Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.1
More on Models and
Numerical Procedures
Chapter 20
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.2
Models to be Considered
? Constant elasticity of variance
(CEV)
? Jump diffusion
? Stochastic volatility
? Implied volatility function (IVF)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.3
CEV Model (p456)
– When a = 1 we have the Black-
Scholes case
– When a > 1 volatility rises as stock
price rises
– When a < 1 volatility falls as stock
price rises
dzSSdtqrdS a???? )(
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.4
CEV Models Implied Volatilities
?imp
K
a < 1
a > 1
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.5
Jump Diffusion Model (page 457)
? Merton produced a pricing formula
when the stock price follows a diffusion
process overlaid with random jumps
? dp is the random jump
? k is the expected size of the jump
? l dt is the probability that a jump occurs
in the next interval of length dt
dpdzdtkSdS ???l??? )(/
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.6
Jumps and the Smile
? Jumps have a big effect on the
implied volatility of short term options
? They have a much smaller effect on
the implied volatility of long term
options
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.7
Time Varying Volatility
? Suppose the volatility is ?1 for the first
year and ?2 for the second and third
? Total accumulated variance at the end
of three years is ?12 + 2?22
? The 3-year average volatility is
22
2 2 2 12
12
23 2 ;
3
? ? ?? ? ? ? ? ? ?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.8
Stochastic Volatility Models (page 458)
? When V and S are uncorrelated a
European option price is the Black-
Scholes price integrated over the
distribution of the average variance
VL
S
dzVdtVVadV
dzVdtqr
S
dS
a????
???
)(
)(
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.9The IVF Model
(page 460)
S d ztSdttqtrdS
S d zS d tqrdS
),()]()([
)(
????
????
by r e p l a c e d is
m o d e l u s u a l T h e p r i c e s,
o p t i o n o b s e r v e d m a t c h e se x a c t l y t h a t p r i c e
a s s e t t h e f o r p r o c e s s a c r e a t e to d e s i g n e d
is m o d e l f u n c t i o n v o l a t i l i t y i m p i e d T h e
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.10
The Volatility Function
The volatility function that leads to the
model matching all European option
prices is
)(
)]()([)(2)],([
222
2
KcK
KctqtrKctqtctK
m k t
m k tm k tm k t
??
?????????
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.11
Strengths and Weaknesses of the
IVF Model
? The model matches the probability
distribution of stock prices assumed by
the market at each future time
? The models does not necessarily get
the joint probability distribution of stock
prices at two or more times correct
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.12
Numerical Procedures
Topics:
? Path dependent options using trees
? Lookback options
? Barrier options
? Options where there are two stochastic
variables
? American options using Monte Carlo
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.13
Path Dependence,
The Traditional View
? Backwards induction works well for
American options,It cannot be used for
path-dependent options
? Monte Carlo simulation works well for
path-dependent options; it cannot be
used for American options
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.14Extension of Backwards
Induction
? Backwards induction can be used for some
path-dependent options
? We will first illustrate the methodology using
lookback options and then show how it can
be used for Asian options
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.15
Lookback Example (Page 462)
? Consider an American lookback put on a stock where
S = 50,? = 40%,r = 10%,dt = 1 month & the life of
the option is 3 months
? Payoff is Smax-ST
? We can value the deal by considering all possible
values of the maximum stock price at each node
(This example is presented to illustrate the methodology,A more efficient
ways of handling American lookbacks is in Section 20.6.)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.16Example,An American Lookback
Put Option (Figure 20.2,page 463)
S0 = 50,? = 40%,r = 10%,dt = 1 month,
56.12
56.12
4.68
44.55
50.00
6.38
62.99
62.99
3.36
50.00
56.12 50.00
6.12 2.66
36.69
50.00
10.31
70.70
70.70
