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ACTS 4302
Instructor: Natalia A. Humphreys
SOLUTION TO HOMEWORK 7
Lesson 13: Asian, Barrier, and Compound options.
Sufficient work must be shown to get credit for a correct answer. Partial credit may be given for
incorrect answers which have some positive work.
Problem 1
You are given the following binomial tree for stock prices. Each period is one year.
43.2
36
30
28.8
24
19.2
A 2-year arithmetic annual average strike Asian call option on this stock is priced using this binomial
tree. You are given:
(i) r = 0.05
(ii) δ = 0.03
Determine the price of the option.
(A) 1.24
(B) 1.60
(C) 1.52
(D) 1.79
(E) 2.13
Solution. In the tree u = 1.2 and d = 0.8 at all nodes. The risk-neutral probability of an up is
e0.02 − 0.8
e(r−δ)h − d
=
= 0.5505; 1 − p∗ = 0.4495
u−d
0.4
The stock prices and their averages are:
p∗ =
Scenario Yr 1 Yr 2 Arith Average Payoff
Probability
uu
36
43.2
39.6
3.6
0.55052 = 0.3031
ud
36
28.8
32.4
0
0.5505 · 0.4495 = 0.2474
du
24
28.8
26.4
2.4
0.4495 · 0.5505 = 0.2474
dd
24
19.2
21.6
0
0.44952 = 0.2020
Therefore, the average payoff is
0.3031 · 3.6 + 0.2474 · 2.4 = 1.6849
Discounting for two years, the value of the option is
1.6849 · e−2·0.05 = 1.5245
Problem 2
Copyright ©Natalia A. Humphreys, 2014
ACTS 4302. AU 2014. HW7 SOLUTIONS
You are given the following binomial tree of stock prices. Each period of the tree is one month.
69.12
57.6
48
48
48
40
33.33
40
48
40
33.33
33.33
33.33
27.78
23.15
A 3-month geometric monthly average strike Asian put option is priced using this tree. You are given:
(i) r = 0.03
(ii) δ = 0.01
Calculate the value of the price of the option.
(A) 2.23
(B) 2.25
(C) 2.27
(D) 2.46
(E) 2.59
Key: A
Solution. In the tree u = 1.2 and d = 1.2−1 = 0.8333 at all nodes. The risk-neutral probability of an
up is
1
e0.02· 12 − 0.8333
0.1683
e(r−δ)h − d
=
=
= 0.4591
p =
u−d
1.2 − 0.8333
0.3667
1 − p∗ = 0.5409
∗
The stock prices and their averages are:
Scenario Per 1 Per 2 Per 3 Geom Average Payoff Probability
uuu
48
57.60 69.12
57.60
0
0.0968
uud
48
57.60
48
51.01
3.0076
0.1140
udu
48
40
48
45.17
0
0.1140
udd
48
40
33.33
40
6.6667
0.1343
duu
33.33
40
48
40
0
0.1140
dud
33.33
40
33.33
35.42
2.0886
0.1343
ddu
33.33 27.78 33.33
31.37
0
0.1343
ddd
33.33 27.78 23.15
27.78
4.6293
0.1583
Therefore, the average payoff is
3.0076 · 0.1140 + 6.6667 · 0.1343 + 2.0886 · 0.1343 + 4.6296 · 0.1583 = 2.2516
Discounting for three months, the value of the option is
2.2516 · e−0.25·0.03 = 2.2348 ≈ 2.23
Page 2 of 8
Copyright ©Natalia A. Humphreys, 2014
ACTS 4302. AU 2014. HW7 SOLUTIONS
Problem 3
St is the price of a nondividend paying stock at time t. You are given:
(i) The stock follows the Black-Scholes framework.
(ii) S0 = 55
(iii) α = 0.10
(iv) σ = 0.25
√
(v) U = S1 S2
Calculate the standard deviation of ln U .
