Transcript OPTION PRICING MODEL

OPTION PRICING MODEL
• Black-Scholes (BS) OPM.
• Cox, Ross, and Rubinstien Binomial Option
Pricing Model: BOPM
BOPM
• Model is based on constructing a replicating
portfolio (RP).
• The RP is a portfolio whose cash flows
match the cash flows of a call option.
• By the law of one price, two assets with the
same cash flows will in equilibrium be
equally priced; if not, arbitrage
opportunities would exist.
SINGLE-PERIOD BOPM
• The single-period BOPM assumes there is
one period to expiration (T) and two
possible states at T -- up state and a down
state.
• The model is based on the following
assumptions:
ASSUMPTION 1
• In one period the underlying stock will
either increase to equal a proportion u times
its initial value (So) or decrease to equal a
proportion d times its initial value.
• Example: assume current stock price is
$100 and u = 1.1 and d = .95.
Binomial Stock Movement
Su  uS0  11
. ($100)  $110
S0
Sd  dS0 .95($100)  $95
Assumption 2
• Assume there is a European call option on
the stock that expires at the end of the
period.
• Example: X = $100.
Binomial Call Movement
Cu  IV  Max[uS0  X ,0]  $10
C0
Cd  IV  Max[dS0  X ,0]  0
Assumption 3
• Assume there is a risk-free security in
which investors can go short or long.
• Rf = period riskfree rate.
• rf = (1 + Rf).
• Example: Rf = .05 and rf = 1.05.
Replicating portfolio
• The replicating portfolio consist of buying
Ho shares of the stock at So and borrowing
Bo dollars.
• Replicating Portfolio is therefore a
leveraged stock purchase.
• Given the binomial stock price movements
and the rate on the risk-free security, the
RP’s two possible values at T are known.
Replicating Portfolio:
H0 (uS0 )  Borf
H0 S0  B0
H0 (dS0 )  B0rf
Constructing the RP
• The RP is formed by solving for the Ho and
Bo values (Ho* and Bo*) which make the
two possible values of the replicating
portfolio equal to the two possible values of
the call (Cu and Cd).
• Mathematically, this requires solving
simultaneously for the Ho and Bo which
satisfy the following two equations.
Solve for Ho and Bo where:
• Equations:
H 0 (uS0 )  B0 rf  Cu
H 0 (dS0 )  B0 rf  Cd
Solution
• Equations:
Cu  Cd
H 
uS0  dS0
Cu (dS0 )  Cd (uS0 )
*
B0 
rf (uS0  dS0 )
*
0
Example
• Equations
$10  0
H 
 .6667
$110 $95
$10($95)  0($110)
*
B0 
 $60.32
1.05($110 $95)
*
0
Example
• A portfolio consisting of Ho* = .6667
shares of stock and debt of Bo* = $60.32,
would yield a cash flow next period of $10
if the stock price is $10 and a cash flow of 0
if the stock price is $95.
• These cash flows match the possible cash
flow of the call option.
RP’s Cashflows
• End-of-Period CF:
.6667($110)  $60.32(105
. )  $10
.6667($95)  $60.32(105
. )0
Law of One Price
• By the law of one price, two assets which
yield the same CFs, in equilibrium are
equally priced.
• Thus, the equilibrium price of the call is
equal to the value of the RP.
Equilibrium Call Price
• BOPM Call Price:
C H S B
*
0
*
0 0
*
0
Example:
C  (.6667)$100  $60.32  $6.35
*
0
Arbitrage
• The equilibrium price of the call is
governed by arbitrage.
• If the market price of the call is above
(below) the equilibrium price, then an
arbitrage can be exploited by going short
(long) in the call and long (short) in the
replicating portfolio.
Arbitrage: Overpriced Call
• Example: Market call price = $7.35
• Strategy:
– Short the Call
– Long RP
• Buy Ho*= .6667 shares at $100 per share.
• Borrow Bo* = $60.32.
• Strategy will yield initial CF of $1 and no
liabilities at T if stock is at $110 or $95.
Initial Cashflow
Short call
Long Ho shares at So
Borrow Bo
CFo
$7.35
-$66.67=(.6667)$100
$60.32
$1
End-of-Period CF
Position
Su = $110
Cu = $10
Sd = $95
Cd = 0
Call
-10
0
Stock:
.6667 S
$73.34
$63.34
Debt:
$60.32(1.05)
-$63.34
-$63.34
Cashflow
0
0
Arbitrage: underpriced Call
• Example: Market call price = $5.35
• Strategy:
– Long In call
– Short RP
• Sell Ho*= .6667 shares short at $100 per share.
• Invest Bo* = $60.32 in riskfree security.
• Strategy will yield initial CF of $1 and no
liabilities at T if stock is at $110 or $95.
Initial Cashflow
Long call
Short Ho shares at So
Invest Bo in RF
CFo
-$5.35
$66.67=(.6667)$100
-$60.32
$1
End-of-Period CF
Position
Su = $110
Cu = $10
Sd = $95
Cd = 0
Call
10
0
Stock:
.6667 S
-$73.34
-$63.34
Investment:
$60.32(1.05)
$63.34
$63.34
Cashflow
0
0
Conclusion
• When the market price of the call is equal
to $6.35, the arbitrage is zero.
• Hence, arbitrage ensures that the price of
the call will be equal to the value of the
replicating portfolio.
Alternative Equation
• By substituting the expressions for Ho* and
Bo* into the equation for Co*, the equation
for the equilibrium call price can be
alternatively expressed as:
BOPM Equation
Alternative Equation:
1
C 
rf
*
0
 pC
where:
rf  d
p
ud
u
 (1  p) Cd

