slope from

+1

0

. To reach

-1

the slope must fall by 2.

❌ The slope falls at 45, and a fall in slope means options written, not bought.

❌ That counts only the bought 40 call. The two written 45 calls each cost

48 - 45 = 3

❌ Two calls are written at 45, so subtract

2 \times 3

, not

3

❌ The written calls enter with a minus sign. You owe

2 \times 3 = 6

on them, so subtract it:

8 - 6 = 2

❌ That is the peak payoff. Net profit subtracts the $1.20 premium.

❌ The premium was paid, not received. Subtract the $1.20 from the peak payoff of 5.

❌ The $1.20 premium is paid once, for the whole position: 5 − 1.20.

❌ Check the slope change at 45. It goes from

+1

-1

❌ Check the payoff at 48. Evaluate every leg, with the written calls counted twice.

❌ Check the net profit. Take the peak payoff and subtract the premium once.

Show solution

Strikes from the kinks. At 40 the slope rises $0 \to 1$ : buy 1 call. At 45 it falls $1 \to -1$ : write 2 calls. At 50 it rises $-1 \to 0$ : buy 1 call.

\text{Payoff} = \text{Max}[0,\, S_T - 40] - 2\,\text{Max}[0,\, S_T - 45] + \text{Max}[0,\, S_T - 50]

At $S_T = 48$ : $8 - 2 \times 3 + 0 = 2$ (on the falling side, $50 - 48 = 2$ ✓).

Largest net profit: the peak payoff is 5 at $S_T = 45$ , so 5 − 1.20 = $3.80. Below 40 or above 50 the payoff is 0, so the worst outcome is losing the $1.20 premium.

Problem 4 · Reading Option Prices for Volatility

Given: A call's market price rises from $4.10 to $4.60 while the stock price $S_t$ , strike $K$ , time to maturity $\tau$ and interest rate $r$ are unchanged. Separately, in late October 2008 the VIX stood at about $71\%$ , against a historical S&P 500 volatility of roughly $15\%$ to $20\%$ .

What happened to the call's implied volatility?

What did the 71% VIX reading say?

✅ Correct! Price rises with volatility, so a higher price with every other input fixed means a higher implied volatility; the VIX collects that forward-looking reading for S&P 500 options.

❌ Backwards. A higher volatility always means a higher option price, so a higher price cannot come from a lower

\sigma

❌ Something had to change. With

S_t

K

\tau

r

fixed,

\sigma

is the only input left to explain the higher price.

❌ The direction needs no formula. Because price is increasing in

\sigma

, a higher price means a higher

\sigma

; only the exact number needs the model.

❌ That describes historical volatility, which looks back. The VIX is implied from option prices and looks forward.

❌ Volatility is a standard deviation, not a forecast of direction: it measures how much the index is expected to move, up or down.

❌ One-year implied volatility was much lower, because the market expected volatility to decline over the year.

❌ Check the direction. Option price is increasing in

\sigma

❌ Check what the VIX measures.

Show solution

Implied volatility. In a pricing model $C = C(S_t, K, \tau, r, \sigma)$ , and a higher $\sigma$ always means a higher price. With $S_t$ , $K$ , $\tau$ , $r$ unchanged, the only way the model can match a market price that rose from $4.10 to $4.60 is with a larger $\sigma$ : implied volatility rose.

The VIX. It is the 30-day expected volatility of the S&P 500, as an annualized standard deviation, implied by S&P 500 option prices. A reading of 71% compares with the historical range as

\frac{71}{20} = 3.55, \qquad \frac{71}{15} \approx 4.73

so the market was bracing for far more turbulence than usual over the coming month. It says nothing about direction, and one-year implied volatility was much lower.

Solved: 0 / 4