18.1
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Numerical
Procedures
Chapter 18
18.2
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Binomial Trees
? Binomial trees are frequently used to
approximate the movements in the price
of a stock or other asset
? In each small interval of time the stock
price is assumed to move up by a
proportional amount u or to move down
by a proportional amount d
18.3
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Movements in Time dt
(Figure 18.1)
Su
Sd
S
18.4
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
1,Tree Parameters for a
Nondividend Paying Stock
? We choose the tree parameters p,u,and d so
that the tree gives correct values for the mean
& standard deviation of the stock price
changes in a risk-neutral world
er dt = pu + (1– p )d
s2dt = pu 2 + (1– p )d 2 – [pu + (1– p )d ]2
? A further condition often imposed is u = 1/ d
18.5
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
2,Tree Parameters for a
Nondividend Paying Stock
(Equations 18.4 to 18.7)
When dt is small,a solution to the equations is
tr
t
t
ea
du
da
p
ed
eu
d
ds?
ds
?
?
?
?
?
?
18.6
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
The Complete Tree
(Figure 18.2)
S0
S0u
S0d
S0 S
0
S0u2
S0d2
S0u2
S0u3 S0u4
S0d2
S0u
S0d
S0d4
S0d3
18.7
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Backwards Induction
? We know the value of the option
at the final nodes
? We work back through the tree
using risk-neutral valuation to
calculate the value of the option
at each node,testing for early
exercise when appropriate
18.8
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Example,Put Option
S0 = 50; X = 50; r =10%; s = 40%;
T = 5 months = 0.4167;
dt = 1 month = 0.0833
The parameters imply
u = 1.1224; d = 0.8909;
a = 1.0084; p = 0.5076
18.9
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Example (continued)
Figure 18.3
8 9, 0 7
0, 0 0
7 9, 3 5
0, 0 0
7 0, 7 0 7 0, 7 0
0, 0 0 0, 0 0
6 2, 9 9 6 2, 9 9
0, 6 4 0, 0 0
5 6, 1 2 5 6, 1 2 5 6, 1 2
2, 1 6 1, 3 0 0, 0 0
5 0, 0 0 5 0, 0 0 5 0, 0 0
4, 4 9 3, 7 7 2, 6 6
4 4, 5 5 4 4, 5 5 4 4, 5 5
6, 9 6 6, 3 8 5, 4 5
3 9, 6 9 3 9, 6 9
1 0, 3 6 1 0, 3 1
3 5, 3 6 3 5, 3 6
1 4, 6 4 1 4, 6 4
3 1, 5 0
1 8, 5 0
2 8, 0 7
2 1, 9 3
18.10
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Calculation of Delta
Delta is calculated from the nodes at
time dt
41.055.4412.56 96.616.2 ??????
18.11
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Calculation of Gamma
Gamma is calculated from the nodes at
time 2dt
? ?
? ?
1 2
2
0 64 3 77
62 99 50
0 24
3 77 10 36
50 39 69
0 64
11 65
0 03
?
?
?
? ? ?
?
?
? ?
?
?
.,
.
,;
.,
.
.
.
.G a m m a =
1
18.12
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Calculation of Theta
Theta is calculated from the central
nodes at times 0 and 2dt
day c a l e n d a r per,or
y e a rper =T h e ta
0120
3.4
1 6 6 7.0
49.477.3
-
??
?
18.13
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Calculation of Vega
? We can proceed as follows
? Construct a new tree with a volatility of
41% instead of 40%,
? Value of option is 4.62
? Vega is
4 62 4 49 0 13.,,? ?
p e r 1 % c h a n g e i n v o l a t i l i t y
18.14
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Trees and Dividend Yields
? When a stock price pays continuous
dividends at rate q we construct the tree in
the same way but set a = e(r – q )dt?
