BinomialModelEquityAssetsBlackScholesEqn

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The Binomial Model

1 In this lecture. . .

G a simple model for an asset price random walkG delta hedging

G no arbitrage

G

the basics of the binomial method for valuing optionsG

risk neutrality


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2 Introduction

We have seen a model for equities and other assets that is based on the

mathematical theory of stochastic calculus. There is another, equally

popular, approach that leads to the same partial differential equation,

the BlackScholes equation, in a way that some people find moreaccessible, it can be made equally rigorous. This approach, via

the binomial model for equities, is the subject of this lecture.


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3 Equities can go down as well as up

In the binomial model we assume that the asset, which initially has

the value 6 , can, during a timestep s W , either rise to a value X 6 or

fall to a value Y 6 , with Y X . The probability of a rise is

S and so the probability of a fall is > S .

S

uS

vS

t

1.One timestep in a binomial random walk.


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G The three constants X , Y and S are chosen to give the binomial walk

the same drift and standard deviation as that given by the stochastic

differential equation model.

Having only these two equations for the three parameters gives us

one degree of freedom in this choice. This degree of freedom is often

used to give the random walk the further property that after an up and

a down movement (or a down followed by an up) the asset returns

to its starting value, 6 . This gives us the requirement that

X Y

(1)


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For the random walk to have the correct drift we need

S X > S Y H

{ s W

Rearranging this equation we get

S

H

{ s W

> Y

X > Y

(2)

Then for the random walk to have the correct standard deviation we

need

S X

> 5 ;

0

{

s W

(3)

Equations (1), (2) and (3) can be solved to give

X

K

0

> { s W

0

{

s W

L

6

?

0

> { s W

0

{

s W

@

>


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4 The binomial tree

The binomial model allows the stock to move up or down a pre-

scribed amount over the next timestep. If the stock starts out with

value6

then it will take either the valueX 6

orY 6

after the next

timestep.We can extend the random walk to the next timestep. After two

timesteps the asset will be at eitherX

6, if there were two up moves,

X Y 6, if an up was followed by a down or vice versa, or

Y

6, if there

were two consecutive down moves. After three timesteps the assetcan be at

X

6,

X

Y 6, etc.


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The resulting structure is called the binomial tree.

S

u4S

u3vS

u2v

2S

uv3S

v4S

2.The binomial tree.


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S

u4

S

u3vS

u2v2S

uv3S

v4S

3.Binomial tree: schematic.


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The top and bottom branches of the tree at expiry can only be

reached by one path each, either all up or all down moves. Whereasthere will be several paths possible for each of the intermediate val-

ues at expiry.

GTherefore the intermediate values are more likely to be reached

than the end values.

GThe binomial tree contains within it an approximation to the prob-

ability density function for the lognormal random walk.


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5 An equation for the value of an option

GSuppose, for the moment, that we know the value of the option at

the timeW s W .

For example, this time may be the expiry of the option. Now con-

struct a portfolio at timeW

consisting of one option and a short po-

sition in a quantity d of the underlying. At time W this portfolio has

value

h 9 > d 6

where the value 9 is to be determined.


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At time W s W the portfolio takes one of two values, depending on

whether the asset rises or falls. These two values are

9

> @ : $ and ' > > @ Y 6

Since we assume that we know 9 , ' > , : , ; , $ and @ , the values

of both of these expressions are known, and, in particular, depend on@ .

Having the freedom to choose@

, we can make the value of this

portfolio the same whether the asset rises or falls.


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This is ensured if we make

'

> @ X 6 9

>

> @ Y 6

This gives us the choice

@

9

> '

>

X > Y 6

(4)

when the new portfolio value is

h s h 9

>

: '

> '

>

: > ;

'

>

>

; '

> '

>

X > Y


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Since the value of the portfolio has been guaranteed, we can say

that its value must coincide with the value of the original portfolioplus any interest earned at the risk-free rate; this is the no-arbitrage

argument. Thus

s h U h s W

After some manipulation this equation becomes

9

9

> '

>

: > ;

: '

>

> ; '

7 I 9 : > Y

(5)

G This, then, is an equation for 9 given 9

, and 9>

, the optionvalues at the next timestep, and the parameters : and ; describing

the random walk of the asset.


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To I 9 we can write (5) as

0

7 I 9

' 5

'

> 5

'

>

(6)

where

S

H

7 I 9

> Y

X > Y

(7)

InterpretingS

as a probability, this is just risk neutrality again.

GAnd (6) is the statement that the option value at time 9 is the present

value of the risk-neutral expected value at any later time.

Supposing that we know'

and'

> we can use (6) to find'

. But

do we know ' and ' > ?


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6 Valuing back down the tree

We certainly know ' and ' > at expiry, time % , because we know

the option value as a function of the asset then, this is the payoff

function. If we know the value of the option at expiry we can find

the option value at the time % > I 9 for all values of $ on the tree.But knowing these values means that we can find the option values

one step further back in time.

