Transcript Document

2. Lattice Methods
2.3 Dealing with Options on Dividend-paying Stocks
(Hull, Sec. 17.3, page 401)
Math4143 W08, HM Zhu
With Continuous Dividend Yields
Assume there is a constant dividend yield D0
paid on the underlying. Then
dS
  dX   r  D0  dt
S
To accommodate the constant dividend yield in the tree model,
1. Replace r by r  D0 in the parameters u, d , and p in the tree
construction of stock prices. For example, in the case u  1 / d ,
it becomes:
 t
u =e
, d =e
 t
, p
e
r  D0   t
d
ud
1
For the case when p  , it becomes:
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u =e
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r  D0   t

1  e
2
t

1 ,
.
d =e
r  D0   t

1  e
2
t

1 .
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With Continuous Dividend Yields
2. Use this tree to calculate the possible value of option at expiry.
Then work back down the tree to caculate the present value of
the option at previous time points is obtained using
Vnm  E e?  tVnm1 
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With Continuous Dividend Yields
2. Use this tree to calculate the possible value of option at expiry.
Then work back down the tree to caculate the present value of
the option at previous time points is obtained using
Vnm  E e r tVnm1 
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With Discrete Dividend Yields
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With Dollar Dividend
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With Dollar Dividend
A better procedure:
– Draw the tree for the stock price less the present
value of the dividends
– Create a new tree for the stock price 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
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With Dollar Dividend
Assume that there is one ex-dividend date,  ,
during the life of the option.
-- Construct a tree for the uncertain component S * , i.e.,
 r  it 

S

De
, if it  

S*  
if it  
 S ,
Using the volatility  * of S * , a tree can be constructed in usual
way to model S * .
-- To model S , we simply add back the present value of future
dividend, i.e.,
 r  it 
* n in

S
u
d

De
,

Sni   0 * n i  n
 S0 u d ,
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if it  
, n  0,1,
if it  
,i
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Example 1:
American put
option on a stock:
S0 = 52;
K = 50;
D = $2.06
r =10%;
* = 40%;
T = 5 months;
 = 3.5 months
t = 1 month
Example 1: Tree model for S*
89.07
0.00
+$2.05
79.35
0.00
70.70
0.00
62.99
0.64
56.12
2.16
50.00
4.49
70.70
0.00
62.99
0.00
56.12
1.30
50.00
3.77
44.55
6.96
56.12
0.00
50.00
2.66
44.55
6.38
39.69
10.36
44.55
5.45
39.69
10.31
35.36
14.64
35.36
14.64
31.50
18.50
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28.07
21.93
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2. Lattice Methods
2.4 Comments and Extensions of the model
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Trees for Options on Indices, Currencies and
Futures Contracts
(Hull, Sec. 11.9, Sec. 17.2)
As with Black-Scholes:
– For options on stock indices, replace the
continuous dividend yield D0 with the dividend
yield on the index
– For options on a foreign currency, D0 equals
the foreign risk-free rate rf
– For options on futures contracts D0 = r
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Time Dependent Interest Rate and
Dividend Yield (page 409)
• Making interest rate r or dividend yield D a function
of time does not affect the geometry of the tree. The
probabilities on the tree become functions of time
• Discounting factor becomes a function of time as
well
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Time Dependent Volatility (page 409)
• Changing  at each time step does affect the
geometry of the tree. (The probabilities on the tree
become functions of time)
• Or we can make  a function of time by making the
lengths of the time steps inversely proportional to
the variance rate.
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Trinomial Tree
(see Technical Note 9, www.rotman.utoronto.ca/~hull)
ue
 3 t
d  1/ u
2

  1
t
 r   
pu 
2 
2  6
12 
2
pm 
3
2

  1
t
 r   
pd  
2 
2  6
12 
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Su
pu
S
pm
S
pd
Sd
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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
• Numerically efficient over a binomial or trinomial tree
• Figlewski and Gao, “The adaptive mesh model: a
new approach to efficient option pricing”, J. of
Financial Ecomonics, 53:313-351 (1999)
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Further Reading
1. L. Clewlow and C. Strickland. Implementing Derivatives
Models. Wiley, Chichester, West Sussex, England, 1998
(relationship between finite differences and trinomial trees)
2. D J Higham. Nine Ways to Implement the Binomial Method
for Option Valuation in Matlab. SIAM review, 44:661-677,
2002 (issues of implementing binomial trees)
3. J C Hull. Option, Futures, and Other Derivatives. Prentice
Hall. (classic reference)
4. G. Levy. Computational Finance. Numerical Methods for
Pricing Financial Instruments. Elsevier ButterworthHeinemann, oxford, 2004 (implied lattices and efficient
implementations)
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