ACT 4000, MIDTERM #1
ADVANCED ACTUARIAL TOPICS
FEBRUARY 8, 2007
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. Consider a single-period binomial market. There are two assets available
for trade. The first asset pays 2 in the upstate and 1 in the downstate. The second
asset pays 2 in the upstate and 3 in the downstate. The price of asset 1 is PI = 1.39.
The price of asset 1 is P2 = 2.29. The market is arbitrage-free.
(1) [6 points] Compute the price of a risky asset that pays 10 in the upstate and 0
in the downstate.
(2) [4 points] What is the implicit effective interest rate in this model?
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 = 1.25, d = 0.85 and the interest-rate is 5% effective per period.
(i) (5 points) Compute the price of a European put option on the stock which expires
in 4 periods and has a strike price of 8.5.
(ii) (5 points) Compute the price of an American put option on the stock which
expires in 4 periods and has a strike price of 8.5 and describe the optimal exercise
policy for this American put option.
2
#1
ACT 4000 - MIDTERM
Question 3. In North America, it is common. to issue investment products with
return guarantees. For instance, a product may offer the returns of the S&P 500
index except the investor is guaranteed a 0% return over the investment horizon and
the maximum total return the investor will be credited over the investment horizon
is 30%.
You are the pricing actuary for a large insurance company. Your company has decided
to sell an investment product for which the investments will be credited a return equal
to that on an index that will experience either a 20% gain or a 10% loss at the end
of one period (i.e. the single-period binomial model). The continuously compounded
interest-rate (force of interest) for the period is 0.08. The customer is guaranteed a
5% return and you are to set the maximum total return the customer will receive
over the period (denoted a) so that your insurance company will break even on the
product.
Determine
a.
Suppose that the exchange rate is 0.79 €I$. Let r($) = 3%, r(€) = 5%, u = 1.1850, d = 0.8581,
T = 15 months, n = 3, and K = 0.8 €.
(a) What is the price of a 15-month European call?
(b) What is the price of a 15-month European put?
The price of a 6-month dollar-denominated
call option on the euro with a $0.90
strike is $0.0404. The price of an otherwise equivalent put option is $0.0141. The
annual continuously compounded dollar interest rate is 5%.
a. What is the 6-month dollar-euro
forward price?
b. If the euro-denominated
annual continuously
is 3.5%, what is the spot exchange rate?
compounded
interest rate
A one-period arbitrage-free model has two assets with the following price dynamics.
Asset
1
12
Asset
2
10
10.4
8
2.5
A call option on asset 1 with a strike price of 9 has a price of 1.8.
(i) Compute the state prices implicit in this model.
(ii) Compute the risk neutral probabilities for this model.
(iii) Compute the implied short rate for this model. [Compute the implied short rate as a
force of interest.]
(iv) The price of a call option on asset 2 is 2.4. What is the strike price of this call
option?
7
Suppose call and put prices are given by
Strike
Call premium
Put premium
50
18
7
55
14
10.75
60
9.50
14.45
Find the convexity violations. What spread would you use to effect arbitrage?
Demonstrate that the spread position is an arbitrage.
8
10 .
Suppose the S&P 500 futures price is 1000, a
and n = 3.
=
If )
30%, r
=
5%, 8
=
5%, T
a. What are the prices of European calls and puts for K
you find the prices to be equal?
=
b. What are the prices of American calls and puts for K
= $1000?
c. What are the time-O replicating
=
I,
$1 OOO? Why do
portfolios for the European call and put?
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9. 10
.
7.
(a) We can calculate the price of the call currency option in a very similar way to our previous
calculations. Please pay attention to the fact that we have a strike price and exchange rate in
Euros, therefore the foreign interest rate is the $ interest rate!
For the European call option, we have:
Time (yrs)
0.416667
0.79
0.095828
0.936117
0.17665
-
0.677895
0.03079
0.833333
1.25
1.10926
0.31197
1.314427
0.51443
0.803277
0.06837
0.95185
0.15185
0.581698
0.689288
o
o
0.499152
o
The value of the option is 0.096€.
(b) for the European put option, we have:
1.25
Time (yrs)
0.416667
0.833333
0.79
0.086435
0.936117
1.10926
1.314427
0.031
o
o '
0.677895
0.13698
0.803277
0.05858
0.95185
o
0.581698
0.20903
0.689288
0.11071
0.499152
0.30085
The value of the option is 0.086€.
Question 9.6.
a)
We can use put-call-parity
+ C (K,
'*
'*
Fo. T
=
T) - P (K, T)
FO,T = erT
to determine the forward price:
PV (forward
=
- PV (strike)
e-rT FO.T _ Ke-rT
+ Ke-rT]
[+C (K, T) -. P (K, T)
= eO.05*O.5[$0.0404
= $0.92697.
price)
- $0.0141 + $0.ge-O.05*O.5]
b)
Given the forward price from above and the pricing formula for the forward price, we can
find the current spot rate:
FO.T
-¢}
xo
=
= xoe(r-rf)T
=
Fo. Te-(r-rf)T
$0.92697e-(O.05-0.035)O.5
=
$0.92.
'.¢'quations(9.17) and (9.18) of the textbook are violated. To see this, let us calculate the values.
,b;>'t
!~ve:
. (K_1_) _-_C_( K_z_)
Kz 'ch
=
K)
_18_-_1_4= 0.8
55 - 50
C (Kz) - C (K3)
and
14 - 9.50
60 - 55
K3 - Kz
= 0.9,
violates equation (9. I7) and
10.75 - 7
55 - 50
= 0.75
and
P (K3) - P (Kz)
14.45 - 10.75
60 - 55
K3 - K2
= 0.74,
}1 violates equation (9.18) .
.a!culate lambda in order to know how many options to buy and sell when we construct the
rfly spread that exploits this form of mispricing.Because the strike prices are symmetric around
~~mbdais equal to 0.5.
I..,
Pdore, we use a call and put butterfly spread to profit from these arbitrage opportunities.
Transaction
"Buy 1 50 strike call
.Sell 2 55 strike calls
,Buy 1 60 strike call
I'~',"
TOTAL
r~
t=O
ST
-18
+28
-9.50
+0.50
< 50
50 :::
ST :::
o
ST -
50
55 ::: ST
ST - 50
o
o
110 - 2
o
o
o
50 2: 0
o
60 -
ST -
55
:::
60
ST
> 60
ST X ST
110 - 2
ST -
ST
2: 0
50
o
60
x
ST
-
t=O
21.50
-7
Transaction50
0ST
02STx-<STST
-14.45
+0.05
0ST-x-:S
ST
260
50
-:S
ST
ST
50
:S
055
60
110
S
ST
T- ~
~
:S110
60
55
>60
Please note that we initially receive money and have non-negative future payoffs. Therefore ,
have found an arbitrage possibility, independent of the prevailing interest rate.
we
Question 10.18.
a)
We have to use the formulas of the textbook to calculate the stock tree and the prices of the
options. Remember that while it is possible to calculate a delta, the option price is just the value
of B, because it does not cost anything to enter into a futures contract. In particula~, this yields the
following prices: For the European call and put, we have: premium = 122.9537. The prices must
be equal due to put-call-parity.
b)
We can calculate for the American call option: premium
put option: premium = 124.3347.
c) .' We have the following time
For the European call option:
e replicating
portfolios:
Buy 0.5371 futures contracts.
Borrow 122.9537
For the European put option:
Sell 0.4141 futures contracts.
Borrow 122.9537
= 124.3347
and for the American