Calculate the Arbitrage Profits - Arbitrageur - Statistics Assignment Help

Assignment Task

1. Suppose that USD-sterling spot and forward exchange rates are as follows:
• Spot 1.6080
• 90-day forward 1.6056
• 180-day forward 1.6018

What opportunities are open to an arbitrageur in the following situations? Calculate the arbitrage profits. Assume the risk-free interest rate is zero.
a. A 180-day European call option to buy £1 for $1.57 costs 2 cents?
Borrow 0.02 to buy call, short 180-day forward.
If ST > 1.57, exercise call to buy sterling at 1.57 and sell for 1.6018 under the forward contract,
Thus, profit is 0.0318 – 0.02 = 0.0118.
If ST < 1.57, not exercise the call, buy sterling at ST and sell at 1.6018 under the forward contract.
Thus, profit is greater than 0.0118.

b. A 90-day European put option to sell £1 for $1.64 costs 2 cents?
Long put for 0.02, long 90-day forward.
If ST < 1.64, put exercised, buy sterling at 1.6056 under forward and sell at 1.64 using put.
Thus, profit is 0.0344 – 0.02 = 0.0144.
If ST > 1.64, put not exercised, buy sterling at 1.6056 under forward and sell at market price ST.
Thus, profit is greater than 0.0144.

2. Assume the stock price follows a geometric Brownian motion, where current price S0=100, expected return μ=0 and volatility σ=0.80. The risk-free interest rate r=0.02 p.a.
a. Suppose that an investor short-sells 10,000 shares of the stock, what is the amount of loss that the investor is 97.5% certain that will not be exceeded in 3 months’ time?
We can be 97.5% certain that the stock price will not exceed
V = 100*exp (-0.5*0.82

*0.25 + 1.96*√0.25*0.8) = 202.1795

Thus, loss in this scenario is given by (202.1795 – 100)*100,000 = 1,021,795

b. Suppose that an investor sells a forward contract written on the stock with a maturity of 3 months. Assume the forward contract is for 10,000 shares of the stock. What is the amount of loss that the investor is 97.5% certain that will not be exceeded in 3 months’ time?
Current forward price is F0 = 100e0.02*0.25 = 100.50125. Given that ST = 202.1795 in 3 months, the loss is given by (202.1795 – 100.50125) *10,000 = 1,016,782

c. Suppose that an investor sells 10,000 put options written on the stock with strike $75 and a maturity of 3 months. What is the amount of loss that the investor is 97.5% certain that will not be exceeded in 3 months’ time?
In this case, the worst-case scenario is that ST = 42.148 in 3 months, the payoff from the short put is -(75 – 42.148) *10,000 = -328,521. Premium initially received is 46,242, therefore, the net loss is given by 328,521 – 46,242e0.02*0.25 = $282,048

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