A 2-month European put option on a non-dividend paying stock is currently selling for $2. The stock price is $47, the strike price is $50, and the risk-free rate is 6% per year (with continuous compounding) for all maturities. Does this create any arbitrage opportunity? Why? Design a strategy to take advantage of this opportunity and specify the profit you make.

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Business

Eliezer

14 June, 01:43

A 2-month European put option on a non-dividend paying stock is currently selling for $2. The stock price is $47, the strike price is $50, and the risk-free rate is 6% per year (with continuous compounding) for all maturities. Does this create any arbitrage opportunity? Why? Design a strategy to take advantage of this opportunity and specify the profit you make.

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Leyla Hutchinson · Answer 1

Yes, this create any arbitrage opportunity

Explanation:

In this case the present value of the strike price is 50e^{0.06x2/12} = 49.50.

And since $2<49.50-47, an arbitrageur should borrow $49 at 6% for two months, buy the stock, and buy the put option. This generates a profit in all circumstances.

If the stock price is above $50 in two months, the option expires worthless, but the stock can be sold for at least $50. A sum of $50 received in two months has a present value of $49.50 today. The strategy therefore generates profit with a present value of at least $0.5.

If the stock price is below $50 in one month the put option is exercised and the stock owned is sold for exactly $50 (or $49.5 in present value terms). The trading strategy therefore generates a profit of exactly $0.5 in present value terms.