ACT4000, MIDTERM #1
ADVANCED ACTUARIAL TOPICS
FEBRUARY 9, 2009
HAL W. PEDERSEN
You have 70 minutes to complete this exam. When the invigilator instructs you to
stop writing you must do so immediately. If you do not abide by this instruction you
will be penalised. All invigilators have full authority to disqualify your paper if, in
their judgement, you are found to have violated the code of academic honesty.
Each question is worth 10 points. Provide sufficient reasoning to back up your answer
but do not write more than necessary.
This exam consists of 8 questions. Answer each question on a separate page of the
exam book. Write your name and student number on each exam book that you use
to answer the questions. Good luck!
Question 1. You are considering investments in a single-period binomial market.
(As you know, this means that we are currently at time 0 and at time 1 the world will
be in one of two states which we will call the “upstate” and the “downstate”.) There
are two assets available for trade. The first asset currently sells for 10 and at time 1
will be worth 12 in the upstate and 8 in the downstate. The second asset currently
sells for 10 and at time 1 will be worth 15 in the upstate and 2 in the downstate.
(i) (3 points) What is the current price of an asset that at time 1 pays 1 in the upstate
and 0 in the downstate?
(ii) (2 points) What is the current price of an asset that at time 1 pays 0 in the
upstate and 1 in the downstate?
(iii) (5 points) If you do not want any risk, is it possible for you to deposit 100 at
time 0 and receive a certain payoff at time 1? Explain how this can be done or why it
cannot be done. If this can be done, what is the effective interest-rate you will earn
over the period?
Question 2. An equity securities market model follows a multi-period binomial
model. At each node of the binomial tree the current stock price S will branch
to uS in the upstate and dS in the downstate. You are given that the initial stock
price is 10, u = 1.35, d = 0.85 and the interest-rate is 5% effective per period. The
stock does not pay dividends.
(i) (4 points) Compute the price of a European put option on the stock which expires
in 4 periods and has a strike price of 9.0.
(ii) (6 points) Compute the price of an American put option on the stock which
expires in 4 periods and has a strike price of 9.0 and describe the optimal exercise
policy for this American put option.
1
2
ACT4000 – MIDTERM #1
Question 3. The insurance company you work for has recently began issuing a “stock
index GIC.” The essence of the contract is that the investor places an amount of
principal in an account for two years and the investor is guaranteed some minimum
effective return over the two-year period. The investor’s returns are based on the
returns on the TSE 35 index. As pricing actuary, you are told that the product is to
guarantee a 0% return and you are to set the maximum total return the investor will
receive over the two years so that the insurance company will break even.
The continuously compounded risk-free interest rate is 4% and the stock index is
currently at 50 and will go to either 25 or 75 at the end of the two years.
(i) (5 points) Compute the maximum total return the investor will receive over the
two years so that the insurance company will break even. If this break even return
does not exist then explain why.
(ii) (2 points) What are the embedded options in this contract?
(iii) (3 points) Write a general algebraic expression for the cost to the insurance
company per $1 invested of providing a maximum return of R, for each R > 0?
Question 4. Assume the Black-Scholes option pricing model. Consider a standard
European call option on a stock. The strike price of the option is equal to the current
price of the stock (i.e. the option is at-the-money). The option has one year to
maturity and the stock does not pay dividends.
Is the option’s delta greater than 0.5, less than 0.5, or equal to 0.5. Justify your
answer.
Question 5 (Text Question 12.7, page 407). You are given the following data.
•
•
•
•
•
•
S = $100
K = $95
σ = 30%
r = 0.08
δ = 0.03
T = 0.75
Compute the Black-Scholes price of a call.
ACT4000 – MIDTERM #1
3
Question 6 (Text Question 13.2, page 439). You are given the following data assuming a Black-Scholes model.
•
•
•
•
S = $40
σ = 30%
r = 0.08
δ=0
Suppose you sell a 40-strike put with 91 days to expiration.
(i) (5 points) What is delta?
(ii) (5 points) If the option is on 100 shares, what investment is required for a deltahedged portfolio?
Question 7 (Text Question 10.1, page 338). You are given the following data.
•
•
•
•
•
S = $100
K = 105
r = 8% (continuously compounded)
T = 0.5
δ=0
You are given u = 1.3 and d = 0.8.
For a single-period binomial model compute the premium, ∆ and B for a European
call option.
Question 8. For a two-period binomial model, you are given the following data.
• Each period is one year.
• The current price for a non-dividend paying stock is $20.
