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A U T U M N 2009
G E N E R A L F O R M U L A T I O N O F B I N O M I A L M O D E
L S
Numerical Methods in Finance (Implementing Market Models)
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Finbarr Murphy 2007
Lecture Objectives
General Binomial Models
Move from a 1-step model to a general understanding
Understand the difference between CRR89 and JR83
Multiplicative versus additive Binomial Processes
Hedging
Understand how Binomial trees can be used to calculate hedge
sensitivities
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Agenda
Page
Finbarr Murphy 2007
Computing Hedge Sensitivities
General Formulation of the BinomialModel
1
2
2
12
3
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General Formulation of the Binomial Model
Recall Eq. 1.2.1
If we assume a risk neutral mean and variance in the
above GBM process, we have
We would like the parameters of our binomial process
(u, dandp) to match the parameters of this risk
neutral stochastic process (rand )
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SdzSdtdS +=
SdzrSdtdS +=
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General Formulation of the Binomial Model
We have some freedom how the binomial parameters
are chosen as there are three an only two GBMparameters.
Cox, Ross and Rubinstein (1979) (CRR) assumed that
the up probability equalled the down probability = 1/2
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2
1
2
2
2
1
2
1
=
=
=
+
p
ed
eu
ttr
ttr
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General Formulation of the Binomial Model
An alternative proposed by Jarrow and Rudd
(1983)(JR) sets the jump sizes equal giving us
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( )
( )
t
r
p
ed
eu
t
t
+=
=
=
2
2
1
2
1
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General Formulation of the Binomial Model
These solutions work well for small steps but are
impractical for large time steps. Why?
We need to reformulate and use the natural log of the
asset price x = ln(S)
Applying Its Lemma, it can be shown that
where
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Edzvdtdx +=
2
2
1
= rv
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General Formulation of the Binomial Model
The binomial process forxis shown below
X can go up to x+xu with a probability ofpu or down
to x+xdwith a probability ofpd
The is theAdditive Binomial Process
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t
x
x+xd
x+xu
pu
pd
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General Formulation of the Binomial Model
The mean and variance of this discrete binomial
process over t is given by
We can now set the jump sizes to be equal (xu = -
xd) or set the probabilities to be equal (pu = pd)
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222222 ][
][
tvtxpxpxE
tvxpxpxE
dduu
dduu
+=+=
=+=
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General Formulation of the Binomial Model
Equal probabilities (EQP) implies
leaving
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22222
2
1
2
1
2
1
2
1
tvtxx
tvxx
du
du
+=+
=+
222
222
342
1
2
3
34
2
1
2
1
tvttvx
tvttvx
d
u
=
+=
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General Formulation of the Binomial Model
Equal jump sizes proposed by Trigeoris (1992) (TRG)
implies
leaving
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22222
)()(
tvtxpxp
tvxpxp
du
du
+=+
=+
x
tvp
tvtx
u
+=
+=
2
1
2
1
222
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General Formulation of the Binomial Model
The multiplicative and additive binomial process
described thus far will be implemented using MatLab
Once we can implement the simplest model
(European Call Option on a non dividend paying stock
with a multiplicative binomial process), we can easily
extend for the remaining binomial processes.
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Agenda
Page
Finbarr Murphy 2007
Computing Hedge Sensitivities
General Formulation of the BinomialModel
1
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2
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Computing Hedge Sensitivities
Clearly, the underlying asset price has a significant
bearing on the option price
Firstly, consider the gearing of an option. An
asset/stock price is 50 and an underlying call option
on this asset is 8.00
With 1000 we could buy 20 shares or 125 options
If the asset price moved from 50 to 51, we would
make a profit of 20 or 2%
But this stock price rise might move the option from
8 to 8.50 giving us a profit of 62.50 or 6.25%
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Computing Hedge Sensitivities
The gearing here is 6.25/2.00 or 3.125
The gearing tell us how much leverage the option
offers, and tells us something about the risk!
Gearing is relatively unsophisticated
Delta tells is the rate of change of the option price
relative to the stock price.
In our simple example
= 0.75/1.00 = 0.75
Or 75%
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Computing Hedge Sensitivities
More mathematically, we can say
Knowing delta will allow us to hedge out the
underlying asset price risk
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0,11,1
0,11,1
SS
CC
S
C
S
Cdelta
=
==
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Computing Hedge Sensitivities
Gamma, is the rate of change of Delta to the
underlying asset price
The delta and gamma discussed are forward values.
To calculate today's delta and gamma, we need to
extend the tree back one and two steps respectively.
Discuss
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( ) ( )[ ] ( ) ( )[ ]
( )0,22,2
0,21,20,21,21,22,21,22,2
2
2
2
1SS
SSCCSSCC
S
Cgamma
==
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Computing Hedge Sensitivities
Vega is the rate of change of the option price relative
to a small change in the volatility
is typically chosen to be 1%
Rho is the rate of change of the option price relativeto a small
change in the interest rate
r is typically chosen to be 1bp (0.0001)18M
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+
=
2
)()( CCCvega
rrrCrrC
rCrho
+
=
2)()(
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Recommended Texts
Required/Recommended
Clewlow, L. and Strickland, C. (1996) Implementing
derivativemodels, 1st ed., John Wiley and Sons Ltd.
Chapter 2
Additional/Useful
Hull, J. (2009) Options, futures and other derivatives, 7th
ed.,Prentice HallChapters 11
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