option pricing model
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
OPTION PRICING MODELOPTION PRICING MODEL
• Black-Scholes (BS) OPM.
• Cox, Ross, and Rubinstien Binomial Option Pricing Model: BOPM
• Black-Scholes (BS) OPM.
• Cox, Ross, and Rubinstien Binomial Option Pricing Model: BOPM
BOPMBOPM
• 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.
• 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 BOPMSINGLE-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:
• 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 1ASSUMPTION 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.
• 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
S uSu 0 11. ($100) $110S uSu 0 11. ($100) $110
S0S0
S dSd 0 95. ($100) $95S dSd 0 95. ($100) $95
Assumption 2Assumption 2
• Assume there is a European call option on the stock that expires at the end of the period.
• Example: X = $100.
• Assume there is a European call option on the stock that expires at the end of the period.
• Example: X = $100.
Binomial Call Movement
C IV Max uS Xu [ , ] $100 0C IV Max uS Xu [ , ] $100 0
C0C0
C IV Max dS Xd [ , ]0 0 0C IV Max dS Xd [ , ]0 0 0
Assumption 3Assumption 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.
• 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 portfolioReplicating 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.
• 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:
H uS B ro f0 0( ) H uS B ro f0 0( )
H S B0 0 0H S B0 0 0
H dS B rf0 0 0( ) H dS B rf0 0 0( )
Constructing the RPConstructing 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.
• 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:Solve for Ho and Bo where:
• Equations:• Equations:
df
uf
CrBdSH
CrBuSH
000
000
)(
)(
SolutionSolution
• Equations:• Equations:
HC C
uS dS
BC dS C uS
r uS dS
u d
u d
f
00 0
00 0
0 0
*
* ( ) ( )
( )
HC C
uS dS
BC dS C uS
r uS dS
u d
u d
f
00 0
00 0
0 0
*
* ( ) ( )
( )
ExampleExample
• Equations• Equations
32.60$)95$110($05.1
)110($0)95($10$
6667.95$110$
010$
*0
*0
B
H
32.60$)95$110($05.1
)110($0)95($10$
6667.95$110$
010$
*0
*0
B
H
ExampleExample
• 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.
• 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 CashflowsRP’s Cashflows
• End-of-Period CF:• End-of-Period CF:
. ($110) $60. ( . ) $10
. ($95) $60. ( . )
6667 32 105
6667 32 105 0
. ($110) $60. ( . ) $10
. ($95) $60. ( . )
6667 32 105
6667 32 105 0
Law of One PriceLaw 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.
• 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 PriceEquilibrium Call Price
• BOPM Call Price:• BOPM Call Price:
C H S B
Example
C
0 0 0 0
0 6667 32 35
* * *
*
:
(. )$100 $60. $6.
C H S B
Example
C
0 0 0 0
0 6667 32 35
* * *
*
:
(. )$100 $60. $6.
Arbitrage 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.
• 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 CallArbitrage: 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.
• 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 CashflowInitial Cashflow
Short call
Long Ho shares at So
Borrow Bo
CFo
Short call
Long Ho shares at So
Borrow Bo
CFo
$7.35
-$66.67=(.6667)$100
$60.32
$1
$7.35
-$66.67=(.6667)$100
$60.32
$1
End-of-Period CFEnd-of-Period CF
Position Su = $110Cu = $10
Sd = $95Cd = 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 CallArbitrage: 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.
• 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 CashflowInitial Cashflow
Long call
Short Ho shares at So
Invest Bo in RF
CFo
Long call
Short Ho shares at So
Invest Bo in RF
CFo
-$5.35
$66.67=(.6667)$100
-$60.32
$1
-$5.35
$66.67=(.6667)$100
-$60.32
$1
End-of-Period CFEnd-of-Period CF
Position Su = $110Cu = $10
Sd = $95Cd = 0
Call 10 0
Stock:.6667 S
-$73.34 -$63.34
Investment:$60.32(1.05)
$63.34 $63.34
Cashflow 0 0
ConclusionConclusion
• 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.
• 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 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:
• 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 EquationBOPM Equation
Alternative Equation:Alternative Equation:
Cr
pC p C
where
pr d
u d
fu d
f
0
11* ( )
:
Cr
pC p C
where
pr d
u d
fu d
f
0
11* ( )
:
BOPM EquationBOPM Equation
Example:Example:
C
where
p
0
1
1056667 3333 0 35
105 95
11 956667
*
.. ($10) (. )( ) $6.
:
. .
. ..
C
where
p
0
1
1056667 3333 0 35
105 95
11 956667
*
.. ($10) (. )( ) $6.
:
. .
. ..
NoteNote
• 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.
• 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.
RealismRealism
• To make the BOPM more realistic, we need to – extend the model from a single period to
multiple periods, and – estimate u and d.
• 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 BOPMMultiple-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.
• 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 ExampleTwo-Period Example
• Using our previous example, suppose we subdivide the period to expiration into two periods.
• Assume:– u = 1.0488, d = .9747,– So = $100, n = 2,– Rf = .025, X = $100
• Using our previous example, suppose we subdivide the period to expiration into two periods.
