11 black scholes

8/3/2019 11 Black Scholes

1/19

Lecture 11: A Brief Introduction to Continuous Time Option Pricing

Readings:o Ingersoll Chapters 14 17

o Cochrane Chapter 17

o Shimko Finance in Continuous Time: A Primer (from which these notesare largely drawn)

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We will spend some time here building up the tools we need to develop the Black-Scholes Partial Differential Equation. This will be done in a relatively informal way andyou should consult other texts if you wish to pursue these issues in more depth.

First we need to introduce an Ito Process. Ill build this idea up slowly so bear with me

if you are already familiar with the concept.

Definition: A stochastic process, defined byB(0) = 0 (more generally B(0) = B0 a fixed starting point) and

B(t + 1) = B(t) + (t + 1) t {0, 1, 2, }where the innovations in B are independent standard normal random variables:

(t + 1) ~ iidN(0,1) t, is a special version of a random walk special in that it hasnormally distributed increments.

This is a simple example of a discrete time stochastic process where we see a newrealization of the process B(t) at each point in time, i.e. at each time t.

The realization at any time tof the process can be arbitrarily high or low. At each time tthe innovations in the process B are unpredictable (and normally distributed). In otherwords, as with all random walks, the expected value of a future realization of the processas of date tis simply B(t). The expected change in the process is always zero and thevariance of the change depends on how far into the future you are trying to forecast.Over one period the variance is 1. Over five periods (you know B(11) and areforecasting B(16)) the variance is 5 (the expectation is still: E11(B(16)) = B(11)).

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Now suppose we observe the process more frequently than at each fixed time interval.

Let = 1/n for some arbitrary integern > 1. We want to describe a process with thesame characteristics as the random walk described above but observed more frequently:

B(t + ) = B(t) + (t + ), with B(0) = 0 and B = B(t + ) B(t) = (t + ) ~ iid

N(0, )

Overn periods of length this new process has the same expected change (or drift inthis example there is none) and the same variance as the original has over one fixed timeinterval or period.

Finally let dt, a very small increment of time (so n is very large). Define small

heuristically by letting dt be the smallest positive real number such that dt

= 0 whenever

> 1.

Then:B(0) = 0

B(t + dt) = B(t) + (t + dt), t [0, T]where (t + dt) ~ iidN(0, dt)

Define dB(t) = B(t + dt) B(t) = (t + dt), as the increments in the process B(t).

dB(t) may be thought of as a normally distributed random variable with mean 0 andvariance dt. It is often referred to as white noise. The process B(t) is a standard Wienerprocess.

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Some properties of dB(t) that follow by construction are:(1) E[dB(t)] = 0 the expected change in B(t) is zero

E[dB(t)] = E[B(t + dt) B(t)] = E[ (t + dt)] = 0 since (t + dt) ~ N(0, dt)

(2) E[dB(t) dt] = 0E[dB(t) dt] = E[dB(t)]dt = 0 since dt is a constant.

(3) E[(dB(t) dt)2] = 0E[(dB(t) dt)2] = dt2 E[dB(t)2] = dt2 Var(dB(t)) = dt2(dt) = 0 since dt

= 0 for all >

1.

(4) Var[dB(t) dt] = 0Var[dB(t) dt] = dt2 Var[dB(t)] = dt2 dt = 0 as above.

Note: (2) and (3) or (4) imply dB(t)dt = 0 since its expectation and variance are both zero.

(5) E[dB(t)2] = dt

E[dB(t)2] = Var[dB(t)] = dt since dB(t) = ((t + dt) and (t + dt) ~ N(0, dt) andsince

E[ (t + dt)] = 0 implies that E[ (t + dt)2] = Var[ (t + dt)]

(6) Var[dB(t)2] = 0

Var[dB(t)2] = E[dB(t)4] (E[dB(t)2])2 = E[ (t + dt)4] dt2 = 3dt2 dt2 = 0

Follows since if (t + dt) ~ N(0, 2), then E[ (t + dt)4] = 3 4.

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Similar to the note above, (5) and (6) imply that dB(t)2 = dt, a constant.

