The price of a European call that expires in 3 months and has a strike price of

Question

The price of a European call that expires in 3 months and has a strike price of $30 is $3. The underlying stock price is $31. The stock pays no dividends. Risk-free interest rate is 10% per annum with continuous compounding.

Suppose that the price of a European put with the same maturity and strike price is $1. Describe the arbitrage opportunity and compute the arbitrage profit.

Now suppose that the stock pays a dividend of $0.50 in 2 months. Find the price of the European put (with 3-month maturity and strike price of $30 (typo corrected)) using the put-call parity with dividends.

Explanation / Answer

To avoid arbitrage opportunity, Put-Call parity equation must hold.

P + S = C + PV(X)

Buy a put and buy a stock equal to buy a call and buy a bond.

LHS = P + S = 1 + 31 = 32

RHS = 3 + 30*exp(-0.10*3/12) = 32.26

Since, LHS RHS, parity equation does not hold.

In fact, RHS > LHS

To get arbitrage profit, buy cheaper and sell costlier.

Buy a call, buy a bond, sell stock (short position), and write a Put.

Of course, they are at same exercise price and same maturity.

Arbitrage profit = 32.26 – 32 = $0.26

Now if parity equation with dividends holds, find put price.

P + S = C + PV(X) + PV(Div)

P = 3 + 30*exp(-0.10*3/12) + 0.50*exp(-0.10*2/12) – 31 = 1.75

Put price = $1.75

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