0.00
62.99 56.12
6.87 0.00
56.12
56.12 50.00
11.57 5.45
44.55
35.36
50.00
14.64
50.00
5.47 A
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.17Why the Approach Works
This approach works for lookback options because
? The payoff depends on just 1 function of the path followed
by the stock price,(We will refer to this as a,path
function”)
? The value of the path function at a node can be calculated
from the stock price at the node & from the value of the
function at the immediately preceding node
? The number of different values of the path function at a
node does not grow too fast as we increase the number of
time steps on the tree
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.18
Extensions of the Approach
? The approach can be extended so that there
are no limits on the number of alternative
values of the path function at a node
? The basic idea is that it is not necessary to
consider every possible value of the path
function
? It is sufficient to consider a relatively small
number of representative values of the
function at each node
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.19Working Forward
? First work forwards through the tree
calculating the max and min values of
the,path function” at each node
? Next choose representative values of
the path function that span the range
between the min and the max
– Simplest approach,choose the min,the
max,and N equally spaced values
between the min and max
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.20
Backwards Induction
? We work backwards through the tree in the
usual way carrying out calculations for each of
the alternative values of the path function that
are considered at a node
? When we require the value of the derivative at
a node for a value of the path function that is
not explicitly considered at that node,we use
linear or quadratic interpolation
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.21Part of Tree to Calculate
Value of an Option on the
Arithmetic Average
(Figure 20.2,page 464)
S = 50.00
Average S
46.65
49.04
51.44
53.83
Option Price
5.642
5.923
6.206
6.492
S = 45.72
Average S
43.88
46.75
49.61
52.48
Option Price
3.430
3.750
4.079
4.416
S = 54.68
Average S
47.99
51.12
54.26
57.39
Option Price
7.575
8.101
8.635
9.178
X
Y
Z
0.5056
0.4944
S=50,X=50,?=40%,r=10%,T=1yr,
dt=0.05yr,We are at time 4dt
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.22Part of Tree to Calculate Value of
an Option on the Arithmetic
Average (continued)
Consider Node X when the average of 5
observations is 51.44
Node Y,If this is reached,the average becomes
51.98,The option price is interpolated as 8.247
Node Z,If this is reached,the average becomes
50.49,The option price is interpolated as 4.182
Node X,value is
(0.5056× 8.247 + 0.4944× 4.182)e–0.1× 0.05 = 6.206
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.23
A More Efficient Approach
for Lookbacks (Section 20.6,page 465)
? ?
?
?
D ef i ne
w her e i s t he M A X s toc k pr i c e
C ons tr uc t a tr ee f o r ( )
U s e t he t r ee t o v al ue t he l oo k bac k
opti on i n " s toc k pr i c e uni ts " r ather
than dol l ar s
Y t
F t
S t
F t
Y t
( )
( )
( )
( )
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.24
Using Trees with Barriers
(Section 20.7,page 467)
? When trees are used to value
options with barriers,
convergence tends to be slow
? The slow convergence arises
from the fact that the barrier is
inaccurately specified by the tree
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.25
True Barrier vs Tree Barrier for a
Knockout Option,The Binomial Tree Case
Barrier assumed by tree
True barrier
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.26
True Barrier vs Tree Barrier for a Knockout
Option,The Trinomial Tree Case
Barrier assumed by tree
True barrier
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.27
Alternative Solutions
to the Problem
? Ensure that nodes always lie on the
barriers
? Adjust for the fact that nodes do not
lie on the barriers
? Use adaptive mesh
In all cases a trinomial tree is
preferable to a binomial tree
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.28
Modeling Two Correlated
Variables (Section 20.8,page 472)
APPROACHES:
1.Transform variables so that they are not
correlated & build the tree in the transformed
variables
2.Take the correlation into account by adjusting
the position of the nodes
