Finance-Theory · Unit 12 · Video 4 · Interactive Practice

Writing Options and the Straddle: A Bet on Volatility

IKey Formulas

Profit at expiration	Position	Break-even and extremes
$C - \text{Max}[0,\ S_T - K]$	Written (short) call	$S_T = K + C$ ; best case $+C$ ; loss unbounded as $S_T$ rises
$P - \text{Max}[0,\ K - S_T]$	Written (short) put	$S_T = K - P$ ; best case $+P$ ; worst case at $S_T = 0$ is $-(K - P)$
$\lvert S_T - K\rvert - (C + P)$	Long straddle: call $+$ put, same strike	$S_T = K \pm (C + P)$ ; worst case $-(C + P)$ at $S_T = K$
$(C + P) - \lvert S_T - K\rvert$	Written straddle	$S_T = K \pm (C + P)$ ; best case $C + P$ at $S_T = K$ ; worst downside at $S_T = 0$ is $-(K - C - P)$

Key Insight: An option is a contract between two parties, so at expiration the holder's payoff and the writer's payoff sum to zero. A straddle pays $\text{Max}[0,\ S_T - K] + \text{Max}[0,\ K - S_T] = \lvert S_T - K\rvert$ , the size of the move whatever its direction, and it loses only inside a band $2(C + P)$ wide: the premiums decide how large a move the bet needs.

IIFour Positions, One Strike

Each written position is its long position turned upside down: whatever the holder gains, the writer loses.

Position Premium (illustrative) C = 4 S_T 60

Long call (pays $C$ )	if $S_T < K$	if $S_T = K$	if $S_T > K$
Payoff	$0$	$0$	$S_T - K$
Profit	$-C$	$-C$	$S_T - K - C$

Long put (pays $P$ )	if $S_T < K$	if $S_T = K$	if $S_T > K$
Payoff	$K - S_T$	$0$	$0$
Profit	$K - S_T - P$	$-P$	$-P$

Short call (receives $C$ )	if $S_T < K$	if $S_T = K$	if $S_T > K$
Payoff	$0$	$0$	$-(S_T - K)$
Profit	$C$	$C$	$C - (S_T - K)$

Short put (receives $P$ )	if $S_T < K$	if $S_T = K$	if $S_T > K$
Payoff	$-(K - S_T)$	$0$	$0$
Profit	$P - (K - S_T)$	$P$	$P$

IIIThe Straddle: Bought or Written

A call plus a put at one strike pays $\lvert S_T - K\rvert$ ; the premiums $C + P$ set how far the stock must move.

Side C (illustrative) 4 P (illustrative) 3 S_T 70

💡 In the autumn of 2008 volatility was extraordinarily high, so both premiums were large and the band was wide: a straddle bought then paid only if markets became more volatile still. That price was set where supply equals demand, so neither side had a free lunch.

IVFind the Break-Evens

Where does each position's profit cross zero?

Position Guess 50 Second guess 80

VQuiz Questions

Problem 1 · The Writer's Side of a Call

Given: you write a call with strike $K = 80$ and receive a premium of $6 per share. At expiration the stock is at $95. Find your profit per share.

✅ Correct! The holder exercises and collects 15 from you; the premium of 6 you received up front covers part of it, leaving a loss of $9.

❌ Not quite. That is the holder's profit. The writer's payoff is the mirror image,

-\text{Max}[0,\ S_T - K]

, so the holder's gain is your loss.

❌ Not quite. That is your payoff, not your profit. You were paid the premium up front, and it offsets part of what you owe.

❌ Not quite. The premium was received by the writer, so it is added to the payoff, not subtracted from it.

❌ Not quite. Use the writer's profit

C - \text{Max}[0,\ S_T - K]

Show solution

The stock finished above the strike, so the holder exercises. Per share, the writer's payoff and profit are

-\text{Max}[0,\ 95 - 80] = -15, \qquad C - \text{Max}[0,\ S_T - K] = 6 - 15 = -9.

A loss of $9 per share. The writer's break-even is $K + C = 80 + 6 = 86$ : above $86 every extra dollar on the stock is a dollar lost.

Problem 2 · The Call Paid Off, So the Straddle Did?

Given: you buy a straddle at $K = 70$ , paying $5 for the call and $3 for the put. At expiration the stock is at $76, so the call finishes in the money. Find your profit per share.

