Question

A stock currently sells for $32. A 6-month call option with a strike price of $35 has a price of $2.27. The price of the put option that satisfies the put-call-parity is $5.5229.Assuming a 4% continuously compounded risk-free rate and a 6% continuous dividend yield:

a) What are the arbitrage opportunities if the price of the call option in question 5 was $2?

b)What if this price was $3?

Answer 1

Solution.>

The put call parity with dividend yield is:

C + X * e^-r*t = P + S * e^-q*t

Part a)

Exercise price (X) : $35

Call option price (C) : $2

Put option price (P) : $5.5229

Risk-free rate (r) : 4%

Dividend yield (q) : 6%

Current market price (S) : $32

Time to maturity (t) : 0.5 years

Let’s plug these values in the put-call parity equation:

2 + 35* e^(-0.04*0.5) = 5.5229 + 32* e^(-0.06*0.5)

36.307 = 36.577

As we can see, the right hand side is greater than the left hand side by (36.577 - 36.307) = 0.27

To make use of this arbitrage opportunity, we will buy the call and sell the put.

1. Sell the put: We sell a put option and receive the $5.5229 premium. We also short sell the stock and receive $31.0541. The total cash inflow is $36.577.

2. Buy the call: We payout a total of $36.307 for the fiduciary call option. That is we pay $2 as premium for the call option and invest 34.307 in a bond for 6 months at 4%.

3. Net cash inflow: Our net cash inflow is 36.577 - 36.307) = $0.27

Part b)

Exercise price (X) : $35

Call option price (C) : $3

Put option price (P) : $5.5229

Risk-free rate (r) : 4%

Dividend yield (q) : 6%

Current market price (S) : $32

Time to maturity (t) : 0.5 years

Let’s plug these values in the put-call parity equation:

3 + 35* e^(-0.04*0.5) = 5.5229 + 32* e^(-0.06*0.5)

37.307 = 36.577

As we can see, the left hand side is greater than the right hand side by (37.307 - 36.577) = 0.73

To make use of this arbitrage opportunity, we will sell the call and buy the put.

1. Buy the put: We payout a total of $36.577 for the put option. That is we pay $5.5229 as premium for the put option and buy the stock at $31.0541.

2. Sell the call: We sell a call option and receive the $3 premium. We will also get $34.307 from the bond.

3. Net cash inflow: Our net cash inflow is (37.307 - 36.577) = $0.73

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