Solution. Let
Q1 =
S(1)
S(2)
and Q2 =
S(0)
S(1)
Then
S1 S2
S1
· S0 ·
·
= S02 · Q21 · Q0
S1 S2 = S0 ·
S0
S0 S1
Qi are independent and are each lognormal with parameters m and v.
1
1
U = (S1 S2 ) 2 = S0 · Q1 · Q02
ln U = ln S0 + R, where R = ln Q1 +
1
ln Q2
2
1
3
E[R] = m + m = m
2
2
1
5
V ar[R] = V ar[U ] = v 2 + v 2 = v 2 = 0.078125
4√ 4 √
p
p
√
5
5
V ar[ln U ] = V ar[R] =
v=
· 0.25 = 0.2795 (v = σ t)
2
2
Note that
3
E[ln U ] = ln 55 + (0.1 − 0.5 · 0.252 ) = 4.1105
2
Problem 4
For 3-month options on a stock with strike price 40, you are given:
(i) The stock’s price is 40.
(ii) δ = 0.03.
(iii) A European put option has premium 1.70.
(iv) A down-and-in call option with barrier 35 and strike price 40 has premium 0.95.
(v) The continuously compounded risk-free interest rate is 6%.
Determine the premium for a 3-month down-and-out call option with barrier 35 and strike price 40.
(A) 0.45
(B) 0.75
(C) 1.05
(D) 1.35
(E) 2.95
Key: C
Solution. By the parity relationship for barrier options having the same barrier:
Knock-in option + Knock-out option = Ordinary option
Therefore,
Cdown-and-out = C − Cdown-and-in
By PCP for a dividend paying stock with continuous dividends:
C(S, K, T ) − P (S, K, T ) = S0 e−δT − Ke−rT ⇒ C = 1.7 + 40 · e−0.03·0.25 − 40 · e−0.06·0.25 = 1.9966
Hence,
Cdown-and-out = 1.9966 − 0.95 = 1.0466 ≈ 1.05
Page 3 of 8
Copyright ©Natalia A. Humphreys, 2014
ACTS 4302. AU 2014. HW7 SOLUTIONS
Problem 5
For a stock with price 40, you have the following portfolio of barrier options, all expiring in 3 months:
(i) An up-and-in call, barrier 45, exercise price 40.
(ii) An up-and-in call, barrier 50, exercise price 40.
(iii) An up-and-out call, barrier 45, exercise price 40.
(iv) An down-and-in call, barrier 35, exercise price 45.
(v) An down-and-in put, barrier 35, exercise price 40.
(vi) A down deferred rebate, barrier 32, payoff 5.
During the 3 month period, the minimum stock price is 31 and the maximum stock price is 47. The
final price is 44.
Determine the total payoff at the end of 3 months.
Solution. The following table shows payoff for each option:
Option Payoff
(i)
4
(ii)
0
(iii)
0
(iv)
0
(v)
0
(vi)
5
Thus, the total payoff is 9.
Brief explanations:
(i) max(S) = 47 > barrier of 45 ⇒ barrier is met ⇒ C = max(0, 44 − 40) = 4.
(ii) barrier is not met ⇒ C = 0.
(iii) barrier is met, out option ⇒ C = 0.
(iv) min(S) = 31 > barrier of 35 ⇒ barrier is met ⇒ C = max(0, 44 − 45) = 0.
(v) max(0, 40 − 44) = 0. Problem 6
For a stock following the Black-Scholes framework:
(i) The current price is 90.
(ii) The stock pays a dividend of 2 at the end of 3 months.
(iii) The continuously compounded risk-free interest rate is 4%.
(iv) σ = 0.3.
Let S(t) be the price of the stock at time t.
Calculate the current value to the buyer of an agreement to pay the buyer the maximum of 100 and
the stock price at the end of 4 months.