3.Take the correlation into account by adjusting
the probabilities
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.29
Monte Carlo Simulation and
American Options
? Two approaches:
– The least squares approach
– The exercise boundary parameterization approach
? Consider a 3-year put option where the initial
asset price is 1.00,the strike price is 1.10,the
risk-free rate is 6%,and there is no income
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.30Sampled Paths
Path t=0 t=1 t=2 t=3
1 1.00 1.09 1.08 1.34
2 1.00 1.16 1.26 1.54
3 1.00 1.22 1.07 1.03
4 1.00 0.93 0.97 0.92
5 1.00 1.11 1.56 1.52
6 1.00 0.76 0.77 0.90
7 1.00 0.92 0.84 1.01
8 1.00 0.88 1.22 1.34
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.31
The Least Squares Approach
(page 474)
? We work back from the end using a least
squares approach to calculate the
continuation value at each time
? Consider year 2,The option is in the money
for five paths,These give observations on S
of 1.08,1.07,0.97,0.77,and 0.84,The
continuation values are 0.00,0.07e-0.06,
0.18e-0.06,0.20e-0.06,and 0.09e-0.06
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.32
The Least Squares Approach
continued
? Fitting a model of the form V=a+bS+cS2
we get a best fit relation
V=-1.070+2.983S-1.813S2
for the continuation value V
? This defines the early exercise decision
at t=2,We carry out a similar analysis at
t=1
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.33
The Least Squares Approach
continued
In practice more complex functional
forms can be used for the continuation
value and many more paths are
sampled
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.34
The Early Exercise Boundary
Parametrization Approach (page 477)
? We assume that the early exercise boundary
can be parameterized in some way
? We carry out a first Monte Carlo simulation
and work back from the end calculating the
optimal parameter values
? We then discard the paths from the first
Monte Carlo simulation and carry out a new
Monte Carlo simulation using the early
exercise boundary defined by the parameter
values,
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.35
Application to Example
? We parameterize the early exercise
boundary by specifying a critical asset
price,S*,below which the option is
exercised.
? At t=1 the optimal S* for the eight paths
is 0.88,At t=2 the optimal S* is 0.84
? In practice we would use many more
paths to calculate the S*
20.1
More on Models and
Numerical Procedures
Chapter 20
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.2
Models to be Considered
? Constant elasticity of variance
(CEV)
? Jump diffusion
? Stochastic volatility
? Implied volatility function (IVF)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.3
CEV Model (p456)
– When a = 1 we have the Black-
Scholes case
– When a > 1 volatility rises as stock
price rises
– When a < 1 volatility falls as stock
price rises
dzSSdtqrdS a???? )(
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.4
CEV Models Implied Volatilities
?imp
K
a < 1
a > 1
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.5
Jump Diffusion Model (page 457)
? Merton produced a pricing formula
when the stock price follows a diffusion
process overlaid with random jumps
? dp is the random jump
? k is the expected size of the jump
? l dt is the probability that a jump occurs
in the next interval of length dt
dpdzdtkSdS ???l??? )(/
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.6
Jumps and the Smile
? Jumps have a big effect on the
implied volatility of short term options
? They have a much smaller effect on
the implied volatility of long term
options
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.7
Time Varying Volatility
? Suppose the volatility is ?1 for the first
year and ?2 for the second and third
? Total accumulated variance at the end
of three years is ?12 + 2?22
? The 3-year average volatility is
22
2 2 2 12
12
23 2 ;
3
? ? ?? ? ? ? ? ? ?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.8
Stochastic Volatility Models (page 458)
? When V and S are uncorrelated a
European option price is the Black-
Scholes price integrated over the
distribution of the average variance
VL
S
dzVdtVVadV
dzVdtqr
S
dS
a????
???
)(
)(
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.9The IVF Model
(page 460)
S d ztSdttqtrdS
S d zS d tqrdS
),()]()([
)(
????