These properties are important in that they demonstrate that the variance of a function ofa random variable can vanish. When this is true, the expectation sign is redundant:E[f(dB(t))] = f(dB(t)) if Var[f(dB(t))] = 0.

These properties lead to the following multiplication rules that will come in handy:dt2 = 0 this essentially says dt is smalldB(t) dt = 0 and dB(t)2 = dt these just eliminate the redundant expectations operatorsince the variances of these functions of the random variable dB( ) are zero.

Standard Brownian Motion or Standard Wiener Process

Definition: A standard Brownian motion, denoted B(t), is a stochastic process definedby:

B(0) = B0 (= 0) a.s. (with probability 1)

has increments Bs Bt ~ N(0, s t) s,t with s > t

112010,...,,,0 nn ttttttt BBBBBBBB are independently distributed for any

0 < t0 < t1 < t2 < < tn-1 < tn T. So the increments are independent normalrandom variables. (Or simply, dB(t) - a standard Wiener process - is thedifferential representation.) B is continuous in each sample path. Continuousmeans you can draw the sample path without lifting your pen from the paper.This is true because while dB(t) is a random variable it is of infinitesimalmagnitude (no jumps).

Alternative representation for B(t): Integral form: +=t

dBBtB0

0 )()(

Notes:(a) B is nowhere differentiable. The intuition for this is that for any point in a samplepath, the change to the right and to the left are independent random variables.

(b) Et[Bs] = Et[Bt + (Bs Bt)] = Bt + E[Bs Bt] = Bt The forecast of Bs made at time tisalways Bt.

(c) Vart[Bs] = Vart[Bt + Bs Bt] = s t - since Bt is known at time t. This tells us aboutthe volatility of the realization around the guess made in note (b).

(d) Vart[Bs] as s

Despite all our time developing the idea, the standard Brownian motion is not a goodmodel for stock price movements. We want a process that allows for a drift in prices, i.e.a generalization of a standard Brownian motion for which the expected change, over any

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future interval of the process is non-zero (we would like to have an expected change inprice, an expected return, given the observed behavior of prices).Ito Process

Consider two processes(1) a standard Brownian motion volatility, but zero expected change (no drift)

and(2) a process that is a constant change for each increment of time f(t) = t for someconstant a drift but no volatility.

Now add the two together this is a simple version of an Ito process.

Definition: A stochastic process X defined by:X(0) = X0

dXt = (Xt, t)dt + (Xt, t)dBt where dBt is the instantaneous increment of astandard Brownian motion

is an Ito process. Note for notational simplicity I am writing Xt and Bt rather than X(t)and B(t).

: [0, T] is the drift of the process

: [0, T] is the diffusion

In the simple example given above (Xt, t) = and (Xt, t) = 1 Xt and t. In general

it need not be this simple (with and being constants) but it is always true (since

both Xt and tare known at each time t) that the values (Xt, t) and (Xt, t), whichdetermine the drift and diffusion of the process over the next instant in time, are known at

time t.

Some examples:

(1) if (Xt, t) = 0 and (Xt, t) = 1 then we have a standard Brownian motion

(2) if (Xt, t) = (a constant) and (Xt, t) = (a constant) then gives the drift or

expected change for each increment of time. can enhance or diminish (depending on

whether > 1 or < 1) the changes in the Brownian motion.

This is an arithmetic Brownian motion, it allows for negative realizations and theexpected growth is linear (constant absolute growth), not the limited liability and theexponential expected growth that stock prices exhibit. For the arithmetic Brownianmotion:

P(0) = P0P(1) = P0 + (1)P(2) = P1 + (2) = P0 + (1) + (2)

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Linear growth: example = 1

Stock prices on the other hand exhibit exponential expected growth:P(0) = P0P(1) = P0(1 + r)

P(2) = P1(1 + r) = P0(1 + r)2

Notes on the arithmetic Brownian motion(a) X can be positive or negative and arbitrarily large in either direction

(b) Xs Xt ~ N( (s t), 2 (s t))

(c) Vart(Xs) as s

1 2

21

1011001%

Growth

100%Growth

f(t) = t

99 100

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(3) Geometric Brownian motion

tttt XtXandXtX == ),(),(

so that

t

t

t

tttt dBdtX

dXordBXdtXdX +=+=

a process with a constant expected return over time and a constant variance of return.This is a simplified but more natural model of stock prices:

Notes:(a) if X starts positive it remains positive.(b) X has an absorbing barrier at 0.(c) the conditional distribution of Xs given Xt is lognormal. Ln(Xs) is normally

distributed and the conditional mean of Ln(Xs) for s > t is Ln(Xt) + (s t) 2(s t)

and the conditional standard deviation of Ln(Xs) is ts . The conditional expected

value of Xs is)( ts

teX

, to find expected future price inflate current price by the

continuously compounded expected rate of return.

