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Transcript
Brock University
Faculty of Business
Department of Finance, Operations, and Information Systems
FNCE 4P17
Derivatives, Part II
Winter 2009
Instructor: Hatem Ben Ameur
Office: TA 318
Email: [email protected]
Phone: (905) 688–5550 x 5874
Class time: Wednesdays 15:30–17:00 and Fridays 14:00–15:30 (TA 307)
Description:
A derivative is a contract whose return depends on the price movements of some
underlying assets. There are three main families of derivative contracts: options, futures,
and swaps. They all have the ability to reduce risk; thus, are widely used for hedging
purposes.
This course covers advanced topics related to options, futures, and swaps. We focus on
their structure, evaluation, and hedging properties. Complex products, such as exotic
options, callable bonds, and swaptions are considered.
Materials:
The following textbook is required, and will be used frequently during the term.
1. Robert W. Kolb and James A. Overdahl, 2007, Futures, Options, and Swaps, 5th
Edition, Blackwell.
The following textbooks offer useful alternate explanations, and are listed in increasing
order of difficulty.
2. John C. Hull, 2005, Fundamentals of Futures and Options, 6th Edition, Prentice Hall.
3. Don M. Chance and Robert Brooks, 2007, An Introduction to Derivatives and Risk
Management, 7th Edition, Thomson.
4. John C. Hull, 2008, Options, Futures, and Other Derivative Securities, 7th Edition,
Prentice Hall.
Monitoring the news about financial derivatives is recommended as an important
complement to classroom work. Selected articles from business newspapers such us The
Financial Times, The Globe and Mail, The National Post, and The Wall Street Journal
will be discussed in class.
Grading:
The grading policy is based on:
• four assignments each worth 5%;
• two midterm exams each worth 30%;
• and a quiz worth 20%.
The midterm exams are cumulative. The assignments will improve your skills, and help
you prepare for the exams. For each assignment, you will be given selected problems to
solve, articles to comment on, and empirical experiments to implement. A quiz is planned
at the last week.
Office hours will be fixed during the first class.
In every aspect of the course, students must comply with the Brock University Honour
Code.
Outline:
Week
1–2
3
4
5–6
7
8
9
10
11–12
Chapter
Risk-Neutral Pricing
Option Sensitivities
Pricing American Options
Options on Stock Indexes, Foreign Currencies,
and Futures – First Midterm Exam
Pricing Corporate Securities
Exotic Options
Interest-Rate Options – Second Midterm Exam
Long-Term T-Bond Futures Contract
Swap Contracts and Swaptions – Quiz
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
Derivatives, Part II
Hatem Ben Ameur
Brock University – Faculty of Business
Finance, Operations, and Information
Systems
FNCE 4P17
Winter 2010
1
Hatem Ben Ameur
1
Derivatives, Part II
Brock University, FNCE 4P17
Risk-Neutral Pricing
Topics Covered:
1
The Present-Value Principle
2
Pricing in the Binomial Model
3
Risk-Neutral Pricing of …nancial assets
4
Pricing in the Black and Scholes Model
2
Hatem Ben Ameur
1.1
Derivatives, Part II
Brock University, FNCE 4P17
The Present-Value Principle
Let t be the present time and T an investment horizon.
The discount factor from time T to time t, indicated by
t;T , is the present value at time t of one dollar to be
received with certainty at time T .
The compound factor from time t to time T is the future
value to be received with certainty at time T of one dollar
invested at time t. In a perfect market, the compound
factor is the inverse of the discount factor:
1
ct;T =
.
t;T
In a perfect market, an asset whose future cash ‡ows and
their occurrence dates are known in advance with certainty can be evaluated using the present-value principle:
PVt =
X
tn>t
3
t;tn CFtn ,
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
where t;tn , for tn > t, is the discount factor from
time tn to time t, CFtn the future cash ‡ow (in dollars) promised at time tn, and PVt the asset’s present
value at time t (in dollars).
For this equation to hold, the asset’s cash ‡ows and their
occurrence dates have to be known in advance with certainty.
Exercise: Can a Treasury bond be evaluated using the
PV principle? Give the assumptions under which a stock
can be evaluated using the PV principle. Does the PV
principle apply for a corporate bond, a European call option, or a forward/futures contract?
Exercise: The nominal interest rate is …xed at rnom =
4% (per year), and interest is compounded semi-annually.
Give the semi-annual interest rate r:5 (in % per six months),
which is relevant to compound interest, the compound
factor c0;1, and the discount factor 0;1. Compute the
equivalent annual interest rate r1 (in % per year), which
4
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
would apply if interest were compounded annually. This is
the e¤ective interest rate. Compute again the compound
factor c0;1 and the discount factor 0;1. Compute the
equivalent monthly interest rate r1=12 (in % per month)
and quarterly interest rate r0:25 (in % per quarter). Compute again the compound factor c0;1 and discount factor
0;1 . Recognize that r:5 , r1 , r1=12 and r:25 are all equivalent rates. The continuously compounded interest rate
rc (in % per year) is a nominal rate, which would apply if interest were compounded at each second (or even
a fraction of a second). Compute again the compound
factor c0;1 and the discount factor 0;1. Recall that
a n
! ea, when n ! 1,
1+
n
where a 2 R and n 2 N.
An arbitrage opportunity is an investment strategy that
guarantees a riskless pro…t.
Suppose the PV principle applies for a …nancial asset. If
the asset is not traded at its (fair) present value, arbitrage
5
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
would be possible. Borrow at the risk-free rate, and buy
the asset if it is undervalued. Short sell the asset, and
save at the risk-free rate if the asset is overvalued.
1.2
Pricing in the Binomial Model
We consider a market for a saving account (the riskless
asset) and a stock (the risky asset). Trading activities
take place only at the current time t0 = 0 and at horizon
t1 = T . No trading is allowed in between. All positions
are then closed at the horizon.
In addition, the stock price is assumed to move from its
current level S0 according to a one-period binomial tree:
S0
up
S1 = uS0
p
%
&
1 p
.
S1down = dS0
t0 = 0
t1 = T
6
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
The stock price can rise by a factor of u or drop by a
factor of d, where u > d.
The probabilities p and 1 p de…ne the physical probability measure P , under which investors evaluate likelihoods
and make decisions in the real world.
The parameters u and d can be seen as volatility parameters. The greater is u d, the higher the volatility of
the stock return.
Example: Assume that the stock price is currently quoted
at $100, and can either increase by 25% or decrease by
20%. The factors u and d are
u = 1:25, and d = 0:8.
7
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
In this case, the one-period binomial tree is
up
S0
%
&
t0 = 0
S1 = uS0 = 1:25
100 = 125
.
S1down = dS0 = 0:8
100 = 80
t1 = T
The price can move upward from $100 to $125, or downward from $100 to $80.
The (periodic) risk-free rate is indicated by r, and is expressed in % over the time period [t0, t1].
The binomial tree is arbitrage free if, and only if, the
following property holds:
d < 1 + r < u.
up
Indeed, in the case of an upward movement S1 = uS0,
the rate of return on the stock u 1 (in % per period)
8
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
must exceed the risk-free rate r (in % per period). Otherwise, the stock would be overpriced, and an arbitrage opportunity would arise. Similarly, the risk-free rate r must
exceed the rate of return on the stock under a downward
movement, that is, d 1.
Example (continued): The risk-free rate is r = 7%
(per period). Is the model arbitrage free?
The binomial tree is simple, but viable. It is used here to
go through the fundamentals of options pricing.
The goal now is to characterize the (present) value of a
European call option C0 in the one-period binomial tree,
as a function of the stock price S0, the option strike price
X , the volatility parameters u and d, and the risk-free
rate r.
Consider a European call option on a stock with a maturity date T = t1 and a strike price X .
9
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
The hedge portfolio consists of holding
shares of the
stock and a single signed call option on that stock with
an initial value of
H0 =
S0
C0 ,
S1
C1 .
and a terminal value of
H1 =
For a speci…c level of , the hedge portfolio is riskless,
and, by the no-arbitrage principle, must earn the risk-free
rate. A formula for the (present) value of the call option
can therefore be derived.
The value of the hedge portfolio moves along the binomial
tree as follows:
up
H0
%
&
up
H1 =
S1
up
C1
.
H1down =
10
S down
C1down
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
The hedge portfolio is riskless if, and only if,
up
H1 = H1down.
Solving for
gives
up
=
C1
up
S1
C1down
S1down
=
C
,
S
which depends on the known parameters S0, X , u, and
d.
Given the hedge parameter , the hedge portfolio, as a
riskless investment, should earn the risk-free rate:
H0 =
S0
C0
up
H1
H1down
H1
=
=
=
.
1+r
1+r
1+r
Solving for C0 gives
up
p C1 + (1 p ) C1down
C0 =
,
1+r
11
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
where
p =
1+r d
2 (0; 1) .
u d
The probabilities p and 1 p de…ne the so-called riskneutral probability measure P , which is not related in
any way to the physical probability measure P .
In sum, the value of the European call option can be
expressed as a weighted average (an expectation) of its
promised cash ‡ow, which is discounted at the risk-free
rate:
C1
j S0 ,
C0 = E
1+r
where E [:] is the expectation sign under the risk-neutral
probability measure P , C0 the call-option value at time
t0, and C1 the call-option value at time t1.
The pricing formula discounts the risky cash ‡ow of the
call option by the risk-free rate as if investors were risk
neutral, while they are not. This is done via a major
12
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
correction. We move from the real world, seen under the
physical probability measure P , to a risk-neutral world,
seen under the risk-neutral probability measure P .
Example (continued): Consider a European call option
on the previously mentioned stock with a strike price of
X = $100.
1. Compute the risk-neutral probabilities for upward and
downward movements.
2. Draw the one-period binomial tree for the call option.
3. Compute the hedge ratio.
4. Use the formula to compute the value of the call
option.
13
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
5. Check that the hedge portfolio earns the risk-free
rate.
6. Compute in two di¤erent ways the value of its associated European put option.
The risk-neutral probabilities are:
1+r d
u d
1 + 7% 0:8
=
1:25 0:8
= 0 :6
p =
and
1
p = 0:4.
The one-period binomial tree for the stock, the call op-
14
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
tion, and the hedge portfolio are:
up
S0 = 100
C0 = 14:02
= 0:556
H0 = 42:02
t0 = 0
%
&
S1 = 125
up
C1 = 25
up
H1 = 0:56
125
S1down = dS0 = 0:8
C1down = 0
H1down = 0:56 80
t1 = T
where
0 :6
25 + 0:4 0
C0 =
,
1 + 7%
25 0
,
=
125 80
H0 = 0:56 100 14:02.
The call-option value veri…es:
C0 =
S0
H0,
C1 =
S1
H1,
and
15
25 = 45
100 = 80 ,
0 = 45
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
where C1 is the call-option value at time t1, that is, the
call-option payo¤ in this example.
In other words, one can fully replicate the call option
using a strategy that consists of holding
shares of the
underlying stock and borrowing at the risk-free rate.
If the signer decides to replicate the option, he insures
the promised payment to the option holder at maturity.
A market in which one can fully replicate all derivative
contracts is called a complete market. The binomial tree
and the Black and Scholes model are two examples.
Exercise: Compute the present value of the stock in the
binomial tree.
The stock price, discounted at the risk-free rate, is said
to verify the martingale property.
