The Binomial Model: Autumn 2 0 0 9
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A U T U M N 2 0 0 9
THE BINOMIAL MODEL
MSc
COMPUTATIONAL FINANCE
Numerical Methods in Finance (Implementing Market Models)
©Finbarr Murphy 2007
Lecture Objectives Binomial Models Understand how Binomial Models can be used to value — Options on an asset paying a known dividend yield — Options on an asset paying a known discrete proportional dividend — Options on an asset paying a known cash dividend Examine how time varying volatility can be incorporated into a
MSc
COMPUTATIONAL FINANCE
binomial model
©Finbarr Murphy 2007
Agenda Page
Assets Paying a Continuous Yield
1
2 2
Assets Paying a Proportional Discrete Dividend 5 3
Assets Paying a Cash Dividend
17
MSc
COMPUTATIONAL FINANCE
Time Varying Volatility
9
3
©Finbarr Murphy 2007
Assets Paying A Continuous Dividend Yield For the purposes of option pricing, we can assume
that some assets pay a continuous dividend yield. Some examples include Stock Indices Futures
MSc
COMPUTATIONAL FINANCE
Commodities
If we assume a risk neutral mean and variance in the
GBM process, we have
dS = rSdt + σSdz 4
©Finbarr Murphy 2007
Assets Paying A Continuous Dividend Yield If the continuous dividend yield is δ, we can replace
the r with (r- δ) giving
dS = (r − δ ) Sdt + σSdz It’s simply a matter of making this replacement
MSc
COMPUTATIONAL FINANCE
throughout the calculations
5
©Finbarr Murphy 2007
Agenda Page
Assets Paying a Continuous Yield
1
2 2
Assets Paying a Proportional Discrete Dividend 5 3
Assets Paying a Cash Dividend
17
MSc
COMPUTATIONAL FINANCE
Time Varying Volatility
9
6
©Finbarr Murphy 2007
Assets Paying A Proportional Discrete Dividend A Proportional Discrete Dividend might include share
splits and rights issue Assuming that a company will issue a 10% stock
MSc
COMPUTATIONAL FINANCE
dividend to all shareholders, we can assume that the asset/stock price will reduce by 10%
7
©Finbarr Murphy 2007
Assets Paying A Proportional Discrete Dividend The stock price reduction will take place on the ex-
dividend date (note: ex-dividend and record dates!) Let δˆ be the proportional dividend amount and τ be
the ex-dividend date
MSc
COMPUTATIONAL FINANCE
Prior to the ex-dividend date, the asset price remains
unchanged After the dividend date, the asset price reduces by
(1− δˆ )
8
©Finbarr Murphy 2007
Assets Paying A Proportional Discrete Dividend At this point we will look at how an American Put is
calculated where there is a known discrete proportional dividend Assume that the dividend date corresponds to a node
MSc
COMPUTATIONAL FINANCE
on the tree
9
©Finbarr Murphy 2007
Assets Paying A Proportional Discrete Dividend If the time iΔt is before ex-dividend, then the value of
the asset is unchanged On or after iΔt, the value of the asset becomes
(
)
S 1 − δˆ u j d i − j
(
S 1 − δˆ u 3
j=4
Su 2
COMPUTATIONAL FINANCE MSc
j=3 j=2
j=-4
(
Su
j=1 j=0
)
(
S
)
Sd Sd
j=-1
(
j=-2
2
(
)
(
)
(
S 1 − δˆ
S 1 − δˆ d
(
)
)
S 1 − δˆ d 2
S 1 − δˆ d 3
j=-3
)
S 1 − δˆ u 2
S 1 − δˆ u
S
)
S 1 − δˆ u 4
(
)
S 1 − δˆ d 4 i=0
i=1
i=2
i=3
i=4
10
©Finbarr Murphy 2007
Assets Paying A Proportional Discrete Dividend If we assume equal jumps Δxu=Δxd This is easier to code than equal probabilities (I
think!)
