Lecture #9: BlackScholes option
pricing formula
· Brownian Motion
The first formal mathematical model of
financial asset prices, developed by
Bachelier (1900), was the continuoustime
random walk, or Brownian motion. This
continuoustime process is closely related to
the discretetime versions of the random
walk.
· The discretetime
random walk
PPkk = P = Pk1 k1 + + k,k, k k = = ( () with probability ) with probability (1 (1
), P), P00 is fixed. Consider the following continuous is fixed. Consider the following continuous
time process Ptime process Pnn(t), t (t), t [0, T], which is constructed [0, T], which is constructed
from the discrete time process Pfrom the discrete time process Pkk, k=1,..n as , k=1,..n as
follows: Let h=T/n and define the process follows: Let h=T/n and define the process
PPnn(t) = P(t) = P[t/h][t/h] = P = P[nt/T][nt/T] , t , t [0, T], where [x] denotes [0, T], where [x] denotes
the greatest integer less than or equal to x. Pthe greatest integer less than or equal to x. Pnn(t) is (t) is
a left continuous step left continuous step function.
We need to adjust , such that Pn(t) will
converge when n goes to infinity. Consider
the mean and variance of Pn(T):
E(Pn(T)) = n(21)
Var (Pn(T)) = 4n(1)
2
We wish to obtain a continuous time version
of the random walk, we should expect the
mean and variance of the limiting process
P(T) to be linear in T. Therefore, we must
have
n(21) T
4n(1) 2 T
This can be accomplished by setting
= ½*(1+h /), =h
· The continuous time
limit
It cab be shown that the process P(t) has the following three It cab be shown that the process P(t) has the following three
properties:properties:
1. For any t1. For any t11 and t and t22 such that 0 such that 0 t t1 1 < t< t2 2 T: T:
P(t P(t11)P(t)P(t22) ) (((t(t22tt11), ),
22(t(t22tt11))))
2. For any t2. For any t11, t , t 2 , 2 , tt33, and t, and t4 4 such that 0 such that 0 t t1 1 < t< t2 2 tt1 1 < t< t2 2 t t3 3 < t< t44
T, the increment T, the increment
P(t P(t22) P(t) P(t11) is statistically independent of the increment P) is statistically independent of the increment P
(t(t44) P(t) P(t33).).
3. The sample paths of P(t) are continuous. 3. The sample paths of P(t) are continuous.
P(t) is called arithmetic Brownian motion or Winner process. P(t) is called arithmetic Brownian motion or Winner process.
If we set If we set =0, =0, =1, we obtain standard Brownian =1, we obtain standard Brownian
Motion which is denoted as B(t). Accordingly, Motion which is denoted as B(t). Accordingly,
P(t) = P(t) = t + t + B(t) B(t)
Consider the following moments:
E[P(t) | P(t0)] = P(t0) +(tt0)
Var[P(t) | P(t0)] =
2(tt0)
Cov(P(t1),P(t2) =
2 min(t1,t2)
Since Var[ (B(t+h)B(t))/h ] = 2/h, therefore,
the derivative of Brownian motion, B’(t)
does not exist in the ordinary sense, they are
nowhere differentiable.
· Stochastic differential
equations
Despite the fact, the infinitesimal increment of Brownian Despite the fact, the infinitesimal increment of Brownian
motion, the limit of B(t+h) = B(t) as h approaches to an motion, the limit of B(t+h) = B(t) as h approaches to an
infinitesimal of time (dt) has earned the notation dB(t) and infinitesimal of time (dt) has earned the notation dB(t) and
it has become a fundamental building block for it has become a fundamental building block for
constructing other continuous time process. It is called constructing other continuous time process. It is called
white noise. For P(t) define earlier we have dP(t) = white noise. For P(t) define earlier we have dP(t) = dt + dt +
dB(t). This is called stochastic differential equation. The dB(t). This is called stochastic differential equation. The
natural transformation dP(t)/dt = natural transformation dP(t)/dt = + + dB(t)/dt doesn’t dB(t)/dt doesn’t
male sense because dB(t)/dt is a not well definedmale sense because dB(t)/dt is a not well defined
(althrough dB(t) is).(althrough dB(t) is).