16
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
The multi-period binomial tree is a simple extension of
the one-period binomial tree. Both are arbitrage free
under the same condition, that is,
d < 1 + r < u.
In a multi-period binomial tree, the stock price moves
according to a Markov process, and veri…es the martingale
property.
The call-option value at time t can then be expressed as
follows:
Ct+1
Ct = E
j Ft
1+r
Ct+1
=E
j St ,
1+r
where Ft represents the information available to investors
up to time t.
This formula holds for both European call and put options. Pricing European options in a multi-period binomial tree is done as follows:
17
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
1. Start at maturity, and evaluate the option;
2. Use the risk-neutral pricing formula, and evaluate the
option backward in time;
3. Stop at the origin.
Exercise (continued): Extend the one-period binomial
tree to a three-period binomial tree. Evaluate the European call option, and its associated put option. Check
that the call-put parity holds.
1.3
Risk-Neutral Pricing
In a risky environment, it is possible to extend the presentvalue (PV) evaluation principle, as long as the market
model is arbitrage free. The martingale property is a
18
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
su¢ cient condition for the market model to preclude arbitrage.
The present value of an asset can be expressed as follows:
2
3
X
4
PVt = E
t;tn CFtn j Ft5 ,
tn>t
where E [ ] is the expectation sign under the risk-neutral
probability measure, t;tn , for tn > t, the discount factor
from time tn to time t, CFtn the cash ‡ow promised by
the asset at time tn, and Ft the information available
to investors up to time t. This formula supports market
risk, credit risk, and random payment dates.
This formula discounts the risky cash ‡ows of a …nancial
asset using their associated risk-free rates, while higher
rates should be used. An adjustment is therefore made:
the expectation is computed under the risk-neutral probability measure, rather than the physical probability measure.
This is the risk-neutral pricing formula, which holds as
long as the market model is arbitrage free.
19
Hatem Ben Ameur
1.4
Derivatives, Part II
Brock University, FNCE 4P17
Pricing in the Black and Scholes Model
Black and Scholes considered a frictionless market for a
stock (the risky asset) and a saving account (the riskless
asset) in which trading takes place continuously.
The stock price process fStg is assumed to follow a geometric Brownian motion characterized by
ST = SteR ,
where R is the continuously compounded rate of return
on the stock over [t, T ]. The annualized rate of return
on the stock is R= (T t) (in % per year). The rates of
return on the stock over successive and non-overlapping
time intervals are independent.
The Black and Scholes model turns out to be arbitrage
free. Under the risk-neutral probability measure P , the
rate of return on the stock R is random, and follows a
normal distribution:
p
2
R= r
=2 (T t) +
T tZ ,
20
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
where r is the continuously compounded risk-free rate (in
% per year), the volatility of stock returns (per year),
T t the option’s remaining time to maturity (in years),
and Z a random variable that follows the standard normal
distribution N (0, 1).
The stock price process is said to be lognormal, since
the natural logarithm of a future stock price, given the
current stock price, follows a normal distribution.
Black and Scholes used a hedge portfolio, and derived a
formula for a European call option on a stock paying no
dividend:
Ct =
=
h
E e r(T t) max (0, ST X ) j St
N (d1) St Xe r(T t)N (d2) ,
i
where
d1
d2
ln (St=X ) + r + 2=2 (T
p
=
T t
p
= d1
T t.
21
t)
, and
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
Here, St is the current stock price, X the option’s strike
price, the volatility of stock returns, T t the option’s
remaining time to maturity, r the risk-free rate (in % per
year), and N ( ) the cumulative function of the standard
normal distribution.
"
Please establish the second term of
the Black and Scholes formula.
#
The hedge portfolio used in the Black and Scholes model
consists of a short position on the call option and a long
position on
shares of the underlying stock. This portfolio is similar to the one used in the binomial model
with the di¤erence that it has to be continuously adjusted. The Black and Scholes model is not realistic, but
remains very useful in practice.
1.5
Assignment 1
Exercise 1: This exercise shows how to replicate a European call option in the Black and Scholes model using
22
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
the
hedging strategy, based on the underlying stock
and the saving account.
Consider 1; 000 European call options on a stock paying
no dividend. The size of each option contract is of 100.
The initial stock price is St = $49, the strike price X =
$50, the risk-free rate r = 5% (per year), the volatility
= 20% (per year), and the time to maturity T t =
20=52 = 0:3846 (in years).
1
Check that the call-option value is Ct = $240; 052:73.
2
Compute the
coe¢ cient at the start.
The stock price at maturity is ST = $57 14 ; thus, the call
option expires in the money.
3 Compute the call-option payo¤ paid by the seller to
the buyer at maturity.
23
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
The path of the stock price, observed once a week, was as
follows: week0 49 (in dollars), week1 48 81 , week2 47 38 ,
week3 50 41 , week4 51 34 , week5 53 18 , week6 53, week7
51 78 , week8 51 38 , week9 53, week10 49 78 , week11 48 12 ,
week12 49 87 , week13 50 38 , week14 52 81 , week15 51 78 ,
week16 52 87 , week17 54 78 , week18 54 85 , week19 55 78 ,
week20 57 41 .
4 Use the
hedging strategy, rebalanced weekly, to
replicate the European call option. Use Excel, and …ll in
the following hedging table. Explain.
Week
#
0
1
...
19
20
Stock
Price
49
48 18
Delta
Shares
Purch.
Cost
Cum.
Cost
Interest
($000)
55 78
57 14
5 Repeat this exercise for the following path of stock
prices: week0 49 (in dollars), week1 49 34 , week2 52,
24
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
week3 50, week4 48 38 , week5 48 41 , week6 48 34 , week7
49 58 , week8 48 14 , week9 48 41 , week10 51 18 , week11 51 12 ,
week12 49 87 , week13 49 78 , week14 48 43 , week15 47 12 ,
week16 48, week17 46 41 , week18 48 81 , week19 46 85 , week20
48 18 .
6
Simulate a random path for the stock price and
do again question 4 . De…ne your code parameters as
follows: s for St, X for the option strike price, sigma for
, Tau for the remaining time to maturity T
t, r for
the risk-free rate, a and b for the lower and upper bound
of the interval [a, b], respectively, and delta for the hedge
ratio .
7
Set the number of observation dates N as an additional parameter. Do again question 6 with increasing
N 2 f20, 40, 80, 160g. De…ne the hedge error e (epsilon) and show that e ! 0 when N ! 1. This question is not mandatory, but very instructive. It is worth
an additional 10 points in the …rst mid-term exam (the
deadline for question 7 ).
25
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
Exercise 2: In the Black and Scholes model, the stock
price follows a lognormal process, which is a Markov
process. The following transition parameter keeps track
of the behaviour of the stock price.
1. Show that the following transition parameter can be
expressed as
P (ST 2 [a, b] j St)
0
ln (b=St)
r
@
p
=N
0
[email protected]
ln (a=St)
r
p
T
2 =2
T
t)
(T
t
2 =2
t
(T
t)
1
1
A
A,
where N (:) is the cumulative distrubution function
of N (0; 1).
2. Comment on the case where a = X and b = 1.
Give an example, and support your argument by selected numbers.
26
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
3. Set St = $100 and r = 4%. Fill in the following
Tables for
P (ST 2 [a; b] j St) .
Set T t = 1 (year) when changes and = 20%
(par year) when T t changes. Intrepret your results.
T
[a; b]
[70; 80]
[80; 90]
[90; 100]
[100; 110]
[110; 120]
10%
20%
30%
:5
t
1
2
.
27
Hatem Ben Ameur
2
Derivatives, Part II
Sensitivity Analysis
Topics Covered:
1. Single-Option Sensitivity Analysis
2. Portfolio Sensitivity Analysis
28
Brock University, FNCE 4P17
Hatem Ben Ameur
2.1
Derivatives, Part II
Brock University, FNCE 4P17
Single-Option Sensitivity Coe¢ cients
The value of an option contract, v , depends on several inputs, including the underlying asset price S , the volatility
of asset returns , the option’s remaining time to maturity T t, and the risk-free rate r.
The sensitivity of an option with respect to its underlying
asset price is
v
,
S
where S is a small change in the underlying asset price,
and v is the associated change in the option value. All
other inputs are assumed to be …xed.
A sensitivity coe¢ cient cannot directly be computed from
real-life observations. Indeed, the inputs that a¤ect the
option value change together over time. A sensitivity coe¢ cient can, however, be computed via a market model,
especially when a closed-form solution is available for the
29
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
option value. The Black and Scholes model is an example.
The delta coe¢ cient, indicated by , is the …rst partial
derivative of v with respect to S , that is,
=
@v
@S
v
,
S!0 S
when the right-hand quantity exists.
= lim
Exercise: The call-option parameters are S = $100,
X = $100, = 0:3 (per year), T t = 180 (days), and
r = 8% (per year). Use Table 14.1, and approximate
at S = $100. Approximate the change in the call-option
value if the stock price drops by $0:5.
The gamma coe¢ cient, indicated by
30
, is the second
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
partial derivative of v with respect to S , that is,
@ 2v
=
@S 2
@ @v
=
@S @S
@
,
=
@S
when the right-hand quantity exists.
Table 14.4 and Table 14.5 provide and , as functions
of some relevant parameters, for European call and put
options in the Black and Scholes model.
[Please explain Figure 14.1 on page 478.]
Exercise: Comment on Figure 14.4 on page 484, and
Figure 14.13 on page 491 in conjunction with Figure 14.1
on page 478. Please adjust the shape of the curve of .
31
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
The …rst-order Taylor expansion is
S,
v
where S is a small change in the stock price. All other
parameters are kept …xed.
Exercise: Use a Taylor expansion of order one, and approximate the change in the call-option value if the stock
price drops by $0:5. Compare with the true value.
The second-order Taylor expansion, which is more accurate than the …rst-order one, is
1
( S )2 .
2
Again, all the parameters that a¤ect the option value are
kept …xed, except the stock price.
v
S+
Exercise: Use a second-prder Taylor expansion, and approximate the change in the call-option value if the stock
price drops by $0:5. Compare with the true value. Use
Table 14.6 on page 480.
32
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
The vega coe¢ cient, indicated by V , is the …rst partial
derivative of v with respect to , that is,
V=
@v
@
= lim
!0
v
,
where
is a small change in the volatility of stock
returns. Here, all the parameters that impact the option
value are kept …xed, except the volaltility of stock returns.
Exercise: How does a call-option value behave as a function of the volatility of stock returns? Answer the same
question for a put-option value. Use Table 14.4, Table 14.5, and Figure 14.8. Check if Figure 14.8 is correct.
Exercise: Approximate the change in the put-option value
if the volatility rises by 1%. Compare with the true value.
Use Table 14.6 on page 480.
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
The theta coe¢ cient, indicated by , is the …rst partial
derivative of v with respect to t, that is,
@v
@ (T t)
v
,
= lim
t!0 t
where t is a small change in time, for example, one day
from the present time. Here, all parameters that impact
the option value are kept …xed, except the time index.
=
@v
=
@t
Exercise: How does the call-option value behave as a
function of the time index? Use Table 14.4.
The call-option value decreases over time. This is the
time-decay phenomenon for a call option.
[Please explain the time-decay phonomenon from Figure 14.1.]