(d) The variance of a forecast of Xs tends to as s tends to .

(4) Mean Reverting Process

tttt XtXandXktX == ),()(),(

where k, , and are all greater than zero.

If = 1 this is called an Ornstein Uhlenbeck process.

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For Xt > the drift is negative and for Xt < the drift is positive. This type of processis used a lot in the modeling of interest rates.

Notes:(a) X is always positive if it starts positive(b) as X approaches 0 the drift is positive and the volatility is zero

(c) as s the variance of a forecast of Xs is finite

(d) if = (as pictured above) the distribution of Xs given Xt (s > t) is non central 2

with mean + )()( tskt eX

and variance2)(

2)(2)( )1) (()) ((

22 tskk

tsktskkt eeeX

+

We now turn to Itos lemma:If we have Xt an Ito process and Yt =f(Xt) what does dYt look like? In other words, weknow dXt has a drift and a diffusion what are they for dY t?

Clearly this is an important question for derivative pricing if we think of Xt as the price ofthe underlying asset then for the rightf( ), Yt =f(Xt) is the price of the derivative.

Itos Lemma (Univariate Case)

Let X be an Ito Process defined by tttt dBdtdX +=

where the dependence of and on (Xt, t) is suppressed for notational convenience.

Letf: [0, T] . Then Yt =f(Xt, t) is an Ito Process defined by

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tttxttx xttttxt d BtXfd ttXftXftXfd Y ),(]),(),(),([2

21 +++=

Typically it is assumed that the functionf(.) is twice continuously differentiable in bothXt and t, however, we only require thatfx,fxx, andft exist and are continuous.

Intuition: Take a 2nd order Taylor series expansion off(Xt + dt, t + dt) around (Xt, t) thendYt =f(Xt +dt, t + dt) f(Xt, t).

)(),(2

)) (,()) (,(

),(),(),(),(

21

22

122

1

r e s i d u a l Rd td XtXf

d ttXfd XtXf

d ttXfd XtXftXfd ttXf

ttx t

tttttxx

ttttxtd tt

++

++

++=++

Now the famous line, it can be shown that the residual R0 as dt 0.

Consider the different terms and use the multiplication rules. We know what dXt, dt, anddt2 are but what are (dXt)

2 and dXtdt?

dtd t

d tdBdBd t

dBdtdX

tt

tttttt

tttt

22

2222

22

00

2

)()(

=++=

++=

+=

000)( =+=+= dtdBdtdtdX tttt

Then collecting terms from the Taylors series expansion

tttxttxxttttx

ttttdttt

dBtXfd ttXftXftXf

tXfd ttdXXftXfd ttXfdY

),(]),(),(),([

),(),(),(),(2

21 +++=

++=+=+

Note that the residual includes only higher order terms and the multiplication rulestherefore imply that it indeed vanishes. This finishes a heuristic proof of the lemma.

Note that in ordinary calculus dX is small enough so that dX 2 vanishes. In stochastic

calculus dX is a random variable so dX2 does not vanish (instead it converges to dtt2 )

but terms of higher order, dX3 or dXdt do vanish.

Examples:

(1) Consider an Ito process Yt = Bt2 for t 0, where Bt is a standard Brownian motion.

Find dYt. We first identify Xt then f() and finally computefx,fxx, andft in order to use thelemma.

==

====

tttt

tttttt

XTfandXtXfY

dBdXsoBXand

],0[:),(

10

2

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then0),(2),(2),( === tXfandtXfXtXf tttxxttx

and we arrive at

tttt

ttt

tttxttxxttttxt

BXdBBdtdBXd tX

dBtXfd ttXftXftXfdY

=+=

+++=

+++=

since2112]12002[

),(]),(),(),([

221

22

1

So Yt is an Ito process with a drift ( ) of 1 and a diffusion ( ) of 2Bt.