The moments of dB(t):
E[dB(t)] =0
Var[dB(t)] = dt
E[ dB dB ] = dt
Var[dB dB ] = o(dt)
E[dB dt] = 0
Var[dB dt ] = o(dt)
If we treat terms of order of o(dt) as
essentially zero, the (dB)2 and dBdt are both
nonstochastic variables.
| dB dt
dB | dt 0
dt | 0 0
Using th above rule we can calculate (dP)2
= 2dt. It is not a random variable!
· Geometric Brownian
motion
If the arithmetic Brownian motion P(t) is
taken to be the price of some asset, the price
may be negative. The price process p(t)=
exp(P(t)), where P(t) is the arithmetic
Brownian motion, is called geometric
Brownian motion or lognormal diffusion.
· Ito’s Lemma
Although the first complete mathematical theory of Brownian Although the first complete mathematical theory of Brownian
motion is due to Wiener(1923), it is the seminal motion is due to Wiener(1923), it is the seminal
contribution of Ito (1951) that is largely responsible for the contribution of Ito (1951) that is largely responsible for the
enormous number of applications of Brownian motion to enormous number of applications of Brownian motion to
problems in mathematics, statistics, physics, chemistry, problems in mathematics, statistics, physics, chemistry,
biology, engineering, and of course, financial economics. biology, engineering, and of course, financial economics.
In particular, Ito constructs a broad class of continuous In particular, Ito constructs a broad class of continuous
time stochastic process based on Brownian motion – now time stochastic process based on Brownian motion – now
known as Ito process or Ito stochastic differential known as Ito process or Ito stochastic differential
equations – which is closed under general nonlinear equations – which is closed under general nonlinear
transformation. transformation.
Ito (1951) provides a formula – Ito’s lemma
for calculating explicitly the stochastic
differential equation that governs the
dynamics of f(P,t):
df(P,t) = f/P dP + f/t dt + ½ 2f/P2 (dP)2
· Applications in
Finance
A lognormal distribution for stock price
returns is the standard model used in
financial economics. Given some
reasonable assumptions about the random
behavior of stock returns, a lognormal
distribution is implied. These assumptions
will characterize lognomal distribution in a
very intuitive manner.
Let S(t) be the stock's price at date t. We
subdivided the time horizon [0 T] into n
equally spaced subintervals of length h. We
write S(ih) as S(i), i=0,1,…,n. Let z(i) be
the continuous compounded rate of return
over [ (i1)h ih], ie S(i)=S(i1)exp(z(i)),
i=1,2,..,n. It is clear that
S(i)=S(0)exp[z(1)+z(2)+…+z(i)].
The continuous compounded return on the
stock over the period [0 T] is the sum of the
continuously compounded returns over the
n subintervals.
Assumption A1. The returns {z(j)} are i.. A1. The returns {z(j)} are i...
Assumption A2. E[z(t)]=Assumption A2. E[z(t)]=h, where h, where is the is the
expected continuously compounded return per unit expected continuously compounded return per unit
.
Assumption A3. var[z(t)]=Assumption A3. var[z(t)]=.
Technically, these assumptions ensure that as the Technically, these assumptions ensure that as the
time decrease proportionally, the behavior of the time decrease proportionally, the behavior of the
distribution for S(t) dose not explode nor distribution for S(t) dose not explode nor
degenerate to a fixed point. degenerate to a fixed point.
Assumption 13 implies that for any infinitesimal Assumption 13 implies that for any infinitesimal
time subintervals, the distribution for the time subintervals, the distribution for the
continuously compounded return z(t) has a normal continuously compounded return z(t) has a normal
distribution with mean distribution with mean h, and variance h, and variance 22h. This h. This
implies that S(t) is lognormally that S(t) is lognormally distributed.