As with Black-Scholes:-
For options on stock indices,q equals the
dividend yield on the index-
For options on a foreign currency,q equals
the foreign risk-free rate-
For options on futures contracts q = r
18.15
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Binomial Tree for Dividend
Paying Stock
? Procedure:-
Draw the tree for the stock price less
the present value of the dividends-
Create a new tree by adding
the present value of the dividends at
each node?
This ensures that the tree recombines and
makes assumptions similar to those when
the Black-Scholes model is used
18.16
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Extensions of Tree Approach
? Time dependent interest rates
? The control variate technique
18.17
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Alternative Binomial Tree
Instead of setting u = 1/d we can set
each of the 2 probabilities to 0.5 and
ttr
ttr
ed
eu
ds?ds?
ds?ds?
?
?
)2/(
)2/(
2
2
18.18
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Trinomial Tree (Page 409)
6
1
212
3
2
6
1
212
/1
2
2
2
2
3
?
?
?
?
?
?
?
?
? s
?
s
d
??
?
?
?
?
?
?
?
?
?
? s
?
s
d
?
??
ds
r
t
p
p
r
t
p
udeu
d
m
u
t
S S
Sd
Su
pu
pm
pd
18.19
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Adaptive Mesh Model
? This is a way of grafting a high
resolution tree on to a low resolution
tree
? We need high resolution in the region of
the tree close to the strike price and
option maturity
18.20
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Monte Carlo Simulation
When used to value European stock options,this
involves the following steps:
1,Simulate 1 path for the stock price in a risk neutral
world
2,Calculate the payoff from the stock option
3,Repeat steps 1 and 2 many times to get many sample
payoff
4,Calculate mean payoff
5,Discount mean payoff at risk free rate to get an
estimate of the value of the option
18.21
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Sampling Stock Price Movements
(Equations 18.13 and 18.14,page 411)
? In a risk neutral world the process for a stock
price is
? We can simulate a path by choosing time
steps of length dt and using the discrete
version of this
where e is a random sample from f(0,1)
tStSS des?d??d ?
dS S dt S dz? ??? s
18.22
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
A More Accurate Approach
(Equation 18.15,page 411) ? ?
? ?
? ? tt
etSttS
tttSttS
dzdtSd
des?ds??
?d?
dse?ds????d?
s?s???
or
is th i s of v er s i o n d i s c r e te T he
U s e
2/?
2
2
2
)()(
2/?)(ln)(ln
2/?ln
18.23
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Extensions
When a derivative depends on several
underlying variables we can simulate
paths for each of them in a risk-neutral
world to calculate the values for the
derivative
18.24
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Sampling from Normal
Distribution (Page 412)
? One simple way to obtain a sample
from f(0,1) is to generate 12 random
numbers between 0.0 & 1.0,take the
sum,and subtract 6.0
18.25
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
To Obtain 2 Correlated
Normal Samples
Ob ta i n i n d ependent n o r m a l sa m p l e s
a n d a n d se t
A p r o ce d u r e kn o w n a C h o l e sky ' s
d e co m p o si t i o n ca n b e u se d w h e n
sa m p l e s a r e r e q u i r e d f r o m m o r e th a n
tw o n o r m a l v a r i a b l e s
e e
e
e ? ?
1 2
1 1
2 1 2
2
1
?
? ? ?
x
x x
18.26
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Standard Errors in Monte
Carlo Simulation
The standard error of the estimate of
the option price is the standard
deviation of the discounted payoffs
given by the simulation trials divided by
the square root of the number of
observations.
