Solving for the takeover price from a call value

Question

A share trades at $30 on Monday. Overnight there is a $20\%$ chance the company is acquired. If acquired, Tuesday's share price is $50 or \text{\char36} $x$ with equal probability; if not acquired, it is $0 with probability $60\%$ and $10 with probability $40\%$ . A fairly priced option giving the right (but not the obligation) to buy one share at $40 on Tuesday costs $15. Find $x$ .

Short answer

$x=180$ . The option pays only in the acquisition branch: $(50-40)$ or $(x-40)$ , each with probability $0.1$ . Fair value means $15=0.1\cdot10+0.1(x-40)$ , so $x=180$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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