Question

Suppose a stock is currently priced at $50 a share, and in one period, it will be worth either $45 or $55. There is European PUT options on the stock with one period to expiration and an exercise price of $50. The riskless interest rate over the period is 4 percent. Call options do not exist.

a) Create a riskless portfolio and show that it is riskless.

b) Calculate the current equilibrium put price, current intrinsic value and current time value.

c) Suppose that the current price of the option is $2.90. Is there a profitable arbitrage? If yes, design the arbitrage and calculate the profits.

d) Suppose that the current price of the option is $1.00. Is there a profitable arbitrage? If yes, design the arbitrage and calculate the profits.

e) Consider another put option with an exercise price of $47.50. Create a riskless portfolio and show that it is riskless. Calculate the current equilibrium put price current intrinsic value and current time value.

f) Consider another put option with an exercise price of $52.50. Create a riskless portfolio and show that it is riskless. Calculate the current equilibrium put price current intrinsic value and current time value.

g) Suppose that the put option with an exercise price of $52.50 is an American option rather than European. Should you exercise it today? Explain.

Answer 1

a) Let the riskless portfolio have A unit of shares and 1 unit of put option

now, If the stock goes up , value of portfolio = A*55+0 ( payoff of put option = 0 if stock goes up)

If the stock goes down , value of portfolio = A*45+5 ( payoff of put option = 5 if stock goes down)

For riskless portfolio , payoff under both situations should be same

55*A = 45*A+ 5

=> A = 0.5

So, the riskless portfolio comprises of long 0.5 shares and 1 unit of put option

This portfolio is riskless as the payoff is the same for any market situation in future

b) Value of riskless portfolio in future = 0.5*55 = 27.5

So, value of riskless portfolio today = 27.5/1.04 = 26.44231

Now, value of portfolio today = A*50+ P (where p is the value of put option)

=> 0.5*50 + P = 26.44231

=> P = 1.44231

So, current equilibrium price of put option = $1.44231

Current Intrinsic value = max(K-St,0) = max(50-50,0) = $0

Current Time value = Price - Intrinsic value = 1.44231 -0 = $1.44231

c) If the put option price is $2.90 , arbitrage is possible and the steps are

1. Today, Short Sell 1 share and  2 put options and get $50+2.9*2 =$55.8

2. Today invest the amount at risk free rate for 1 period at 4% to get 55.8*1.04 = $58.032 after one period

3. After 1 period, if stock price = 55 , put options will be worthless, buy the stock at $55 and return to the lender.

Arbitrage profit = $3.032

If stock price =$45, both put options will be exercised, buy these two stocks at $50 each and sell one in the market at$45 and return the other. net payment to be done=$50+$50-$45 = $55, Arbitrage profit = $3.032

So, in both cases, arbitrage profit of $3.032 can be made at the end of the period

d) If the put option price is $1 , arbitrage is possible and the steps are

1. Today, Buy 1 share and  2 put options for $50+1*2 = $52 by borrowing $52 for one period at 4%

2. The amount to be repaid =52*1.04 = $54.08

3. After 1 period, if stock price = 55 , put options will be worthless, Sell the stock at $55 and repay the loan. Arbitrage profit = $55 -$ 54.08 = $0.92

If stock price =$45, sell oe put option just before expiry to get $5, use other put option to sell the stock at $50.  and thus get $55, repay the loan and make an Arbitrage profit = $55-$54.08 = $0.92

So, in both cases, arbitrage profit of $0.92 can be made at the end of the period