• Assume:– u = 1.0488, d = .9747,– So = $100, n = 2,– Rf = .025, X = $100
110$02 SuSuu
110$02 SuSuu
S udSud 0 23$102.S udSud 0 23$102.
S d Sdd 20 $95S d Sdd 2
0 $95
88.104$uSS 0u 88.104$uSS 0u
S dSd 0 47$97.S dSd 0 47$97.
S0 $100S0 $100
Method for Pricing CallMethod 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.
• 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 ExpirationStep 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
• 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 CdStep 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 = $7.32At Su = $104.88, Cu = $7.32• Using Single-Period Model• Using Single-Period Model
C H S B
C
where
HC C
S S
BC udS C u S
r S S
u u u u
u
uuu ud
uu ud
uuu ud o
f uu ud
( )$104. $97. $7.
:
( ) ( )
( )$97.
1 87 56 32
1
5602
C H S B
C
where
HC C
S S
BC udS C u S
r S S
u u u u
u
uuu ud
uu ud
uuu ud o
f uu ud
( )$104. $97. $7.
:
( ) ( )
( )$97.
1 87 56 32
1
5602
At Sd = $97.47, Cd = $1.48At Sd = $97.47, Cd = $1.48
• Using Single-Period Model• Using Single-Period Model
C H S B
C
where
HC C
S S
BC d S C udS
r S S
d d d d
d
dud dd
ud dd
dud dd o
f ud dd
(. )$97. $28. $1.
:
.
( ) ( )
( )$28.
3084 47 58 48
3084
582
0
C H S B
C
where
HC C
S S
BC d S C udS
r S S
d d d d
d
dud dd
ud dd
dud dd o
f ud dd
(. )$97. $28. $1.
:
.
( ) ( )
( )$28.
3084 47 58 48
3084
582
0
Step 3: Find CoStep 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.
• 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 = $5.31At So = $100, Co = $5.31
• Using Single-Period Model• Using Single-Period ModelC H S B
C
where
HC C
S S
BC dS C uS
r S S
u d
u d
u d o
f d d
0 0 0 0
0
0
00
7881 50 31
7881
50
* * *
*
*
*
(. )$100 $73. $5.
:
.
( ) ( )
( )$73.
C H S B
C
where
HC C
S S
BC dS C uS
r S S
u d
u d
u d o
f d d
0 0 0 0
0
0
00
7881 50 31
7881
50
* * *
*
*
*
(. )$100 $73. $5.
:
.
( ) ( )
( )$73.
10$
110$02
uu
uu
C
SuS10$
110$02
uu
uu
C
SuS
S udS
Cud
ud
0 23
23
$102.
$2.
S udS
Cud
ud
0 23
23
$102.
$2.
S d S
Cdd
dd
20
0
$95S d S
Cdd
dd
20
0
$95
32.7$C
88.104$uSS
u
0u
32.7$C
88.104$uSS
u
0u
S dS
Cd
d
0 47
48
$97.
$1.
S dS
Cd
d
0 47
48
$97.
$1.
S
C
0
0 31
$100
$5.*
S
C
0
0 31
$100
$5.*
ModelModel
Point: Multiple-Period ModelPoint: 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.
• 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 StrategyPoint: 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.
• 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 DividendsPoint: 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.
• 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
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.
– 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 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.
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
1 1 0
1 1 0 9 5 3 5l n ( . ) . ( . )
S 0 1 0 0
S
gd
d
9 5
0 9 5 0 5 1 3 5l n ( . ) . ( . )
S
g
d d
d d
9 0 2 5
9 5 1 0 2 6 2 52
.
l n ( . ) . ( . )
S
gu d
u d
1 0 4 5 0
1 1 9 5 0 4 4 0 5
.
l n ( ( . ) ( . ) ) . ( . )
S
gu u
u u
1 2 1
1 1 1 9 0 6 2 52l n ( . ) . ( . )
S
g
u u u
u u u
1 3 3 1 0
1 1 2 8 5 9 1 2 53
.
l n ( . ) . ( . )
S
g
u u d
u u d
1 1 4 9 5
1 1 9 5 1 3 9 3 3 7 52
.
l n ( ( . ) ( . ) ) . ( . )
S
g
u d d
u d d
9 9 2 7 5
9 5 1 1 0 0 7 3 3 7 52
.
l n ( ( . ) ( . ) ) . ( . )
S
gd d d
d d d
8 5 7 4
9 5 1 5 3 9 1 2 52
.
l n ( . ) . ( . )
E g
V g
( ) .
( ) .1
1
0 2 2
0 0 5 4
E g
V g
( ) .
( ) .1
1
0 4 4
0 1 0 8
E g
V g
( ) .
( ) .1
1
0 6 6
0 1 6 2
B i n o m i a l p r o c e s s
u d q S 1 1 9 5 5 1 0 00. , . , . ,
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
Binomial Distribution
• 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
Note Sk g: ( ) . [. . ] . [ . . ]13 35 095 022 5 0513 022 0
Binomial Distribution
• The expected value and variance of the logarithmic return after two periods are .044 and .0108:
E g
V g
( ) . (. ) . (. ) . ( . ) .
