numerical procedures chapter 18
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Numerical Procedures Chapter 18. 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. - PowerPoint PPT PresentationTRANSCRIPT
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18.1
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
Numerical Procedures
Chapter 18
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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
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18.3
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
Movements in Time t(Figure 18.1)
Su
Sd
S
p
1 – p
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18.4
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
1. Tree Parameters for aNondividend 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 t = pu + (1– p )d
2t = pu 2 + (1– p )d 2 – [pu + (1– p )d ]2
• A further condition often imposed is u = 1/ d
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18.5
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
2. Tree Parameters for aNondividend Paying Stock
(Equations 18.4 to 18.7)
When t is small, a solution to the equations is
tr
t
t
ea
du
dap
ed
eu
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18.6
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
The Complete Tree(Figure 18.2)
S0
S0u
S0d S0 S0
S0u2
S0d2
S0u2
S0u3 S0u4
S0d2
S0u
S0d
S0d4
S0d3
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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
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18.8
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
Example: Put Option
S0 = 50; X = 50; r =10%; = 40%;
T = 5 months = 0.4167;
t = 1 month = 0.0833
The parameters imply
u = 1.1224; d = 0.8909;
a = 1.0084; p = 0.5076
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18.9
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
Example (continued)Figure 18.3
89.070.00
79.350.00
70.70 70.700.00 0.00
62.99 62.990.64 0.00
56.12 56.12 56.122.16 1.30 0.00
50.00 50.00 50.004.49 3.77 2.66
44.55 44.55 44.556.96 6.38 5.45
39.69 39.6910.36 10.31
35.36 35.3614.64 14.64
31.5018.50
28.0721.93
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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 t
41.055.4412.56
96.616.2
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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 2t
1 2
2
0 64 377
62 99 500 24
377 10 36
50 39 690 64
11650 03
. .
.. ;
. .
..
..Gamma = 1
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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 2t
day calendar per .or
yearper =Theta
0120
3.41667.0
49.477.3
-
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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 49 013. . . per 1% change in volatility
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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 )t
• 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
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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
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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
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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
)2/(
)2/(
2
2
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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
rt
p
p
rt
p
udeu
d
m
u
t
S S
Sd
Su
pu
pm
pd
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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
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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
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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 t and using the discrete version of this
where is a random sample from (0,1)tStSS ˆ
dS S dt S dz
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18.22
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
A More Accurate Approach(Equation 18.15, page 411)
ttetSttS
tttSttS
dzdtSd
or
is this of version discrete The
Use
2/ˆ
2
2
2
)()(
2/ˆ)(ln)(ln
2/ˆln
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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
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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 (0,1) is to generate 12 random numbers between 0.0 & 1.0, take the sum, and subtract 6.0
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18.25
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
To Obtain 2 Correlated Normal Samples
Obtain independent normal samples
and and set
A procedure known a Cholesky's
decomposition can be used when
samples are required from more than
two normal variables
1 2
1 1
2 1 221
x
x x
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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.
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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
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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
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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
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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
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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 ƒi,j as the value of ƒ at time it when the stock price is jS
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18.32
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
Finite Difference Methods(continued)
In ƒ ƒ ƒ
ƒ
we set ƒ ƒ ƒ
ƒ ƒ ƒ ƒ ƒ or
ƒ ƒ ƒ ƒ
, ,
, , , ,
, , ,
trS
SS
Sr
S S
S S SS
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 22
2
1 1
2
2
1 1
2
2
1 1
2
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18.33
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
Implicit Finite Difference Method (Equation 18.25, page 420)
If we also set ƒ ƒ ƒ
we obtain the implicit finite difference method.
This involves solving simultaneous equations
of the form:
ƒ ƒ ƒ ƒ
, ,
, , , ,
t t
a b c
i j i j
j i j j i j j i j i j
1
1 1 1
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18.34
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
Explicit Finite Difference Method (Equation 18.32, page 422)
ƒƒƒƒ
:form the of equations solving involves This
method difference finite explicit the obtain we
point )( the at arethey as point )( the at
same the be to assumed are and If
,,,,
2
11*
1*
11*
2
1
jijjijjijji cba
i,j,ji
SfSf
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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
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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)
ƒi , j ƒi +1, j
ƒi +1, j –1
ƒi +1, j +1
ƒi +1, jƒi , j
ƒi , j –1
ƒi , j +1
Implicit Method Explicit Method
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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
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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)
when
- when
where & are easily calculated constants &
* is the solution to
2 2
C S c S AS
SS S
S X S S
A
S
S X c S e N d SSq T t
( ) ( )*
*
*
* ( *) ( *)*( )
2
12
2
1
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18.39
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
The Barone Adesi & Whaley Analytic Approximation for American Put Options
when
when
where & are easily calculated constants &
* * is the solution to
1 1
P S p S AS
SS S
X S S S
A
S
X S p S e N d SSq T t
( ) ( )**
**
**
** ( **) ( **)**( )
1
11
1
1