ACT370 - PROBLEM SET 1
1. A stock that pays no dividends is currently priced at W! œ 100. The delivery price for a oneyear forward contract on the stock is J!ß" œ 105. Find the payoff and the profit at time 1 for the
following scenarios:
(a) long position on the forward contract, W" œ "!!,
(b) long position on the forward contract, W" œ "#!,
(c) short position on the forward contract, W" œ "!!,
(d) short position on the forward contract, W" œ "#!.
2. A stock is currently priced at W! œ "!!. The continuously compounded risk free rate of
interest is 8%. Find the prepaid forward price, the forward price and the annualized forward
premium for a one-year forward contract on one share of the stock in each of the following
situations.
(a) The stock pays no dividends.
(b) The stock pays a dividend of 1 in 6 months and 1.5 in one year just before delivery.
(c) The stock pays continuous dividends at a rate of 2.5%.
3. Describe how to construct a synthetic short position on a stock with a zero-coupon bond and a
forward contract on the stock. The short position will be covered at time X .
4. The current spot price of platinum is $420 per ounce. The one year continuously
compounded risk free rate of interest is 5%.
(a) Find the one year forward price of platinum per ounce, assuming the price is consistent with
the existence no arbitrage opportunities.
(b) Suppose that the one year forward price for platinum is $450 per ounce. Construct a strategy
from which an investor can obtain a riskless profit.
5. A bond with face amount $100,000 matures in 30 years and has an annual coupon rate of 10%
with semi-annual coupons. The bond is priced at a yield rate of 6% compounded semiannually.
The current risk free rate of interest is quoted as a nominal annual rate of 4% compounded
semiannually. Find the prepaid forward price and the delivery price of the bond for a one year
forward contract on the bond, with delivery to take place after the coupon is paid.
6. At time 0, the share price of ABC stock was $95.25. At that time, a call option on a share of
ABC stock with a strike price of $120 expiring one year later had a premium of $1.10. Risk free
interest rate is 5% annual effective
(a) Determine the payoff and the profit on the purchased call option on the expiry date for each
of the following stock values at time 1: 90 , 110 , 130 .
(b) Determine the payoff and the profit on the written call option on the expiry date for each of
the following stock values at time 1: 90 , 110 , 130 .
Determine the share price needed on expiry to obtain a profit on the purchased call option of at
least 20.
7. At time 0, the share price of ABC stock was $95.25. At that time, a put option on a share of
ABC stock with a strike price of $120 expiring one year later had a premium of $38.40. Risk free
interest rate is 5% annual effective
(a) Determine the payoff and the profit on the purchased put option on the expiry date for each of
the following stock values at time 1: 90 , 110 , 130 .
(b) Determine the payoff and the profit on the written put option on the expiry date for each of
the following stock values at time 1: 90 , 110 , 130 .
Determine the share price needed on expiry to obtain a profit on the purchased put option of at
least 20.
8. You own a car with a replacement value of $40,000. A one-year insurance policy on the car
has a premium of $1000 and has a deductible of $500. Insurance costs for damage to the car are
paid, if necessary, at the end of the year. Describe the insurance policy as an option with an
appropriately defined strike price and payoff and profit at the end of the year. Ignore any effect
of interest and depreciation in the value of the car over the year.
Probems 9-16 sse the following information. Price of XYZ stock at time 0 is 20. Annual
effective interest is at rate 5%. Call and put option (European) values for various strike prices
are:
Strike Price Call Price
Put Price
15
6.46
0.75
17
5.16
1.35
19
4.06
2.16
20
3.59
2.64
21
3.17
3.17
23
2.45
4.36
25
1.89
5.70
It is assumed that XYZ stock pays no dividends.
9.(a) Formulate the payoff and profit on a floor on XYZ stock consisting of being long in one
share of the stock and purchasing a put with strike price 25 and expiring at time 1. Assume that
the price of the stock and the put both must be borrowed at time 0.
(b) Suppose that at time 0 you borrow "Þ)* and purchase a call with strike price 25. Formulate
the profit at time 1.
10. A floor is consists of being long in one share of the stock and purchasing a put with strike
price 20. Suppose that another investor borrows $B at time 0, purchases a call with strike price
20 and invests the rest at 5%. What amount B results in the same profit at time 1 as the floor?