( ) . [. . ] . [. . ] . [ . . ] .2
22 2 2
25 1906 5 0440 25 1026 044
25 1906 044 5 0440 044 25 1026 044 0108
Note Sk g: ( ) . [. . ] . [. . ] . [ . . ]23 3 325 1906 044 5 0440 044 25 1026 044 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 g nE g
V g nV gn
n
( ) ( )
( ) ( )
1
1
Binomial Distribution
• Note: If q = .5, then skewness is zero.
Sk g nSk g
If q Sk gn
n
( ) ( )
. ( )
1
5 0
Binomial Distribution
• 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
n q 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
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.
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.
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
/ /
/ /
TermsTerms
t
VeA
eA
Time to expiration as a proportion of a year.
Annualized variance of the stock’s logarithmic return.
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: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
( )( )
( )(. ) .
.
.
[. (. )]/ [ / ]
[. (. )]/ [ / ]
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 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
BOPM ValuesSecond Example
BOPM ValuesSecond Example
n u d rf Co*
2 1.1 .95 1.02469 $7.11
4 1.0649 .95987 1.01227 $6.91
52 1.0153 .9865 1.00939 $6.89
n u d rf Co*
2 1.1 .95 1.02469 $7.11
4 1.0649 .95987 1.01227 $6.91
52 1.0153 .9865 1.00939 $6.89
1t,05.R,103923.,00108.V,044. Af
Ae
Ae
Ae 1t,05.R,103923.,00108.V,044. A
fAe
Ae
Ae
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
Summary of the BOPMSummary 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 multiple-period model.
• Estimation of u and d.
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 multiple-period model.
• Estimation of u and d.
BOPM and the B-S OPMBOPM and the B-S OPM
– The BOPM for large n is a practical, realistic model.
– As n gets large, the BOPM converges to the B-S 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.
– The BOPM for large n is a practical, realistic model.
– As n gets large, the BOPM converges to the B-S 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 FormulaB-S OPM Formula
• B-S Equation:• B-S Equation:
C S N dX
eN d
dS X R T
T
d d T
RT0 0 1 2
10
2
2 1
5
* ( ) ( )
ln( / ) ( . )
C S N dX
eN d
dS X R T
T
d d T
RT0 0 1 2
10
2
2 1
5
* ( ) ( )
ln( / ) ( . )
Terms: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.
– 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) termN(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) 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) termN(d) term
n d d
d d
d
N d n d d
N d n d d
( ) . [ . ( )
. ( ) . ( )
. ( ) ]
( ) ( );
( ) ( );
1 5 1 196854
115194 000344
019527
0
1 0
2 3
4 4
n d d
d d
d
N d n d d
N d n d d
( ) . [ . ( )
. ( ) . ( )
. ( ) ]
( ) ( );
( ) ( );
1 5 1 196854
115194 000344
019527
0
1 0
2 3
4 4
B-S OPM ExampleB-S OPM Example
• ABC call: X = 50, T = .25, S = 45, = .5, Rf = .06• ABC call: X = 50, T = .25, S = 45, = .5, Rf = .06
88.2)3131(.e
50)4066)(.45(C
)d(Ne
X)d(NSC
3131.)4864.(N)d(N
4066.)2364.(N)d(N
4864.25.5.2364.d
2364.25.5.
25).)5(.06(.)50/45ln(d
)25)(.06(.*0
2RT10*0
2
1
2
2
1
88.2)3131(.e
50)4066)(.45(C
)d(Ne
X)d(NSC
3131.)4864.(N)d(N
4066.)2364.(N)d(N
4864.25.5.2364.d
2364.25.5.
25).)5(.06(.)50/45ln(d
)25)(.06(.*0
2RT10*0
2
1
2
2
1
B-S FeaturesB-S Features
• Model specifies the correct relations between the call price and explanatory variables:
• Model specifies the correct relations between the call price and explanatory variables:
C f S X T R0* ( , , , , )
C f S X T R0
* ( , , , , )
Dividend AdjustmentsDividend 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:
• 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:
C Max C S t X D C S T X
Where
S SD
e
t ex dividend time
Ad d
d Rt
0
0
[ ( , , ), ( , , )]
:
.. .
*
*
*
C Max C S t X D C S T X
Where
S SD
e
t ex dividend time
Ad d
d Rt
0
0
[ ( , , ), ( , , )]
:
.. .
*
*
*
Implied VarianceImplied 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.
– 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 StudyB-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.
• 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 StudiesMacBeth-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-the-
money calls and overprice out-of-the money calls.
– the B-S model was good at pricing on-the-money calls with some time to expiration.
• 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-the-
money calls and overprice out-of-the money calls.
– the B-S model was good at pricing on-the-money calls with some time to expiration.
Bhattacharya StudiesBhattacharya 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.
• 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.
ConclusionConclusion
• 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.
• 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.