(2) )exp(3 tt BtY ++=)exp(3),(10 tttttttt XttXfdBdXBX ++=====

1),exp(),exp( === ttxxtx fXfXf

tttt d BBd tBd Y )e x p () ]e x p (1[ 21 ++=

(3) tt dBdtdX += an arithmetic Brownian motion, can also be written tt BtX +=

(since is a constant). Consider a process defined by )exp(0 tt XSS = with S0 > 0.

St =f(Xt, t) where the functionf: [0, T] is defined byf(Xt, t) = S0exp(Xt).

0)exp()exp( 00 === ttxxtx fXSfXSf

and

ttt

ttttt

d BSd tSd BXSd tXSXSd S

++=

++=

])[ ()e x p (])e x p ()e x p ([

22

1

0

2

02

1

0

and we see that S is a geometric Brownian motion.

Xt is normally distributed so Ln(St) = Ln(S0)+Xt is normal with mean Ln(S0) + tand

variance 2t. The log of S has normal increments so S has lognormal increments.

Solving a simple problem:

Start with a deterministic example. Suppose that a security with current value Vguarantees a flow of cash payments at the rate $1dt every instant of time forever. This isa continuous-time equivalent of a risk-free perpetuity paying $1 each discrete period. Ifthe instantaneous risk-free rate is a constant r, what is the current value of the security?

First write the law of motion for V using Itos lemma. Here it is simple, because V doesnot depend on any stochastic variable: V = V(t). Thus dV = Vtdt.

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Calculate the expected capital gain from owning the security: E[dV] = Vtdt.

Calculate the expected cash flow to the security: $1dt.

Thus the total expected return on the security is: Vtdt + 1dt = [Vt + 1]dt.

Now, set the total expected return equal to the risk free dollar return: rVdt = [Vt + 1]dt.Divide both sides by dt: rV = Vt + 1

and we get a differential equation whose value V must satisfy. Then we just have tosolve the equation for V some how. Here just guess that V cannot depend on tsince it isthe value of a constant, risk-free perpetuity. Thus Vt = 0 and the solution is V = 1/r. Thisis equivalent to the present value of a risk-free perpetuity in discrete time but here ris theinstantaneous risk free rate.

Now, suppose X follows a geometric Brownian motion: tttt dBXdtXdX += . Asecurity with value V collects Xdt continuously forever. V represents the current value

of a perpetuity whose continuous cash flowXdtgrows at the average exponential rate .Suppose further that the risk of the cash flow variation is considered diversifiable risk.The economy is risk neutral and the instantaneous risk free rate is a constant, r. What isthe value of this security?

V = V(Xt, t) however, just as above, since its perpetual (and so looks the same today asit does in 2 years) V cant actually depend on tonly on X, so V = V(X t). What is V(X)?

Using Itos lemma:

txx xx d BXVd tXVXVd V ++= ][22

21

The expected capital gain on this security is E[dV]

dVXVXd VE x xx += ][][22

21

The expected cash flow isXdtby the construction of the security.

The total expected return on the security is the sum of these components:

d tXVXVXR e tu r nT o ta lE x p e c te d x xx ++= ][22

21

Since the risk of the security is diversifiable, to avoid arbitrage it must be that this totalreturn equals rVdt(a riskless return on the securitys value). Setting them equal weobtain a differential equation for V:

][ 2221 XVXVXr V x xx ++=

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To solve the differential equation, guess that doubling the current level ofX(current cash

flow and so the expected future cash flow) will double V. Then V= Xfor some

constant . This implies that Vx = , and Vxx = 0. Substituting these into thedifferential equation we get:

)(

1

=+=

r

orXXXr and so

)( =

r

XV

This is the equation for the present value of a perpetuity growing at a constant rate ofwhen the riskless rate is r.