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
Exercise: A call-option holder knows with certainty that
he will lose value over time. Is it pro…table to hold a call
option? Explain.
All the parameters that impact the option value are kept
constant, except the stock price and the time index. Then,
the …rst-order Taylor expansion for a multivariate function
can be written as
t.
S+
v
Exercise: Approximate the change in the call-option value
if the stock price rises by $0.5 over one trading day. There
are 252 trading days per year. Compare with the true
value. Use Table 14.6 on page 480.
Finally, the rho coe¢ cient, indicated by , is the …rst
partial derivative of v with respect to r, that is,
@v
=
@r
= lim
r!0
35
v
,
r
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
where r is a small change in the risk-free rate. Here, all
parameters that impact the option value are kept …xed,
except the risk-free rate.
Exercise: Approximate the change in the call-option value
if the stock price rises by $0.5 over one trading day, and
the risk-free rate drops by 1%. Compare with the true
value. Use Table 14.6 on page 480.
The single-option sensitivity coe¢ cients, as de…ned above,
can be extended to all derivative contracts.
2.2
Portfolio Sensitivity Coe¢ cients
Consider a potfolio of derivative contracts and their underlying assets. The value of this portfolio is a linear
combination of its basic components. For each component, compute the sensitivity coe¢ cient. Then, apply
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
the same linear combination, and obtain the associated
portfolio-sensitivity coe¢ cient.
Consider the hedge portfolio considered in the previous
chapter:
P =
S
C,
where
2 R is the delta coe¢ cient computed at S .
The delta coe¢ cient of this portfolio is
@P
=
@S
=
@S @C
@S
@S
1
= 0.
The hedge portfolio is therefore insensitive to small changes
in the stock price. This portfolio is called
neutral.
A
neutral portfolio is not necessarily
neutral, V neutral,
neutral, or
neutral; however,
neutral portfolios and the like can be obtained. For example, a portfolio combining two call options and their common underlying asset can be made
neutral. Indeed, the
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
value of the portfolio is
P = nsS + n1C1 + n2C2,
where nS , n1, and n2, are the number of shares, …rst
call option, and second call option, respectively. Solve
the following equation for nS , n1, and n2:
(
ns + n1 1 + n2 2 = 0
,
0 + n1 1 + n2 2 = 0
and determine the associated
neutral portfolio.
Example: Set ns = 1, 1 = 0:6151, 2 = 0:4365,
1 = 0:0181, and 2 = 0:0187. Show that n1 =
5:1917 and n2 = 5:0251. Identify the associated
neutral portfolio.
Following the same rules, a sensitivity analysis can be
done for simple and complex strategies that combine derivative contracts and their underlying assets.
A long straddle is a portfolio of a long position on a call
option and a long position on a put option, both with
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
the same parameters. The pro…t function of the straddle,
under the assumption that it is held till maturity, is
= CT
C0 + PT
= max (0, ST
X)
P0
C0 + max (0, X
ST )
P0,
where CT and C0 are the call-option values at maturity
and at the start, respectively; and PT and P0 are the putoption values at maturity and at the start, respectively.
Similarly, the pro…t function of the straddle, under the
assumption that it is held up to time t, is
= Ct
C0 + Pt
P0,
where Ct and Pt are functions of the stock price at time
t, among other parameters.
[Please explain Figure 14.15 on page 496.]
Exercise: Consider the straddle associated to the exercise price X = $100, described in Table 14.7 on page 495.
Consider the straddle value at the start, that is, C0 + P0,
as a function of the stock price, the volatility of stock
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
returns, the remaining time to maturity, and the risk-free
rate. Compute its sensitivity coe¢ cients, and interpret
your results. How will the straddle behave if the stock
price drops by $1 and the volatility rises by 2% over the
next three trading days?
A strangle is similar to a straddle except that its associated call and put options have di¤erent strike prices.
[Please explain Figure 14.16 on page 497.]
Exercise: Consider the strangle de…ned in Figure 14.16
on page 497. Answer the same questions as for the straddle.
A butter‡y spread is a portfolio that employs three call
options associated to the same parameters except for
their strike prices.
Consider the butter‡y spread taken from Table 14.7 on
page 495, which consists of a long position on the call
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
option with a strike price of X1 = $90, a long position
on the call option with a strike price of X3 = $110, and
a short position on two call options with a strike price of
X2 = $100.
[Please explain Figure 14.17 on page 499.]
Exercise: Consider the butter‡y spread in Figure 14.17
on page 499. Answer the same questions as for the straddle.
Consider two call options associated to the same parameters except for their strike prices. A bull spread is a
portfolio of a long position on the call option with the
lowest strike price and a short position on the call option
with the highest strike price.
Exercise: Consider the bull spread de…ned in Figure 14.18
on page 500. Answer the same questions asked for the
straddle.
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Hatem Ben Ameur
2.3
Derivatives, Part II
Brock University, FNCE 4P17
Assignment 1 Part 2
Do exercises no 1, 2, 3, 4, 9, and 16 on pages 505–506.
Exercise: Consider the call option in Table 14.6.
1 Use the formulas in Table 14.1–Table 14.4 and compute the call-option sensitivity coe¢ cients in Table 14.6.
2 Use Excel and draw Figure 14.1, Figure 14.9, and
Figure 14.13. Comment your results.
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Hatem Ben Ameur
3
Derivatives, Part II
Brock University, FNCE 4P17
Pricing American Options
Topics Covered:
1. Exercise Value and Holding Value
2. Pricing American Options in the Black and Scholes
Model
(a) The American Call Option
(b) Transition Parameters for Pricing American Options
3. Pricing American Options in the Binomial Model
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Hatem Ben Ameur
3.1
Derivatives, Part II
Brock University, FNCE 4P17
Exercise Value and Holding Value
The American version of a European option gives its
holder the additional right to exercise the option early,
before maturity.
Thus, an American option is worth more than its associated European option. The di¤erence in value, called
the early exercise premium, stems from the early-exerce
feature.
Let Vt (s) be the value of an American option, and vt (s)
be the value of its associated European option at time t
when St = s. Uppercase letters are used for American
options, and lowercase letters are used for their European
counterparts.
Let Vte (s) and Vth (s) be the exercise value and the holding value of an American option at time t when St = s,
respectively. The exercise value is also called the intrinsic
value.
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
Example: The exercise value of a vanilla call option is
Cte (s) = max (0, s
X ) , for t
T,
where St = s is the current underlying asset price, and
X is the exercise price. Similarly, the exercise value of a
vanilla put option is
Pte (s) = max (0, X
s) , for t
T.
The holding value of an American option is
Vth (s)
=E
h
t;u Vu j St
i
=s ,
where E [ ] is the expectation sign under the risk-neutral
probability measure, u is the …rst decision date after t,
u = t is the discount factor from time u to time t, and
St = s is the stock price at time t. The holding value
veri…es
Vth (s)
0, for all s.
with the convention that
VTh (s) = 0, for all s.
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
The holding value represents the future potentialities of
the contract, discounted back to the current date. Except for a few exceptions, the holding value cannot be
computed in a closed form, and must be approximated
in some way. The reason is that Vu itself depends on an
expectation, which itself depends on an expectation, and
so on. In sum, the holding value is a high-dimensional
integral. Computing the holding value is the most challenging issue in pricing American options.
The value of an American option is
Vt (s) = max Vte (s) , Vth (s) , for all s,
and the optimal exercise policy consists of exercising the
option at time t when the underlying asset is at St = s
if, and only if,
Vte (s) > Vth (s) ,
and holding the contract for at least another moment if,
and only if,
Vte (s)
Vth (s) .
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
This is to say that the option holder acts in an optimal
way.
A European option can be seen as a particular case of an
American option, where the exercise values veri…es
Vte (s) = 0, for t < T and all s,
and
VTe (s) = VT (s) , for all s.
The following general results hold:
Vt (s)
Vth (s)
vt (s) , for all s,
and
Vt (s)
Vte (s) , for all s.
The exercise premium at time t when St = s is de…ned
as Vt (s) vt (s) 0.
[Please explain Figure 15.1 on page 514.]
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
The time value of an American option at time t when
St = s is de…ned as
Vt (s)
Vte (s)
0.
The larger the time value of an American option, the
longer is its early exercise.
3.2
Pricing American Options in the Black
and Scholes Model
3.2.1
The American Call Option
Call-option values admit the following bounds:
ct
St
Xe r(T t),
and
Ct
St
48
X.
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
An American call option on a stock that pays no dividend
cannot be exercised optimally before maturity.
Indeed, one has
Cte = St
X
St
Xe r(T t)
ct
Cth.
Exercising the call option before maturity is clearly suboptimal. When the underlying stock pays dividends, however, it may be optimal to exercise the call option early,
just before an ex-dividend date.
"
Please draw two paths for the stock price,
with and without dividend paying.
#
Consider a call option on a dividend-paying stock. Let
D1; : : : ; DN be the dividends, and t1; : : : ; tN their associated ex-dividend dates over the option’s life [t, T ].
Fisher Black approximated the American call-option value
as follows.
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
1. Adjust downward the current stock price by the PV
of all dividends:
St = St
N
X
Dne r(tn t).
n=1
This is somewhat equivalent to adjust Stn downward
by Dn at tn, for n = 1; : : : ; N .
2. Compute the European call-option value using the
Black and Scholes formula:
c = BS (St , X , , T
t, r) .
3. For each ex-dividend date tm, adjust the call-option’s
strike price downward by the PV of all remaining
dividends, including the one that is about to go exdividend, that is,
Xm = X
X
Dne r(tn tm).
n m
This is equivalent to adjust upward the stock price
by the same amount.
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
4. Compute the American call-option value using the
Black and Scholes formula, under the assumption of
an early exercise just before tm, that is,
cm = BS (St , Xm, , tm
t, r ) .
5. The approximate call-option value and its optimal
exercise policy are given by
cb = max fc; c1; : : : ; cN g .
The intuition behind this approximation is that the American call-option can be exercised only prior to an exdividend date or at maturity.
Example: The parameters of the American call option
are St = $60, X = $60,
= 20% (per year), T
t = 180 (days), r = 9% (per year), D1 = D2 = $2,
t1 = 60 (days), t2 = 150 (days). Approximate its value
and identify its optimal exercise policy. Check that the
(approximate) optimal exercise policy, as given by Black’s
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
approximation, consists of exercising the call option early,
at time t2.
The optimal exercise policy, provided by Black’s approximation, is deterministic while it must be expressed as a
function of time and the stock price as follows: exercise
the call option early at the ex-dividend date tn if, and
only if, the stock price is su¢ ciently high at that time.
Merton extended the Black and Scholes model to a …nancial asset that pays dividends continuously at a …xed
rate (in % per year). Think of the underlying asset as
a stock index. The extension is easily done by adjusting
downward the current stock price as follows:
St = St e
(T t) .
Similarly, in the Black-Scholes-Merton model, an American call option can be exercised early, at any time before
maturity if, and only if, the stock price is su¢ ciently high.
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
Along the same lines, an American put option on a stock
can be exercised early at any time before maturity, whether
or not the stock pays dividends. The optimal exercise policy is as follows: exercise the put early if, and only if, the
stock price is su¢ ciently low.
[Please explain Figure 15.3 and Figure 15.4 on page 524.]
