Transcript Binomial Trees in Practice

Binomial Trees in
Practice
Chapter 16
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.1
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
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.2
Movements in Time Dt
(Figure 16.1)
Su
S
Sd
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.3
Risk-Neutral Valuation
We choose the tree parameters p , u ,
and d so that the tree gives correct
values for the mean and standard
deviation of the stock price changes in a
risk-neutral world.
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.4
1. Tree Parameters for a
Nondividend Paying Stock

Two conditions are
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
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.5
2. Tree Parameters for a
Nondividend Paying Stock
(Equations 16.4 to 16.7)
When Dt is small a solution to the equations is
ue
s Dt
d  e s
Dt
ad
p
ud
a  e r Dt
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.6
The Complete Tree
(Figure 16.2)
S0u 3
S 0u 4
S0u 2
S0u
S0
S0d
S0u
S0
S 0d
S0d 2
S0d 3
S0u 2
S0
S 0d 2
S 0d 4
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.7
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
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.8
Example: Put Option
S0 = 50; K = 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
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.9
Example (continued)
Figure 16.3
89.07
0.00
79.35
0.00
70.70
0.00
70.70
0.00
62.99
0.00
62.99
0.64
50.00
2.66
50.00
3.77
50.00
4.49
56.12
0.00
56.12
1.30
56.12
2.16
44.55
5.45
44.55
6.38
44.55
6.96
39.69
10.31
39.69
10.36
35.36
14.64
35.36
14.64
31.50
18.50
28.07
21.93
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.10
Calculation of Delta
Delta is calculated from the nodes at time
Dt
2.16  6.96
Delta 
 0.41
5612
.  44.55
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.11
Calculation of Gamma
Gamma is calculated from the nodes at
time 2Dt
0.64  3.77
3.77  10.36
D1 
 0.24; D 2 
 0.64
62.99  50
50  39.69
D1  D 2
Gamma =
 0.03
1165
.
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.12
Calculation of Theta
Theta is calculated from the central nodes
at times 0 and 2Dt
3.77  4.49
Theta =
  4.3 per year
01667
.
or - 0.012 per calendar day
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.13
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  013
.
per 1% change in volatility
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.14
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
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.15
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
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.16
Extensions of Tree Approach


Time dependent interest rates
The control variate technique
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.17
Alternative Binomial Tree
Instead of setting u = 1/d we can set
each of the 2 probabilities to 0.5 and
ue
( r  s 2 / 2 ) Dt  s Dt
d e
( r  s 2 / 2 ) Dt  s Dt
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.18
Monte Carlo Simulation



Monte Carlo simulation can be
implemented by sampling paths through
the tree randomly and calculating the
payoff corresponding to each path
The value of the derivative is the mean of
the PV of the payoff
See Table 16.1 on page 366
Fundamentals of Futures and Options Markets, 5th Edition, Copyright © John C. Hull 2004
16.19