????
by r e p l a c e d is
m o d e l u s u a l T h e p r i c e s,
o p t i o n o b s e r v e d m a t c h e se x a c t l y t h a t p r i c e
a s s e t t h e f o r p r o c e s s a c r e a t e to d e s i g n e d
is m o d e l f u n c t i o n v o l a t i l i t y i m p i e d T h e
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.10
The Volatility Function
The volatility function that leads to the
model matching all European option
prices is
)(
)]()([)(2)],([
222
2
KcK
KctqtrKctqtctK
m k t
m k tm k tm k t
??
?????????
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.11
Strengths and Weaknesses of the
IVF Model
? The model matches the probability
distribution of stock prices assumed by
the market at each future time
? The models does not necessarily get
the joint probability distribution of stock
prices at two or more times correct
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.12
Numerical Procedures
Topics:
? Path dependent options using trees
? Lookback options
? Barrier options
? Options where there are two stochastic
variables
? American options using Monte Carlo
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.13
Path Dependence,
The Traditional View
? Backwards induction works well for
American options,It cannot be used for
path-dependent options
? Monte Carlo simulation works well for
path-dependent options; it cannot be
used for American options
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.14Extension of Backwards
Induction
? Backwards induction can be used for some
path-dependent options
? We will first illustrate the methodology using
lookback options and then show how it can
be used for Asian options
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.15
Lookback Example (Page 462)
? Consider an American lookback put on a stock where
S = 50,? = 40%,r = 10%,dt = 1 month & the life of
the option is 3 months
? Payoff is Smax-ST
? We can value the deal by considering all possible
values of the maximum stock price at each node
(This example is presented to illustrate the methodology,A more efficient
ways of handling American lookbacks is in Section 20.6.)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.16Example,An American Lookback
Put Option (Figure 20.2,page 463)
S0 = 50,? = 40%,r = 10%,dt = 1 month,
56.12
56.12
4.68
44.55
50.00
6.38
62.99
62.99
3.36
50.00
56.12 50.00
6.12 2.66
36.69
50.00
10.31
70.70
70.70
0.00
62.99 56.12
6.87 0.00
56.12
56.12 50.00
11.57 5.45
44.55
35.36
50.00
14.64
50.00
5.47 A
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.17Why the Approach Works
This approach works for lookback options because
? The payoff depends on just 1 function of the path followed
by the stock price,(We will refer to this as a,path
function”)
? The value of the path function at a node can be calculated
from the stock price at the node & from the value of the
function at the immediately preceding node
? The number of different values of the path function at a
node does not grow too fast as we increase the number of
time steps on the tree
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.18
Extensions of the Approach
? The approach can be extended so that there
are no limits on the number of alternative
values of the path function at a node
? The basic idea is that it is not necessary to
consider every possible value of the path
function
? It is sufficient to consider a relatively small
number of representative values of the
function at each node
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.19Working Forward
? First work forwards through the tree
calculating the max and min values of
the,path function” at each node
? Next choose representative values of
the path function that span the range
between the min and the max
– Simplest approach,choose the min,the
max,and N equally spaced values
between the min and max
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.20
Backwards Induction
? We work backwards through the tree in the
usual way carrying out calculations for each of
the alternative values of the path function that
are considered at a node
? When we require the value of the derivative at
a node for a value of the path function that is
not explicitly considered at that node,we use
linear or quadratic interpolation
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.21Part of Tree to Calculate
Value of an Option on the
Arithmetic Average
(Figure 20.2,page 464)
S = 50.00
Average S
46.65
49.04
51.44
53.83
Option Price
5.642
5.923
6.206
6.492
S = 45.72
Average S
43.88
46.75
49.61
52.48
Option Price
3.430
3.750
4.079
4.416
S = 54.68
Average S
47.99
51.12
54.26
57.39
Option Price
7.575
8.101
8.635
9.178
X
Y
Z
0.5056
0.4944
S=50,X=50,?=40%,r=10%,T=1yr,
dt=0.05yr,We are at time 4dt
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.22Part of Tree to Calculate Value of
an Option on the Arithmetic
Average (continued)
Consider Node X when the average of 5
observations is 51.44
Node Y,If this is reached,the average becomes
51.98,The option price is interpolated as 8.247
Node Z,If this is reached,the average becomes
50.49,The option price is interpolated as 4.182
Node X,value is
(0.5056× 8.247 + 0.4944× 4.182)e–0.1× 0.05 = 6.206
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.23
A More Efficient Approach
for Lookbacks (Section 20.6,page 465)
? ?
