Contracts and Pricing III

ExpectationExponential DistributionOptions Pricing

Problem

After being caught in an embarrassing high-profile data fabrication scandal, Calambya University decides to stop sharing data with the national college ranking organization, effective in 2023. Rival school Mahogany University is overjoyed and offers each student a tuition refund worth $X$ thousand dollars, where $X = Calambya’s 2023 Ranking - Mahogany’s 2023 Ranking$ .

One Mahogany student, Justin, offers to sell the following contract to his friend Kevin for $0.70$ thousand dollars, before $X$ is known: Justin will give Kevin the right but not the obligation to purchase Justin's tuition refund for only $0.5$ thousand dollars.

Suppose $X \sim Exp (β)$ , where $β$ is the scale. Compute $β$ to the nearest tenth under the assumption that the contract is priced fairly.

Original Problem Link: https://www.quantguide.io/questions/contracts-and-pricing-iii

Solution

Step 1: What is this contract, really?

Kevin is buying the right, but not the obligation, to buy Justin's refund of $X$ thousand dollars for a strike of $0.5$ thousand dollars. That's exactly a call option on $X$ with strike $K = 0.5$ .

He only exercises it when $X > 0.5$ , in which case he pockets $X - 0.5$ risk-free (buy the refund for $0.5$ , and it's worth $X$ ). If $X \leq 0.5$ , he just walks away.

So the payoff at expiry is

Payoff (X) = max (X - 0.5, 0)

and Kevin pays $0.70$ upfront, before $X$ is known, to hold this.

Step 2: "Fair pricing" — two equivalent checkpoints

There are two natural moments to reason about fairness from.

Step 0 — the instant Kevin signs the contract. Fair pricing should mean his net expected profit (payoff minus what he paid) is exactly $0$ . If it weren't, one side would have a built-in edge, and the price wouldn't really be fair to both parties.

Step 1 — right after signing, ignoring the sunk premium. He's already paid $0.70$ . So "fair" here just means his expected payoff going forward should equal the $0.70$ he handed over.

These sound like different conditions. They aren't — let's write both out and check.

View A (net profit, from Step 0):

R_{1} (X) = {- 0.7 (X - 0.5) - 0.7 = X - 1.2 X \leq 0.5 X > 0.5

Fair pricing $⟹ E [R_{1} (X)] = 0$ .

View B (raw payoff, from Step 1):

R_{2} (X) = {0 X - 0.5 X < 0.5 X \geq 0.5

Fair pricing $⟹ E [R_{2} (X)] = 0.7$ , since his expected take should just match what he already paid.

Step 3: These are the same equation

Notice that

R_{1} (X) = R_{2} (X) - 0.7 for every X .

(Check both pieces: for $X \leq 0.5$ , that's $0 - 0.7 = - 0.7$ ✓. For $X > 0.5$ , that's $(X - 0.5) - 0.7 = X - 1.2$ ✓.)

E [R_{1} (X)] = E [R_{2} (X)] - 0.7

Setting $E [R_{1} (X)] = 0$ immediately gives $E [R_{2} (X)] = 0.7$ — View B's condition, exactly. Same equation, just shifted by the premium.

So we can just work with the cleaner one:

E [max (X - 0.5, 0)] = 0.7

Step 4: Using memorylessness of the exponential

We're told $X \sim Exp (β)$ with $β$ as the scale, so $E [X] = β$ .

The exponential is memoryless: conditional on $X > 0.5$ , the excess $X - 0.5$ is again distributed $Exp (β)$ . So

E [max (X - 0.5, 0)] = P (X > 0.5) \cdot E [X - 0.5 ∣ X > 0.5] = e^{- 0.5/ β} \cdot β

Our fairness condition becomes

β e^{- 0.5/ β} = 0.7

Step 5: Solve numerically

This is transcendental in $β$ , so plot $y = β e^{- 0.5/ β}$ against $y = 0.7$ on Desmos and read off where they cross:

β \approx 1.10

Sanity check: $1.10 \times e^{- 0.5/1.10} = 1.10 \times e^{- 0.4545} \approx 1.10 \times 0.6348 \approx 0.698 \approx 0.70$ ✓.