· Lognormal distribution
At time t < t+h
lnSt+h ~ [lnSt+(
2/2)h,]
where (m,s) denotes a normal distribution
with mean m and standard deviation s.
· Continuously compounded return
ln(St+h/St) ~ [(
2/2)h,]
· Expected returns
Et[ln(St+h/St)] = (
2/2)h
Et[St+h/St] = exp(h)
· Variance of returns
Vart[ln(St+h/St)] =
2h
Vart[St+h/St] = exp(2h)(exp(
2h)1)
· Estimation of
n+1: number of stock observationsn+1: number of stock observations
SSjj: stock price at the end of jth interval, j=1,…n: stock price at the end of jth interval, j=1,…n
h: length of time intervals in yearsh: length of time intervals in years
LetLet
u ujj = ln[S = ln[Sjj+D+Djj)/S)/Sj1j1]]
u = (uu = (u11+…+u+…+unn)/n is an estimator for ()/n is an estimator for (
22/2)h, s={ /2)h, s={
[(u[(u11u)u)
22+…+(u+…+(unnu)u)
22]/(n1)}]/(n1)}1/21/2 is an estimator for is an estimator for
hh1/21/2..
Example: Daily returnsExample: Daily returns
DayDay Closing priceClosing price DividendDividend Daily ReturnDaily Return
07/0407/04
08/0408/04
09/0409/04
10/0410/04
11/0411/04
14/0414/04
15/0415/04
16/0416/04
17/0417/04
18/0418/04
21/0421/04
22/0422/04
00
00
00
00
00
00
00
00
00
00
00
00
DayDay Closing priceClosing price DividendDividend Daily ReturnDaily Return
23/0423/04
24/0424/04
25/0425/04
28/0428/04
29/0429/04
30/0430/04
01/0501/05
02/0502/05
05/0505/05
MeanMean
.
AnnualizedAnnualized
Annualized Annualized
Mean(250 d)Mean(250 d)
. (250 d) . (250 d)
00
00
00
00
00
00
00
00
00
% %
% %
· Fundamental
equation for derivative securities
Stock price follows Ito process:Stock price follows Ito process:
dS = dS = (S,t)dt + (S,t)dt + (S,t)dz(S,t)dz
At this point, we assume At this point, we assume (S,t) =(S,t) =S, and S, and (S,t)= (S,t)=
SS
Let C(S,t) be a derivative security, according to Let C(S,t) be a derivative security, according to
Ito’s lemma, the process followed by C is Ito’s lemma, the process followed by C is
dC = [ dC = [C/C/S S (S,t) + (S,t) + C/C/t + ½ t + ½ 22C/C/SS22
22(S,t)]dt + [(S,t)]dt + [C/C/S S (S,t)]dz (S,t)]dz
Consider a portfolio P, combination of S and C Consider a portfolio P, combination of S and C
to eliminate uncertainty: P = - C + to eliminate uncertainty: P = - C + C/C/S S , the S S , the
dynamics of P isdynamics of P is
dP = -dC + dP = -dC + C/C/S dS,S dS,
dP = - [dP = - [C/C/S S (S,t) + (S,t) + C/C/t + ½ t + ½ 22C/C/SS22
22(S,t)]dt -[(S,t)]dt -[C/C/S S (S,t)]dz + (S,t)]dz + C/C/S[S[(S,t)dt (S,t)dt
+ + (S,t)dz] (S,t)dz]
Collecting terms involving dt and dz together we get Collecting terms involving dt and dz together we get
dP = - [dP = - [C/C/S S (S,t) + (S,t) + C/C/t + ½ t + ½ 22C/C/SS22
22(S,t) -(S,t) -C/C/S S (S,t)dt]dt -[(S,t)dt]dt -[C/C/S S (S,t) - (S,t) -
C/C/S S (S,t)]dz(S,t)]dz
or dP = - [or dP = - [C/C/t + ½ t + ½ 22C/C/SS22 22(S,t)]dt (S,t)]dt
The portfolio is a riskfree portfolio, hence it The portfolio is a riskfree portfolio, hence it
should earn risk free return, . should earn risk free return, .
