HullFund9eCh18ProblemSolutions.doc

CHAPTER 18
Binomial Trees in Practice
Practice Questions Problem 18.8.
Consider an option that pays off the amount by which the final stock price
exceeds the average stock price achieved during the life of the option. Can
this be valued from a binomial tree using backwards induction? No! This is an example of a path-dependent option. The payoff depends on
the path followed by the stock price as well as on its final value. The
option cannot be valued by starting at the end of the tree and working
backward, because the payoff at a final branch depends on the path used to
reach it. European options for which the payoff depends on the average
stock price can be valued using Monte Carlo simulation, as described in
Section 18.6. Problem 18.9.
A nine-month American put option on a non-dividend-paying stock has a
strike price of $49. The stock price is $50, the risk-free rate is 5% per
annum, and the volatility is 30% per annum. Use a three-step binomial tree
to calculate the option price. In this case, [pic], [pic], [pic], [pic], [pic], and [pic]. Also
[pic]
The output from DerivaGem for this example is shown in the Figure S18.1.
The calculated price of the option is $4.29. Using 100 steps the price
obtained is $3.91.
[pic]
Figure S18.1 Tree for Problem 18.9 Problem 18.10.
Use a three-time-step tree to value a nine-month American call option on
wheat futures. The current futures price is 400 cents, the strike price is
420 cents, the risk-free rate is 6%, and the volatility is 35% per annum.
Estimate the delta of the option from your tree. In this case [pic], [pic], [pic], [pic], [pic], and [pic]. Also
[pic]
The output from DerivaGem for this example is shown in the Figure S18.2.
The calculated price of the option is 42.07 cents. Using 100 time steps the
price obtained is 38.64. The option's delta is calculated from the tree is
[pic]
When 100 steps are used the estimate of the option's delta is 0.483. [pic]
Figure S18.2 Tree for Problem 18.10 Problem 18.11.
A three-month American call option on a stock has a strike price of $20.
The stock price is $20, the risk-free rate is 3% per annum, and the
volatility is 25% per annum. A dividend of $2 is expected in 1.5 months.
Use a three-step binomial tree to calculate the option price. In this case the present value of the dividend is [pic] . We first build a
tree for [pic], [pic], [pic], [pic], and [pic] with [pic]. This gives
Figure S18.3. For nodes between times0 and 1.5 months we then add the
present value of the dividend to the stock price. The result is the tree in
Figure S18.4. The price of the option calculated from the tree is 0.674.
When 100 steps are used the price obtained is 0.690.
[pic] Figure S18.3 First tree for Problem 18.11 [pic]
Figure S18.4 Final Tree for Problem 18.11
Problem 18.12.
A one-year American put option on a non-dividend-paying stock has an
exercise price of $18. The current stock price is $20, the risk-free
interest rate is 15% per annum, and the volatility of the stock is 40% per
annum. Use the DerivaGem software with four three-month time steps to
estimate the value of the option. Display the tree and verify that the
option prices at the final and penultimate nodes are correct. Use DerivaGem
to value the European version of the option. Use the control variate
technique to improve your estimate of the price of the American option. In this case [pic], [pic], [pic], [pic], [pic], and [pic]. The parameters
for the tree are
[pic]
The tree produced by DerivaGem for the American option is shown in Figure
S18.5. The estimated value of the American option is $1.29. [pic]
Figure S18.5 Tree to evaluate American option for Problem 18.12
[pic] Figure S18.6 Tree to evaluate European option in Problem 18.12 As shown in Figure S18.6, the same tree can be used to value a European put
option with the same parameters. The estimated value of the European option
is $1.14. The option parameters are [pic], [pic], [pic], [pic] and [pic]
[pic] [pic]
The true European put price is therefore
[pic]
This can also be obtained from DerivaGem. The control variate estimate of
the American put price is therefore 1.29 + 1.10 -1.14 = $1.25. Problem 18.13.
A two-month American put option on a stock index has an exercise price of
480. The current level of the index is 484, the risk-free interest rate is
10% per annum, the dividend yield on the index is 3% per annum, and the
volatility of the index is 25% per annum. Divide the life of the option
into four half-month periods and use the binomial tree approach to estimate
the value of the option. In this case [pic], [pic], [pic], [pic] [pic], [pic], and [pic]
[pic]
The tree produced by DerivaGem is shown in the Figure S18.7. The estimated
price of the option is $14.93.