Solution. In general,
max(S, K) = K + max(S − K, 0)
The agreement pays
max(S(t), 100) = 100 + max(S(t) − 100, 0)
The present value of the above maximum is
1
100e−0.04· 3 + C
Page 4 of 8
Copyright ©Natalia A. Humphreys, 2014
ACTS 4302. AU 2014. HW7 SOLUTIONS
The pre-paid forward on the stock is
F P (S) = S0 − P V (Div) = 90 − 2e−0.04·0.25 = 90 − 1.9801 = 88.02
Using F P (S) for S and δ = 0 in the Black-Scholes Formula for a stock, we get:
C = SN (d1 ) − Ke−rt N (d2 ), where
ln 88.02
+ 0.5 · 0.32
ln (S/K) + (r − δ + 0.5σ 2 )t
100 + 0.04q
√
d1 =
=
σ t
0.3 13
1
3
=−
0.0993
= −0.5732
0.1732
N (d1 ) = N (−0.57) = 1 − N (0.57) = 1 − 0.7157 = 0.2843
√
d2 = d1 − σ t = −0.5732 − 0.1732 = −0.7464
N (d2 ) = N (−0.75) = 1 − N (0.75) = 1 − 0.7734 = 0.2266
1
Ke−rt = 100e−0.04· 3 = 98.6755
C = 88.02 · 0.2843 − 98.6755 · 0.2266 = 25.0241 − 22.3599 = 2.6642
Therefore, the current value to the buyer of an agreement to pay the buyer the maximum of 100 and
the stock price at the end of 4 months is:
1
100e−0.04· 3 + 2.6642 = 98.6755 + 2.6642 = 101.3397
Problem 7
For a stock you are given:
(i) The current price is 50.
(ii) The continuous dividend rate is 0.02
(iii) The continuously compounded risk-free interest rate is 6%.
Compound options with strike price 5.00 and 3-month expiry allow buying an option on the stock
expiring 1 year from now with a strike price of 55.
The following table has prices for 3 of the 4 compound options.
Premium for . . .
Call Put
Call on . . .
1.8
Put on . . .
1.25
3.6
Determine the price of the Put on Put option.
(A) −1.54
(B) −0.27
(C) 0.26
(D) 1.25
(E) 3.05
Key: C
Solution. Parity relationships for compound options:
P utOnP ut = CallOnP ut − P + xe−rt1
C = CallOnCall − P utOnCall + xe−rt1
P = C − Se−δt + Ke−rt
Using the values given:
C = 1.8 − 1.25 + 5e−0.06·0.25 = 5.4756
P = 5.4756 − 50e−0.02·1 + 55e−0.06·1 = 5.4756 − 49.0099 + 51.7971 = 8.2627
P utOnP ut = 3.6 − 8.2627 + 5e−0.06·0.25 = 3.6 − 8.2627 + 4.9256 = 0.2629 ≈ 0.26
Page 5 of 8
Copyright ©Natalia A. Humphreys, 2014
ACTS 4302. AU 2014. HW7 SOLUTIONS
Problem 8
For a European call-on-call option:
(i) The price of the underlying stock is 45.
(ii) The stock’s annual volatility is 0.3.
(iii) The continuous annual dividend rate is 2%
(iv) The continuously compounded risk-free rate is 5%.
(v) For the call-on-call option, the premium is 0.75, time to expiry is 3 months, and the strike price
is 5.
(vi) For the underlying call option, time to expiry is 6 months and the strike price is 45.
(vii) Options are priced using the Black-Scholes formula.
Determine the premium for a European put-on-call option with the same underlying asset and strike
price.
(A) 1.35
(B) 1.61
(C) 2.27
(D) 2.31
(E) 2.85
Key: B
Solution. We are given: S0 = 45, σ = 0.3, δ = 2%, r = 5%, CC = 0.75, t1 = 0.25, x = 5, K = 45, t =
0.5. We need to find P C. By definition of a compound option, for t0 < t1 < T , CC allows buying a
call option at time t1 and P C allows selling a call option at time t1 .