(
)
S 1 − δˆ u j d i − j
((
j=4
COMPUTATIONAL FINANCE
j=2
MSc
j=3
j=-4
j=1 j=0
x + ∆x x
)
)
(
)
Sd Sd
j=-2
2
(
)
S
j=-1
(
S 1 − δˆ u 4 ln 1 − δˆ ln −1 ( x + 2∆x ) + ∆x x + 2∆x S 1 − δˆ u 2 S 1 − δˆ u
(
)
(
)
(
S 1 − δˆ
S 1 − δˆ d
(
)
)
S 1 − δˆ d 2
S 1 − δˆ d 3
j=-3
)
(
)
S 1 − δˆ d 4 i=0
i=1
i=2
i=3
i=4
11
©Finbarr Murphy 2007
Agenda Page
Assets Paying a Continuous Yield
1
2 2
Assets Paying a Proportional Discrete Dividend 5 3
Assets Paying a Cash Dividend
17
MSc
COMPUTATIONAL FINANCE
Time Varying Volatility
9
12
©Finbarr Murphy 2007
Assets Paying A Cash Dividend Cash dividends pose a problem because the cash
amount is absolute S = €100, Δt = 1/3, σ = 0.2%, r = 6%, δ = €3, τ=2/3 x + 2 ∆x Sx = 138. ln( e3543 − D ) + ∆x
MSc
COMPUTATIONAL FINANCE
S = 100 x = ln(S )
S =e
x + ∆x
+ 2∆xx+2 ∆x Sx = ln( e x +2 e − D− D )
Sx = ln( e x −e xD− D )
x + 2 ∆x Sx = 109. ln( e6554 − D ) − ∆x x ln( e9565 − D ) + ∆x Sx = 108. x xS ==ln( − D ) − ∆x 86.e3556
S = e x −∆x
x −2 − 2∆xx−2 ∆x
Sx = ln( e e − D− D )
x − 2 ∆x xS ==ln( − D ) + ∆x 85.e6566 x − 2 ∆x xS ==ln( 67.e8889 − D ) − ∆x
13
©Finbarr Murphy 2007
Assets Paying A Cash Dividend This becomes a non recombining tree or a bushy tree A bushy tree is a tree in which the number of
MSc
COMPUTATIONAL FINANCE
branches increases exponentially relative to observation times; branches never recombine
14
©Finbarr Murphy 2007
Assets Paying A Cash Dividend We want to avoid bushy trees as these are
computationally expensive We get around this by combining future cash flows
into the current stock price The current value of the stock, S is €100
MSc
COMPUTATIONAL FINANCE
This price includes the expected δ = €3 in τ = six
months (say)
S = Sˆ + δe − rτ
So, Therefore
Sˆ = S − δe − rτ = 100 − 3e −0.06×0.5 = 97.0887 15
©Finbarr Murphy 2007
Assets Paying A Cash Dividend Now, project Sˆ forward using the general additive
method Add in the discounted cashflows The sum is an asset price tree that recombines
MSc
COMPUTATIONAL FINANCE
Verify this with the attached Excel sheet.
Extend the Excel sheet to price the American put Week3_2.xls
16
©Finbarr Murphy 2007
Assets Paying A Cash Dividend See the following Excel Sheet
MSc
COMPUTATIONAL FINANCE
Future Cash Flows (discounted)
Sˆ
values
S values 17
©Finbarr Murphy 2007
Assets Paying A Cash Dividend There is one important point to note. The volatility of Sˆ
will be different from that ofS In particular, we use volatility
σˆ instead of σ
Further information and a function is available in
MSc
COMPUTATIONAL FINANCE
MatLab [AssetPrice, OptionValue] = binprice(Price, Strike, Rate, Time,
Increment, Volatility, Flag, DividendRate, Dividend, ExDiv) [Price, Option] = binprice(100, 100, 0.06, 1, 1/3, 0.2, 0, 0, 3,
1.5)
18
©Finbarr Murphy 2007
Agenda Page
Assets Paying a Continuous Yield
1
2 2
Assets Paying a Proportional Discrete Dividend 5 3
Assets Paying a Cash Dividend
18
MSc
COMPUTATIONAL FINANCE
Time Varying Volatility
9
19
©Finbarr Murphy 2007
Time Varying Volatility Up to now, we have assumed that volatility is
constant. Intuition alone suggests that volatility varies over time We can adapt our binomial model to accommodate
time varying volatility σi where i = 0,…,T
MSc
COMPUTATIONAL FINANCE
To ensure a recombining binomial tree, we fix the
