12
Introduction CHAPTER 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
change.
Problem 12.9.
A stock price is currently $50. It is known that at the end of two months it will be either $53
or $48. The risk-free interest rate is 10% per annum with continuous compounding. What is
the value of a two-month European call option with a strikeprice of $49? Use no-arbitrage
arguments.
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:
0.8 50 f
where f is the value of the option. Since the portfolio must earn the risk-free rate of interest
(0.8 50 f )e0.102/12 = 38.4
i.e.,
f = 2.23
The value of the option is therefore $2.23.
This can also be calculated directly from equations (12.2) and (12.3). u = 1.06 , d = 0.96 so
that
e0.102/12 0.96
p=
= 0.5681
1.06 0.96
and
f = e 0.102/12 0.5681 4 = 2.23
Problem 12.10.
A stock price is currently $80. It is known that at the end of four months it will be either $75
or $85. The risk-free interest rate is 5% per annum with continuous compounding. What is
the value of a four-month European put option with a strikeprice of $80? Use no-arbitrage
arguments.
At the end of four months the value of the option will be either $5 (if the stock price is $75)
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
riskless. The current value of the portfolio is:
0.5 80 + f
where f is the value of the option. Since the portfolio is riskless
(0.5 80 + f )e0.054/12 = 42.5
i.e.,
f = 1.80
The value of the option is therefore $1.80.
This can also be calculated directly from equations (12.2) and (12.3). u = 1.0625 , d = 0.9375
so that
e0.054/12 0.9375
p=
= 0.6345
1.0625 0.9375
1 p = 0.3655 and
f = e 0.054/12 0.3655 5 = 1.80
Problem 12.11.
A stock price is currently $40. It is known that at the end of three months it will be either $45
or $35. The risk-free rate of interest with quarterly compounding is 8% per annum. Calculate
the value of a three-month European put option on the stock with an exercise price of $40.
Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same
answers.
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 option and + shares so that the initial
investment is positive.)
The value of the portfolio is either 35 + 5 or 45 . If:
35 + 5 = 45
i.e.,
= 0.5
the value of the portfolio is certain to be 22.5. For this value of the portfolio is therefore
riskless. The current value of the portfolio is
40 + f
where f is the value of the option. Since the portfolio must earn the risk-free rate of interest
( 40 0.5 + f ) 1.02 = 22.5
Hence
f = 2.06
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 p + 35(1 p ) = 40 1.02
i.e.,
10 p = 5.8
or:
p = 0.58
The expected value of the option in a risk-neutral world is:
0 0.58 + 5 0.42 = 2.10
This has a present value of
2.10
= 2.06
1.02
This is consistent with the no-arbitrage answer.
Problem 12.12.
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
probability of an up move, p , is given by
e0.053/12 0.95
p=
= 0.5689
1.06 0.95
There is a payoff from the option of 56.18 51 = 5.18 for the highest final node (which
corresponds to two up moves) zero in all other cases. The value of the option is therefore
5.18 0.5689 2 e 0.056 /12 = 1.635
This can also be calculated by working back through the tree as indicated in Figure S12.1.
The value of the call option is the lower number at each node in the figure.
Figure S12.1 Tree for Problem 12.12
Problem 12.13.
For the situation considered in Problem 12.12, what is the value of a six-month European
put option with a strike price of $51? Verify that the European call and European put prices
satisfy putcall parity. If the put option were American, would it ever be optimal to exercise
it early at any of the nodes on the tree?
The tree for valuing the put option is shown in Figure S12.2. We get a payoff of
51 50.35 = 0.65 if the middle final node is reached and a payoff of 51 45.125 = 5.875 if
the lowest final node is reached. The value of the option is therefore
(0.65 2 0.5689 0.4311 + 5.875 0.43112 )e 0.056 /12 = 1.376
This can also be calculated by working back through the tree as indicated in Figure S12.2.
The value of the put plus the stock price is from Problem 12.12
1.376 + 50 = 51.376
The value of the call plus the present value of the strike price is
1.635 + 51e 0.056/12 = 51.376
This verifies that putcall parity holds
To test whether it worth exercising the option early we compare the value calculated for the
option at each node with the payoff from immediate exercise. At node C the payoff from
immediate exercise is 51 47.5 = 3.5 . Because this is greater than 2.8664, the option should
be exercised at this node. The option should not be exercised at either node A or node B.