By the parity relationships for compound options:
CallOnCall(S, K, x, σ, r, t1 , t, δ) − P utOnCall(S, K, x, σ, r, t1 , t, δ) = C(S, K, σ, r, t, δ) − xe−rt1
and
C = Se−δt N (d1 ) − Ke−rt N (d2 )
The underlying call value is
ln 45 + 0.05 − 0.02 + 0.5 · 0.32
ln (S/K) + (r − δ + 12 σ 2 )t
q
√
= 45
d1 =
σ t
0.3 12
1
2
=−
0.0375
= 0.1768
0.2121
N (d1 ) = N (0.1768) = 0.5702
√
d2 = d1 − σ t = 0.1768 − 0.2121 = −0.0354, N (d2 ) = N (−0.0354) = 0.4859
Se−δt = 45e−0.02·0.5 = 44.5522, Ke−rt = 45e−0.05·0.5 = 43.8889
C = 44.5522 · 0.5702 − 43.8889 · 0.4859 = 4.0763
By the parity relationships for compound options:
P utOnCall(S, K, x, σ, r, t1 , t, δ) = CallOnCall(S, K, x, σ, r, t1 , t, δ) − C(S, K, σ, r, t, δ) + xe−rt1 =
= 0.75 − 4.0763 + 5 · e−0.05·0.25 = 0.75 − 4.0763 + 4.9379 = 1.6116 ≈ 1.61
Problem 9
A 6-month European put-on-call option is modeled with a 1-year 2-period binomial tree with u =
1.2, d = 0.8. You are given:
(i) The price of the underlying stock is 50.
(ii) The continuous annual dividend rate is 0.03.
(iii) The strike price of the underlying call is 55.
(iv) The underlying call expires in 1 year.
(v) The strike price of the put-on-call is 0.6.
(vi) The continuously compounded risk-free rate is 0.05.
Determine the premium of the put-on-call option.
(A) 0
(B) 0.26
(C) 0.28
(D) 4.15
(E) 7.67
Key: C
Page 6 of 8
Copyright ©Natalia A. Humphreys, 2014
ACTS 4302. AU 2014. HW7 SOLUTIONS
Solution. The stock and the call trees are
72
60
50
48
40
32
17
Cu
0
C
Cd
0
The risk-neutral probability is:
e(r−δ)h − d
e0.02·0.5 − 0.8
=
= 0.5251; 1 − p∗ = 0.4749
u−d
0.4
Calculate the value of the underlying call:
p∗ =
Cu = e−0.05·0.5 · 0.5251 · 17 = 8.7067, Cd = 0
C = e−0.05·0.5 · 0.5251 · 8.7067 = 4.4592
Thus, the call tree is:
17
8.7067
4.4592
0
0
0
And the corresponding 6-month put is:
0
PC
0.6
Page 7 of 8
Copyright ©Natalia A. Humphreys, 2014
ACTS 4302. AU 2014. HW7 SOLUTIONS
Therefore,
P C = e−0.05·0.5 · 0.4749 · 0.6 = 0.2779 ≈ 0.28
Problem 10
You are given the following binomial tree for stock prices. Each period is one year.
101.4
78
60
54.6
42
29.4
You are given that r = 0.05 and δ = 0.02.
A two-year up-and-in barrier call option with barrier 70 and strike price 65 is priced using this tree. It
is assumed that the price movements within each period are monotonic.
Determine the price of this call option.
(A) 0
(B) 2.33
(C) 8.83
(D) 9.99
(E) 11.29
Key: D
Solution. u = 1.3 and d = 0.7 at all nodes. Then
0.3305
e0.05−0.02 − 0.7
=
= 0.5508, 1 − p∗ = 0.4492
p∗ =
1.3 − 0.7
0.6
The option only pays if the stock price first goes up to 78, but then it pays only at 101.4 node. The
payoff is 36.4 at this node. It is zero at all other nodes. Calculating the expected value of 1 path and
discounting:
Cui = e−2·0.05 · 36.4 · 0.55082 = 9.9922 ≈ 9.99 Page 8 of 8