Question

1) An eight-month European put option on a dividend-paying stock is currently selling for $3. The stock price is $30, the strike price is $32, and the risk-free interest rate is 8% per annum. The stock is expected to pay a dividend of $2 three months later and another dividend of $2 six months later. Explain the arbitrage opportunities available to the arbitrageur by demonstrating what would happen under different scenarios.

2) The volatility of a non-dividend-paying stock whose price is $40, is 35%. The risk-free rate is 6% per annum (continuously compounded) for all maturities. Use a two-step tree to calculate the value of a derivative that pays off [max(?!−52,0)]" where is the stock price in six months?
3) A stock is expected to pay a dividend of $0.60 per share in one month, in four months and in seven months. The stock price is $25, and the risk-free rate of interest is 6% per annum with continuous compounding for all maturities. You have just taken a long position in an eight-month forward contract on the stock. Six months later, the price of the stock has become $29 and the risk-free rate of interest is still 6% per annum. What is the value your position six months later?

4) Suppose that the term structure of interest rates is flat in England and Germany. The GBP interest rate is 5% per annum and the EUR rate is 4% per annum. In a swap agreement, a financial institution pays 8% per annum in GBP and receives 6% per annum in EUR. The exchange rate between the two currencies has changed from 1.2 EUR per GBP to 1.15 EUR per GBP since the swap’s initiation. The principal in British pounds is 15 million GBP. Payments are exchanged every year, with one exchange having just taken place. The swap will last three more years. What is the value of the swap to the financial institution in terms of euros? Assume all interest rates are continuously compounded.

5) The premium of a call option with a strike price of $45 is equal to $5 and the premium of a call option with a strike price of $50 is equal to $3.5. The premium of a put option with a strike price of $45 is equal to $3. All these options have a time to maturity of 3 months. The risk-free rate of interest is 8%. In the absence of arbitrage opportunities, what should be the premium of a put option with a strike price of $50?

6) A financial institution has just bought 9-month European call options on the Chinese yuan. Suppose that the spot exchange rate is 14 cents per yuan, the exercise price is 15 cents per yuan, the risk-free interest rate in the United States is 3% per annum, the risk-free interest rate in China is 5% per annum, and the volatility of the yen is 10% per annum. Calculate vega of the financial institution’s position. Check the accuracy of your vega estimate by valuing the option at a volatility of 10% and 10.1% sequentially.
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7) A fund manager has a portfolio worth $55 million with a beta of 1.37. The manager is concerned about the performance of the market over the next five months and plans to use six-month futures contracts on the S&P 500 to hedge the risk. The current level of the index is 3,000, one contract is on 250 times the index, the risk-free rate is 5% per annum, and the dividend yield on the index is 3% per annum. The current 6-month futures price is 3,030. The fund manager takes a position in S&P 500 index futures to eliminate half of the exposure to the market over the next five months. Calculate the effect of your strategy on the fund manager’s returns if the level of the market in five months is 2,950 and one-month futures price is 1% higher than the index level in five months.

8) Suppose that zero interest rates with continuous compounding are as follows:

Maturity (months) Rate (% per annum) 3 6.0 6 6.2 9 6.4 12 6.5 15 6.6 18 6.7

Assume that a bank can borrow or lend at the rates above. What is the value of an FRA where it will earn 6.9% (per annum with quarterly compounding) for a three-month period starting in fifteen months on a principal of $1,500,000?

Answer 1

Risk free interest rate per month = 8%/12 =0.006667

Present value of Dividends = 2/(1+0.006667*3) + 2/(1+0.006667*6) = $3.88

Adjusted Stock price = $30- $3.88 =$26.12

Intrinsic value of put option = $32-$26.12 =$5.88

As the Put option is available for $3 it is cheaper than its intrinsic value and may present arbitrage opportunities

Arbitrage will work as follows

1. Today borrow $30+$3 =$33 and purchase the stock as well as the put option. Of the amount borrowed , borrow $3.85 for a period of 6 months so that it matures to an amount of $4 after 6 months

and remaining amount of $29.15 for 8 months , maturity amount to be repaid after 8 months= 29.15*(1+0.08*8/12) = $30.71

.2. Get $2 and $2 as dividend after 3 and 6 months and repay $4 after 6 months

3. After 8 months,

If stock price > $32

Sell the stock at the market price which is more than $32 and repay the loan of $30.71, make at least $1.29 as arbitrage profit

If stock price < $32

Sell the stock at $32 using the put option and repay the loan of $30.71, make exactly $1.29 as arbitrage profit

So, in all possible situations, arbitrage profit can be made.