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CHAPTER 21 Basic Numerical Procedures
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CHAPTER 21 Basic Numerical Procedures
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CHAPTER 21Basic Numerical Procedures
Practice Questions
Problem 21.1.
Which of the following can be estimated for an American option by constructing a single
binomial tree: delta, gamma, vega, theta, rho?
Delta, gamma, and theta can be determined from a single binomial tree. Vega is determined
by making a small change to the volatility and recomputing the option price using a new tree.
Rho is calculated by making a small change to the interest rate and recomputing the option
price using a new tree.
Problem 21.2.
Calculate the price of a three-month American put option on a non-dividend-paying stock
when the stock price is $60, the strike price is $60, the risk-free interest rate is 10% per
annum, and the volatility is 45% per annum. Use a binomial tree with a time interval of one
month.
In this case, S0 60 , K 60 , r 01 , 045 , T 025 , and t 00833 . Also
u e t
e 045 00833
11387
1
d 08782
u
a e r t e 0100833 10084
ad
p 04998
ud
1 p 05002
The output from DerivaGem for this example is shown in the Figure S21.1. The calculated
price of the option is $5.16.
Figure S21.1: Tree for Problem 21.2
,Problem 21.3.
Explain how the control variate technique is implemented when a tree is used to value
American options.
The control variate technique is implemented by
1. Valuing an American option using a binomial tree in the usual way ( f A ) .
1. Valuing the European option with the same parameters as the American option using
the same tree ( f E ) .
2. Valuing the European option using Black-Scholes-Merton (=fBSM ). The price of the
American option is estimated as fA + fBSM − fE.
Problem 21.4.
Calculate the price of a nine-month American call option on corn futures when the current
futures price is 198 cents, the strike price is 200 cents, the risk-free interest rate is 8% per
annum, and the volatility is 30% per annum. Use a binomial tree with a time interval of three
months.
In this case F0 198 , K 200 , r 008 , 03 , T 075 , and t 025 . Also
u e03 025 11618
1
d 08607
u
a 1
ad
p 04626
ud
1 p 05373
The output from DerivaGem for this example is shown in the Figure S21.2. The calculated
price of the option is 20.34 cents.
Figure S21.2: Tree for Problem 21.4
Problem 21.5.
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 using the
binomial tree approach? Explain your answer.
, A binomial tree cannot be used in the way described in this chapter. This is an example of
what is known as a history-dependent option. The payoff depends on the path followed by the
stock price as well as its final value. The option cannot be valued by starting at the end of the
tree and working backward since the payoff at the final branches is not known
unambiguously. Chapter 27 describes an extension of the binomial tree approach that can be
used to handle options where the payoff depends on the average value of the stock price.
Problem 21.6.
“For a dividend-paying stock, the tree for the stock price does not recombine; but the tree for
the stock price less the present value of future dividends does recombine.” Explain this
statement.
Suppose a dividend equal to D is paid during a certain time interval. If S is the stock price
at the beginning of the time interval, it will be either Su D or Sd D at the end of the time
interval. At the end of the next time interval, it will be one of ( Su D)u , ( Su D)d ,
( Sd D)u and ( Sd D )d . Since ( Su D) d does not equal ( Sd D )u the tree does not
recombine. If S is equal to the stock price less the present value of future dividends, this
problem is avoided.
Problem 21.7.
Show that the probabilities in a Cox, Ross, and Rubinstein binomial tree are negative when
the condition in footnote 8 holds.
With the usual notation
ad
p
ud
ua
1 p
ud
If a d or a u , one of the two probabilities is negative. This happens when
e ( r q ) t e t
or
e( r q ) t e t
This in turn happens when (q r ) t or (r q) t Hence negative probabilities
occur when
| (r q) t |
This is the condition in footnote 8.
Problem 21.8.
Use stratified sampling with 100 trials to improve the estimate of in Business Snapshot
21.1 and Table 21.1.
In Table 21.1 cells A1, A2, A3,..., A100 are random numbers between 0 and 1 defining how
far to the right in the square the dart lands. Cells B1, B2, B3,...,B100 are random numbers
between 0 and 1 defining how high up in the square the dart lands. For stratified sampling we
could choose equally spaced values for the A’s and the B’s and consider every possible
combination. To generate 100 samples we need ten equally spaced values for the A’s and the
B’s so that there are 10 10 100 combinations. The equally spaced values should be 0.05,
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