BOPM Equation
Example:
1
C 
.6667($10)  (.3333)(0)  $6.35

105
.
where:
105
. .95
p
.6667
11
. .95
*
0
Note
• In the alternative expression, p is defined as
the risk-neutral probability of the stock
increasing in one period.
• The bracket expression can be thought of as
the expected value of the call price at T.
• Thus, the call price can be thought of as the
present value of the expected value of the
call price.
Realism
• To make the BOPM more realistic, we need
to
– extend the model from a single period to
multiple periods, and
– estimate u and d.
Multiple-Period BOPM
• In the multiple-period BOPM, we subdivide
the period to expiration into a number of
subperiods, n.
– As we increase n (the number of subperiods),
• we increase the number of possible stock prices
at T, which is more realistic, and
• we make the length of each period smaller,
making the assumption of a binomial process
more realistic.
Two-Period Example
• Using our previous example, suppose we
subdivide the period to expiration into two
periods.
• Assume:
– u = 1.0488,
– So = $100,
– Rf = .025,
d = .9747,
n = 2,
X = $100
Suu  u S0  $110
2
Su  uS0  $104.88
S0  $100
Sud  udS0  $102.23
Sd  dS0  $97.47
Sdd  d 2 S0  $95
Method for Pricing Call
• Start at expiration where you have three
possible stocks prices and determine the
corresponding three intrinsic values of the
call.
• Move to period 1 and use single-period
model to price the call at each node.
• Move to period one and use single-period
model to price the call in current period.
Step 1: Find IV at Expiration
• Start at expiration where you have three
possible stocks prices and determine the
corresponding three intrinsic values of the
call.
– Cuu = Max[110-100,0] = 10
– Cud = Max[102.23-100,0] = 2.23
– Cdd = Max[95-100,0] = 0
Step 2: Find Cu and Cd
• Move to preceding period (period 1) and
determine the price of the call at each stock
price using the single-period model.
• For Su = $104.88, determine Cu using
single-period model for that period.
• For Sd = $97.47, determine Cd using
single-period model for that period.
At Su = $104.88, Cu = $7.32
• Using Single-Period Model
Cu  Hu Su  Bu
Cu  (1)$104.87  $97.56  $7.32
where:
Cuu  Cud
Hu 
1
Suu  Sud
Bu
Cuu ( udS0 )  Cud ( u 2 So )

