Question

A call on a stock with stock price $25.93 has a strike price $22 and expiration 230 days from today, available for a premium of $7. The stock also has a put with the same strike price and expiration as the above call, available for a put premium of $3.56. Both options are European, there are no dividends paid on the stock, and the interest rate for the expiration period is 8% per year. How much could you make per arbitrage portfolio

Answer 1

a]

As per the put-call parity equation, C + (K/(1 + r)^t) = P + S,

where C = price of call option,

P = price of put option,

S = current stock price

K = strike price of option

r = risk free rate

t = time to expiration in years

We plug in the values into the equation :

C + (K/(1 + r)^t) = P + S

7 + (22/(1 + 8%)^230/365) = 3.56 + 25.93

7 + 20.96 = 3.56 + 25.93

27.96 = 29.49

We can see that the left hand side is lower than the right hand side by (29.49 - 27.96) = 1.53

To make an arbitrage profit, we buy the left hand side (fiduciary call) and sell the right hand side (protective put).

• Buy fiduciary call: We pay $7 as premium for the call option and invest $20.96 (present value of strike price for 230 days at 8% - this is seen in the left hand side of the put-call parity equation above) in a zero-coupon bond for 230 days at 8%. Total amount paid for the fiduciary call option = $20.96 + $7 = $27.96
• Sell the protective put: We sell a put option and receive the $3.56 premium. We also short sell the stock and receive $25.93. The total cash inflow is $25.93 + $3.56 = $29.49
• Net cash inflow: Our net cash inflow is ($29.49 - $27.96) = $1.53

When the options expire in 230 days, there are two scenarios. The spot price of the stock will either be above $22 or below $22.

If the stock price at expiration is above $22

• Receive $22 from the bond
• Exercise call option, pay $22 and receive the stock
• Deliver the stock to cover the short sale
• The put option expires without being exercised
• No net income or loss
• Arbitrage profit = net cash inflow at the time of entering the strategy = $1.53

If the stock price at expiration is below $22

• Receive $22 from the bond
• Put option is exercised. You pay $22 and take delivery of the stock
• Deliver the stock to cover the short sale
• The call option expires without being exercised
• No net income or loss
• Arbitrage profit = net cash inflow at the time of entering the strategy = $1.53

Arbitrage profit = $1.53