Solution of Derivative

In this case our initial investment is 91 how do we

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Unformatted text preview: vestment. In this case, our initial investment is $91. How do we find the future value? We use the current risk-free interest rate and multiply the initial investment by it. However, as our bond is default-free, and does not bear coupons, the effective annual interest rate is exactly the 9.9% we have calculated before. Therefore, the future value of $91 is $91 × (1 + 0.0989 ) = $100 , and our profit in six months is zero! Question 2.8. We saw in question 2.7. b) that there is no advantage in buying either the stock or the forward contract if we can borrow to buy a stock today (so both strategies do not require any initial cash) and if the profit from this strategy is the same as the profit of a long forward contract. The profit of a long forward contract with a price for delivery of $53 is equal to: $S T − $53 , where ST is the (unknown) value of one share of XYZ at expiration of the forward contract in one year. If we borrow $50 today to buy one share of XYZ stock (that costs $50), we have to repay in one year: 7 Part 1 Insurance, Hedging, and Simple Strategies $50 × (1 + r ) . Our total profit in one year from borrowing to buy one share of XYZ is therefore: $S T − $50 × (1 + r ) . Now we can equate the two profit equations and solve for the interest rate r: $S T − $53 ⇔ ⇔ ⇔ = $S T − $50 × (1 + r ) $53 $53 −1 $50 r = $50 × (1 + r ) = r = 0.06 Therefore, the 1-year effective interest rate that is consistent with no advantage to either buying the stock or forward contract is 6 percent. Question 2.10. a) Figure 2.7 depicts the profit from a long call option on the S&R index with 6 months to expiration and a strike price of $1,000 if the future price of the option premium is $95.68. The profit of the long call option is: max[0, ST − $1,000] − $95.68 ⇔ max[−$95.68, S T − $1,095.68] where ST is the (unknown) value of the S&R index at expiration of the call option in six months. In order to find the S&R index price at which the call option diagram intersects the x-axis, we have to set the above equation equal to zero. We get: ST − $1,095.68 = 0 ⇔ S T = $1,095.68 . This is the only solution, as the other part of the maximum function, -$95.68, is always less than zero. b) The profit of the 6 month forward contract with a forward price of $1,020 is: $S T − $1,020 . In order to find the S&R index price at which the call option and the forward contract have the same profit, we need to set both parts of the maximum function of the profit of the call option equal to the profit of the forward contract and see which part permits a solution. First, we see immediately that $ST − $1,020 = $ST − $1,095.68 does not have a solution. But we can solve the other leg: $S T − $1,020 = −$95.68 ⇔ ST = $924.32 , which is the value given in the exercise. Question 2.12. a) Long Forward The maximum loss occurs if the stock price at expiration is zero (the stock price cannot be less than zero, because companies have limited liability). The forward then pays 0 – Forward price. The maximum gain is unlimited. The stock price at expiration could theoretically grow to infinity, there is no bound. We make a lot of money if the stock price grows to infinity (or to a very large amount). 8 Chapter 2 An Introduction to Forwards and Options b) Short Forward The profit for a short forward contract is forward price – stock price at expiration. The maximum loss occurs if the stock price raises sharply, there is no bound to it, so it could grow to infinity. The maximum gain occurs if the stock price is zero. c) Long Call We will not exercise the call option if the stock price at expiration is less than the strike price. Consequently, the only thing we lose is the future value of the premium we paid initially to buy the option. As the stock price can grow very large (and without bound), and our payoff grows linearly in the terminal stock price once it is higher than the strike, there is no limit to our gain. d) Short Call We have no control over the exercise decision when we write a call. The buyer of the call option decides whether to exercise or not, and he will only exercise if he makes a profit. As we have the opposite side, we will never make any money at the expiration of the call option. Our profit is restricted to the future value of the premium, and we make this maximum profit whenever the stock price at expiration is smaller than the strike price. However, the stock price at expiration can be very large and has no bound, and as our loss grows linearly in the terminal stock price, there is no limit to our loss. e) Long Put We will not exercise the put option if the stock price at expiration is larger than the strike price. Consequently, the only thing we lose whenever the terminal stock price is larger than the strike is the future value of the premium we paid initially to buy the option. We will profit from a decline in the stock prices. However, stock prices cannot be smaller than zero, so our maximum gain is restricted to strike price l...
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