The Multi-Variate Case:

Now introduce a second Ito process Y to the system so we have:

2

1

),,(),,(

),,(),,(

tttttt

tttttt

dBtYXdttYXdY

dBtYXdttYXdX

+=

+=

Define E[dB1dB2] = dt as the correlation between B1 and B2. We can also show thatE[(dB1dB2)2] = 0, which says the variance of dB1dB2 is zero. Therefore dB1dB2 =

E(dB1dB2) = dt.

Now suppose that Wt = f(Xt, Yt, t). The multivariate extension of Itos lemma has thefollowing Ito differential:

]2[22

21

tyyttx ytxxttytxt dYfdYdXfdXfd tfdYfdXfdW +++++=

terms of higher order than dX2 or dY2 vanish as before.Now we have to expand our list of multiplication rules:

00)()(221212221 ====== dtdtdBdtdBdtdBdBdtdBdB

Then

dtdBdtdBdtdXdY

dtdBdtdY

dtdBdtdX

=++=

=+=

=+=

))((

)(

)(

21

2222

2212

Making these substitutions into the Ito differential we have:212

212

21 ][ d Bfd Bfd tffffffd W yxy yx yx xtyx +++++++=

Again if B1 and B2 were deterministic, terms of higher order than dB1 and dB2 wouldvanish. Since B1 and B2 are stochastic variables terms of higher order than (dB1)2 and(dB2)2 vanish.

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Application:Let X and Y be two real valued Ito processes with:

2

1

dBdtdY

dBdtdX

+=

+=

Show that Z = XY is an Ito process, that is show that it can be written:dBdtdZ +=

First note XYtYXfZ == ),,( so1,,0,0,0,, ====== xyyyxxtyx fandfffXfYf .

Using Itos lemma

dtXdYYdX

dtdBdtXdBdtY

XdBYdBdtXY

dBfdBfdtffffffdZ yxyyxyxxtyx

++=

++++=

++++=

+++++++=

)()(

][

][

21

21

2122

122

1

And the result follows by simple substitution.

Stochastic Integrals and Trading GainsTrading Strategies:

A trading strategy is a stochastic process that specifies at each state and each date theholdings of each of the available securities.

Formally: t : N is a measurable function with respect to F t (the information set, a

subset of , that is available at time t). then is : [0, T] N , the

specification of t for each t.

Trading Gains for Brownian Prices:

Let B be a Brownian motion (representing the price processes for the assets) and let bea trading strategy.

The gains of trading strategy are given by the stochastic integral of with respect toB:

T

ttdB

0

where B may be an N dimensional Brownian motion

This integral is stochastic since B is stochastic.

Lets consider some simple examples:(1) Suppose that = where is a constant for all t, then:

)( 00

BBdB T

T

tt =

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(2) Consider a piecewise constant strategy on [0, T] so for some 0 = t 0 < t1 < < tk =

T t = (tm-1) t [tm-1, tm) m = 1, , k

Then is called a simple trading strategy and

=

=

k

m

ttm

T

tt mmBBtdB

1

1

0

))((1

(3) The stochastic integral is also defined for trading strategies that are not simple the

idea is to find a simple trading strategy m that approaches the strategy in the sense

that

0)]()([0

2 =

T

mm

dtttELim

The stochastic integral ttdB is then defined by

0)()(

2

00

=

T

t

T

tmm

dBtdBtELim

Informally we can think of this as

"00

=T

m

T

ttLimdB

Trading Gains for Ito Prices

t

0 t1 t2 t3 T

15

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Let S be an Ito process (security price process) defined by:

tttt dBdtdS

SS

+=

= 00

Then the gains of trading strategy are given by the stochastic integral of with

respect to S:

+=T

ttt

T

tt

T

tt dBdtdS000

an integral with one nonstochastic and one stochastic

term.

The Black-Scholes ModelReview:Binomial option pricing formula single period2 states of nature

3 securities1 risky stock2 riskless bond3 call option on the stock

the idea is that you replicate the call option using the stock and the bonda simple portfolio of the 2 securities spans the 2 states of nature

Binomial option pricing formula multi (T) period2T states of nature3 securities

1 risky stock

2 riskless bond3 call option on the stockthe idea is that you replicate the option with the stock and the bond at each date2 long-lived securities and a dynamic trading strategy span the 2T states of nature

Black-Scholes Model (continuous time)Infinite number of states of nature3 securities

1 risky stock2 riskless bond

3 call option on the stockthe idea is that you replicate the option with the stock and the bond using a continuoustrading strategy2 long-lived securities and a continuous trading strategy span an infinite number of statesof nature

The Available Securities

Stock the price of the risky stock is assumed to follow a geometric Brownian motion

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tttt dBSdtSdS

SS

+=>= 000

Bond the price of the riskless bond follows the process

dtrd tt

=>= 000

where ris the (assumed constant) instantaneous riskless rate of return.