space step Δx but vary the probabilities pu/pd and time steps Δt
20
©Finbarr Murphy 2007
Time Varying Volatility Recall the binomial process
dx = vdt + σdz 1 2 v=r− σ 2 The mean and variance of this discrete binomial
MSc
COMPUTATIONAL FINANCE
process over Δti is given by
E[∆x] = p ∆xu + p ∆xd = vi ∆ti 2 u 2 d 2 2 2 2 E[∆x ] = pi ∆xu + pi ∆xd = σ i ∆ti + vi ∆ti u i
d i
Note the i subscripts denoting time specific instances 21
©Finbarr Murphy 2007
Time Varying Volatility This gives us
)
(
MSc
COMPUTATIONAL FINANCE
1 2 4 2 2 u ∆ti = 2 − σ i ± σ i + 2vi ∆x pi ∆xu 2vi 1 vi ∆ti pi = + 2 2∆x If Δx is set as
∆x = σ ∆t + v ∆t 2
Given
1 σ = N
2
N
∑σ i =1
i
and
2
1 v= N
N
∑v i =1
i 22
©Finbarr Murphy 2007
Time Varying Volatility Δt approximates the average time step when the tree
is built but individual Δti’s can vary See OHP Interest rates vary between 4 and 6%
MSc
COMPUTATIONAL FINANCE
Volatility varies between 14 and 17% It is difficult to choose timesteps that correspond to
cashflow dates and exercise dates.
23
©Finbarr Murphy 2007
Recommended Texts Required/Recommended Clewlow, L. and Strickland, C. (1996) Implementing derivative
models, 1st ed., John Wiley and Sons Ltd. — Chapter 2
Additional/Useful Brandimarte, P. (2006) Numerical methods in finance and
MSc
COMPUTATIONAL FINANCE
economics: A matlab-based introduction, 2nd Revised ed., John Wiley & Sons Inc. — Chapter 7
24
THE BINOMIAL MODEL
MSc
COMPUTATIONAL FINANCE
Numerical Methods in Finance (Implementing Market Models)
©Finbarr Murphy 2007
Lecture Objectives Binomial Models Understand how Binomial Models can be used to value — Options on an asset paying a known dividend yield — Options on an asset paying a known discrete proportional dividend — Options on an asset paying a known cash dividend Examine how time varying volatility can be incorporated into a
MSc
COMPUTATIONAL FINANCE
binomial model
©Finbarr Murphy 2007
Agenda Page
Assets Paying a Continuous Yield
1
2 2
Assets Paying a Proportional Discrete Dividend 5 3
Assets Paying a Cash Dividend
17
MSc
COMPUTATIONAL FINANCE
Time Varying Volatility
9
3
©Finbarr Murphy 2007
Assets Paying A Continuous Dividend Yield For the purposes of option pricing, we can assume
that some assets pay a continuous dividend yield. Some examples include Stock Indices Futures
MSc
COMPUTATIONAL FINANCE
Commodities
If we assume a risk neutral mean and variance in the
GBM process, we have
dS = rSdt + σSdz 4
©Finbarr Murphy 2007
Assets Paying A Continuous Dividend Yield If the continuous dividend yield is δ, we can replace
the r with (r- δ) giving
dS = (r − δ ) Sdt + σSdz It’s simply a matter of making this replacement
MSc
COMPUTATIONAL FINANCE
throughout the calculations
5
©Finbarr Murphy 2007
Agenda Page
Assets Paying a Continuous Yield
1
2 2
Assets Paying a Proportional Discrete Dividend 5 3
Assets Paying a Cash Dividend
17
MSc
COMPUTATIONAL FINANCE
Time Varying Volatility
9
6
©Finbarr Murphy 2007
Assets Paying A Proportional Discrete Dividend A Proportional Discrete Dividend might include share
splits and rights issue Assuming that a company will issue a 10% stock
MSc
COMPUTATIONAL FINANCE
dividend to all shareholders, we can assume that the asset/stock price will reduce by 10%
7
©Finbarr Murphy 2007