• u = 1.2840, where u is one plus the rate of capital gain on the stock per period
if the stock price goes up.
• d = 0.8607, where d is one plus the rate of capital loss on the stock per period
if the stock price goes down.
• The continuously compounded risk-free interest rate is 5%.
Calculate the price of an American call option on the stock with a strike price of $22.
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Question 2
u
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cont. int_rate
delta
h
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0.8500
0.0488
0.0000
1.0000
p*
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n
K
4
9.00
American
Put Price
0.68467 European
Put Price
0.60470
American
Call Price
3.20038 European
Call Price
3.20038
Stock Prices
7
r=ln(1.05)
(i) Answer = 0.60470
(ii) Answer = 0.68467
Exercise at period 3 if stock has fallen three times.
6
(Exercise when intrinsic value is greater than value of future cash flows if not exercised.)
10
9
8
33.21506
20.91319
13.16756
8.29069
5.22006
44.84033
28.23280
17.77621
11.19243
7.04708
4.43705
60.53445
38.11428
23.99788
15.10978
9.51356
5.99002
3.77150
81.72151
51.45428
32.39714
20.39820
12.84331
8.08653
5.09152
3.20577
4
5
6
7
5
4
10.00
13.50000
8.50000
18.22500
11.47500
7.22500
24.60375
15.49125
9.75375
6.14125
0
1
2
3
3
2
1
0
S_0
110.32404
69.46328
43.73614
27.53757
17.33847
10.91681
6.87355
4.32779
2.72491
148.93745
93.77543
59.04379
37.17572
23.40693
14.73770
9.27929
5.84252
3.67862
2.31617
201.06556
126.59683
79.70912
50.18722
31.59936
19.89589
12.52704
7.88740
4.96614
3.12683
1.96874
8
9
10
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
Put Intrinsic Value
European Put Payoff
10
9
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
6
7
8
9
10
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
9
10
8
7
6
0.00000
0.50000
0.00000
0.00000
1.77500
0.00000
0.00000
0.00000
2.85875
0.00000
0.00000
0.00000
0.70931
3.77994
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1
2
3
4
5
5
4
3
2
1
0
0
American Put Prices
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
10
9
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
6
7
8
9
10
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
8
7
6
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.70931
3.77994
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1
2
3
4
5
5
4
3
2
1
0
0
European Put Prices
10
9
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
6
7
8
8
7
6
0.68467
0.13235
1.10994
0.00000
0.23161
1.78798
0.00000
0.00000
0.40532
2.85875
0.00000
0.00000
0.00000
0.70931
3.77994
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0
1
2
3
4
5
5
4
3
2
1
0
Price
Call Intrinsic Value
10
9
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
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0.00000
0.00000
0.00000
0.00000
0.00000
6
7
8
9
10
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
8
7
6
0.60470
0.13235
0.96999
0.00000
0.23161
1.54308
0.00000
0.00000
0.40532
2.43018
0.00000
0.00000
0.00000
0.70931
3.77994
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0
1
2
3
4
5
5
4
3
2
1
0
Price
European Call Payoff
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
9
10
10
9
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
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0.00000
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0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
5
6
7
8
8
7
6
5
4.50000
0.00000
9.22500
2.47500
0.00000
15.60375
6.49125
0.75375
0.00000
24.21506
11.91319
4.16756
0.00000
0.00000
1
2
3
4
4
3
2
1
0
0
American Call Prices
9
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
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5
6
7
8
9
10
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
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0.00000
0.00000
0.00000
0.00000
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8
9
10
8
7
6
5
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
24.21506
11.91319
4.16756
0.00000
0.00000
1
2
3
4
4
3
2
1
0
0
European Call Prices
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
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8
9
10
10
9
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5
6
7
7
6
5
3.20038
5.85781
1.69546
10.06173
3.54335
0.60482
16.03232
6.91982
1.58764
0.00000
24.21506
11.91319
4.16756
0.00000
0.00000
0
1
2
3
4
4
3
2
1
0
10
Price
10
9
8
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5
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3.20038
5.85781
1.69546
10.06173
3.54335
0.60482
16.03232
6.91982
1.58764
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24.21506
11.91319
4.16756
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(j;'"
c:..
.
<::..
II
"
J-- ~-
./
."~"
s
I .5
)
·s
cal/
-
f~+
:;: (S -
C
I~JL-J) -f
-
(-)
-
5/+
/ ,,/
/ ./
o
-
(I - . '5) :::
-.5
Ddt6;;
J,
f'J(rl,)
t-(
•.