We can derive this price process as follows. By definition, for any t,tr

te=

0 .

Differentiate this with respect to t:

t

trt rredt

d

==

0 (constant exponential growth) then dtrd tt =

Option a European call option that matures at time T and has an exercise price equal tok. The price process of the European call option is denoted C, where ),( tSC t .

A derivation of the Black-Scholes PDE

1st a trading strategy is self financing if it generates no dividends (positive or negative)for any time tin 0 < t< T. Let at be the holdings of the stock at time tand bt be theholdings of the riskless bond. Then the trading strategy (a, b) is self financing if

tttt

tt

bSadbdSabSa +=+++ 00

0000 for any time tin 0 < t< T.

Assume that C(St, t), the function relating the price of the call option to the stock price (orthe call price process), is twice continuously differentiable. Then, using Itos lemma

tttstts sttttst d BStSCd tStSCtSCStSCd C ),(]),(),(),([)1(22

21 +++=

ClaimSuppose that there exists a self financing strategy (a, b) with

TTTTT bSaTSC +=),( (so the time T payoffs on the strategy and the call are thesame)

then in the absence of arbitrage opportunities the current (t = 0) price of the European calloption on the stock is given by 00000 )0,( bSaSC += .

ProofAssume for contradiction that

00000 )0,( bSaSC +> .

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Then the trading strategy (a, b, -1) (long the strategy and short the call option) has a t= 0

payoff (cost) of 0)0,( 00000 >+ bSaSC and the time T payoff of this position is zero

hence it is an arbitrage opportunity.

If we instead assume that

00000 )0,( bSaSC +< .Then the trading strategy (-a, -b, 1) (short the strategy and long the call option) is anarbitrage. So the replicating portfolios (trading strategys) initial cost must be the initialoption price if the strategy is self financing.

The same argument applied at each date tin the interval [0, T] implies that],0[),()2( TtbSatSC ttttt +=

It follows that

ttttttt

tttttt

ttttt

dBSadtrbSa

dtrbdBSdtSa

dbdSadC

++=

++=

+=

)()3(

)(

so Ct follows an Ito process as defined in (3)

Using (1) and (3) and matching coefficients in dB t we see it must be true that

tttts SaStSC =),(or

),( tSCa tst = (the amount of the stock held at each time tis the options delta at thattime)

Now using (2)

t

ttt

tSatSCb

= ),(

or

t

ttst

t

StSCtSCb

),(),()4(

=

which shows that a self financing trading strategy (a, b) for which (2) holds does exist.

Matching coefficients in dt for (1) and (3) gives

tttttts stttts rbSaStSCtSCStSC +=++22

21 ),(),(),(

The 1st terms on each side of the above equation are equivalent so we can write

222

1 ),(),( tts stttt StSCtSCrb +=

and using (4) we find

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0),(),(),(),( 2221 =+++ tts sttttst StSCtSCStSr CtSr C

This is the Black-Scholes partial differential equation. The boundary condition is of

course:)0,max()(),( kSkSTSC TTT

+

This PDE can be solved in several ways none of which we will pursue. By directcalculation of the derivatives you can verify that the Black-Scholes formula satisfies thePDE.

( )

tTdd

tT

tTrk

SLn

d

where

dNtTrkdNStSC

t

tt

=

++

=

=

12

2

1

21

)(2

)())(exp()(),(

and N(.) is the cumulative standard normal distribution function

Note that as conjectured C(St,t) is twice continuously differentiable.

Finally note that this formula has the same form as the binomial models we examined, thecall price is the stock price times the option delta less the discounted value of the exerciseprice times a factor determined by the distribution of the stock price process (i.e. less the

amount borrowed to form the replicating portfolio).

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11 black scholes

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