Assets Paying A Proportional Discrete Dividend The stock price reduction will take place on the ex-
dividend date (note: ex-dividend and record dates!) Let δˆ be the proportional dividend amount and τ be
the ex-dividend date
MSc
COMPUTATIONAL FINANCE
Prior to the ex-dividend date, the asset price remains
unchanged After the dividend date, the asset price reduces by
(1− δˆ )
8
©Finbarr Murphy 2007
Assets Paying A Proportional Discrete Dividend At this point we will look at how an American Put is
calculated where there is a known discrete proportional dividend Assume that the dividend date corresponds to a node
MSc
COMPUTATIONAL FINANCE
on the tree
9
©Finbarr Murphy 2007
Assets Paying A Proportional Discrete Dividend If the time iΔt is before ex-dividend, then the value of
the asset is unchanged On or after iΔt, the value of the asset becomes
(
)
S 1 − δˆ u j d i − j
(
S 1 − δˆ u 3
j=4
Su 2
COMPUTATIONAL FINANCE MSc
j=3 j=2
j=-4
(
Su
j=1 j=0
)
(
S
)
Sd Sd
j=-1
(
j=-2
2
(
)
(
)
(
S 1 − δˆ
S 1 − δˆ d
(
)
)
S 1 − δˆ d 2
S 1 − δˆ d 3
j=-3
)
S 1 − δˆ u 2
S 1 − δˆ u
S
)
S 1 − δˆ u 4
(
)
S 1 − δˆ d 4 i=0
i=1
i=2
i=3
i=4
10
©Finbarr Murphy 2007
Assets Paying A Proportional Discrete Dividend If we assume equal jumps Δxu=Δxd This is easier to code than equal probabilities (I
think!)
(
)
S 1 − δˆ u j d i − j
((
j=4
COMPUTATIONAL FINANCE
j=2
MSc
j=3
j=-4
j=1 j=0
x + ∆x x
)
)
(
)
Sd Sd
j=-2
2
(
)
S
j=-1
(
S 1 − δˆ u 4 ln 1 − δˆ ln −1 ( x + 2∆x ) + ∆x x + 2∆x S 1 − δˆ u 2 S 1 − δˆ u
(
)
(
)
(
S 1 − δˆ
S 1 − δˆ d
(
)
)
S 1 − δˆ d 2
S 1 − δˆ d 3
j=-3
)
(
)
S 1 − δˆ d 4 i=0
i=1
i=2
i=3
i=4
11
©Finbarr Murphy 2007
Agenda Page
Assets Paying a Continuous Yield
1
2 2
Assets Paying a Proportional Discrete Dividend 5 3
Assets Paying a Cash Dividend
17
MSc
COMPUTATIONAL FINANCE
Time Varying Volatility
9
12
©Finbarr Murphy 2007
Assets Paying A Cash Dividend Cash dividends pose a problem because the cash
amount is absolute S = €100, Δt = 1/3, σ = 0.2%, r = 6%, δ = €3, τ=2/3 x + 2 ∆x Sx = 138. ln( e3543 − D ) + ∆x
MSc
COMPUTATIONAL FINANCE
S = 100 x = ln(S )
S =e
x + ∆x
+ 2∆xx+2 ∆x Sx = ln( e x +2 e − D− D )
Sx = ln( e x −e xD− D )
x + 2 ∆x Sx = 109. ln( e6554 − D ) − ∆x x ln( e9565 − D ) + ∆x Sx = 108. x xS ==ln( − D ) − ∆x 86.e3556
S = e x −∆x
x −2 − 2∆xx−2 ∆x
Sx = ln( e e − D− D )
x − 2 ∆x xS ==ln( − D ) + ∆x 85.e6566 x − 2 ∆x xS ==ln( 67.e8889 − D ) − ∆x
13
©Finbarr Murphy 2007
Assets Paying A Cash Dividend This becomes a non recombining tree or a bushy tree A bushy tree is a tree in which the number of
MSc
COMPUTATIONAL FINANCE
branches increases exponentially relative to observation times; branches never recombine
14
©Finbarr Murphy 2007
Assets Paying A Cash Dividend We want to avoid bushy trees as these are
computationally expensive We get around this by combining future cash flows
into the current stock price The current value of the stock, S is €100
MSc
COMPUTATIONAL FINANCE
This price includes the expected δ = €3 in τ = six
months (say)
S = Sˆ + δe − rτ
So, Therefore
Sˆ = S − δe − rτ = 100 − 3e −0.06×0.5 = 97.0887 15
©Finbarr Murphy 2007
Assets Paying A Cash Dividend Now, project Sˆ forward using the general additive
method Add in the discounted cashflows The sum is an asset price tree that recombines
MSc
COMPUTATIONAL FINANCE
Verify this with the attached Excel sheet.