dP/P = - [dP/P = - [C/C/t + ½ t + ½ 22C/C/SS22 22(S,t)]dt / [- C + (S,t)]dt / [- C +
C/C/S S] = r dt, rearranging terms leads to the S S] = r dt, rearranging terms leads to the
well known BS partial differential equation: well known BS partial differential equation:
C/C/t + r St + r SC/C/S + ½ S + ½ 22SS2222C/C/SS22 – rC = 0 – rC = 0
This is the fundamental partial differential equation This is the fundamental partial differential equation
for derivatives. The solution for an specific for derivatives. The solution for an specific
derivative is determined by boundary conditions. derivative is determined by boundary conditions.
For example, the European call option is For example, the European call option is
determined by boundary condition: cdetermined by boundary condition: cTT = max(0,S = max(0,STT
K). K).
· Risk neutral pricing
The drift term The drift term does not appear in the fundamental does not appear in the fundamental
equation. Rather, the reiskfree rate r is there. equation. Rather, the reiskfree rate r is there.
Under risk neutral measure, the stock price Under risk neutral measure, the stock price
dynamics is dS = rSdt + dynamics is dS = rSdt + .
If interest rate is constant as in BS, the European If interest rate is constant as in BS, the European
option can be priced as option can be priced as
c = exp[r(Tt)] Ec = exp[r(Tt)] E**[max(0,S[max(0,STTK)] K)]
where Ewhere E* * denotes the expectation under risk neutral denotes the expectation under risk neutral
.
· The BlackScholes Formula for
European Options (with dividend yield q)
c = exp[r(Tt)] [0,] max(0,STK)g(ST)dST
where g(ST) is the probability density function
of the terminal asset price. By using Ito’s
lemma, we can show
ln(ST) ~ N(lnS + (r ½
2)(Tt), (Tt)1/2)
c=SeqTN(d1)Xe
rTN(d2)
p=XerTN(d2)Se
qTN(d1)
wherewhere
dd11=[ln(S/X)+(rq+=[ln(S/X)+(rq+
22/2)T]/(/2)T]/(TT1/21/2), d), d22=d=d11TT
1/21/2
Example: X=$70, Maturity date = June 27 (Evaluate on May Example: X=$70, Maturity date = June 27 (Evaluate on May
5: T=53/365 = )5: T=53/365 = )
____________________________________________________________________________________________
S==
X==
T==
r==
. = . =
q = =
European option prices: Call = Put = European option prices: Call = Put =
· Implied volatility
The volatility that makes the model price
equal its market price.
Assume that the call and put options in the
above example are traded at and
, respectively.
Call implied volatility:
Put implied volatility:
Stock Price Price
Part A Part A
maturity 22 days, r=%maturity 22 days, r=%
Type of Type of
optionsoptions
Strike priceStrike price Mean option Mean option
priceprice
Implied Implied
volatility(%) volatility(%)
CallCall
CallCall
CallCall
PutPut
PutPut
9090
9595
100100
9090
9595
4 3/44 3/4
1 3/81 3/8
1/21/2
1/21/2
2 7/82 7/8
Part BPart B
maturity 50 days, r=%maturity 50 days, r=%
Type of Type of
optionsoptions
Strike priceStrike price Mean option Mean option
priceprice
Implied Implied
volatility(%) volatility(%)
CallCall
CallCall
CallCall
PutPut
PutPut
9090
9595
100100
9090
9595
5 3/85 3/8
2 3/42 3/4
3 3/43 3/4
The prices are midpoint prices. The implied
volatility seems to depend upon whether the
option is in/out or atthemoney. The
implied volatility for calls seems to differ
from the implied volatility for puts.