11.(a) Formulate the payoff and profit on a cap on XYZ stock consisting of a short position in
one share of the stock along with a purchased call option with strike price 25 and expiring at
time 1. Assume that the difference between the price of the shorted stock and the call is invested
at time 0.
(b) Suppose that at time 0 you borrow 5.70 and buy a put with strike price 25 expiring at time 1.
Formulate the profit at time 1.
12.(a) Formulate the payoff and profit of a covered call with strike price 19.
(b) Formulate the payoff and profit on a written put with strike price 19.
13.(a) Formulate the payoff and profit if the stock is sold short and 20 is invested for the year.
(b) Formulate the payoff and profit of a purchased put and a written call both with strike
price 17, assuming that the net premium is invested for the year.
(c) The payoff on a short forward contract with delivery price 17 and delivery at time 1 is
"(  W" . The no-arbitrage delivery price for a forward contract in the stock for delivery at time 1
is #!Ð"Þ!&Ñ œ #". The payoff on a short forward contract with delivery at time 1 at price 21 is
#"  W" . Find the cost at time 0 for a short forward contract for delivery at time 1 with delivery
price 17.
(d) The payoff at time 1 on the combination of a purchased put and written call, both with strike
17, is "(  W" . Verify that the price of this combination at time 0 is the same as the price of the
short forward with delivery price 17. This is a synthetic short forward with delivery price 17.
14. Using the XYZ option prices, create a long synthetic forward contract for delivery at time 1
with delivery price 23. Find the price at time 0 of this contract. Verify that this synthetic forward
contract price is the present value of the difference between 23 and 21.
15.(a) Establish the algebraic relationship between the premiums of 1 year puts and calls on
XYZ stock using the put-call parity relationship. Verify that the put and call values for XYZ
stock satisfy the relationship.
(b) Suppose that a put with strike price 24 on XYZ stock has a premium of 5.01. Find the
premium on a call with strike price 24.
(c) For and XYZ option strike price O , the call premium is 2.62 and the put premium is 4.05.
Find O .
16. A floor position has a minimum profit at time 1 of  &Þ%". Determine the strike price of the
put that is part of the floor.
A) 15
B) 17
C) 19
D) 20
E) 21
ACT370 - PROBLEM SET 1 SOLUTIONS
1. For a forward contract, profit œ payoff.
Payoff at time 1 on long forward position œ W"  J!ß" , so (a)  &, (b) 15.
Payoff at time 1 on short forward position œ J!ß"  W" , so (a) &, (b)  "&.
T
2. (a) J!ß"
œ W! œ "!! ß J!ß" œ W! /< œ "!)Þ$# ,
annualized forward premium œ <  $ œ Þ!).
T
(b) J!ß"
œ W!  pv of dividends œ "!!  /Þ!)ÐÞ&Ñ  "Þ&/Þ!) œ *(Þ'&,
T <
J!ß" œ J!ß"
/ œ "!&Þ(* ,
J
annualized forward premium œ X" † 68Ð W!ßX Ñ œ 68Ð "!&Þ(*
"!! Ñ œ Þ!&'$.
!
T
(c) J!ß"
œ W! /$ œ *(Þ&$ ß J!ß" œ W! /<$ œ "!&Þ'& ,
annualized forward premium œ <  $ œ Þ!&& .
3. The equation forward contract œ stock purchase  zero-coupon bond can be written as
 stock purchase œ  forward contract  zero-coupon bond .
The left side of the equation represents a short position on the stock. The right side of the
equation represents a short position on a forward contract and a short position on a zero-coupon
bond for W! . A short position in the stock has a payoff at time X of  WX .
A short position on the forward contract has a payoff at time X of W! /<X  WX .
A short position on the zero-coupon bond is borrowing made at time 0, and has a payoff at time
X of  W! /<X (repay the loan). The combined payoff of the short forward and short zero is
W! /<X  WX  W! /<X œ  WX .
4. (a) O œ W! /<X œ %#!/Þ!& œ %%"Þ&$ Þ
(b) The no arbitrage price is $441.53 (from part (a)). A riskless profit can be obtained in the
following way:
(i) take a short position on a one year forward contract with forward (delivery) price 450
(ii) borrow 420 at the risk free rate for one year and buy one ounce of platinum
(iii) at the end of the year, deliver the platinum and receive 450; the loan is paid back with
amount 420/Þ!& œ %%"Þ&$, and a riskless profit of %&!  %%"Þ&$ œ )Þ%( is made.
'!
