HullFund8eCh12ProblemSolutions.pdf

CHAPTER 12

Introduction to Binomial Trees

Practice Questions

Problem 12.8.

Consider the situation in which stock price movements during the life of a European option

are governed by a two-step binomial tree. Explain why it is not possible to set up a position

in the stock and the option that remains riskless for the whole of the life of the option.

The riskless portfolio consists of a short position in the option and a long position in

shares. Because changes during the life of the option, this riskless portfolio must also

At the end of two months the value of the option will be either $4 (if the stock price is $53) or

$0 (if the stock price is $48). Consider a portfolio consisting of:

shares

1 option

The value of the portfolio is either 48 or 53 4 in two months. If

48 53 4

i.e.,

0 8

the value of the portfolio is certain to be 38.4. For this value of the portfolio is therefore

riskless. The current value of the portfolio is:

The value of the option is therefore $2.23.

This can also be calculated directly from equations (12.2) and (12.3). 1 06u , 0 96d so

that

e

p

and

0 10 2 12

0 5681 4 2 23f e

or $0 (if the stock price is $85). Consider a portfolio consisting of:

shares

1 option

(Note: The delta, of a put option is negative. We have constructed the portfolio so that it

is +1 option and shares rather than 1 option and shares so that the initial

investment is positive.)

The value of the portfolio is either 85 or 75 5 in four months. If

85 75 5

i.e.,

0 5

the value of the portfolio is certain to be 42.5. For this value of the portfolio is therefore

1 80f

The value of the option is therefore $1.80.

This can also be calculated directly from equations (12.2) and (12.3). 1 0625u ,

0 9375d so that

0 05 4 12

0 9375

0 6345

1 0625 0 9375

1 0 3655p

At the end of three months the value of the option is either $5 (if the stock price is $35) or $0

(if the stock price is $45).

Consider a portfolio consisting of:

shares

1 option

(Note: The delta, , of a put option is negative. We have constructed the portfolio so that it

is +1 option and shares rather than

1

investment is positive.)

35 5

45

riskless. The current value of the portfolio is

(40 0 5 ) 1 02 22 5f

Hence

i.e., the value of the option is $2.06.

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

45 35(1 ) 40 1 02p p

i.e.,

10 5 8p

or:

1 02

A stock price is currently $50. Over each of the next two three-month periods it is expected to

go up by 6% or down by 5%. The risk-free interest rate is 5% per annum with continuous

compounding. What is the value of a six-month European call option with a strike price of

$51?

A tree describing the behavior of the stock price is shown in Figure S12.1. The risk-neutral

e

p

There is a payoff from the option of 56 18 51 5 18 for the highest final node (which