Question

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A stock is traded at $50 a share, and there will be no dividend on the...

A stock is traded at $50 a share, and there will be no dividend on the stock in 6 months. The 6-month European calls and puts on the stock with same strike price of $45 are traded at $8 and $2, respectively. The six-month risk-free rate is 5% with continuous compounding. Is there an arbitrage opportunity? If yes, construct an arbitrage portfolio and clearly explain how arbitrage profits are created. If no, why?

Expert Solution

a]

As per the put-call parity equation, C + (K/ert) = 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/ert) = P + S

8 + (45/e(0.05*(6/12))) = 2 + 50

8 + 43.67 = 2 + 50

51.67 = 52

We can see that the left hand side is lower than the right hand side by (52 - 51.67) = 0.33

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 $8 as premium for the call option and invest $43.67 (present value of strike price for 6 months at 5% - this is seen in the left hand side of the put-call parity equation above) in a zero-coupon bond for 6 months at 5%. Total amount paid for the fiduciary call option = $43.67 + $8 = $51.67
Sell the protective put: We sell a put option and receive the $2 premium. We also short sell the stock and receive $50. The total cash inflow is $50 + $2 = $52.
Net cash inflow: Our net cash inflow is ($52 – $51.67) = $0.33

When the options expire in 6 months, there are two scenarios. The spot price of the stock will either be above $45 or below $45.

If the stock price at expiration is above $45

Receive $45 from the bond
Exercise call option, pay $45 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 = $0.33

If the stock price at expiration is below $45

Receive $45 from the bond
Put option is exercised. You pay $45 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 = $0.33

In either case, arbitrage profit = $0.33

jeff jeffy answered 2 years ago

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