18.27
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Application of Monte Carlo
Simulation
? Monte Carlo simulation can deal with
path dependent options,options
dependent on several underlying
state variables,and options with
complex payoffs
? It cannot easily deal with American-
style options
18.28
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Determining Greek Letters
For ?:
1.Make a small change to asset price
2.Carry out the simulation again using the same
random number streams
3.Estimate ? as the change in the option price
divided by the change in the asset price
Proceed in a similar manner for other Greek letters
18.29
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Variance Reduction
Techniques
? Antithetic variable technique
? Control variate technique
? Importance sampling
? Stratified sampling
? Moment matching
? Using quasi-random sequences
18.30
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Representative Sampling
Through the Tree
? We can sample paths randomly through a
binomial or trinomial tree to value an option
? An alternative is to choose representative
paths
? Paths are representative if the proportion of
paths through each node is approximately
equal to the probability of the node being
reached
18.31
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Finite Difference Methods
? Finite difference methods aim
to represent the differential
equation in the form of a
difference equation
? Define ?i,j as the value of ?
at time idt when the stock
price is jdS
18.32
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Finite Difference Methods
(continued)
In
? ? ?
w e s et
? ? ?
? ? ? ? ?
or
? ? ? ?
,,
,,,,
,,,
?
?
?
?
s
?
?
?
?
?
?
?
?
t
rS
S
S
S
r
S S
S S S
S
S S
i j i j
i j i j i j i j
i j i j i j
? ? ?
?
?
?
?
?
??
?
?
?
?
?
?
? ?
? ?
? ?
? ?
1
2
2
2
2 2
2
2
1 1
2
2
1 1
2
2
1 1
2
?
? ?
?
?
18.33
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Implicit Finite Difference
Method (Equation 18.25,page 420)
If w e a l so set
? ? ?
w e o bta i n the i m pli c i t f i nit e dif f er en ce m eth od,
T his i nv ol v es sol v i n g si m ult an eo us eq ua t i on s
of the f or m:
? ? ? ?
,,
,,,,
?
? t t
a b c
i j i j
j i j j i j j i j i j
?
?
? ? ?
?
? ? ?
1
1 1 1
?
18.34
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Explicit Finite Difference
Method (Equation 18.32,page 422)
????
:f o r m t h e of e q u a t i o n s s o l v i n g i n v o l v e s T h i s
m e t h o d d i f f e r e n c e f i n i t e e x p l i c i t t h e o b ta i n we
p o i n t )( t h e at a r et h e y as p o i n t )( t h e at
s a m e t h e be to a s s u m e d a r e a n d If
,,,,
2
11
*
1
*
11
*
2
1
?????
???
?
????
jijjijjijji
cba
i,j,ji
SfSf
18.35
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Implicit vs Explicit Finite
Difference Method
? The explicit finite difference method is
equivalent to the trinomial tree approach
? The implicit finite difference method is
equivalent to a multinomial tree
approach
18.36
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Implicit vs Explicit
Finite Difference Methods
(Figure 18.16,page 422)
?i,j ?i +1,j
?i +1,j –1
?i +1,j +1
?i +1,j?i,j
?i,j –1
?i,j +1
Implicit Method Explicit Method
18.37
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
Other Points on Finite
Difference Methods
? It is better to have ln S rather than S as
the underlying variable
? Improvements over the basic implicit
and explicit methods:
- Hopscotch method
- Crank-Nicolson method
18.38
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
The Barone Adesi & Whaley Analytic
Approximation for American Call Options
Appendix 18A,page 433)
? ?? ?
w h e n
- w he n
w h e r e & a r e e a sil y ca lculat e d co n sta n ts &
* is the so lution to
2 2
C S
c S A
S
S
S S
S X S S
A
S
S X c S e N d S
S
q T t
( )
( )
*
*
*
* ( *) ( *)
*
( )
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
? ? ? ?
? ?
2
1
2
2
1
g
g
g
18.39
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
The Barone Adesi & Whaley Analytic
Approximation for American Put Options
? ?? ?
w h e n
w h e n
w h e r e & a r e e a sil y c a lcu la t e d c o n st a n ts &
* * is t h e s o lu tio n t o
1 1
P S
p S A
S
S
S S
X S S S
A
S
X S p S e N d S
S
q T t
( )
( )
* *
* *
* *
* * ( * *) ( * *)
* *
( )
?
?
?
?
?
?
?
? ?
? ?
?
?
?
?
?
? ? ? ? ?
? ?
1
1
1
1
1
g
g
g