Question

The current stock price is $65 and it will pay $3 dividends in 1 month and 4 months. A European at-the-money put option with 5 months to maturity is traded for $6.A European at-the-money call option with 5 months to maturity is $5. The spot rate for 1, 4, 5 months are 13%,11% and9.5%(c.c).

1.Is there an arbitrage opportunity? If yes describe the investment strategy that pays off today and has for sure zero cash flows at any time in the future.

2.What level in the interest rate ensures that markets are arbitrage free?

Answer 1

Assumptions:

1. Dividend is $3 each for Month 1 and 4.
2. All Spot Interest Rates are risk free and annualized.
3. Strike Price is not given for arbitrage. Hence its been assumed as $67 for 5 months.

The Values provided in the question are as follows:

Formula for the Put Call Parity to compute strike price for European option-

Put-Call Parity – dividend paying stock (discrete dividend)

S-PV(Div) = C-P + e^{-r T}

where-

S-PV(Div)= Stock's existing price excluding dividend's present value.

C= Call Option value

P= Put Option value

e^{-r T}= PV of Strike Price

Putting values in the formula-

59.14(w1) = 5-6 + e * {1/1+ (0.095*5 month/12 months)}

= (59.14 + 1) * 1.03958 = e

Exercise Price/Strike Price= 62.52

Workings:

a) Net Stock Price

1) Is there an arbitrage opportunity? If yes describe the investment strategy that pays off today and has for sure zero cash flows at any time in the future.

Both Put and Call options are 'In the money'; maening the amount that an option is in the money is called the Intrinsic value meaning the option is at least worth that amount.

If Put-call parity doesnt hold, there will be arbitrage opportunity

Arbitrage Investment strategy for Risk free profit-

Suppose strike price is 67.

For example one trader has already the stock at time t = 0 which is in portfolio A.

Step1: At time t = 0 he sells a put and a stock (so portfolio A), so he obtains p + S Dollar.

Step2: Further, he buys a call and put the rest of the money to a deposit with r% interest rate.

The cashflow is p + S − c = 6 + 65 − 5 = 66.

Step3: Invested for 3 months this gives 66 {1/1+ (0.095*5 month/12 months)}]= $68.612.

Result:

Case a)- If ST > K = 67, he exercises the call, so he buys one share for the price K and own one share to claim profit of $1.612 ($68.612 - $67).

Case b)- If ST < K = 67, then the put will be exercised from the other party. Then he must buy one share from the other party for the price K, so he has one share and a profit of $1.612.

Riskless profit is $3.612 for 67 Strike Price.

2.What level in the interest rate ensures that markets are arbitrage free?

When intrinsic value of strike price($67) validates the market interest rates, then market would be arbitrage free.

When Invested $66 for 3 months result into $67-

=[(67 / 66) -1}* 12 months /5 months]

= 3.67%

Hence for assumed strike price for arbitrage; interest rate is 3.67%