Several approximations have been proposed for pricing
American options. Some of them are available in the
textbook. All of them su¤er from two major disadvantages. They are neither accurate nor general.
3.2.2
Transition Probabilities for Pricing American
Options
Transition probabilities under the risk-neutral probability measure completely characterize the dynamics of the
stock price. The following numerical procedure can be
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
used for pricing American options as long as the transition probabilities are known in a closed form.
n
o
Let G = a0 = 0; : : : ; ap; ap+1 = 1 be a grid points
for the stock price. Any choice for the ai, for i =
1; : : : ; p, will work as long as they cover a large domain
and verify
a ! 0, when p ! 1.
Consider the transition probabilities
Stn+1 2 [ai, ai] j Stn = ak , for all i and k,
which are known in a closed form in the Black-ScholesMerton model, among other models.
P
Consider an American option characterized by its exercise
value Vte, which can be exercised early at the decision
dates t0; : : : ; tN = T . Assume that the option-value
function is known at a decision date tn+1 and the grid
points G . This is not a strong assumption since the option
value is known everywhere at maturity. The numerical
procedure acts backward in time from tn+1 to time tn,
as follows.
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
1. Interpolate the value function at time tn+1 by a
piecewise constant approximation:
Vbtn+1 Stn+1 =
p
X
n+1
i
i=0
I Stn+1 2 [ai, ai] ,
where I ( ) is the indicator function, and
Vbtn+1 (ai), for all i.
n+1
i
=
2. Move backward in time from time tn+1 to time tn,
and consider the holding value
Vthn (ak )
=E
h
e r(tn+1 tn)Vtn+1 Stn+1
j Stn = ak ,
which is not known in closed form.
3. Approximate the holding value function as follows:
Vethn (akh)
i
r(t
t
)
b
n+1 n V
=E e
tn+1 Stn+1 j Stn = ak
=
P
e r(tn+1 tn)
X n+1
i
i
Stn+1 2 [ai, ai] j Stn = ak .
55
i
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
4. Approximate the option value at tn and at the grid
points G using
Vetn (ak ) = max Vten (ak ) , Vethn (ak ) , for all ak in G ,
and identify the optimal exercise policy.
5. Interpolate Vetn (ak ), for all ak in G , to Vbtn (s), for
all s > 0.
6. Follow the same procedure, and go from time tn to
time tn 1, backward in time to time t0.
7. Reach any level of desired precision by increasing the
grid size p since
Vbt0 St0 ! Vt0 St0 , when p ! 1.
This numerical procedure has been proposed by Ben Ameur
et al for American call and put options in European Journal of Operational research, and for American Asian options in Management Science. It is used by several …nancial institutions all over the world.
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Hatem Ben Ameur
3.3
Derivatives, Part II
Brock University, FNCE 4P17
Pricing American Options in the Binomial Model
The binomial model can be adjusted to be consistent with
the Black-Scholes-Merton model:
VtBM ! VtBSM, when
t ! 0.
The convergence speed is believed to be 0:5, meaning
that the error
e =j VtBM
VtBSM j ,
is cut by half if the time increment t is divided by 4.
The …rst part of this result is hard to prove, and the
second is an open question.
For the binomial model to be consistent with the BlackScholes-Merton model, set
u=e
p
t,
1
e(r ) t
d = , and p =
u
u d
57
d
.
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
Example: Consider an American call option on a stock
that pays dividends continuously at a …xed rate
=
13:75% (per year). The option parameters are as follows:
St0 = $60, X = $60, = 20% (per year), T t = 180
(days), r = 9% (per year), and t = 90 (days). Compute its early exercise premium, and identify its optimal exercise policy. Please check that u = 1:104412,
d = 0:905460, p = 0:416663, and that the two-period
binomial tree for the stock price is
S0 = 60
%
&
S1u = 66:26
S1d = 54:32
58
%
&
%
&
S2uu = 73:18
S2ud = S2du = 60 .
S2dd = 49:19
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
If the stock pays discrete dollar dividends over the option’s life, Black’s intuition applies to adjust the binomial
tree, though with a minor modi…cation.
1. Adjust downward the stock price at the present date
t0 by the PV of all dividends over the option’s life:
St0 = St0
PV all dividends.
This is somewhat equivalent to adjust St downward
by Dt, for all ex-dividend dates.
2. Create the binomial tree as usual.
3. At each decision tn, adjust upward the stock price
by the PV of all remaining dividends (including the
one that is to be paid).
4. Go backward through the adjusted binomial tree, and
ompute options values as usual.
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Hatem Ben Ameur
3.4
Derivatives, Part II
Brock University, FNCE 4P17
Assignment 2 Part 1
Read the chapter, do the examples, and exercises no 1–7
on page 537. This part is to be prepared for the midterm
exam, but should not be handed in as part of Assignment 2.
Do exercises no 8, 17, 19.
Exercise (Bonus – worth 10 points in the second
mid-term exam): This exercise is to be handed in after the reading week. Use VB-Excel. In all cases, your
program must be fully automated.
1 Implement Black’s procedure for pricing an American
call option on a stock that pays discrete dollar dividends.
Let the number of ex-dividend dates ‡exible. Solve the
example shown in the textbook.
2 Implement the analytical approximation of American option prices, and draw Figure 15.3 and Figure 15.4.
Solve the examples shown in the textbook.
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
3 Implement the binomial tree on a stock that pays
dollar dividends. Let the number of ex-dividend dates
‡exible. Solve the examples in the textbook. Report the
binomial trees reported in the textbook.
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Hatem Ben Ameur
4
Derivatives, Part II
Exotic Options
Topic Covered:
1. Forward-Start options
2. Instalment Options
3. Compound Options
4. Chooser Options
5. Barrier Options
6. Binary Options
7. Asset-or-Nothing Options
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Brock University, FNCE 4P17
Hatem Ben Ameur
Derivatives, Part II
8. Lookback Options
9. Asian Options
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Brock University, FNCE 4P17
Hatem Ben Ameur
4.1
Derivatives, Part II
Brock University, FNCE 4P17
Forward-Start Options
Standard call and put options are known as vanilla options. Other option contracts, known as exotic options,
are traded on exchanges and in over-the-counter markets.
They are supposed to match hedgers’needs and requirements. Forward-start options are examples.
Under a forward-start option, the holder pays the premium before the option starts at parity. There are three
key dates: the date the contract is signed t, the date the
option starts T1, and the date the option matures T2.
Date T1 is called the grant date.
Forward-start options are often used in compensation packages for executives.
Proposal: The presentation on executive options will
cover forward-start options, as well as other option contracts used in compensation packages.
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
Forward-start options can be evaluated using the riskneutral evaluation principle. Consider a European forwardstart option in the Black and Scholes model. The value
of the European forward-start option at the start is
vt
=E
h
e r(T1 t)vTBS
ST1 ; X
1
= ST1 ; ; T2
T1; r j St ,
since it starts at parity. On the other hand, one has
vTBS
ST1 ; X = ST1 ; ; T2
1
= ST1 f ( ; T2 T1; r ) ,
T1; r
where f is a function of the volatility of the stock logreturns , the option’s life T2
T1, and the risk-free
rate, but not the stock price ST1 . Using the martingale
property, the value of the European forward-start option
is
vt = E
= St
h
i
r(T
t)
1
e
ST1 j St
f
f BS ( ; T2 T1; r )
= vtBS (St; St; ; T2
T1; r ) ,
whether it is a call or a put option.
65
( ; T2
i
T1; r )
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
If the stock underlying the option pays continuous dividends at a constant dividend rate (in % per year), the
martingale property is adjusted as follows:
E
h
e r(T1 t)ST1 j St
i
=e
(T1 t) S ,
t
and the value of the European forward-start option is
consequently adjusted as follows:
vt =
=
=
h
i
r(T
t)
1
E e
ST1 j St
f ( ; T2 T1; r;
e (T1 t)St f BSM ( ; T2 T1; r; )
e (T1 t)vtBSM (St; St; ; T2 T1; r ) .
)
All in all, compute the value of a European forward-start
option at the grant date as if the stock price were equal
to the current stock price, and then discount this value
back from the grant date to the start date at the dividend
rate.
Valuing American forward-start options can be done backward in time through a stochastic dynamic program, as
it can be done for American vanilla options.
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Hatem Ben Ameur
4.2
Derivatives, Part II
Brock University, FNCE 4P17
Instalment Options
The holder of a European instalment option must pay
instalments at certain decision dates to keep the option
alive. Therefore, at each decision date, the holder must
choose between the following:
1. to pay the instalment, which keeps the option alive
till the next decision date;
2. not to pay the instalment, which puts an end to the
contract.
The main bene…ts for the holder of an instalment option are twofold. First, risk management with instalment
options is ‡exible. Like American options, instalment options are well suited for hedging cash ‡ows whose timing
is uncertain. Second, the hedging cost structure with instalment options is ‡exible too. Instead of paying lump
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
sums for hedging instruments, corporations may enter an
instalment option at a low initial cost, and adjust the instalment schedule according to their cash forecasts and
liquidity constraints. This feature is particularly attractive
for corporate treasurers who massively hedge interest-rate
and currency risks with forwards, futures, or swaps, because option contracts imply an entry cost that may be
incompatible with a temporary cash shortage.
Consider an instalment option with positive instalments
0 ; : : : ; N , to be paid at the decision dates t0 = 0; : : : ;
tN = T . The initial instalment 0 plays the role of the
option’s premium. By the risk-neutral evaluation principle, the net holding value of this instalment option at
decision date tn is
vtnh
(s )
n
= E [e r(tn+1 tn)vtn+1 (Stn+1 ) j Stn = s]
n,
and the overall value is
vtn (s) = max 0, vtnh
(s) , for n = 0; : : : ; N
n
68
1.
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
The value of the instalment option at maturity is
vtN (s) =
(
max (0; s X ) , for an instalment call option
.
max (0; X s) , for an instalment put option
The exit strategy at time tn is as follows: put an end to
the option’s life if, and only if,
(
Stn < an, for an instalment call option
,
Stn > bn, for an instalment put option
where an and bn are two thresholds that are functions of
the option parameters. The threshold is determined by
solving for s = Stn the following equation:
vtnh
(s) = 0.
n
One way of pricing the instalment option is via backward
iteration: from the known function vtN , compute vtN 1 ,
then from vtN 1 compute vtN 2 , and so on, down to vt0 .
Neither the value function vtn , for n = 0; : : : ; N 1, nor
the threshold vn (an or bn) is known in a closed form.
They must be approximated in some way. Stochastic
69
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
dynamic programming can be used (Ben Ameur et al.,
2006, European Journal of Operational Research).
Example: Consider an instalment call option with only
one intermediate instalment 1 to be paid at t1 < t2 =
T . The net holding value at the decision date t1 is
r(T t1 ) c (S ) j S = s]
cnh
t1
1
T T
t1 (s) = E [e
= E [e r(T t1) max (0; ST X ) j St1 = s]
= cBSM
St1 ; X; ; T t1; r;
1,
t1
1
where X is the option’s strike price, and the overall value
is
ct1 (s) = max 0; cBSM
St1 ; X; ; T
t1
t1; r;
1
.
This is the payo¤ from a European call option with maturity t1 and strike price 1 on a call option with maturity
T and exercise price X , the underlying call option being
written on a dividend-paying stock. Options on options
are also known as compound options. There are mainly
four types of compound options: calls on calls, calls on
puts, puts on calls, and puts on puts.