 $97.56
r f ( Suu  Sud )
At Sd = $97.47, Cd = $1.48
• Using Single-Period Model
Cd  Hd Sd  Bd
Cd  (.3084)$97.47  $28.58  $1.48
where:
Hd
Cud  Cdd

.3084
Sud  Sdd
Bd
Cud ( d 2 S0 )  Cdd ( udSo )

 $28.58
rf ( Sud  Sdd )
Step 3: Find Co
• Substitute the Cu and Cd values
(determined in step 2) into the equations for
Ho* and Bo*, then determine the current
value of the call using the single-period
model.
At So = $100, Co = $5.31
• Using
C0*  Single-Period
H0* S0  B0* Model
C0*  (.7881)$100  $73.50  $5.31
where:
H
*
0
*
0
B
Cu  Cd

.7881
Su  Sd
Cu ( dS0 )  Cd ( uSo )

 $73.50
rf ( Sd  Sd )
S uu  u S 0  $110
2
Model
Cuu  $10
Su  uS0  $104.88
S0  $100
Cu  $7.32
Sud  udS0  $102.23
Cud  $2.23
C0*  $5.31
Sd  dS0  $97.47
Cd  $1.48
Sdd  d 2 S0  $95
Cdd  0
Point: Multiple-Period Model
• Whether it is two periods or 1000, the
multiple-period model determines the price
of a call by determining all of the IVs at T,
then moving to each of the preceding
periods and using the single-period model
to determine the call prices at each node.
• Such a model is referred to as a recursive
model -- Mechanical.
Point: Arbitrage Strategy
• Like the single-period model, arbitrage
ensures the equilibrium price. The arbitrage
strategies underlying the multiperiod model
are more complex than the single-period
model, requiring possible readjustments in
subsequent periods.
• For a discussion of multiple-period
arbitrage strategies, see JG, pp. 158-163.
Point: Impact of Dividends
• The model does not factor in dividends. If a
dividend is paid and the ex-dividend date
occurs at the end of any of the periods, then
the price of the stock will fall. The price
decrease will cause the call price to fall and
may make early exercise worthwhile if the
call is America.
Point: Adjustments for Dividends
and American Options
– The BOPM can be adjusted for dividends by
using a dividend-adjusted stock price (stock
price just before ex-dividend date minus
dividend) on the ex-dividend dates. See JG,
pp.192-196.
– The BOPM can be adapted to price an
American call by constraining the price at each
node to be the maximum of the binomial value
or the IV. See JG, pp. 196-199.
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.
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:
Sn
•
g  ln
n
F
I
G
HS JK
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.
Binomial process
S uuu  133.10
u  11
. , d .95, q  .5, S0  100
g uuu  ln(11
. 3 )  .2859 (.125)
S uu  121
g uu  ln(11
. 2 )  .1906 (.25)
S uud  114.95
S u  110
g u  ln(11
. )  .0953 (.5)
g uud  ln((11
. 2 )(.95))  .1393 (.375)
S ud  104.50
S 0  100
g ud  ln((11
. )(.95))  .0440 (.5)
S d  95
g d  ln(.095)  .0513 (.5)
S udd  99.275
g udd  ln((.952 )(11
. ))  .0073 (.375)
S dd  90.25
g dd  ln(.952 )  .1026 (.25)
S ddd  85.74
g ddd  ln(.952 )  .1539 (.125)
E ( g1 )  .022
E ( g1 )  .044
E ( g1 )  .066
V ( g1 )  .0054
V ( g1 )  .0108
V ( g1 )  .0162
Binomial Process
• Note: When n = 1, there are two possible prices and
logarithmic returns:
F
uS I
lnG J ln(u)  ln(11
. )  .095
HS K
F
dS I
lnG J ln(d )  ln(.95)  .0513
HS K
0
0
0
0
Binomial Process
• When n = 2, there are three possible prices and
logarithmic returns:
F
u S I
lnG J ln(u )  ln(11
. )  .1906
HS K
F
udS I
lnG J ln(ud )  ln((11
. )(.95))  .044
HS K
F
d S I
nG J ln(d )  ln(.95 )  .1026
HS K
2
0
2
2
2
2
0
0
0
2
0
0
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
n!
pnj 
q j (1  q ) n  j
(n  j )! j !
Binomial Distribution
• Using the binomial probabilities, the
expected value and variance of the
logarithmic return after one period are .022
and .0054:
E ( g1 )  .5(.095)  .5( .0513)  .022
V ( g1 )  .5[.095.022]2  .5[ .0513.022]2  .0054
Note: Sk(g1 )  .5[.095.022]3  .5[.0513.022]3  0
Binomial Distribution
• The expected value and variance of the
logarithmic return after two periods are .044
and .0108:
E(g2 )  .25(.1906)  .5(.0440)  .25( .1026)  .044
V(g2 )  .25[.1906.044]2  .5[.0440.044]2  .25[ .1026.044]2  .0108
Note: Sk(g2 )  .25[.1906.044]3  .5[.0440.044]3  .25[.1026.044]3  0
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:
E ( gn )  nE ( g1 )
V ( gn )  nV ( g1 )
Binomial Distribution
• Note: If q = .5, then skewness is zero.
Sk(g n )  nSk(g1 )
If q  .5  Sk(g n )  0
Binomial Distribution
• Note: The expected value and variance of
the logarithmic return are also equal to
E ( gn )  n[q ln u  (1  q) ln d ]
V ( gn )  n q (1  q)[ln(u / d )]
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  estimated mean of the loagarithmic return.
Ve  Estimated var iance of the log arithmic return.
Objective: Solve for u and d where:
n[q ln u  (1  q ) ln d ]   e
n q (1  q )[ln(u / d )]2  Ve
Or given q  .5:
n[.5 ln u  .5 ln d ]   e
n (.5) 2 [ln(u / d )]2  Ve
Derivation of u and d formulas
Solution:
ue
Ve / n   e / n
d  e  Ve / n   e / n
where:
 e and Ve  mean and var iance for a
period equal in length to n.
For the mathematical derivation, see JG: 180  181.
Annualized Mean and Variance
• e and Ve are the mean and variance for a length of
time equal to the option’s expiration.
• Often the annualized mean and variance are used.
• 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.
• If annualized parameters are used in the formulas
for u and d, then they must be multiplied by the
proportion t, where t is the time to expiration
expressed as a proportion of a year.
Equations
ue
d e
A
tVe

