Transcript OPTION PRICING MODEL
ESTIMATING u AND d
Binomial Distribution Derivation of the Estimating Formula for u an d
Estimating 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 resulting equations that satisfy this objective are:
Equations
u
e tV e A
/
n
e A
/
n d
e
tV e A
/
n
e A
/
n
Terms
t
time to
exp
iration
exp
ressed as a proportion of a year
.
e A
,
V e A
annualized mean and
var log
arithmic return
.
Logarithmic Return
• The logarithmic return is the natural log of the ratio of the end-of-the-period stock price to the current price: ln
S n S
0
Example
: ln $110 $100 ln $95 $100 .
0953 .
0513
Annualized 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.
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:
Estimates of u and d:
e A V e A
4
e q
4
V e A
u
e
[.
(.
0 )]/ 6 0 6 016836
d
e
[.
25 016836 )]/ 6 0 6 .
9739
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%.
n 6 u
BOPM Values
d rf Co* 1.02684
.9739
1.0037
$3.25
30 100 1.01192
.9882
1.00074
$3.34
1.00651
.9935
1.00022
$3.35
0 ,
V e A
.
016836 ,
R f A
.
0927 ,
t
.
25
R p R
) 1
u 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:
u
e tV e A
/
n d
e
tV e A
/
n
1 /
u
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:
g n
ln F
S
H
n
0 I K
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.
Binomial process u
d
.
95 ,
q
.
5
S g u u
1 1% .
09 53
S u
g u
11 2 1906 25 )
S u
g u
11% 11
S u
g u
ln(.
095 ) .
(. )
S u
g u
S u
g u
ln(.
95 2 ) .
1026 (.
25 )
S uuu
g uuu
11 3 2859 125 )
S uud
g uud
11 2 ln(( . )(.
95 1393 375 )
S udd
g udd
ln((.
95 2 .
(.
)
E g
1 1 022 0054
E g
1 1 044 0108
S ddd
g ddd
ln(.
95 2 ) .
(.
)
E g
1 1 066 0162
Binomial Process
• Note: When n = 1, there are two possible prices and logarithmic returns: ln ln F F
uS
0 H
S
0
dS
0 H
S
0 I K I K .
095 ln(.
95 ) .
0513
Binomial Process
• When n = 2, there are three possible prices and logarithmic returns: ln ln
n
F H F G 2
u S S S
2
d S S
0 0
udS
0 0 0 0 I K I K I K ln( ln( ln(
d u ud
2 2 ) ) ) 11 2 ln(.
95 2 ) .
1906 )) .
044 .
1026
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
p nj
(
n
n
!
j q j
( 1
q
)
Binomial Distribution
• Using the binomial probabilities, the expected value and variance of the logarithmic return after one period are .022 and .0054:
E g
1 ) 1 ) ) .
] 2 0513 ) .
022 .
] 2 .
0054
Binomial Distribution
• The expected value and variance of the logarithmic return after two periods are .044 and .0108:
E g
1 ) 1 ) ) .
] 2 ) 1026 ) .
] 2 .
044 .
] 2 .
0108
Binomial Distribution
• 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: ( (
n
)
n
)
nE g
1 ) 1 )
Binomial Distribution
• Note: The expected value and variance of the logarithmic return are also equal to
n n
) )
u n q
( 1
q
( 1
q
2
Deriving 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.
Deriving the formulas for u and d
Let
:
e V e
estimated mean of the loagarithmic return
.
Estimated
var
iance of the
log
arithmic return
.
Objective Solve for u and d where
:
u
( 1
q
e n q
( 1
q
2
V e Or given q
n
5
n
5 2
u
5 2
e
V e
Derivation of u and d formulas
Solution:
u
e d
e
e
/
n
e
/
n where
:
e and V e
mean and
var
iance for a period equal in length to n
.
For the mathematical derivation see JG
: .
Annualized Mean and Variance
Equations
u
e tV e A
/
n
e A
/
n d
e
tV e A
/
n
e A
/
n