5. The bond price at time 0 is "!!ß !!!@Þ!$
 &!!!+'!lÞ!$ œ "&&ß $&"Þ"$ .
The prepaid forward price at time 0 is the value of the asset minus the present value of the
coupon payments up to the time of delivery. This is
#
"&&ß $&"Þ"$  &!!!Ð@Þ!#  @Þ!#
Ñ œ "%&ß '%$Þ$$ .
The forward price for delivery at the end of the year after the coupon is paid is
"%&ß '%$Þ$$Ð"Þ!#Ñ# œ "&"ß &#(Þ$# .
At the end of the year the bond would have 29 years until maturity, so the forward price of
151,527.32 corresponds to a yield to maturity of 6.1642% compounded semiannually from the
forward date one year from now until the maturity date 20 years after that.
6.(a) Profit œ Payoff  accumulated premium
œ Payoff  "Þ"!Ð"Þ!&Ñ œ 7+BÖW"  "#!ß !×  "Þ"' .
Asset price: *!
""!
"$!
Payoff:
!
!
"!
Profit:
 "Þ"'
 "Þ"'
)Þ)%
(b) Profit œ Payoff  accumulated premium œ Payoff  %"Þ'! .
Asset price: *!
""!
"$!
Payoff:
!
!
 "!
Profit:
"Þ"'
"Þ"'
 )Þ)%
For a profit of 20, we must have 7+BÖW"  "#!ß !×  "Þ"' œ #! , so that
W"  "#! œ #"Þ"' , and W" œ "%"Þ"' .
7.(a) Profit œ Payoff  accumulated premium œ 7+BÖ"#!  W" ß !×  %!Þ$# .
Asset price: *!
""!
"$!
Payoff:
$!
"!
!
Profit:
 "!Þ$#
 $!Þ$#
 %!Þ$#
(b) Profit œ Payoff  accumulated premium œ Payoff  %!Þ$# .
Asset price: *!
""!
"$!
Payoff:
 $!
 "!
!
Profit:
"!Þ$#
$!Þ$#
%!Þ$#
For a profit of 20, we must have 7+BÖ"#!  W" ß !×  %!Þ$# œ #! , so that
"#!  W" œ '!Þ$# and W" œ &*Þ') .
8. If damage is greater than 500, up to 40,000, the insurance pays the damage amount minus 500.
Another way of describing this is that if the value of the car after damage, say W" , is between 0
and 39,500, the insurance pays the amount by which the value of the car is below 39,500, so the
insurance pays 7+BÖ$*ß &!!  W" ß !× . This is the payoff at the end of the year of a purchased
option with a strike price of 39,500. The profit on the insurance policy would be
7+BÖ$*ß &!!  W" ß !×  "!!!.
9.(a) The payoff at time 1 is payoff on stock  payoff on purchased put
#&  W" if W" Ÿ #&
#& if W" Ÿ #&
œ W"  œ
œœ
œ 7+BÖ#&ß W" × .
!
W" if W"  #&
if W"  #&
The total cost at time 0 is $25.70, and the accumulated value at time 1 is $26.99. The profit at
time 1 on the floor created by the combination of being long on the stock and having a purchased
 "Þ**
if W" Ÿ #&
put is payoff  #'Þ** œ œ
W"  #'Þ** if W"  #&
(b) The amount owed at time 1 is "Þ)*Ð"Þ!&Ñ œ "Þ*). The profit at time 1 is
 "Þ*)
if W" Ÿ #&
œ W  #&  "Þ*) œ W  #'Þ*) if W  #& .
"
"
"
This is the same profit as the floor (.01 difference due to roundoff).
10Þ The cost of the floor at time 0 is #!  #Þ'% œ ##Þ'%, so 23.77 is owed at time 1.
W  Ð#!  W" Ñ  #$Þ(( œ  $Þ((
The profit at time 1 on the floor is œ "
W"  #$Þ((
if W" Ÿ #!
.
if W"  #!
The price at time 0 of the call with strike 20 is 3.59, and B  $Þ&* is invested at 5%.
The amount owing at time 1 is "Þ!&B. The profit at time 1 is
!
if W" Ÿ #!
$Þ((
if W" Ÿ #!
ÐB  $Þ&*ÑÐ"Þ!&Ñ  "Þ!&B  œ
œ œ
.
W"  #! if W"  #!
W"  #$Þ(( if W"  #!