GThus we work our way back down the tree until we get to the root.

This root is the current time and asset value, and thus we find the

option value today.


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We are valuing a European call option, strike price 100 and expiry

in four months. Todays asset price is 100, the volatility is 20%. Theinterest rate is zero. We use a timestep of one month so that there are

four steps until expiry. Using these numbers we have

I 9 ,

X , Y and

S

.

As an example, after one timestep the asset takes either the value

@ or @ . Working back

from expiry, the option value at the timestep before expiry when6

is given by

H

> @

@ > @


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Working right back down the tree to the present time, the option

value when the asset is 100 is 6.14. Compare this with the theo-retical, continuous-time solution (given by the BlackScholes call

value) of 6.35. The difference is entirely due to the size and number

of the timesteps. The larger the number of timesteps, the greater the

accuracy.


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6.14

6.16

6.18

6.2

6.22

6.24

6.26

6.28

6.3

6.32

6.34

6.36

0 50 100 150 200

4.Option price as a function of number of timesteps.


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7 The greeks

The greeks are defined as derivatives of the option value with respect

to various variables and parameters.

G It is important to distinguish whether the differentiation is with

respect to a variable or a parameter.

If the differentiation is only with respect to the asset price and/or

time then there is sufficient information in our binomial tree to esti-

mate the derivative. The options delta, gamma and theta can all be

estimated from the tree.

On the other hand, if you want to examine the sensitivity of the

option with respect to one of the parameters, then you must perform

another binomial calculation. This applies to the options vega and

rho for example.

Let me take these two cases in turn.


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From the binomial model the options delta is defined by

9

> '

>

X > Y 6

We can calculate this quantity directly from the tree, using the option

value at the two points marked D, together with todays asset price

and the parameters X and Y .

D

D

G

G

G

5.Calculating the Greeks.


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Estimating the other type of greeks, the ones involving differenti-

ation with respect to parameters, is slightly harder. They are harderto calculate in the sense that you must perform a second binomial

calculation.

The vega is the sensitivity of the option value to the volatility

# 9

#

Suppose we want to find the option value and vega when the volatil-

ity is 20%. The most efficient way to do this is to calculate the op-

tion price twice, using a binomial tree, with two different values of . Calculate the option value using a volatility of D t , for a small

number t , call the values you find 9D

. The vega is approximated by

'

> '

>

t


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8 Early exercise

American-style exercise is easy to implement in a binomial setting.

The algorithm is identical to that for European exercise with one

exception. We use the same binomial tree, with the same X , Y and S ,

but there is a slight difference in the formula for9

. We must ensurethat there are no arbitrage opportunities at any of the nodes.

Introduce the notation 6 QM

to mean the asset price at the Q th timestep,

at the node M from the bottom, T M T Q .

In our lognormal world we have

6

Q

M

6 X

M

Y

Q > M

where 6 is the current asset price. Also introduce 9 QM

as the option

value at the same node. Our ultimate goal is to find 9

knowing the

payoff, i.e. knowing9

1

M

for all T M T 0

where0

is the number

of timesteps.


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In the American option problem, arbitrage occurs if the option

value goes below the payoff at any time. If our theoretical valuefalls below the payoff then it is time to exercise. If we do then exer-

cise the option its value and the payoff must be the same. If we find

that

9

3

M

> '

Q

M

: > ;

0

> 7 I 9

: '

3

M

> ; '

Q

M

: > ;

M Payoff $ 3M

then we use this as our new value.


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But if

'

Q

M

> '

3

M

: > ;

0

> 7 I 9

: '

3

M

> Y 9

Q

M

X > Y

Payoff 6 3M

we should exercise, giving us a better value of

9

Q

M

Payoff 6

Q

M

26


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We can put these two together to get

9

Q

M

P D [

U

'

3

M

> '

Q

M

: > ;

0

> 7 I 9

: '

3

M

> Y 9

Q

M

X > Y

Payoff

$

3

M

This ensures that there are no arbitrage opportunities. This modifi-

cation is easy to code.

27


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9 The continuous-time limit

Equation (5) and the BlackScholes equation are more closely re-

lated than they may at first seem. Recalling that the BlackScholes

equation is in continuous time, we examine (5) ass W .

First of all, we have that

: z P

S

I 9

P

I 9 c c c

and

; z > P

5

I 9

P

I 9 ? ? ?

Next we write

9 9 6 W 9

9 X 6 W s W and 9 > ' Y 6 W s W

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Expanding these expressions in Taylor series for smalls W

and sub-

stituting into (4) we find that

d z

# 9

# 6

as s W

Thus the binomial delta becomes, in the limit, the BlackScholes

delta.

Similarly, we can substitute the expressions for 9 , 9 and ' > into

(5) to find

'

9

P

$

'

$

7

'

$

> U 9

This is the BlackScholes equation. Again, the drift rate { has dis-

appeared from the equation.

BinomialModelEquityAssetsBlackScholesEqn

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