estimating u and d

Binomial Distribution

Derivation of the Estimating Formula

for u an d

ESTIMATING u AND d

Estimating u and dEstimating u and d

The estimating equations for determining u and d are obtained by mathematically solving for the u and d values which make the statistical characteristics of a binomial distribution of the stock’s logarithmic returns equal to the characteristic's estimated value.

The estimating equations for determining u and d are obtained by mathematically solving for the u and d values which make the statistical characteristics of a binomial distribution of the stock’s logarithmic returns equal to the characteristic's estimated value.

The resulting equations that satisfy this objective are:

The resulting equations that satisfy this objective are:

EquationsEquations

u e

d e

tV n n

tV n n

eA

eA

eA

eA

/ /

/ /

u e

d e

tV n n

tV n n

eA

eA

eA

eA

/ /

/ /

Terms

t time to iration ressed

as a proportion of a year

V annualized mean and

iance of the stock s

arithmic return

eA

eA

exp exp

.

,

var '

log .

Logarithmic ReturnLogarithmic Return

• The logarithmic return is the natural log of the ratio of the end-of-the-period stock price to the current price:

• The logarithmic return is the natural log of the ratio of the end-of-the-period stock price to the current price:

ln

:

ln$110

$100.

ln$95

$100.

S

S

Example

n

0

0953

0513

ln

:

ln$110

$100.

ln$95

$100.

S

S

Example

n

0

0953

0513

Annualized Mean and VarianceAnnualized Mean and Variance

• The annualized mean and variance are obtained by multiplying the estimated mean and variance of a given length (e.g, month) by the number of periods of that length in a year (e.g., 12).

• For an example, see JG, pp. 167-168.

• The annualized mean and variance are obtained by multiplying the estimated mean and variance of a given length (e.g, month) by the number of periods of that length in a year (e.g., 12).

• For an example, see JG, pp. 167-168.

Example: JG, pp. 168-169.

• Using historical quarterly stock price data, suppose you estimate the stock’s quarterly mean and variance to be 0 and .004209.

• The annualized mean and variance would be 0 and .016836.

• If the number of subperiods for an expiration of one quarter (t=.25) is n = 6, then u = 1.02684 and d = .9739.

Estimated Parameters:Estimated Parameters:

Estimates of u and d:Estimates of u and d:

eA

eq

eA

eAV V

u e

d e

4 4 0 0

4 4 004209 016836

102684

9739

25 016836 6 0 6

25 016836 6 0 6

( )( )

( )(. ) .

.

.

[. (. )]/ [ / ]

[. (. )]/ [ / ]

eA

eq

eA

eAV V

u e

d e

4 4 0 0

4 4 004209 016836

102684

9739

25 016836 6 0 6

25 016836 6 0 6

( )( )

( )(. ) .

.

.

[. (. )]/ [ / ]

[. (. )]/ [ / ]

Call Price Call Price

The BOPM computer program (provided to each student) was used to value a $100 call option expiring in one quarter on a non-dividend paying stock with the above annualized mean (0) and variance (.016863), current stock price of $100, and annualized RF rate of 9.27%.

The BOPM computer program (provided to each student) was used to value a $100 call option expiring in one quarter on a non-dividend paying stock with the above annualized mean (0) and variance (.016863), current stock price of $100, and annualized RF rate of 9.27%.

BOPM ValuesBOPM Values

n u d rf Co*

6 1.02684 .9739 1.0037 $3.25

30 1.01192 .9882 1.00074 $3.34

100 1.00651 .9935 1.00022 $3.35

n u d rf Co*

6 1.02684 .9739 1.0037 $3.25

30 1.01192 .9882 1.00074 $3.34

100 1.00651 .9935 1.00022 $3.35

0 016836 0927 25, . , . , .V R teA

fA 0 016836 0927 25, . , . , .V R te

AfA

R Rp A t n ( ) /1 1R Rp A t n ( ) /1 1

u and d for Large nu and d for Large n

In the u and d equations, as n becomes large, or equivalently, as the length of the period becomes smaller, the impact of the mean on u and d becomes smaller. For large n, u and d can be estimated as:

In the u and d equations, as n becomes large, or equivalently, as the length of the period becomes smaller, the impact of the mean on u and d becomes smaller. For large n, u and d can be estimated as:

u e

d e u

tV n

tV n

eA

eA

/

/ /1

u e

d e u

tV n

tV n

eA

eA

/

/ /1

Binomial Process

• The binomial process that we have described for stock prices yields after n periods a distribution of n+1possible stock prices.

• This distribution is not normally distributed because the left-side of the distribution has a limit at zero (I.e. we cannot have negative stock prices)

• The distribution of stock prices can be converted into a distribution of logarithmic returns, gn:

• gS

SnnFHGIKJln

0

Binomial Process

• The distribution of logarithmic returns can take on negative values and will be normally distributed if the probability of the stock increasing in one period (q) is .5.