70
Hatem Ben Ameur
4.3
Derivatives, Part II
Brock University, FNCE 4P17
Chooser Options
The holder of a chooser option has the right to determine whether the chooser will become a vanilla call or
put option by a speci…ed choice date. Chooser options
are also known as as-you-like-it options. For simplicity,
assume that the call and put options have the same exercise price and maturity date.
Chooser options may be useful for hedging market risk,
which strongly depends on the occurrence of a future
event.
Example: In 1993, the North American Free Trade Agreement (NAFTA) was being discussed, but had not yet been
agreed upon. NAFTA was known to be bene…cial to the
Mexican peso. Before the NAFTA agreement, chooser
options were e¤ective for hedging the Mexican peso risk.
If NAFTA went through, the chooser option would then
become a call option; otherwise, the chooser option would
become a put option on the Mexican peso.
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
There are mainly three key dates to consider: the evaluation date t, the choice date T1, and the expiry date
T2.
If T1 = t, one has
vt = max cBSM
, pBSM
,
t
t
and if T1 = T2, one has
vT = max (cT , pT )
= (ST X ) I (ST > X ) + (X
X) ,
S T ) I (ST
and, by the risk-neutral evaluation principle,
vt = cBSM
+ pBSM
,
t
t
which is the value of the associated straddle.
Now, if T1 2 (t, T2), one has
vT1
BSM
= max cBSM
T1 , pT1
BSM
r(T2 T1 )
= max cBSM
T1 , cT1 + Xe
ST1 e
= cBSM
T1 + e
)(T2 T1 )
(T2 T1 ) max
72
0, Xe (r
(T2 T1 )
ST1 ,
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
where the right-hand side consists of a portfolio of a call
option with a strike price of X and a maturity date of
T2 and e (T2 T1) put options with a strike price of
Xe (r )(T2 T1) and a maturity date of T1. The riskneutral evaluation principle gives
vt = cBSM
(St; X; ; T2 t; r; ) +
t
(r )(T2 T1 ) ; ; T
e (T2 T1)pBSM
S
;
Xe
t
1
t
4.4
t; r;
Barrier Options
A barrier option acts as a vanilla option under the assumption that the underlying stock price reaches a barrier
along its path. There are four families of barrier options:
down-and-in, down-and-out, up-and-in, and up-and-out
barrier options. They are also known as knock-in and
knock-out options.
[Please discuss the payo¤ from a down-and-in call option.]
73
.
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
A barrier option is a path-dependent option since its …nal payo¤ does not only depend on the stock price at
maturity, but also on its behaviour over the option’s life.
Barrier options are not traded; however, they are useful
at modeling knock-in and knock-out events.
Example: Several corporate securities can be interpreted
as derivatives on a …rm’s value. For example, Merton’s
model of a corporate discount bond assumes that the
…rm can only go bankrupt at maturity, which is a major
limitation. One way to improve Merton’s model is to
de…ne bankruptcy as
Vt < bt, for t 2 [0, T ] ,
where Vt is the …rm’s value at time t, and bt is a barrier related to the debt’s amortization. Instalment and
compound options are also useful at modeling corporate
coupon bonds, since bondholders can put an end to the
corporate bond if certain instalments are not paid.
74
Hatem Ben Ameur
4.5
Derivatives, Part II
Brock University, FNCE 4P17
Binary Options
A binary option pays a certain amount if an event happens, but nothing otherwise. This event is usually related
to the performance of an underlying …nancial asset.
A cash-or-nothing call option pays its holder a …xed amount
if the underlying …nancial asset exceeds a given strike
price X at maturity T . The value of a binary option is
Binary
ct
=
=
=
h
E e r(T t)AI (ST > X )
e r(T t)AP (ST > X )
e r(T t)AN dBSM
,
2
i
where A is the amount to be paid if the binary option
expires in the money, and N dBSM
is the normal dis2
tribution function evaluated at dBSM
. Along the same
2
lines, the value of a cash-or-nothing put option is
Binary
pt
= Ae r(T t)N
75
dBSM
.
2
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
An asset-or-nothing call option promises its holder the
underlying asset under the condition that the underlying
asset exceeds a given strike price at maturity. This is a
plain vanilla call option with X = 0 whose value is
Binary
ct
= St e
(T t) N dBSM
1
.
The value of an asset-or-nothing put option is
Binary
pt
4.6
= St e
(T t) N
dBSM
.
1
Lookback Options
The payo¤ function from a lookback option is based on
extreme values.
The payo¤ from a European option to buy an underlying
asset at the lowest price along its path is
ST
min St,
[0, T ]
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
and the payo¤ from a European option to sell at the
highest price is
max St
[0, T ]
ST .
Under Black, Scholes, and Merton’s model, lookback options can be evaluated in a closed form, since the properties of lognormal processes and their extreme values are
known.
4.7
Asian Options
The payo¤ from an Asian option is based on average
prices, rather than terminal prices. Asian options are useful in markets where there are major players that can manipulate market prices, at least locally in time. Examples
include oil and oil-product markets.
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
For example, a European Asian call option is characterized by the following …nal payo¤:
max 0, S T
St0 +
+ StN
X = max 0,
N +1
!
X .
Other exotic options do exist, such as exchange options,
basket options, rainbow options, and gap options, among
others. They may be helpful to hedge for speci…c risks,
or better analyse fundamental …nancial securities.
4.8
Assignment 2 Part 2
Read the chapter, and do exercises no 1, 5, and 6 on page
616.
78
Hatem Ben Ameur
5
Derivatives, Part II
Brock University, FNCE 4P17
Options on Stock Indices, Currencies, and Futures
Topics Covered:
1. Adjusting Basic Models for Continuous Dividends
2. Options on Stock Indices
3. Options on Foreign Currencies
4. Options on Futures
79
Hatem Ben Ameur
5.1
Derivatives, Part II
Brock University, FNCE 4P17
Options on Stock Indices
Merton extends the Black and Scholes model for a stock
that pays dividends continuously at a …xed rate (in %
per year). From now on, the Black and Scholes model
will be indicated by BS, and the Black-Scholes-Merton
model by BSM.
In BSM, e (T t) shares of stock at time t is equivalent
to one share of stock at time T , the dividends being
reinvested in additional shares over [t, T ]. Thus, BSM
call- and put-option values are obtained by exchanging
St in BS by
St e
(T t) .
For example, the BSM formula for a European call option
is
cBSM
= N (d1) Ste
t
(T t)
80
Xe r(T t)N (d2) ,
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
where
d1 =
=
ln Ste
(T t) =X
p
ln (St=X ) + r
d2 = d1
p
p
+ r + 2 =2 (T
T t
+ 2 =2 (T
T
t)
t
t)
, and
t.
T
The binomial model can also be adjusted for a stock
that pays continuous dividends. The adjustment is done
through risk-neutral probabilities:
e(r ) d
.
p =
u d
Options in BSM admit the following bounds:
cBSM
t
St e
pBSM
t
Xe r(T t)
(T t)
Xe r(T t),
and
81
St e
(T t) .
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
Exercise: Use the no-arbitrage property, and establish
these lower bounds.
The put-call parity for European options in the BSM is
r(T t) = pBSM + S e
cBSM
+
Xe
t
t
t
(T t) ,
and for American options is
St e
(T t)
X
CtBSM
PtBSM
St
Xe r(T t).
Exercise: Use the no-arbitrage property, and establish
the put-call parity for European call and put options.
5.2
Options on Stock Indices
Options on stock indices are traded in exchanges and
over-the-counter markets. Some indices track the movement of the overall stock marker, and others track the
movement of particular industries.
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
Example: The S&P 100 and S&P 500 stock indices are
based on selected 100 and 500 US stocks, respectively.
Options on S&P 100 are of American style, while options
on S&P 500 are of European style.
Since the delivery of a stock index involves high transaction costs, stock-index options are settled in cash using
the following payo¤ function:
Cte = m
max (0; It
X) ,
max (0; X
It) ,
for a call option, and
Pte = m
for a put option. Here, It is the stock-index level at time
t, m is the multiplier, and X is the option’s strike price.
The scalar m plays the role of the option’s size, and is
usually set at 100.
Other exchange-traded options related to stock indices
are the so-called LEAPS, CAPs, and FLEX options.
83
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Derivatives, Part II
Brock University, FNCE 4P17
A stock index is a …nancial asset that provides its holder
with several discrete dividends throughout all over the
year. The assumption that dividends are distributed continuously at a …xed rate is acceptable in this context. The
dividend rate can easily be infered from futures prices, or
estimated from historical observations. Thus, the BSM
formula can be used for pricing.
Exercise: A European call option on a stock index has
the following parameters: It = 350, X = 340,
=
20%, T
t = 150 days, r = 8% (per year), = 4%
(per year). Set m = 1. Check that cBSM
= $25:92.
t
Compute pBSM
.
t
A call option on a stock index can be used to hedge
against upward movements of the stock market, and a
put option on a stock index, to hedge against downward
movements of the stock market.
84
Hatem Ben Ameur
5.3
Derivatives, Part II
Brock University, FNCE 4P17
Foreign Currency Options
Options on foreign currencies are traded both in exchanges
and in over-the-counter markets.
For example, options on the Canadian dollar are traded
on the Philadelphia Stock Exchange.
A foreign currency continuously pays its holder interest
at the foreign risk-free rate. Thus, BMS applies with
= rf , where rf is the foreign risk-free rate (in % per
year).
For example, the European call-option value is
rf (T t)
cBSM
=
N
(
d
e
)
S
t
1
t
Xe r(T t)N (d2) ,
where
d1
d2
ln Ste rf (T t)=X + r + 2=2 (T t)
p
=
T t
ln (St=X ) + r rf + 2=2 (T t)
p
=
, and
T t
p
= d1
T t.
85
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
Example: Consider a European call on the British pound
with the following parameters: St = 1:4 (US$ per British
pound), X = 1:5 (US$ per British pound),
= 50%
(per year), T
t = 200 days, r = 8% (per year), and
rf = 12% (per year). Use a …ve-period binomial tree,
and price the European call option. Compute the value
of the associated put option.
The parameters of the binomial tree are
u=e
t
= 1:1180, d =
1
= 0:847452,
u
and
e(r rf ) t
p =
u d
d
= 0:445579.
Figure 16.1 gives the …ve-period binomial tree. Riskneutral evaluation is then used to value the call option.
The result is cBSM
= 0:1519 (US$ per British pound).
t
Given the call-option value, compute the put-option value.
86
Hatem Ben Ameur
5.4
Derivatives, Part II
Brock University, FNCE 4P17
Options on Futures Contracts
Unlike options on stocks, options on futures contracts are
settled in cash.
Let t, T1, and T2 T1 be the current date, the option’s
maturity date, and the futures’delivery date, respectively.
The futures’delivery date needs not be equal to the option’s maturity date. Suppose that the futures contract
is active at the current date t.
If a call option on a futures contract is exercised at t, the
call-option holder acquires a long position in the underlying futures contract plus a cash amount equal to
m
ft
X ,
where m is the futures’ size, t the current date, ft the
futures price at t, ft the last settlement price, and X
the option’s strike price.