!'-J L A,)
s/1<-)
=
A
\
7'"
.
I
7
z:
Without dividends, the standard Black and Scholes
(1973) pricing formula for the European call opt.ion is given by
c( t)
S(t)N(dd
In
d1
The option's
- e-r(T-t)
(¥) + (r + 4a2)(T
-
- t)
a .JY=t
T- t
aVT - t.
"delta" is given by
where
J( N(d2),
g~~~~
= N(d1).
'
and
With the option
struck at-the-money, S(t) = K, and thus, In e~») = 0 [remember
that In(l) = 0]. All other terms in d1 are positive. Therefore,
d1 > 0, and N(dd
> 0.5 (remember that N(O) = 0.5 and N(·) is
an increasing function of its argument). Thus, an at-the-money
option on a non-dividend-paying stock always has a delta slightly
greater than one-half.
Question 5
Black-Scholes Option Pricing Model
S_0
K
sigma
r
T
delta
Stock Price
Strike Price
Volatility
Interest Rate
Time to Expiration (Years)
Dividend Yield
d_1
d_2
N(d_1)
N(d_2)
Call Price
Put Price
100.00000
95.00000
0.30000
0.08000
0.75000
0.03000
0.07125
0.12254
0.47167
0.21186
0.68142
0.58389
97.77512
89.46763
14.38631
3.85394
Summary
Call Price
Call Delta
Call Gamma
Put Price
Put Delta
Put Gamma
14.38631
0.66626
0.01343
3.85394
-0.31149
0.01343
C(S,K,sigma,r,T,delta) = S_0*exp(-delta*T)*N(d_1)-K*exp(-r*T)*N(d_2)
P(S,K,sigma,r,T,delta) = K*exp(-r*T)*N(-d_2)-S_0*exp(-delta*T)*N(-d_1)
Answer:
Call Price = 14.38631
Question 6
Black-Scholes Option Pricing Model
S_0
K
sigma
r
T
delta
Stock Price
Strike Price
Volatility
Interest Rate
Time to Expiration (Years)
Dividend Yield
d_1
d_2
N(d_1)
N(d_2)
Call Price
Put Price
40.00000
40.00000
0.30000
0.08000
0.24932
0.00000
0.03116
0.03116
0.20805
0.05825
0.58240
0.52323
40.00000
39.21010
2.78040
1.99049
Summary
Call Price
Call Delta
Call Gamma
Put Price
Put Delta
Put Gamma
2.78040
0.58240
0.06516
1.99049
-0.41760
0.06516
C(S,K,sigma,r,T,delta) = S_0*exp(-delta*T)*N(d_1)-K*exp(-r*T)*N(d_2)
P(S,K,sigma,r,T,delta) = K*exp(-r*T)*N(-d_2)-S_0*exp(-delta*T)*N(-d_1)
Answer:
(i) Put Delta = -0.41760
(ii) Short 100*Delta = 41.76 shares & deposit short sale proceeds plus 199.05 put option premiums in the bank.
Question 7
One Period Interest Rate
0.0408
Traded Asset -- Bank Account
Time
Cash Flow Matrix
1
1.0408
1.0408
0
1
100.00
130.00
80.00
0
1
Cash Flow Matrix Inverse
Replicating Portfolio (Bank Account & Stock)
Traded Asset -- Stock
Price of Cash Flows
Time
Cash Flows to Price
Time
25.00
0.00
0
1
Answer:
Delta = 0.5, B=-38.4316, Price=11.5684
Note: One Period Interest Rate = Exp(0.08*0.5)-1
1.0408
1.0408
130.00
80.00
-1.5372631
0.0200000
2.4980525
-0.0200000
-38.4315776
0.5000000
11.5684224
Question 8
u
d
int_rate
delta
h
1.2840
0.8607
0.0500
0.0000
1.0000
p*
0.45020
n
K
2
22.00
American
Put Price
2.50300 European
Put Price
1.96488
American
Call Price
2.05845 European
Call Price
2.05845
Stock Prices
Answer:
American Call Price = 2.05845
10
9
8
69.79995
46.78880
31.36380
21.02401
14.09296
9.44689
89.62314
60.07682
40.27112
26.99482
18.09536
12.12981
8.13094
115.07611
77.13863
51.70812
34.66135
23.23444
15.57468
10.44013
6.99830
5
6
7
7
6
5
20.00
25.68000
17.21400
32.97312
22.10278
14.81609
42.33749
28.37996
19.02386
12.75221
54.36133
36.43987
24.42664
16.37384
10.97583
0
1
2
3
4
4
3
2
1
0
S_0
147.75772
99.04601
66.39322
44.50518
29.83303
19.99789
13.40512
8.98582
6.02344
189.72091
127.17507
85.24890
57.14465
38.30561
25.67729
17.21218
11.53779
7.73409
5.18437
243.60165
163.29279
109.45958
73.37373
49.18440
32.96963
22.10044
14.81452
9.93058
6.65673
4.46219
8
9
10
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
Put Intrinsic Value
European Put Payoff
10
9
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
5
6
7
8
9
10
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
9
10
8
7
6
5
0.00000
4.78600
0.00000
0.00000
7.18391
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1
