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# Binomial tree, Time to harvest, Market risk

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1) Suppose you own a large forest that is due for harvest. The pulp market is picking up, so you know that if you start harvest this year you will forgo rising prices in the future. There is, however, a cost advantage to starting the harvest as soon as possible. The table below shows the present value of the revenue stream and the harvest cost, as functions of the time of harvest. For instance, if you start harvest straight away your revenue stream contribute 10M to your wealth but the costs take 5M away. If you delay harvest till next year, the revenues add 15M to the next year's wealth, and the costs take 6M away, etc. The discount rate is 5%. Use the data below to determine the optimal time to harvest?
Year 0 Year 1 Year 2 Year 3 Year 4 Year 5
PV(Revenue) 10 15 20 23 24 23
PV(Costs) 5 6 7 8 8 8

2)

a) What are the terminal (time 2) payoffs of the options?
b) Derive the risk neutral probabilities for the movement of the binomial tree.Use these to price the call option.
c) would you construct the portfolio of the stock, the call and the risk free asset to exactly replicate the terminal payoffs of the put option?
d) Use the relationship you derived in part c to find the price of the put option.
3) Decompose the risk of each asset into systematic (market) and idiosyncratic (firm specific risk).

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#### Solution Preview

Please see the attached file.

a) What are the terminal (tine 2) payoffs of the options

Exercise Price= 95

Payoff for the call option= Maximum of (Stock Price- Exercise Price,0)

Payoff for the Put option= Maximum of (Exercise Price-Stock Price ,0)

Stock Price Payoff for call option Payoff for put option

130 35 0
110 15 0
90 0 5

Call option Payoffs: Put option Payoffs:

130.00 130.00
35.00 0.00

115.00 115.00

100.00 110.00 100.00 110.00
15.00 0.00

95.00 95.00

90.00 90.00
0.00 5.00

b) Derive the risk neutral probabilities for the movement of the binommial tree.Use these to price the call option

The probabilities have been calculated below and marked on the arrows

130.00

0.6669

115.00

0.6125 0.3331

100.00 110.00

0.3875 0.5944

95.00

0.4056

90.00

Period 1
rf=risk free rate= 7%
&#916;t= 1
u= 1.15 =115/100
d= 0.95 =95/100
r&#916;t= 0.07
e r&#916;t = 1.072508181
p=(e r&#916;t-d)/(u-d)= 0.612540906
1-p= 0.387459094

Period 2
rf=risk free rate= 7% rf=risk free ...

#### Solution Summary

There are 3 problems
1) Optimal time to harvest - capital budgeting question
2) Calculate the values of call and put options using binomial tree
3) Market portfolio - decomposition of risk into systemic and unsystemic risk
The solution provides step by step solution for all the problems

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