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One-step call with interest

Expected valueMedium

A stock at $50$ ends the period at $60$ or $45$ . Cash earns $2%$ over the period. What is the no-arbitrage price of a European call with strike $55$ expiring at the end of the period? Give an exact fraction.

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Hint

Find the weight $q$ that makes the discounted stock fair, or copy the payoff with stock and cash.

Solution

The weight is $q = \frac{50 \times 1.02 - 45}{60 - 45} = \frac{6}{15} = \frac{2}{5}$ .
The call pays $60 - 55 = 5$ in the up state and nothing in the down state.
The price is $\frac{\frac{2}{5} \times 5}{1.02} = \frac{100}{51}$ .

Answer: $\frac{100}{51}$

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