A
/ n  e
/n
tVeA / n  eA / n
Terms
t
Ve A
Time to expiration as a
proportion of a year.
Annualized variance of
the stock’s logarithmic
return.

A
e
Annualized mean of the
stock’s logarithmic
return.
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 to an expiration
of one quarter (t=.25) is n = 6, then u =
1.02684 and d = .9739.
Estimated Parameters:
Estimates of u and d:
  4   (4)(0)  0
A
e
q
e
Ve A  4Ve A  (4)(.004209) .016836
ue
[.25(.016836)]/ 6 [ 0 / 6 ]
d e
 [.25(.016836 )]/ 6 [ 0 / 6 ]
 102684
.
.9739
Example: Working Back
• The estimated annualized mean and
variance are .044 and .0108.
• If the expiration is one year ( t = 1), number
of subperiods to expiration is one (n = 1, h
= 1 year), then u = 1.159 and d = .94187.
• If the expiration is one year (t = 1), the
number of subperiods to expiration is 2 (n =
2, h = ½ year), then u = 1.1 and d = .95.
Call Price
The BOPM computer program (provided to
each student) was used to value a $100 call
option expiring in one quarter on a nondividend paying stock with the above
annualized mean (0) and variance
(.016863), current stock price of $100, and
annualized RF rate of 9.27%.
BOPM 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
  0,Ve .016836, R .0927, t .25
A
R p  (1  R A ) t / n  1
A
f
BOPM Values
Second Example
n
2
u
1.1
d
rf
Co*
.95
1.02469
$7.11
4
1.0649
.95987 1.01227
$6.91
52
1.0153
.9865 1.00939
$6.89
eA  .044, VeA  .00108, eA  .103923, R fA  .05, t  1
R p  (1  R A ) t / n  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
d e
tVeA / n