• The next figure shows a distribution of stock prices and their corresponding logarithmic returns for the case in which u = 1.1, d = .95, and So = 100.

S

gu

u

11%

11 0953 5ln( . ) . (. )

S

gu

u

11%

11 0953 5ln( . ) . (. )

S

gu

u

9 5%

095 0513 5

.

ln(. ) . (. )

S

g

u

u

9 025%

95 1026 252

.

ln(. ) . (. )

S

gu

u

10 45%

11 95 0440 5

.

ln(( . )(. )) . (. )

S

g

u

u

12 1%

11 1906 252

.

ln( . ) . (. )

S

guuu

uuu

13 31%

11 2859 1253

.

ln( . ) . ( . )

S

guud

uud

11 495%

11 95 1393 3752

.

ln(( . )(. )) . (. )

S

g

udd

udd

9 9275%%

95 11 0073 3752

.

ln((. )( . )) . (. )

S

gddd

ddd

8574%

95 1539 1252

.

ln(. ) . (. )

E g

V g

( ) .

( ) .1

1

022

0054

E g

V g

( ) .

( ) .1

1

044

0108

E g

V g

( ) .

( ) .1

1

066

0162

Binomial process

u d q 11 95 5. , . , .

Binomial Process

• Note: When n = 1, there are two possible prices and logarithmic returns:

ln ln( ) ln( . ) .

ln ln( ) ln(. ) .

uS

Su

dS

Sd

0

0

0

0

11 095

95 0513

FHGIKJ

FHGIKJ

Binomial Process

• When n = 2, there are three possible prices and logarithmic returns:

ln ln( ) ln( . ) .

ln ln( ) ln(( . )(. )) .

ln( ) ln(. ) .

u S

Su

udS

Sud

nd S

Sd

20

0

2 2

0

0

20

0

2 2

11 1906

11 95 044

95 1026

FHG

IKJ

FHG

IKJ

FHG

IKJ

Binomial Process• Note: When n = 1, there are two possible prices and

logarithmic returns; n = 2, there are three prices and rates; n = 3, there are four possibilities.

• The probability of attaining any of these rates is equal to the probability of the stock increasing j times in n period: pnj. In a binomial process, this probability is

pn

n j jq qnj

j n j

!

( )! !( )1


• Using the binomial probabilities, the expected value and variance of the logarithmic return after one period are .022 and .0054:

E g

V g

( ) . (. ) . ( . ) .

( ) . [. . ] . [ . . ] .

1

12 2

5 095 5 0513 022

5 095 022 5 0513 022 0054


• The expected value and variance of the logarithmic return after two periods are .044 and .0108:

E g

V g

( ) . (. ) . (. ) . ( . ) .

( ) . [. . ] . [. . ] . [ . . ] .1

12 2 2

25 1906 5 0440 25 1026 044

25 1906 044 5 0440 044 25 1026 044 0108


• Note: The parameter values (expected value and variance) after n periods are equal to the parameter values for one period time the number of periods:

E g nE g

V g nV gn

n

( ) ( )

( ) ( )

1

1


• Note: The expected value and variance of the logarithmic return are also equal to

E g n q u q d

V g nq q u d

n

n

( ) [ ln ( ) ln ]

( ) ( )[ln( / )]

1

1 2

Deriving the formulas for u and dDeriving the formulas for u and d

The estimating equations for determining u and d are obtained by mathematically solving for the u and d values which make the expected value and variance of a binomial distribution of the stock’s logarithmic returns equal to the characteristic's estimated value.

The estimating equations for determining u and d are obtained by mathematically solving for the u and d values which make the expected value and variance of a binomial distribution of the stock’s logarithmic returns equal to the characteristic's estimated value.

Deriving the formulas for u and dDeriving the formulas for u and d

Let estimated mean of the loagarithmic return

V Estimated iance of the arithmic returne

e

: .

var log .

Objective Solve for u and d where

n q u q d

nq q u d V

Or given q

n u d

n u d V

e

e

e

e

: :

[ ln ( ) ln ]

( )[ln( / )]

. :

[. ln . ln ]

(. ) [ln( / )]

1

1

5

5 5

5

2

2 2

Derivation of u and d formulasDerivation of u and d formulas

Solution:Solution:

u e

d e

where

and V mean and iance for a

period equal in length to n

V n n

V n n

e e

e e

e e

/ /

/ /

:

var

.

For the mathematical derivation see JG, : .180 181

Annualized Mean and VarianceAnnualized Mean and Variance

EquationsEquations

u e

d e

tV n n

tV n n

eA

eA

eA

eA

/ /

/ /

u e

d e

tV n n

tV n n

eA

eA

eA

eA

/ /

/ /

estimating u and d

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