Question

Explain how each trader would set up a strategy to carry out their intended purpose. Be as specific as possible in each case, by providing details about the strategy: the position (whether long or short), the size of the exposure (notional amount), the number of contracts involved (if more than one), the expiration date of the contracts, etc. In each case, explain whether the trader will realize a profit or a loss if the underlying asset/variable increases by 10%. An institutional trader is tasked with seeking out arbitrage opportunities, and is particularly focused on price discrepancies between put and call options on Apple Inc. (Nasdaq: AAPL) stock. The institutional trader finds a European call option trading at $23.00, and a European put option trading at $5.00, when the stock price is $175.00. Both options have an exercise price of $160.00, and a maturity of six months. The risk-free interest rate is 2.0% (continuously compounded). 1. Does put-call parity hold? If not, then describe the arbitrage opportunity (see table 11.3). 2. Will the institutional trader realize a profit or a loss if Apple’s stock price increases by 10% over the next six months?

Table 11.3   Arbitrage opportunites when put-call parity does not hold.    Stock price = $31; interest rate = 10%; call price = $3. Both put and call    have strike price of $30 and three months to maturity.
Three-month put price = $2.25	Three-month put price = $1
Action now:	Action now:
Buy call for $3	Borrow $29 for 3 months
Short put to realize $2.25	Short call to realize $3
Short the stock to realize $31	Buy put for $1
Invest $30.25 for 3 months	Buy the stock for $31
Action in 3 months if Sr > 30:	Action in 3 months if Sr > 30:
Receive $31.02 from investment	Call exercised: sell stock for $30
Exercise call to buy stock for $30	Use $29.73 to repay loan
Net profit = $1.02	Net profit = $0.27
Action in 3 months if Sr < 30:	Action in 3 months if Sr < 30:
Receive $31.02 from investment	Exercise put to sell stock for $30
Put exercised: buy stock for $30	Use $29.73 to repay loan
Net profit = $1.02	Net profit = $0.27

Answer 1

The put call parity, C + PV(x) = P + S, where C is the call price (premium on the call option), PV(x) is the present value of strike price, P is the put price and S is the spot price of the underlying.

Here, C = 23

PV (x) = 160*e^(-0.02*6/12) = 158.41

P = 5

S = 175

C+PV = 181.41

P+S = 180

Hence, there is no put-call parity and arbitrage opportunity.

1. Borrow $157 for 6 months

2. Buy put at realize $5

3. Buy stock at $175

4. Short call to realize $23

Action after 6 months if Sr > 160

1. Sell stock at 160 (using call option).

2. Return loan with interest = $158.58

Total profit = $160-$158.58 = $1.42

Action after 6 months if Sr < 160

1. Exercise put to sell stock at $160

2. Return loan with interest = $158.58

Total profit = $1.42

If the stock price increases by 10% over 6 months, the total profit from the stategy will still be $1.42 as profit is independent of underlying spot price.