87
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
Example: Consider a futures call option on copper with
a strike price X = $0:70 (per pound of copper). One
futures contract is on 25,000 pounds of copper. The
futures price for delivery in one month is ft = $0:81
(per pound of copper), and the last settlement price is
ft = $0:80 (per pound of copper). If the call-option
holder exercises his right, he is given a long position on
the futures contract and a cash amount of
25; 000
(0:80
0:70) = $2; 500.
If desired, the position in the futures contract can be
closed out immediately, which would leave the investor
with the following cash amount:
m
ft
X +m
ft
= m (f t X )
= 25; 000 (0:81 0:70)
= $2; 750.
ft
The futures price for delivery at T is related to its underlying asset price:
ft = Ste(c y)(T t),
88
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
where St is the price of the futures’ underlying asset at
t, c the cost of carry, and y the convenience yield. The
cost of carry measures the storage cost plus the interest
required to …nance the asset less the income earned on the
asset. For example, for 1- a non-dividend paying stock,
set c = r and y = 0, 2- a stock index, set c = r
and
y = 0, 3- a foreign currency, set c = r rf and y = 0,
and 4- a commodity held for consumption, the general
formula applies.
For an investment asset underlying the futures contract,
one has
ft = Ste(r
)(T t) ,
which is equivalent to
St = fte (r
)(T t) .
Substituting this expression for St in the BSM formula
gives Black’s formula, which turns out to be independent
of . Indeed, the holding in‡ows and out‡ows are already
included in the futures price.
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
The call-option value is
r(T t)
cB
=
N
(
d
f
e
)
t
1
t
Xe r(T t)N (d2) ,
where
d1
d2
ln fte r(T t)=X + r + 2=2 (T
p
=
T t
ln (ft=X ) + 2 (T t) =2
p
=
, and
T t
p
= d1
T t.
t)
Black’s formula is equivalent to the BSM formula with
St replaced by ft, and by r, as if a futures option were
an option on an underlying asset that continuously pays
dividends at the domestic risk-free rate.
Example: Discuss the option on the stock index futures
on page 546.
European as well as American futures options can be evaluated through a binomial tree, adjusted as follows:
p =
1
u
90
d
.
d
Hatem Ben Ameur
5.5
Derivatives, Part II
Brock University, FNCE 4P17
Assignment 2 Part 2
Read the chapter, do the examples, and exercises no 1–5
on page 564. This part is to be prepared for the midterm
exam, but should not be handed in as part of Assignment 2.
Do exercises no 6 and 8. The question dealing with
Barone-Adesi and Waley’s formula is not mandatory.
Exercise: The sensitivity parameters in the BS model
are known in closed form (see Chapter 2). Give these parameters in closed form in the BSM model, and explain
your approach. Give the associated formulas for options
on stock indices, foreign currencies, and futures. In this
context, single- as well as multiple-asset sensitivity analysis work, modulo some adjustments.
91
Hatem Ben Ameur
6
Derivatives, Part II
Brock University, FNCE 4P17
Pricing Corporate Securities
Topics Covered:
1. Common Stock as a Call Option on the Firm
2. Senior and Junior Debts
3. Callable Bonds
4. Convertible Bonds
5. Warrants
92
Hatem Ben Ameur
6.1
Derivatives, Part II
Brock University, FNCE 4P17
Common Stock as a Call Option
Consider a model for a company with a simple capital
structure, consisting of one common stock and a purediscount bond.
The …rm value Vt at the current time t veri…es
Vt = St + Bt,
where St and Bt are the stock price and bond value at
time t, respectively. This equation holds under the condition that the company is active, that is,
Bt .
Vt
Since the corporate bond bears some credit risk, the PV
evaluation principle cannot be used. Risk-neutral pricing
is a viable alternative. Here, the stock and the corporate
bond are interpreted as option contracts on the …rm.
The following result, established by Merton, is a clue to
pricing corporate securities. The common stock can be
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
seen as a European call option on the …rm with an exercise price and expiry date equal to the bond’s face value
and maturity, respectively. Indeed, if the …rm value at
maturity exceeds the bond face value, that is,
FV,
VT
then
FV
ST = VT
0.
Conversely, if
VT < FV,
then the …rm will go bankrupt, the bondholder will take
control of the …rm, and the stock will end up worthless,
that is,
ST = 0, and BT = VT .
In sum, the stock price at maturity can be expressed as
ST = max (0; VT
94
FV) .
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
This results in interpreting the stock as a call option on
the …rm.
[Please explain Figure 17.1 on page 567.]
The current stock price is then
St = cBS
t; V ; r ) ,
t (Vt; FV; T
where cBS
t is the Black and Scholes formula for a European call option, and V is the volatility associated to
the …rm, but not to the stock.
Exercise: Single-asset sensitivity analysis applies in this
context. Specify the impact on the stock price of a small
change in the …rm value, time to maturity, …rm volatility,
and risk-free rate.
In all cases, the corporate-bond value at maturity is
BT = VT
= VT
= FV
ST
max (0; VT
max (0; FV
95
FV)
VT ) .
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
Two useful interpretations result from the above equations. Holding the corporate bond is equivalent to
1. holding the entire company and selling a European
call option to the stockholder to buy the company
for the bond’s face value,
2. holding a riskless bond and selling a put option to
the stockholder to sell the company for the bond’s
face value.
[Please explain Figure 17.2 on page 568.]
The put-call parity gives
Bt = Vt
= FV
cBS
t (Vt; FV; T
e r(T t)
t;
V ; r)
pBS
t (Vt; FV; T
t;
V ; r) ,
showing that the corporate bond is worth less than its associated riskless bond. The di¤erence in values represents
the credit worthiness of the issuing company. The higher
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
the put-option value, the lower is the credit worthiness of
the issuing company.
The common stock and the corporate bond are interpreted as derivative contracts whose values, St and Bt,
depend on the …rm value Vt and its volatility V . There
are several methods used to approximate Vt and V ,
some of them from St and S .
Exercise: Portfolio-sensitivity analysis applies in this context. Specify the impact on the corporate-bond value of
a small change in the …rm value, time to maturity, …rm
volatility, and risk-free rate.
Moody’s KMV o¤ers a default-prediction model that is
based on option theory and its applications for corporate
securities.
97
Hatem Ben Ameur
6.2
Derivatives, Part II
Brock University, FNCE 4P17
Senior and Subordinated Debts
Debt contracts di¤er by their securitization levels. For example, senior debts are reimbursed …rst (if possible), and
subordinated debts are reimbursed next. Subordinated
debts are also called junior debts.
Consider a company with a simple capital structure that
consists of one common stock, a pure-discount senior
bond with a face value of FVs, and a pure-discount junior
bond with a face value of FVj , both maturing at the same
j
time T . Let Bts and Bt be the senior- and junior-bond
values at time t, respectively. Three scenarios can happen
at maturity:
8
s!
>
V
<
FV
>
T
>
>
>
s = V , and B j = 0
>
>
S
=
0,
B
T
T
>
T
T
>
>
s
j
< FVs
VT < FV + FV !
.
s
s = FVs , and B j = V
>
S
=
0,
B
FV
>
T
T
T
T
>
>
s + FVj !
>
>
V
FV
>
T
>
>
j
>
: ST = VT
FVs + FVj , BTs = FVs, and BT = FVj
98
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
Whatever happens at maturity, these equations can be
summerized as
FVs + FVj
ST = max 0, VT
,
for the stock,
BTs = FVs
max (0, FVs
VT ) ,
for the senior debt, and
j
BT = max (0, VT
FVs) max 0, VT
FVs + FVj
for the junior debt.
In sum, the common stock can be interpreted as a European call option on the …rm with a strike price of FVs+
FVj ; the senior bond as portfolio of a riskless bond with
a face value of FVs and a signed European put option
with a strike price of FVs; and the junior bond as a portfolio of a European call option with a strike price of FVs,
and signed European call option with a strike price of
FVs+FVj . The riskless bond and all option contracts
expire at the corporate bond’s maturity.
[Please explain Figures 17.3 and 17.4 on page 571.]
99
,
Hatem Ben Ameur
6.3
Derivatives, Part II
Brock University, FNCE 4P17
Callable Bonds
A callable bond can be redeemed by the issuer before
maturity for a known call price. A callable can be seen as
a portfolio of a straight bond that contains an embedded
call option at the discretion of the issuer.
Corporate bonds are typically callable, while T-bonds are
seldom callable.
Some callable bonds can be redeemed at any time, and
others at speci…ed dates before maturity. However, the
investor is often protected against the call feature over a
certain protection period, for example, the …rst …ve years
of the bond’s life.
Unlike vanilla call options, call options embedded in bonds
are not traded alone, but rather live within their associated bonds.
100
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Derivatives, Part II
Brock University, FNCE 4P17
The value of a callable bond is always lower than the value
of its associated straight bond, for it is less desirable.
Thus, the value of the embedded call option is
straight
Ct = B t
Bt ,
where Ct is the value at time t of the embedded call
straight
option, Bt
and Bt are the values at time t of the
straight bond and the callable bond, respectively. Here,
capital letters are used since the bond with its embedded
call option can be interpreted as an American derivative
contract.
Clearly, the bond issuer is better to call the bond early
when interest rates are low, and …nance its activities at
a lower cost. The optimal exercise policy for the bond
issuer is as follows: redeem the bond before maturity if,
and only if, the interest rate is su¢ ciently low.
[Please explain Figure 17.5 on page 572.]
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Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
Pricing callable bonds is complex, and can be achieved
through a dynamic program as for American vanilla options. For simplicity, consider a T-bond that can be redeemed at each coupon date.
Assume that the callable bond value is known at a given
future date tn+1.This is not really a limitation since the
bond value is known at maturity, that is, BT = 1 + c,
where c is the last coupon. Risk-neutral evaluation gives
Bnh(r) = E
"
n+1
n
#
Bn+1 rtn+1 j rtn = r ,
where frg is the interest-rate process, n+1= n is the
discount factor over [tn, tn+1], Bnh the bond holding
value at the current decision date tn, and Bn+1 the
(overall) bond-value function at the next decision date
tn+1. Though highly complex, this expectation can be
e¢ ciently computed under several dynamics for the interest rate, which are consistent with the no-arbitrage
principle. See the paper by Ben Ameur et al published in
Journal of Economic, Dynamics, and Control.
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The payo¤ at time tm from the issuer to the investor is
(
cn + c, if the issuer calls
,
Bnh(r) + c otherwise
where cn is the call price at time tn, which is known at
the start.
The issuer must redeem the bond at tn if, and only if,
the holding value exceeds the exercise bene…t, that is,
Bnh(r) > cn,
which is equivalent to
r < rn ,
where r = rtn is the current interest-rate level at time
tn, and rn is a threshold at time tn that identi…es the
optimal redemption policy.
The overall value function of the callable bond at time
tm is
Bn(r ) = min cn, Bnh + c.
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Finally, the embedded call option can be computed using
the equation
straight
Ct = B t
Bt .
Achieving these steps backward in time from maturity to
the start provides the bond value function at each decision
date, and identi…es its optimal redemption policy.
This methodology can be adjusted a bit to accommodate
corporate callable bonds.
"
6.4
Please discuss the callable bond issued by
the Swiss Confederation.
#
Convertible Bonds
A convertible bond can be converted into shares of the
issuing …rm at the discretion of the bond holder.
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The conversion ratio is the number of shares received by
the investor upon conversion, indicated here by N .