[pic] Figure S18.7 Tree to evaluate option in Problem 18.13
Problem 18.14.
How would you use the control variate approach to improve the estimate of
the delta of an American option when the binomial tree approach is used? First the delta of the American option is estimated in the usual way from
the tree. Denote this by [pic]. Then the delta of a European option which
has the same parameters as the American option is calculated in the same
way using the same tree. Denote this by [pic]. Finally the true European
delta, [pic], is calculated using the formulas in Chapter 17. The control
variate estimate of delta is then:
[pic]
Problem 18.15.
How would you use the binomial tree approach to value an American option on
a stock index when the dividend yield on the index is a function of time? When the dividend yield is constant
[pic]
Making the dividend yield, [pic], a function of time makes [pic], and
therefore [pic], a function of time. However, it does not affect [pic] or
[pic]. It follows that if [pic] is a function of time we can use the same
tree by making the probabilities a function of time. The interest rate
[pic] can also be a function of time as described in Section 18.4.
Further Questions Problem 18.16.
An American put option to sell a Swiss franc for dollars has a strike price
of $0.80 and a time to maturity of one year. The volatility of the Swiss
franc is 10%, the dollar interest rate is 6%, the Swiss franc interest rate
is 3%, and the current exchange rate is 0.81. Use a three-time-step tree to
value the option. Estimate the delta of the option from your tree. The tree is shown in Figure S18.8. The value of the option is estimated as
0.0207. and its delta is estimated as
[pic]
[pic] Figure S18.8 Tree for Problem 18.16
Problem 18.17.
A one-year American call option on silver futures has an exercise price of
$9.00. The current futures price is $8.50, the risk-free rate of interest
is 12% per annum, and the volatility of the futures price is 25% per annum.
Use the DerivaGem software with four three-month time steps to estimate the
value of the option. Display the tree and verify that the option prices at
the final and penultimate nodes are correct. Use DerivaGem to value the
European version of the option. Use the control variate technique to
improve your estimate of the price of the American option. In this case [pic], [pic], [pic], [pic], [pic], and [pic]. The parameters
for the tree are
[pic]
The tree output by DerivaGem for the American option is shown in Figure
S18.9. The estimated value of the option is $0.596. The tree produced by
DerivaGem for the European version of the option is shown in Figure S18.10.
The estimated value of the option is $0.586. The Black-Scholes price of the
option is $0.570. The control variate estimate of the price of the option
is therefore
[pic]
Figure S18.9 Tree for American option in Problem 18.17 Figure S18.10 Tree for European option in Problem 18.17
Problem 18.18.
A six-month American call option on a stock is expected to pay dividends of
$1 per share at the end of the second month and the fifth month. The
current stock price is $30, the exercise price is $34, the risk-free
interest rate is 10% per annum, and the volatility of the part of the stock
price that will not be used to pay the dividends is 30% per annum. Use the
DerivaGem software with the life of the option divided into 100 time steps
to estimate the value of the option. Compare your answer with that given by
Black's approximation (see Section 13.10.) DerivaGem gives the value of the option as 1.0349. Black's approximation
sets the price of the American call option equal to the maximum of two
European options. The first lasts the full six months. The second expires
just before the final ex-dividend date. In this case the software shows
that the first European option is worth 0.957 and the second is worth
0.997. Black's model therefore estimates the value of the American option
as 0.997. Problem 18.19. (Excel file)
The DerivaGem Application Builder functions enable you to investigate how
the prices of options calculated from a binomial tree converge to the
correct value as the number of time steps increases. (See Figure 18.4 and
Sample Application A in DerivaGem.) Consider a put option on a stock index
where the index level is 900, the strike price is 900, the risk-free rate
is 5%, the dividend yield is 2%, and the time to maturity is 2 years
a. Produce results similar to Sample Application A on convergence for the
situation where the option is European and the volatility of the index
is 20%.
b. Produce results similar to Sample Application A on convergence for the
situation where the option is American and the volatility of the index
is 20%.
c. Produce a chart showing the pricing of the American option when the
volatility is 20% as a function of the number of time steps when the
control variate technique is used.