期权定价问题悬赏求解

题目如下：

Assume a firm has assets worth $100,and that σ = 30%, r = 8%, and the firm makes no payouts prior to the maturity date ofthe debt.

(a) Construct a binomial lattice withthree periods, representing asset values over each year.

(b) Value the equity of the firmassuming that there exists a three year zero coupon bond with face value 50.

(c) Value the debt of the firm at eachnode in the lattice over the three years.

(d) Assume a two year call optionexists on the risky debt. The strike price of the call option on the bond is42. Compute the price of the call option.

(e) Assume the firm issued a callablebond instead of a straight bond. The terms of the call option are in (d). Thatis the firm retains the right to retire the debt at 42 dollars in the first twoyears. What would be the price of the callable bonds, and what would the equityprice be.

(f) Explain why the callable bonds arecheaper than the otherwise equivalent callable bonds. Also explain when thecall option would be exercised. Can you tell today exactly when the maturitydate is for the callable bond?

急求完整解答，谢谢！

The black (top) values are the values of the underlying. The second (red) values are the values of the two year call option which in this case represents the equity value. So, in this case, the value of the equity at date 0 is 50.9289. Now use the stock prices for the first two years as the “main” lattice. At the expiry date, we can compute the price of the call option with strike 15. These values are shown in green ( third values). We can work backwards, to compute the price of the call option on the equity. This value is 39.876

The current value of the bond is  100-50.93 = 49.07. Here we are assuming that there is only one share of stock traded! So all the prices are in millions. If there were one million shares outstanding, then the units of prices would be in dollars!

This example illustrates how the binomial lattice can be used to price options on options on options. For example we could now price a one year put option on the (green) call option. If the strike price was 50, then the payout would be zero in the up-node and 50-14.617 in the down-node. Once these “boundary” values are known, we can work backwards to find the price today, using the usual risk neutral probabilities. Just out of interest, why might a speculator  want to buy a put option on the call? ( Obviously, the speculator  thinks the call price  will drop after one year!)

Hence the value of the equity is 335,847, and the value of the debt is 664,153

hope it helps

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