Question

A trader wants to buy 500 shares of Stock A and decides to hedge the value of her position with futures contracts on Stock B. Each futures contract of Stock B is on 25 shares. Suppose the spot price of Stock A is $5 per share, and the standard deviation of the change in this price over the life of the hedge is estimated to be $0.33. The futures price of Stock B is $4 per shares and the standard deviation of the change in this over the life of the hedge is $0.43. The coefficient of correlation between the spot price change and futures price change is 0.89.

Now suppose Stock A is dividend-paying, with $1 dividend paid for each 3 months. The spot price of a Stock A is $5, and the risk-free rate of interest is 8% per annum with continuous compounding.

(a) What are the main differences between forwards and futures?

(b) What are the forward price and the initial value of a one-year forward contract on one share of Stock A?

(c) Four months later, the price of the stock is $6 and the risk-free interest rate is still 8%. What are the forward price and the value of the forward contract

A trader wants to buy 500 shares of Stock A and decides to hedge the value of her position with futures contracts on Stock B. Each futures contract of Stock B is on 25 shares. Suppose the spot price of Stock A is $5 per share, and the standard deviation of the change in this price over the life of the hedge is estimated to be $0.33. The futures price of Stock B is $4 per shares and the standard deviation of the change in this over the life of the hedge is $0.43. The coefficient of correlation between the spot price change and futures price change is 0.89.

(a) What is the minimum variance hedge ratio?

(b) What position should the trader enter into?

(c) What is the optimal number of futures contracts when issues associated with daily settlement are considered?

Now suppose Stock A is dividend-paying, with $1 dividend paid for each 3 months. The spot price of a Stock A is $5, and the risk-free rate of interest is 8% per annum with continuous compounding.

(d) What are the main differences between forwards and futures?

(e) What are the forward price and the initial value of a one-year forward contract on one share of Stock A?

(f) Four months later, the price of the stock is $6 and the risk-free interest rate is still 8%. What are the forward price and the value of the forward contract?

Answer 1

a) Forward contract is similar to Futures contract except that Forward contracts are specific and tailor made to the needs of the parties, are more costly , are traded Over the Counter (OTC) and have default risk whereas Futures contract have fixed specifications, less costly , traded on the exchange and do not have default risk.

b) Forward price = (Spot price - present value of dividends) * exp (r*t)

where r is the continuously compounded risk free rate

and t is the time till maturity in years

Assuming that a dividend has just been paid

Present value of dividends = 1*exp(-0.08*3/12)+1*exp(-0.08*6/12)+1*exp(-0.08*9/12)+1*exp(-0.08*12/12) = $3.81

So,  Forward price = (5-3.81)*exp(0.08*1)

=$1.29

Initial value of the forward contract= present value of benefits

Since the forward contract is entered into at the theoretical price of $1.29 , the initial value is 0

c)  Present value of remaining dividends = 1* exp(-0.08*2/12)+1* exp(-0.08*5/12)+ 1* exp(-0.08*8/12) = $2.90

Forward price after 4 months = (6-2.90)*exp(0.08*8/12) = $3.27

Value of the forward contract to the buyer of the forward = (3.27-1.29)*exp(-0.08*8/12) = $1.87

(Value to the seller of the forward contract will be -$1.87)