Extend the Excel sheet to price the American put Week3_2.xls
16
©Finbarr Murphy 2007
Assets Paying A Cash Dividend See the following Excel Sheet
MSc
COMPUTATIONAL FINANCE
Future Cash Flows (discounted)
Sˆ
values
S values 17
©Finbarr Murphy 2007
Assets Paying A Cash Dividend There is one important point to note. The volatility of Sˆ
will be different from that ofS In particular, we use volatility
σˆ instead of σ
Further information and a function is available in
MSc
COMPUTATIONAL FINANCE
MatLab [AssetPrice, OptionValue] = binprice(Price, Strike, Rate, Time,
Increment, Volatility, Flag, DividendRate, Dividend, ExDiv) [Price, Option] = binprice(100, 100, 0.06, 1, 1/3, 0.2, 0, 0, 3,
1.5)
18
©Finbarr Murphy 2007
Agenda Page
Assets Paying a Continuous Yield
1
2 2
Assets Paying a Proportional Discrete Dividend 5 3
Assets Paying a Cash Dividend
18
MSc
COMPUTATIONAL FINANCE
Time Varying Volatility
9
19
©Finbarr Murphy 2007
Time Varying Volatility Up to now, we have assumed that volatility is
constant. Intuition alone suggests that volatility varies over time We can adapt our binomial model to accommodate
time varying volatility σi where i = 0,…,T
MSc
COMPUTATIONAL FINANCE
To ensure a recombining binomial tree, we fix the
space step Δx but vary the probabilities pu/pd and time steps Δt
20
©Finbarr Murphy 2007
Time Varying Volatility Recall the binomial process
dx = vdt + σdz 1 2 v=r− σ 2 The mean and variance of this discrete binomial
MSc
COMPUTATIONAL FINANCE
process over Δti is given by
E[∆x] = p ∆xu + p ∆xd = vi ∆ti 2 u 2 d 2 2 2 2 E[∆x ] = pi ∆xu + pi ∆xd = σ i ∆ti + vi ∆ti u i
d i
Note the i subscripts denoting time specific instances 21
©Finbarr Murphy 2007
Time Varying Volatility This gives us
)
(
MSc
COMPUTATIONAL FINANCE
1 2 4 2 2 u ∆ti = 2 − σ i ± σ i + 2vi ∆x pi ∆xu 2vi 1 vi ∆ti pi = + 2 2∆x If Δx is set as
∆x = σ ∆t + v ∆t 2
Given
1 σ = N
2
N
∑σ i =1
i
and
2
1 v= N
N
∑v i =1
i 22
©Finbarr Murphy 2007
Time Varying Volatility Δt approximates the average time step when the tree
is built but individual Δti’s can vary See OHP Interest rates vary between 4 and 6%
MSc
COMPUTATIONAL FINANCE
Volatility varies between 14 and 17% It is difficult to choose timesteps that correspond to
cashflow dates and exercise dates.
23
©Finbarr Murphy 2007
Recommended Texts Required/Recommended Clewlow, L. and Strickland, C. (1996) Implementing derivative
models, 1st ed., John Wiley and Sons Ltd. — Chapter 2
Additional/Useful Brandimarte, P. (2006) Numerical methods in finance and
MSc
COMPUTATIONAL FINANCE
economics: A matlab-based introduction, 2nd Revised ed., John Wiley & Sons Inc. — Chapter 7
24
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