Transcript OPTION PRICING MODEL

BINOMIAL OPTION PRICING
MODEL
• Chapter 5
• 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 Cash flow
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 Cash flow
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  $104.87
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  $104.87
Cu  $7.32
S0  $100
Sud  udS0  $102.23
C0*  $5.31
Cud  $2.23
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 multi-period
model are more complex than the singleperiod model, requiring possible
readjustments in subsequent periods.
• For a discussion of multiple-period
arbitrage strategies, see JG, pp. 158-163.
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
d e
tVeA / n  eA / n

tVeA / n  eA / n
Terms
t  time to exp iration exp ressed
as a proportion of a year .
 ,Ve  annualized mean and
A
e
A
var iance of the stock ' s
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:
 Sn 

ln 
 S0 
Example:
 $110 
ln 
 .0953
 $100 
 $95 
ln 
  .0513
 $100 
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:
  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
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
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.