Problem 12.21.

A stock price is currently $50. It is known that at the end of six months it will be either $60 or $42. The risk-free rate of interest with continuous compounding is 12% per annum. Calculate the value of a six-month European call option on the stock with an exercise price of $48. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.

At the end of six months the value of the option will be either $12 (if the stock price is $60) or $0 (if the stock price is $42). Consider a portfolio consisting of:

???shares

?1?optionThe value of the portfolio is either 42? or 60??12 in six months. If

42??60??12

i.e.,

??0?6667

the value of the portfolio is certain to be 28. For this value of ? the portfolio is therefore riskless. The current value of the portfolio is: 0?6667?50?f

where f is the value of the option. Since the portfolio must earn the risk-free rate of interest

(0?6667?50?f)e0?12?0?5?28

f?6?96

i.e.,

The value of the option is therefore $6.96.

This can also be calculated using risk-neutral valuation. Suppose that p is the probability of an upward stock price movement in a risk-neutral world. We must have 60p?42(1?p)?50?e0?06 i.e., 18p?11?09 or: p?0?6161

The expected value of the option in a risk-neutral world is:

12?0?6161?0?0?3839?7?3932

This has a present value of

7?3932e?0?06?6?96

Hence the above answer is consistent with risk-neutral valuation.

Problem 12.22.

A stock price is currently $40. Over each of the next two three-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 12% per annum with continuous compounding.

a. What is the value of a six-month European put option with a strike price of $42? b. What is the value of a six-month American put option with a strike price of $42?

a. A tree describing the behavior of the stock price is shown in Figure S12.7. The risk-neutral probability of an up move, p, is given by

e0?12?3?12?0?90p??0?6523

1?1?0?9Calculating the expected payoff and discounting, we obtain the value of the option as [2?4?2?0?6523?0?3477?9?6?0?34772]e?0?12?6?12?2?118

The value of the European option is 2.118. This can also be calculated by working back through the tree as shown in Figure S12.7. The second number at each node is the value of the European option.

b. The value of the American option is shown as the third number at each node on the tree. It is 2.537. This is greater than the value of the European option because it is optimal to exercise early at node C.

40.0002.1182.53744.000 0.8100.810BAC48.4000.0000.00039.6002.4002.40032.4009.6009.60036.0004.7596.000

Figure S12.7 Tree to evaluate European and American put options in Problem 12.22. At each node, upper number is the stock price, the next number is the European put price, and

the final number is the American put price

Problem 12.23.

Using a “trial-and-error” approach, estimate how high the strike price has to be in Problem 12.17 for it to be optimal to exercise the option immediately.

Trial and error shows that immediate early exercise is optimal when the strike price is above 43.2.

This can be also shown to be true algebraically. Suppose the strike price increases by a

relatively small amount q. This increases the value of being at node C by q and the value of being at node B by 0?3477e?0?03q?0?3374q. It therefore increases the value of being at node A by (0?6523?0?3374q?0?3477q)e?0?03?0?551q

For early exercise at node A we require 2?537?0?551q?2?q or q?1?196. This corresponds to the strike price being greater than 43.196.

Problem 12.24.

A stock price is currently $30. During each two-month period for the next four months it is expected to increase by 8% or reduce by 10%. The risk-free interest rate is 5%. Use a two-step tree to calculate the value of a derivative that pays off max[(30?ST)?0]2 where ST is the stock price in four months? If the derivative is American-style, should it be exercised early?

This type of option is known as a power option. A tree describing the behavior of the stock price is shown in Figure S12.8. The risk-neutral probability of an up move, p, is given by

e0?05?2?12?0?9p??0?6020

1?08?0?9Calculating the expected payoff and discounting, we obtain the value of the option as [0?7056?2?0?6020?0?3980?32?49?0?39802]e?0?05?4?12?5?394

The value of the European option is 5.394. This can also be calculated by working back through the tree as shown in Figure S12.8. The second number at each node is the value of the European option.

Early exercise at node C would give 9.0 which is less than 13.2449. The option should therefore not be exercised early if it is American.

32.4000.2785 30.0005.3940BAC27.00013.244934.922D0.00029.1600.705624.30032.49

EF

Figure S12.8 Tree to evaluate European power option in Problem 12.24. At each node, upper

number is the stock price and the next number is the option price

Problem 12.25.

Consider a European call option on a non-dividend-paying stock where the stock price is $40,

the strike price is $40, the risk-free rate is 4% per annum, the volatility is 30% per annum, and the time to maturity is six months.

a. Calculate u, d, and p for a two step tree b. Value the option using a two step tree.

c. Verify that DerivaGem gives the same answer

d. Use DerivaGem to value the option with 5, 50, 100, and 500 time steps.

(a) In this case ?t?0?25 so that u?e0?30?0?25?1?1618, d?1?u?0?8607, and

e0?04?0?25?0?8607p??0?4959

1?1618?0?8607

(b) and (c) The value of the option using a two-step tree as given by DerivaGem is shown in Figure S12.9 to be 3.3739. To use DerivaGem choose the first worksheet, select Equity as the underlying type, and select Binomial European as the Option Type. After carrying out the calculations select Display Tree.

(d) With 5, 50, 100, and 500 time steps the value of the option is 3.9229, 3.7394, 3.7478, and 3.7545, respectively. At each node: Upper value = Underlying Asset Price Lower value = Option Price

Values in red are a result of early exercise.

Strike price = 40

Discount factor per step = 0.9900

Time step, dt = 0.2500 years, 91.25 days

Growth factor per step, a = 1.0101

Probability of up move, p = 0.4959

Up step size, u = 1.1618

Down step size, d = 0.8607 53.99435 13.99435 46.47337 6.871376 4040 3.3739190 34.428320

29.63273

0

Node Time:

0.00000.25000.5000