2
3
4
4
3
2
1
0
0
American Put Prices
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
10
9
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
5
6
7
8
9
10
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
8
7
6
5
0.00000
0.00000
0.00000
0.00000
7.18391
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1
2
3
4
4
3
2
1
0
0
European Put Prices
10
9
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
6
7
8
8
7
6
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
3
4
5
5
4
3
2.50300
0.00000
4.78600
0.00000
0.00000
7.18391
0
1
2
2
1
0
Price
Call Intrinsic Value
10
9
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
6
7
8
9
10
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
8
7
6
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
3
4
5
5
4
3
1.96488
0.00000
3.75706
0.00000
0.00000
7.18391
0
1
2
2
1
0
Price
European Call Payoff
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
10
9
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
6
7
8
9
10
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
9
10
8
7
6
3.68000
0.00000
10.97312
0.10278
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1
2
3
4
5
5
4
3
2
1
0
0
American Call Prices
10
9
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
6
7
8
9
10
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
9
10
8
7
6
0.00000
0.00000
10.97312
0.10278
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1
2
3
4
5
5
4
3
2
1
0
0
European Call Prices
10
9
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
6
7
8
8
7
6
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
3
4
5
5
4
3
2.05845
4.75295
0.04401
10.97312
0.10278
0.00000
0
1
2
2
1
0
Price
10
9
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
6
7
8
8
7
6
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
3
4
5
5
4
3
2.05845
4.75295
0.04401
10.97312
0.10278
0.00000
0
1
2
2
1
0
Price
Question 8 (Alternative Solution)
Solution to (4)
Answer: (C)
First, we construct the two-period binomial tree for the stock price.
Year 0
Year 1
Year 2
32.9731
25.680
20
22.1028
17.214
14.8161
The calculations for the stock prices at various nodes are as follows:
Su = 20 × 1.2840 = 25.680
Sd = 20 × 0.8607 = 17.214
Suu = 25.68 × 1.2840 = 32.9731
Sud = Sdu = 17.214 × 1.2840 = 22.1028
Sdd = 17.214 × 0.8607 = 14.8161
The risk-neutral probability for the stock price to go up is
erh − d
e0.05 − 0.8607
=
= 0.4502 .
u−d
1.2840 − 0.8607
Thus, the risk-neutral probability for the stock price to go down is 0.5498.
p* =
If the option is exercised at time 2, the value of the call would be
Cuu = (32.9731 – 22)+ = 10.9731
Cud = (22.1028 – 22)+ = 0.1028
Cdd = (14.8161 – 22)+ = 0
If the option is European, then Cu = e−0.05[0.4502Cuu + 0.5498Cud] = 4.7530 and
Cd = e−0.05[0.4502Cud + 0.5498Cdd] = 0.0440.
But since the option is American, we should compare Cu and Cd with the value of the
option if it is exercised at time 1, which is 3.68 and 0, respectively. Since 3.68 < 4.7530
and 0 < 0.0440, it is not optimal to exercise the option at time 1 whether the stock is in
the up or down state. Thus the value of the option at time 1 is either 4.7530 or 0.0440.
Finally, the value of the call is
C = e−0.05[0.4502(4.7530) + 0.5498(0.0440)] = 2.0585.
Remark: Since the stock pays no dividends, the price of an American call is the same as
that of a European call. See pages 294-295 of McDonald (2006). The European option
price can be calculated using the binomial probability formula. See formula (11.17) on
page 358 and formula (19.1) on page 618 of McDonald (2006). The option price is
 2
 2
 2
e−r(2h)[   p *2 Cuu +   p * (1 − p*)Cud +  (1 − p*)2 Cdd ]
 2
1 
0
−0.1
2
= e [(0.4502) ×10.9731 + 2×0.4502×0.5498×0.1028 + 0]
= 2.0507