✅ Correct! A move of 6 is smaller than the total premium of 8, so 76 lies inside the loss band from 62 to 78.

❌ Not quite. The put expired worthless, but you still paid $3 for it. Both premiums come off: the cost is

C + P

, not

C

❌ Not quite. Only one leg pays at expiration. Above the strike the put pays

\text{Max}[0,\ 70 - 76] = 0

, so the payoff is

\lvert 76 - 70\rvert = 6

, not

6 + 6

❌ Not quite. That is the payoff. Profit subtracts what both options cost,

C + P

❌ Not quite. Use

\lvert S_T - K\rvert - (C + P)

Show solution

The call pays $\text{Max}[0,\ 76 - 70] = 6$ and the put pays $\text{Max}[0,\ 70 - 76] = 0$ , so

\text{profit} = \lvert 76 - 70\rvert - (5 + 3) = 6 - 8 = -2.

A loss of $2 per share. The break-evens are $K \pm (C + P) = 70 \pm 8$ , that is 62 and 78: the stock must move more than 8 in either direction before the straddle makes money. A call finishing in the money is not enough.

Problem 3 · The Writer's Band and Worst Downside

Given: you write a straddle at $K = 60$ , receiving $5 for the call and $4 for the put. Find the range of stock prices at expiration in which you make money, and your loss per share if the stock falls all the way to zero.

Where do you make money?

What is the loss if the stock falls to zero?

✅ Correct! The writer keeps money within

C + P = 9

of the strike, and the downside is bounded because the stock cannot fall below zero:

-(K - C - P) = -51

❌ Not quite. Those are the break-evens of each option written on its own. In a straddle both premiums are collected, and either leg alone can take them back, so the band is

K \pm (C + P)

on both sides.

❌ Not quite. The band is

2(C + P)

wide in total, so it runs

C + P

either side of the strike, not

2(C + P)

❌ Not quite. The profit is

(C + P) - \lvert S_T - K\rvert

, which hits zero when the move equals the whole of

C + P

, not half of it.

❌ Check the band. Set

(C + P) - \lvert S_T - K\rvert = 0

❌ Not quite. That is the put's payoff owed,

K - 0

. The writer collected both premiums up front, which reduce the loss.

❌ Not quite. The loss is unbounded only on the upside, where the stock has no ceiling. On the downside the stock stops at zero.

❌ Not quite. The premiums were received, so they reduce the loss rather than add to it.

❌ Check the downside. Evaluate

(C + P) - \lvert S_T - K\rvert

S_T = 0

Show solution

The writer's profit is $(C + P) - \lvert S_T - K\rvert$ with $C + P = 5 + 4 = 9$ and $K = 60$ . It is positive when

\lvert S_T - 60\rvert < 9 \iff 51 < S_T < 69.

At $S_T = 0$ the call expires worthless and the put pays its holder 60, so

\text{profit} = 9 - \lvert 0 - 60\rvert = -51 = -(K - C - P).

The writer makes money between $51 and $69, and loses $51 per share if the stock falls to zero. Above $69 the loss grows without limit.

Problem 4 · Which Side of the Straddle?

Given: in a crisis, a call and a put on a stock at $K = 100$ cost $24 in total (illustrative). An investor is convinced the stock will finish at either $80 or $120, a large move either way. Find the side of the straddle that makes money under her belief.

✅ Correct! A move of 20 is large but smaller than the premiums of 24, so both outcomes land inside the band from 76 to 124, where the writer gains.

❌ Not quite. "Large" has to be judged against the premiums. The buyer needs a move of more than

C + P = 24

; a move of 20 leaves a loss of 4.

❌ Not quite. At expiration the holder's profit and the writer's profit sum to zero, so if one side loses the other gains.

❌ Not quite. 20 is the payoff

\lvert S_T - K\rvert

. The buyer paid 24 for it, so the profit is

-4

❌ Not quite. Compare her expected move with the total premium

C + P

Show solution

At either outcome the straddle pays $\lvert 80 - 100\rvert = \lvert 120 - 100\rvert = 20$ . The buyer's and writer's profits are

20 - 24 = -4, \qquad 24 - 20 = +4.

The break-evens are $100 \pm 24$ , that is 76 and 124, and both outcomes she expects lie inside. Under her belief the writer earns $4 per share either way. Buying pays only for someone who expects a move bigger than what the premiums already price, which is why crisis-level premiums make a straddle so hard to profit from.

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