The conversion ratio N is often …xed at the start, such
that
S0 < B0,
N
to preclude immediate conversion.
For simplicity, consider a European convertible discount
bond with maturity T , face value FV, which can be converted into shares at maturity.
At maturity, the …rm will default, or the bond will pay its
holder the larger between its face value and its conversion
value.
In case of default, that is,
VT < FV,
one has
BT = VT , and ST = 0,
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where VT is the …rm value, BT the bond value, and ST
the stock price at maturity. Otherwise, the …rm value
ver…es
FV,
VT
and the bondholder will decide whether to convert the
bond or not. The bond value is
BT = max (FV, N
ST )
= FV + max (0, N
ST
FV) .
Please notice that the known amount FV does not represent a riskless straight bond, but a straight bond issued
by the …rm. Indeed, this payment is uncertain, and will
only be received under the condition that
FV.
VT
All in all, the convertible bond can be interpreted as a
portfolio of an otherwise identical corporate bond and a
call option on N shares of stock with an exercise price
equal to the bond’s face value.
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Proposal: A possible presentation could focus on coupon
corporate bonds, which include both call and conversion
features. This presentation would include callable and
convertible bonds with ‡exible call prices and conversion
factors.
6.5
Warrants
A warrant is similar to a call option except that it is
written by the underlying company, and typically has a
long maturity.
If a warrant is exercised, the underlying company issues
a new share for delivery. A dilution e¤ect takes place,
and stockholders lose some value. Exercising the warrant
makes sense only when the resulting share price exceeds
the exercise price.
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Consider a …rm with a simple capital structure, that is,
M shares of stock and N European warrants. The …rm
value and the stock price just before maturity are related
as follows:
VT
ST .
=M
At maturity, they are related as follows:
VT = M
ST + N
max (0, ST
X) .
If the warrant expires out of the money, the …rm value and
the stock price move along continuous paths; otherwise,
the …rm value remains continuous, while the stock price
jumps downward at maturity. Thus, under exercise at
maturity, one has
VT
=M
ST
= VT
=M
ST + N
wT
=M
ST + N
(ST
108
X) ,
Hatem Ben Ameur
Derivatives, Part II
which can be summarized as
M ST + N
ST =
M +N
Brock University, FNCE 4P17
X
.
The warrant value at maturity is
wT = max (0, ST
M
= max 0,
= max 0,
=
=
M
M +N
1
1+
N
M
X)
ST + N
M +N
M
ST
M +N
max 0, ST
max 0, ST
X
X
M
X
N +M
X
X .
By the law of one price, the warrant value and its associated call-option value are related in the same way:
wt =
1
1+
N
M
cBS
t
N
St + w t , X , S , T
M
t, r ,
where S is the volatility of the stock log-returns, including the warrant. This is a formula for the warrant value
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as a function of the warrant value, which can be solved
numerically.
Proposal: A possible presentation could focus on the relationship between option theory and investment projects,
known as real options. This presentation would include
options to explore, develop, extend, postpone, switch,
and shut down a real project.
6.6
Assignment 3 Part 1
Read the chapter, and do exercises no 1, 2, 3, 5, 6, 7,
and 8 on pages 576 and 577.
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7
Derivatives, Part II
Brock University, FNCE 4P17
The Long-Term T-Bond Futures
Contract (Revisited)
7.1
Futures Contracts
A forward contract commits two parties, one to buy and
the other to sell an underlying asset at a known future
delivery date (maturity) for an agreed-upon delivery price
also known as the forward price. No payment is exchanged up front. Forward contracts are traded over the
counter; therefore, forward-market participants are subject to both market and credit risk.
If prices rise, the long party records a gain since he committed to buying the underlying asset at a lower price,
and the short party records a loss. The opposite happens when prices fall. The cumulative loss recorded by
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one party at maturity can be so great that he can fail to
ful…ll his obligations.
A futures contract acts as a forward contract except that
it is traded on an exchange, and thus is subject to a
margin system and a daily settlement of gains and losses.
Unlike a forward contract, which is settled at maturity, a
futures contract is settled at the end of each trading day,
thereby keeping the loss and counterparty risk at a very
low level.
For each futures contract, the exchange de…nes a set of
terms and conditions related to the contract’s size, quotation unit, minimum price ‡uctuation, grade, trading
hours, delivery terms, daily price limits, and delivery procedures.
First of all, when a futures contract is issued, each party
is invited to open a margin account with his broker, and
to deposit an initial margin. The broker is invited to do
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the same with his clearing …rm, which in turn, does the
same with the clearinghouse.
Next, the trading process continues until the day’s end.
The settlement price is then registered, based on the last
futures price(s). A new settlement price is revealed at the
end of each trading day.
Finally, the futures contract is marked to the market at
the end of each trading day. The contract is closed and
reopened at the new settlement price, and the value of
both parties is consequently reset to zero. The di¤erence
between the new and the last settlement prices, if positive, is subtracted from the seller’s margin account, and
added to the buyer’s margin account. If the di¤erence in
prices is negative, then the opposite is done. This is the
daily settlement system.
When a margin account falls below a certain level, known
as the maintenance margin, the investor receives a margin
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call to bring the margin balance back up to the initial
margin level.
The margin system and daily settlement drastically reduce counterparty risk, since losses, which are monitored
every day, remain limited.
To avoid physical delivery, positions on futures contracts
are usually closed before the delivery month, and thus settled in cash. Liquidity makes it possible to leave futures
markets at any time before maturity.
7.2
The T-Bond Futures Contract: The
Margin and the Daily Settlement Systems
The T-bond futures contract, traded on the CBOT, calls
for the delivery of $100,000 T-bonds with a minimum remaining life of 15 years at the …rst delivery date, within
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a delivery month. Thus, at the …rst delivery date, a deliverable T-bond has at least 15 years to maturity or to
its earliest call date. De…ne the reference T-bond underlying this futures contract as a hypothetical T-bond
with a maturity of 20 years and a coupon rate of 6%
(per year). Delivery months are March, June, September, and December. The initial margin is $2,500, and
the maintenance margin is $2,000.
This futures contract is one of the most widely traded in
the world, mainly because its ability to hedge long-term
interest-rate risk.
In the following Table, we consider a short position on
the CBOT T-bond futures contract, which was taken on
08/01 at the quoted futures price of 97 27. The futures price was thus at $97 + 27=32 per $100 of principal, which corresponded to $97; 843:75 per $100; 000
of principal amount. The seller deposited $2; 500 into
his margin account. The futures price fell to $97; 406:25
at the end of the …rst trading day. This was the new
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settlement price, the old one being $97; 843:75. The
seller recorded a gain, and the buyer recorded a loss. The
absolute di¤erence between the two settlement prices,
that is, $437:50, was subtracted from the buyer’s margin account, and added to the seller’s margin account.
The balance of the seller’s margin account then rose to
$2; 937:50. The futures price rose to $97; 781:25 at
the end of the second trading day. This was the new
settlement price, the old one having been $97; 406:25.
The seller recorded a loss, and the buyer a gain. The
absolute di¤erence between the two settlement prices,
that is, $375:50, was subtracted from the seller’s margin account, and added to the buyer’s margin account.
The balance of the seller’s margin account was then at
$2; 562:50. . . On 08/08, the balance of the seller’s margin account fell to $1250 just after the contract was
marked to the market, which is lower than the maintenance margin of $2000. A margin call of $1250 was
issued the same day, and honoured the next trading day,
on 08/11. On 08/18, the last trading day, the settlement price fell from $100; 781:25 to $100; 500:00, and
the seller then recorded a gain of $281:25. This trader
then closed his position by inversion, left the market, and
withdrew the remaining balance of $3; 031:25.
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7.3
Derivatives, Part II
Brock University, FNCE 4P17
Embedded Options
The hypothetical T-bond is typically not traded on the
market, and thereby cannot be delivered. Thus, the short
trader is given the right to deliver alternative long-term
T-bonds. This is the choosing option, which is also called
the quality option.
To make the delivery fair for both parties, the price received by the short trader is adjusted according to the
quality of the T-bond delivered. This adjustment is made
via a set of conversion factors, which are de…ned by the
CBOT as the prices of the eligible T-bonds at the …rst delivery date under the assumption that interest rates for all
maturities equal 6% per year, and that interest is compounded semi-annually. The T-bond, which is actually
selected for delivery, is known as the cheapest to deliver.
The short trader is given the right to deliver the underlying T-bond within a delivery month. This is the timing
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option. The delivery process takes place according to special features, that is, the delivery sequence and the endof-month delivery rule. The delivery sequence consists
of three consecutive business days: the position day, the
notice day, and the delivery day. During the position day,
the short trader can declare his intention to deliver until
8:00 p.m., while the CBOT closes at 2:00 p.m. (Central Standard Time). This six-hours option is known as
the wild card play. On the notice day, the short trader
has until 5:00 p.m. to state which T-bond will be actually delivered. The delivery then takes place before 10:00
a.m. on the delivery day, against a payment based on the
settlement price on the position day. Finally, during the
last seven business days before maturity, trading on the
T-bond futures contract stop while delivery, based on the
last settlement price, remains possible according to the
delivery sequence. This is the end-of-month option.
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7.4
Derivatives, Part II
Brock University, FNCE 4P17
Evaluation
So far, papers in the literature have considered essentially
the quality option under a ‡at term-structure of interest
rates. The futures contract is then equivalent to a forward
contract.
Let r be the level of interest rate over the contract’s life
[0, T ], and let (c, M ) be the deliverable T-bond with
maturity M and coupon rate c. The theoretical futures
price g0 and the cheapest to deliver (c , M ) are obtained by solving the following equation:
max f(c;M )g0
(c;M )
pT (c:M; r ) = 0,
where f(c;M ) and pT (c:M; r ) are the conversion factor
and the fair value of the T-bond (c, M ) at maturity,
respectively. The conversion factor of the T-bond (c; M ),
de…ned as
f(c;M ) = pT (c; M; 6%) ,
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veri…es
Derivatives, Part II
Brock University, FNCE 4P17
8
>
< > 1,
if c > 6%
f(c;M ) : = 1, if c = 6% .
>
: < 1, if c < 6%
Thus, the conversion factor f(c;M ) adjusts the futures
price g0 upward if a high-quality T-bond (c, M ) is delivered, and downward if a low-quality T-bond (c, M ) is
delivered. No adjustment is made if the reference T-bond
(6%, 20) is delivered. The conversion factors make delivery somewhat fair for both counterparties, but they are
not perfect.
Exercise: Show that the cheapest to deliver bears an extreme coupon. For simplicity, assume that the contract’s
expiry date is a coupon date. Identify the cheapest to deliver as a function of the interest rate r. The last question
is a bonus question, and is not required for the second
midterm exam.
Ben-Abdallah, Ben-Ameur, and Breton (2009) (GERAD
report, and Journal of Banking and Finance) provided a
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full representation of this futures contract, with all its
embedded options. The most important results are:
1. The quality option is the most valuable embedded
option (around 2% of par for viable interest-rate levels);
2. The timing option is the second most valuable embedded option (around 0.2% of par for viable interestrate levels);
3. The end-of-month delivery rule and the wild card
play are less important (around 2 and 0.2 basis points
of par and less, respectively).