This is the same profit, no matter what amount B is borrowed at time 0.
11. The payoff at time 1 is
!
 W"  œ
W"  #&
if W" Ÿ #&
 W"
œ œ
if W"  #&
 #&
if W" Ÿ #&
.
if W"  #&
The proceeds of the short sale at time 0 minus the cost of the call is #!Þ!!  "Þ)* œ 18Þ"". This
is invested for one year at annual effective 5% and grows to $19.02. The profit at expiry on this
"*Þ!#  W" if W" Ÿ #&
cap is œ
.
 &Þ*)
if W"  #&
The amount owed is &Þ(!Ð"Þ!&Ñ œ &Þ**. The profit at time 1 is
#&  W"  &Þ** œ "*Þ!"  W" if W" Ÿ #&
.
œ  &Þ**
if W"  #&
This is the same profit as the cap.
12.(a) A covered call consists of being long in the stock and writing a call.
Payoff on covered call is payoff on stock  payoff on call
if W" Ÿ "*
W
W if W" Ÿ "*
.
œœ "
œœ "
W"  ÐW"  "*Ñ if W"  "*
"* if W"  "*
The cost of this position at time 0 is #!  %Þ!' œ "&Þ*%. The profit at time 1 is
W" if W" Ÿ "*
W"  "'Þ(% if W" Ÿ "*
.
œ "* if W  "*  "&Þ*%Ð"Þ!&Ñ œ œ #Þ#'
if W"  "*
"
 Ð"*  W" Ñ
The payoff on the written put is œ
!
if W" Ÿ "*
W  "*
œœ "
if W"  "*
!
if W" Ÿ "*
.
if W"  "*
The premium received for writing the put at time 0 is 2.16, which grows to #Þ"'Ð"Þ!&Ñ œ #Þ#(.
W  "* if W" Ÿ "*
W  "'Þ($ if W" Ÿ "*
The profit on the written put is #Þ#(  œ "
œœ "
.
!
if W"  "*
#Þ#(
if W"  "*
This is the same (difference of .01 due to roundoff) as the profit on the covered call.
13.(a) At time 1, payoff œ profit œ #"  W" .
(b) Net premium received at time 0 is 5.16  1.3& œ $Þ)", invested for one year, growing
to 4.00. Payoff and profit at time 1 is
!
if W" Ÿ "(
"(  W" if W" Ÿ "(
 % œ "(  W"  % œ #"  W" .
œ  ÐW  "(Ñ if W  "(  œ !
if W"  "(
"
"
(c) The payoff at time 1 is "(  W" œ #"  W"  %, so the price at time 0 of the short forward
contract is the same as the price at time 0 of a short forward with delivery price 21 minus the
present value of 4. The price of the short forward with delivery price 21 is 0 (since 21 is the noarbitrage delivery price), so the price at time 0 of the short forward with delivery price 17 is
 %@ œ  $Þ)".
(d) Price of written call and purchased put is "Þ$&  &Þ"' œ  $Þ)" .
14. The synthetic long forward is a combination of a written put and a purchased call, both with
strike price 23. The cost at time 0 to create this combination is #Þ%&  %Þ$' œ  "Þ*",
which is equal to Ð#"  #$Ñ@Þ!& œ  "Þ*! (roundoff error in option prices).
15.(a) CallÐOß "Ñ  O@Þ!& œ PutÐOß "Ñ  #!
O œ "&ß 'Þ%'  "&@ œ #!Þ(& œ !Þ(&  #!
O œ #$ß #Þ%&  #$@ œ #%Þ$& œ %Þ$'  #! .
(b) CallÐ#%ß "Ñ  #%@ œ &Þ!"  #! p CallÐ#%ß "Ñ œ #Þ"& .
(c) #Þ'#  O@ œ %Þ!&  #! p O œ ##Þ&! .
16. This problem can be solved by trial and error by finding the minimum profit for each strike
price. Alternatively, the profit on a floor is the same as the profit on a purchased call. The
minimum profit at time 1 on a purchased call is  GÐOß "ÑÐ"Þ!&Ñ , where GÐOß "Ñ is the call
premium at time 0. Therefore,  GÐOß "ÑÐ"Þ!&Ñ œ  &Þ%" , from which we get
GÐOß "Ñ œ &Þ"&. The premium for a call with strike price 17 is 5.16 (the difference due to
rounding to nearest .01 for the profit). Answer; B