tVeA / n
 1/ u
Summary of the BOPM
The BOPM is based on the law of one price, in
which the equilibrium price of an option is
equal to the value of a replicating portfolio
constructed so that it has the same cash flows
as the option. The BOPM derivation requires:
• Derivation of single-period model.
• Specification of the mechanics of the multipleperiod model.
• Estimation of u and d.
BOPM and the B-S OPM
– The BOPM for large n is a practical, realistic
model.
– As n gets large, the BOPM converges to the BS OPM.
• That is, for large n the equilibrium value of a call
derived from the BOPM is approximately the
same as that obtained by the B-S OPM.
– The math used in the B-S OPM is complex but
the model is simpler to use than the BOPM.
B-S OPM Formula
• B-S Equation:
X
C  S0 N ( d1 )  RT N ( d 2 )
e
2
ln( S0 / X )  ( R .5 )T
d1 
 T
*
0
d 2  d1  
T
Terms:
– T = time to expiration, expressed as a
proportion of the year.
– R = continuously compounded annual RF rate.
– R = ln(1+Rs), Rs = simple annual rate.
–  = annualized standard deviation of the
logarithmic return.
– N(d) = cumulative normal probabilities.
N(d) term
• N(d) is the probability that deviations less
than d will occur in the standard normal
distribution. The probability can be looked
up in standard normal probability table (see
JG, p.217) or by using the following:
N(d) term
n( d )  1.5[1.196854( d )
.115194( d ) 2 .000344( d ) 3
.019527( d ) 4 ] 4
N ( d )  n( d ); d  0
N ( d )  1  n( d ); d  0
B-S OPM Example
• ABC call: X = 50, T = .25, S = 45,  = .5, Rf = .06
d1 
ln(45 / 50)  (. 06  (. 5) 2 ). 25
  .2364
.5 .25
d 2  .2364 .5 .25   .4864
N(d1)  N(.2364)  .4066
N(d 2 )  N(.4864)  .3131
X
*
C0  S0 N(d1) 
N (d 2 )
RT
e
C*0  (45)(. 4066) 
50
e
(.06)(.25)
(. 3131)  2.88
B-S Features
• Model specifies the correct relations
between the call price and explanatory
variables:
C  f ( S , X , T , R,  )
*
0





Dividend Adjustments
• The B-S model can be adjusted for
dividends using the pseudo-American
model. The model selects the maximum of
two B-S-determined values:
C0A  Max[C ( Sd , t * , X  D), C ( Sd , T , X )]
Where:
Sd  S0 
D
Rt *
e
t *  ex  dividend .. time.
Implied Variance
– The only variable to estimate in the B-S OPM
(or equivalently, the BOPM with large n) is the
variance. This can be estimated using historical
averages or an implied variance technique.
– The implied variance is the variance which
makes the OPM call value equal to the market
value. The software program provided each
student calculates the implied variance.
B-S Empirical Study
• Black-Scholes Study (1972): Conducted
efficient market study in which they
simulated arbitrage positions formed when
calls were mispriced (C* not = to Cm).
• They found some abnormal returns before
commission costs, but found they
disappeared after commission costs.
• Galai found similar results.
MacBeth-Merville Studies
• MacBeth and Merville compared the prices
obtained from the B-S OPM to observed
market prices. They found:
– the B-S model tended to underprice in-themoney calls and overprice out-of-the money
calls.
– the B-S model was good at pricing on-themoney calls with some time to expiration.
Bhattacharya Studies
• Bhattacharya (1980) examined arbitrage
portfolios formed when calls were
mispriced, but assumed the positions were
closed at the OPM values and not market
prices.
• Found: B-S OPM was correctly specified.
Conclusion
• Empirical studies provide general support
for the B-S OPM as a valid pricing model,
especially for near-the-money options.
• The overall consensus is that the B-S OPM
is a useful model.
• Today, the OPM may be the most widely
used model in the field of finance.