There are many reasons why the implied
volatility estimates differ. (why?)
· Option Greeks
Delta: With respect to an increase in stock priceDelta: With respect to an increase in stock price
cc=e=e
qTqTN(dN(d11))
pp=e=e
qTqT[N(d[N(d11)1])1]
Gamma: Delta's change with respect to an increase Gamma: Delta's change with respect to an increase
in sock pricein sock price
cc==pp=N'(d=N'(d11)e)e
qTqT/(S/(STT1/21/2))
Theta with respect to a decrease in maturityTheta with respect to a decrease in maturity
cc=SN'(d=SN'(d11))ee
qTqT/(2T/(2T1/21/2) + qSN(d) + qSN(d11)e)e
qTqTrXerXerTrTN(dN(d22))
pp=SN'(d=SN'(d11))ee
qTqT/(2T/(2T1/21/2) qSN(d) qSN(d11)e)e
qTqT+rXe+rXerTrTN(dN(d22))
Vega: with respect to an increase in volatility
c=p=ST
1/2N'(d1)e
qT
Rho: with respect to an increase in interest
rate
c=XTe
rTN(d2)
p=XTe
rTN(d2)
Example1: X=$70, T=
S=
X=
T=
r=
. =
q =
European Option Prices
d1= N(d1)=
d2= N(d2)=
Call= Put=
Delta=
Gamma=
Theta =
Vega=
Rho=
· Synthetic option
Set aside cash in the amount equal to the model Set aside cash in the amount equal to the model
.
Maintain the stock position equal to the delta of the Maintain the stock position equal to the delta of the
target option.
Cash balance is invested in riskfree assets to earn Cash balance is invested in riskfree assets to earn
.
Close the position at the desired the position at the desired matuirity.
If the model is good, the terminal payoff of this If the model is good, the terminal payoff of this
dynamic strategy should be close to the payoff of dynamic strategy should be close to the payoff of
the target option at the target option at the maturity.
Example: Synthetic put optionExample: Synthetic put option
Day Day ClosinClosin
g price g price
Daily Daily
Return Return
Maturity Maturity Delta Delta Stock Stock
Position Position
Overall Overall
CashCash
07/0407/04
08/0408/04
09/0409/04
10/0410/04
11/0411/04
14/0414/04
15/0415/04
16/0416/04
17/0417/04
18/0418/04
21/0421/04
22/0422/04
23/0423/04
24/0424/04
25/0425/04
28/0428/04
29/0429/04
30/0430/04
01/0501/05
02/0502/05
05/05 05/05
MeanMean
. .
a. . m.
250 d250 d
a. a.
.
250 d 250 d
% %
%%
X==
T==
r==
P= P=
· Duration of an option
An option's is its partial derivative with
respect to a change in the continuously
compounded interest rate. Specifically, the
call option pricing formula (Black and
Scholes) is
c=SN(d1)Xe
rTN(d2)
where
d1=[ln(S/X)+(r+
2/2)T]/(T1/2), d2=d1T
1/2
It follows that
c/r = XTerTN(d2)
and
(c/r)/c = (X/c)TerTN(d2)<0
The total differential of the call option can be
written as
dc = c/r dr + c/S dS
Dc=(dc/dr)/c =(c/c)/r c/S (dS/dr)=
= (X/c)TerTN(d2)(S/c)N(d1)Ds
Consider a call option with strike price of $70 and Consider a call option with strike price of $70 and
maturity of 53 days (53/365= years). The maturity of 53 days (53/365= years). The
current stock price is $, the annual standard current stock price is $, the annual standard
deviation of %. The risk free rate of interest deviation of %. The risk free rate of interest
() is10%. d() is10%. d11=, N(d=, N(d11)=, )=,
dd22=, N(d=, N(d22)=)=
c==
Thus DThus Dcc = (70/) = (70/)
**
+(
If DIf Dss>, then D>, then Dcc is positive. is positive.