?
?
D ef i ne
w her e i s t he M A X s toc k pr i c e
C ons tr uc t a tr ee f o r ( )
U s e t he t r ee t o v al ue t he l oo k bac k
opti on i n " s toc k pr i c e uni ts " r ather
than dol l ar s
Y t
F t
S t
F t
Y t
( )
( )
( )
( )
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.24
Using Trees with Barriers
(Section 20.7,page 467)
? When trees are used to value
options with barriers,
convergence tends to be slow
? The slow convergence arises
from the fact that the barrier is
inaccurately specified by the tree
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.25
True Barrier vs Tree Barrier for a
Knockout Option,The Binomial Tree Case
Barrier assumed by tree
True barrier
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.26
True Barrier vs Tree Barrier for a Knockout
Option,The Trinomial Tree Case
Barrier assumed by tree
True barrier
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.27
Alternative Solutions
to the Problem
? Ensure that nodes always lie on the
barriers
? Adjust for the fact that nodes do not
lie on the barriers
? Use adaptive mesh
In all cases a trinomial tree is
preferable to a binomial tree
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.28
Modeling Two Correlated
Variables (Section 20.8,page 472)
APPROACHES:
1.Transform variables so that they are not
correlated & build the tree in the transformed
variables
2.Take the correlation into account by adjusting
the position of the nodes
3.Take the correlation into account by adjusting
the probabilities
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.29
Monte Carlo Simulation and
American Options
? Two approaches:
– The least squares approach
– The exercise boundary parameterization approach
? Consider a 3-year put option where the initial
asset price is 1.00,the strike price is 1.10,the
risk-free rate is 6%,and there is no income
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.30Sampled Paths
Path t=0 t=1 t=2 t=3
1 1.00 1.09 1.08 1.34
2 1.00 1.16 1.26 1.54
3 1.00 1.22 1.07 1.03
4 1.00 0.93 0.97 0.92
5 1.00 1.11 1.56 1.52
6 1.00 0.76 0.77 0.90
7 1.00 0.92 0.84 1.01
8 1.00 0.88 1.22 1.34
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.31
The Least Squares Approach
(page 474)
? We work back from the end using a least
squares approach to calculate the
continuation value at each time
? Consider year 2,The option is in the money
for five paths,These give observations on S
of 1.08,1.07,0.97,0.77,and 0.84,The
continuation values are 0.00,0.07e-0.06,
0.18e-0.06,0.20e-0.06,and 0.09e-0.06
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.32
The Least Squares Approach
continued
? Fitting a model of the form V=a+bS+cS2
we get a best fit relation
V=-1.070+2.983S-1.813S2
for the continuation value V
? This defines the early exercise decision
at t=2,We carry out a similar analysis at
t=1
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.33
The Least Squares Approach
continued
In practice more complex functional
forms can be used for the continuation
value and many more paths are
sampled
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.34
The Early Exercise Boundary
Parametrization Approach (page 477)
? We assume that the early exercise boundary
can be parameterized in some way
? We carry out a first Monte Carlo simulation
and work back from the end calculating the
optimal parameter values
? We then discard the paths from the first
Monte Carlo simulation and carry out a new
Monte Carlo simulation using the early
exercise boundary defined by the parameter
values,
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
20.35
Application to Example
? We parameterize the early exercise
boundary by specifying a critical asset
price,S*,below which the option is
exercised.
? At t=1 the optimal S* for the eight paths
is 0.88,At t=2 the optimal S* is 0.84
? In practice we would use many more
paths to calculate the S*