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8
8.1
Derivatives, Part II
Brock University, FNCE 4P17
Sample of Exams
First Midterm Exam
Exercise 1 (20%): Consider a three-period binomial
tree for a non-dividend-paying stock with the following
parameters: S0 = 100 (in dollars), u = 1:25, d = 0:8,
r = 7% (per period), and
p =
1+r d
.
u d
1. Show that this market model is arbitrage free.
2. Draw the three-period binomial tree for the stock.
3. Evaluate the European put option on this stock with
a strike price X = 100 (in dollars) and a maturity
T = t3.
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4. Evaluate the associated American put option, and
identify its optimal exercise policy.
5. Evaluate the early-exercise premium associated to
this put option.
6. Consider an Asian put option characterized by the
following exercise value:
Pten = max 0, X
S tn
St0 +
+ Stn
= max 0, X
,
n+1
for n = 0, 1, 2, 3 .
Answer again questions no 3, 4, and 5.
Exercise 2 (20%): Consider two European call options
with the same parameters except for their strike prices.
A bull spread with calls consists of a long position on
the call option c1 with the lower strike price X1, and
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a short position on the call option c2 with the higher
exercise price X2. The current time is indicated by t,
and the maturity by T . The bull spread’s pro…t function
is de…ned as:
= vT
= c1T
vt
c1t
(c2T
c2t) ,
where vt and vT are the values of the bull spread at time
t and T , c1t and c1T are the values of c1 at time t and
T , and c2t and c2T are the values of c2 at time t and T ,
respectively. The following Table provides the sensitivity
parameters of the two call options.
c1
Strike price
Value
Delta
Gamma
Theta
Vega
Rho
90
16:33
0:7860
0:0138
11:2054
20:4619
30:7085
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c2
100
10:30
0:6151
0:0181
12:2607
26:8416
25:2515
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
The current stock price St is at $100, the volatility at
30% (per year), the remaining time to maturity T t at
180 days, and the risk-free rate r at 8% (per year).
1. Plot the curve of the bull spread’s pro…t, as a function of the stock price at maturity ST .
2. Explain why this strategy is called a bull spread.
3. Suppose that the current stock price rises by $0.1.
What is the impact on the bull spread’s value? What
is the impact on the bull spread’s pro…t? Use Delta,
then use Delta and Gamma.
4. Is this bull spread subject to time decay? Explain.
5. Suppose that the current price will rise by $0.1, and
that the volatility will drop by 1% over the next two
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business days. What is the impact on the current
bull spread’s value? What is the impact on the bull
spread’s pro…t?
Exercise 3 (20%): Consider an arbitrage-free market
model for a non-dividend-paying stock and a savings account that grows at the continuous risk-free rate r (in
% per year). Consider a call and a put option on this
stock with an exercise price X and the remaining time to
maturity T
t. Lower-case letters are used to indicate
European-option values, and upper-case letters are used
to indicate American-option values. The …rst …ve questions deal with the American call option, and the last …ve
questions deal with the American put option.
1. Prove that
ct
St
Xe r(T t), for all parameters.
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2. Provide an intuitive explanation for the following
property:
Cth
ct, for all parameters,
and give an example where
Vth > vt, for all parameters.
3. Provide an intuitive explanation for the following
property:
Ct
max (0, St
X ) , for all parameters.
4. Prove that an American call option on a non-dividendpaying stock cannot be exercised optimally before
maturity.
5. Use a graph, and explain why an American call option can be exercised optimally before maturity when
the underlying stock pays dividends before maturity.
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6. Use the same graph, and plot the exercise- and holdingvalue functions of an American put option.
7. Use another graph, and plot the value function of
this American put option.
8. Provide an intuitive explanation why an American
put option can always be exercised optimally before
maturity.
9. Characterize the optimal exercise policy of an American put option at time t 2 [0, T ], as a function
of a threshold at , which is itself a function of the
option’s parameters.
10. Provide an intuitive explanation for the behaviour of
at over time, from the start to maturity.
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Exercise 4 (20%): Consider a …rm with a very simple
capital structure, that is, a share of stock and a pure
discount bond with maturity T and face value FV. The
…rm’s value process is indicated by Vt, the bond’s value
process by Bt, and the stock price by St, for t 2 [0, T ].
The continuously compounded risk-free rate is indicated
by r (in % per year). Examine the …gures, and brie‡y
explain each one.
Exercise 5 (20%): Brie‡y comment on the business
article.
8.2
Second Midterm Exam
Exercise 1 (20%): Consider a three-period binomial
tree for a non-dividend-paying stock with the following
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Brock University, FNCE 4P17
parameters: S0 = $100, u = 1:25, d = 0:8, r = 7%
(per period), and
p =
1+r d
.
u d
All option contracts considered here are of European style,
have an exercise price of X = $100, and expire at time
t3 = T .
1. Draw the three-period binomial tree for the stock.
2. Evaluate the asset-or-nothing call option.
3. Evaluate the lookback call option.
4. Evaluate the up-and-in call option with a barrier of
b = $150.
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Exercise 2 (20%): Merton extended the Black and Scholes model to a stock that continuously pays dividends
at a …xed rate (in % per year). The present value of
all dividends paid over a time window is proportional to
the initial stock price, dividend rate, and time period.
This model, known as the Black, Scholes, and Merton
model, allows one to evaluate European vanilla options
in a closed form by adjusting downward the stock price
as follows:
S0 = S0 e
T,
and using the Black and Scholes formula. For example,
the European call-option value on a dividend-paying stock
is
Xe rT N (d2) ,
c0 = S0 N (d1)
where
d1
d2
ln S0 =X + r + 2=2 T
p
=
T
p
= d1
T.
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Derivatives, Part II
Brock University, FNCE 4P17
1. Show that the adjusted stock price S0 is nothing else
than the stock price S0 net of all dividends paid over
the option’s life.
2. Explain why a foreign currency can be interpreted
as a …nancial asset that continuously pays dividends.
Give Black, Scholes, and Merton’s formula for a European call option on a foreign currency.
3. Give the de…nition of a European call option on a
futures contract on a …nancial asset.
4. Explain why a call option on a futures contract on a
…nancial asset can be evaluated using Black, Scholes,
and Merton’s formula. This result was established by
Black.
5. *** The futures price under Black’s model is assumed to follow a lognormal process, as it is the case
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for the stock in Black and Scholes’model. Explain
why Black’s formula is totally unacceptable for a call
option on the CBOT long-term T-bond futures contract, though frequently used. Use the convergence
principle at the futures contract’s maturity and its
underlying T-bonds’maturities.
Exercise 3 (20%): Consider an instalment option in
the Black and Scholes model. Let t1; : : : ; tN be the sequence of decision dates, and let 1; : : : ; N 1 2 (0, T )
be a sequence of instalments. The up-front payment is
indicated by 0.
1. De…ne a European instalment option.
2. List the pros of instalment options as hedging tools.
3. Show that an instalment call option with only one
instalment 1 to be paid at t1 2 (0, T ) can be
interpreted as a compound call option.
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4. Brie‡y explain how compound options can be used
to analyse portfolios of corporate bonds.
Exercise 4 (30%): Consider a …rm with a simple capital
structure, that is, a common stock and a discount bond
with principal amount P and maturity T . Let Vt, Bt,
and St be the …rm’s value, the bond’s (present) value,
and the stock price at time t, respectively. The …rm plays
the role of the underlying asset in the Black and Scholes
model, with a known initial value V0 and volatility V .
1. Interpret the stock and the bond as derivative contracts on the …rm.
2. Interpret the expression
P (VT
BT ) ,
where P is the physical probability measure. What
happens if the risk-neutral probability measure P
were used.
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3. Prove that
Bt, for t 2 [0, T ] .
Vt
When can the company go bankrupt?
4. Consider now a …rm with a simple capital structure,
that is, a common stock, a short-term discount bond
with principal amount P1 and maturity t1, and a
long-term discount bond with principal amount P
and maturity T . Explain how the stock and the corporate debt can be evaluated using compound options, as functions of the …rm’s value V0 and volatility V . Now, Bt refers to the present value of the
bond portfolio at time t. Explain how and when the
company can go bankrupt, and interpret the expressions
1
= P Vt1
Bt1
and
T
BT j Vt1 > Bt1 .
= P VT
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Brock University, FNCE 4P17
5. This is a numerical investigation that reports the
stock price, the present value of the debt, and the
term-structure of default probabilities, as functions
of the …rm’s value V0 and volatility V . The parameters are as follows:
= 4% (per year), r = 4%
(per year), t1 = 1 year, P1 = $50, t5 = T = 5
years, and PT = $100. The parameter
is the
(instantaneous) rate of return on the …rm under the
physical probability measure, and r is the rate of return on the …rm under the risk-neutral probability
measure.
(a) For each table, comment on the quality of the
corporate debt.
(b) Table 1–Table 3 contain several zeros. Brie‡y
explain.
(c) *** How does the default probability 1 behave
as a function of the volatility V ? Explain.
(d) In Table 1, for V = 20%, one has 1 = 90:3%
and 5 = 13:4%. Interpret this result.
136
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
Table 1: S0, B0, and Default Prob. for V0 = $100
V
t
t0
t0
t1
t2
t3
t4
t5
S0
B0
1
2
3
4
T
0 :1
0:01599
99:98401
0:99609
0
0
0
0:00956
0 :2
1:17348
98:8265
0:90344
0
0
0
0:1342
0 :3
4:63836
95:36164
0:78278
0
0
0
0:26961
0 :4
9:61668
90:38332
0:69246
0
0
0
0:37824
Table 2: S0, B0, and Default Prob. for V0 = $150
V
t
t0
t0
t1
t2
t3
t4
t5
S0
B0
1
2
3
4
T
0:1
20:57029
129:42971
0:081431
0
0
0
0:002756
0 :2
24:75861
125:24139
0:233911
0
0
0
0:084375
137
0 :3
31:47386
118:52614
0:284145
0
0
0
0:206394
0 :4
39:17126
110:82874
0:304564
0
0
0
0:316672
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
Table 3: S0, B0, and Default Prob. for V0 = $200
V
t
t0
t0
t1
t2
t3
t4
t5
S0
B0
1
2
3
4
T
0:1
70:08769
129:91231
0:00002
0
0
0
0:00005
0 :2
70:67994
129:32016
0:01521
0
0
0
0:03499
0 :3
74:29304
125:70696
0:06292
0
0
0
0:14066
0 :4
80:54610
119:45398
0:10908
0
0
0
0:25391
Exercise 5 (15%): This exercise deals with the CBOT
long-term T-bond futures contract.
1. De…ne the options embedded in this futures contract.
2. Is the cheapest to deliver known in advance? Brie‡y
explain.
138
Hatem Ben Ameur
Derivatives, Part II
Brock University, FNCE 4P17
3. Ignore the margin system and the timing option, and
assume a ‡at term-structure of interest rates. The
(fair) value of the T-bond (c, M ) at time t is indicated by pt (c; M; r), where r is the level of interest
rates.
(a) Is the cheapest to deliver known in advance?
Brie‡y explain.
(b) Suppose that only the reference T-bond is available for delivery. What is the futures’ price at
the start?
(c) *** Suppose that only T-bonds (c1, M1) and
(c2, M2) are available for delivery. Give a criterion that identi…es the cheapest to deliver, and
determines the (fair) futures’price at the start.
139