Figure S12.2 Tree for Problem 12.13
Problem 12.14.
A stock price is currently $25. It is known that at the end of two months it will be either $23
or $27. The risk-free interest rate is 10% per annum with continuous compounding. Suppose
ST is the stock price at the end of two months. What is the value of a derivative that pays off
ST2 at this time?
At the end of two months the value of the derivative will be either 529 (if the stock price is
23) or 729 (if the stock price is 27). Consider a portfolio consisting of:
+ :
shares
1 : derivative
The value of the portfolio is either 27 729 or 23 529 in two months. If
27 729 = 23 529
i.e.,
= 50
the value of the portfolio is certain to be 621. For this value of the portfolio is therefore
riskless. The current value of the portfolio is:
50 25 f
where f is the value of the derivative. Since the portfolio must earn the risk-free rate of
interest
(50 25 f )e0.102 /12 = 621
i.e.,
f = 639.3
The value of the option is therefore $639.3.
This can also be calculated from directly equations (12.2) and (12.3). u = 1.08 , d = 0.92 so
that
e0.102 /12 0.92
p=
= 0.6050
1.08 0.92
and
f = e 0.102/12 (0.6050 729 + 0.3950 529) = 639.3
Problem 12.15.
Calculate u , d , and p when a binomial tree is constructed to value an option on a foreign
currency. The tree step size is one month, the domestic interest rate is 5% per annum, the
foreign interest rate is 8% per annum, and the volatility is 12% per annum.
In this case
a = e( 0.050.08)1/12 = 0.9975
u = e0.12
1/12
= 1.0352
d = 1 / u = 0.9660
p=
0.9975 0.9660
= 0.4553
1.0352 0.9660
Further Questions
Problem 12.16.
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 : option
The 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 )e0.120.5 = 28
i.e.,
f = 6.96
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
60 p + 42(1 p) = 50 e0.06
i.e.,
18 p = 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 12.17.
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 M12.1. The riskneutral probability of an up move, p , is given by
e0.123/12 0.90
p=
= 0.6523
1.1 0.9
Calculating the expected payoff and discounting, we obtain the value of the option as
[2.4 2 0.6523 0.3477 + 9.6 0.3477 2 ]e 0.126 /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 S12.3. 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.
40.000
2.118
2.537
44.000
0.810
0.810
B
A
C
36.000
4.759
6.000
48.400
0.000
0.000
39.600
2.400
2.400
32.400
9.600
9.600
Figure S12.3 Tree to evaluate European and American put options in Problem 12.17. 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 12.18.
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 + 0.3477 q )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 12.19.
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 two2
step tree to calculate the value of a derivative that pays off max[(30 ST ), 0] 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 M12.2. The risk-neutral probability of an up move, p , is given by
e0.052 /12 0.9
p=
= 0.6020
1.08 0.9
Calculating 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 0.054 /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 S12.4. 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.
32.400
0.2785
30.000
5.3940
34.922
D 0.000
B
A
C
27.000
13.2449
E
F
29.160
0.7056
24.300
32.49
Figure S12.4 Tree to evaluate European power option in Problem 12.19. At each node, upper
number is the stock price and the next number is the option price
Problem 12.20.
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) This problem is based on the material in Section 12.8. In this case t = 0.25 so that
u = e0.30 0.25 = 1.1618 , d = 1 / u = 0.8607 , and
e0.040.25 0.8607
p=
= 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 M12.3 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
40
40
3.373919
0
34.42832
0
29.63273
0
Node Time:
0.0000
0.2500
0.5000
Figure M12.3 Tree produced by DerivaGem to evaluate European option in Problem 12.20
Problem 12.21.
Repeat Problem 12.20 for an American put option on a futures contract. The strike price and
the futures price are $50, the risk-free rate is 10%, the time to maturity is six months, and the
volatility is 40% per annum.
(a) In this case t = 0.25 and u = e0.40 0.25 = 1.2214 , d = 1 / u = 0.8187 , and
e0.10.25 0.8187
p=
= 0.4502
1.2214 0.8187
(b) and (c) The value of the option using a two-step tree is 4.8604.
(d) With 5, 50, 100, and 500 time steps the value of the option is 5.6858, 5.3869, 5.3981, and
5.4072, respectively.