Lecture 11 Continuous Time Option Pricing

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					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 notes
            are largely drawn)




                                                                               1
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 and
you should consult other texts if you wish to pursue these issues in more depth.

First we need to introduce an “Ito Process.” I’ll build this idea up slowly so bear with me
if you are already familiar with the concept.

Definition: A stochastic process, defined by
B(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) ~ iid N(0,1)  t, is a special version of a random walk – special in that it has
normally distributed increments.

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




The realization at any time t of the process can be arbitrarily high or low. At each time t
the innovations in the process B are unpredictable (and normally distributed). In other
words, as with all random walks, the expected value of a future realization of the process
as of date t is simply B(t). The expected change in the process is always zero and the
variance 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 are
forecasting B(16)) the variance is 5 (the expectation is still: E11(B(16)) = B(11)).



                                                                                            2
Now suppose we “observe” the process more frequently than at each fixed time interval.
Let  = 1/n for some arbitrary integer n > 1. We want to describe a process with the
same 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,)

Over n periods of length Δ this new process has the same expected change (or “drift” – in
this example there is none) and the same variance as the original has over one fixed time
“interval” 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) ~ iid N(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 and
variance dt. It is often referred to as white noise. The process B(t) is a standard Wiener
process.



                                                                                             3
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] = 0
    E[dB(t) dt] = E[dB(t)]dt = 0 since dt is a constant.

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

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




                                                                                              4
These properties are important in that they demonstrate that the variance of a function of
a 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 small
dB(t) dt = 0 and dB(t)2 = dt – these just eliminate the redundant expectations operator
since 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 defined
by:
       B(0) = B0 (= 0) a.s. (with probability 1)

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

        Bt0  B0 , Bt1  Bt0 , Bt2  Bt1 ,..., Btn  Btn 1 are independently distributed for any
       0 < t0 < t1 < t2 < …< tn-1 < tn  T. So the increments are independent normal
       random variables. (Or simply, dB(t) - a standard Wiener process - is the
       differential representation.) B is continuous in each sample path. “Continuous
       means 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 infinitesimal
       magnitude (no jumps).

                                                                                t
       Alternative representation for B(t): Integral form: B(t )  B0   dB( )
                                                                                0
Notes:
(a) B is nowhere differentiable. The intuition for this is that for any point in a sample
path, 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 t is
always Bt.

(c) Vart[Bs] = Vart[Bt + Bs – Bt] = s – t - since Bt is known at time t. This tells us about
the 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 good
model 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
future interval of the process is non-zero (we would like to have an expected change in
price, an expected return, given the observed behavior of prices).


                                                                                                    5
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 some
constant α – 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 a
       standard 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 t are known at each time t) that the values (Xt, t) and (Xt, t), which determine 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 the
expected growth is linear (constant absolute growth), not the limited liability and the
exponential expected growth that stock prices exhibit. For the arithmetic Brownian
motion:

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




                                                                                            6
Linear growth: example α = 1


                                                                              f(t) = t
               101
 1%            100
 Growth




                   2
 100%              1
 Growth
                                                                  99 100
                           1   2

Stock prices on the other hand exhibit exponential expected growth:
       P(0) = P0
       P(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  




                                                                                         7
(3) Geometric Brownian motion
 ( X t , t )  X t and  ( X t , t )  X t
so that
                                   dX t
dX t  X t dt  X t dBt or               dt   dBt
                                    Xt
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) – ½ 2(s – t)
and the conditional standard deviation of Ln(Xs) is  s  t . The conditional expected
value of Xs is X t e ( s t ) , 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
 ( X t , t )  k (  X t ) and  ( X t , t )  X t
where k, , and  are all greater than zero.

If  = 1 this is called an Ornstein – Uhlenbeck process.


                                                                                              8
For Xt >  the drift is negative and for Xt <  the drift is positive. This type of process is
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  = ½ (as pictured above) the distribution of Xs given Xt (s > t) is non central 2
with mean ( X t   )e  k ( s t )  
and variance X t ( k )(e  k ( s t )  e 2k ( s t ) )   (             )(1  e k ( s t ) ) 2
                           2                                        2
                                                                        2k


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

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

Ito’s Lemma (Univariate Case)
Let X be an Ito Process defined by dX t   t dt   t dBt
where the dependence of  and  on (Xt, t) is suppressed for notational convenience.
Let f: [0, T]  . Then Yt = f(Xt, t) is an Ito Process defined by

           dYt  [ f x ( X t , t ) t  f t ( X t , t )  12 f xx ( X t , t ) t2 ] dt  f x ( X t , t ) t dBt



                                                                                                                  9
Typically it is assumed that the function f(.) is twice continuously differentiable in both
Xt and t, however, we only require that fx, fxx, and ft exist and are continuous.

Intuition: Take a 2nd order Taylor series expansion of f(Xt + dt, t + dt) around (Xt, t) then
dYt = f(Xt +dt, t + dt) – f(Xt, t).

 f ( X t  dt , t  dt )  f ( X t , t )  f x ( X t , t )dX t  f t ( X t , t )dt
                         1 2  f xx ( X t , t )(dX t ) 2  1 2  f tt ( X t , t )(dt ) 2
                         2  1 2  f xt ( X t , t )dX t dt  R(residual)

Now the famous line, “it can be shown that” the residual R  0 as dt  0.

Consider the different terms and use the multiplication rules. We know what dXt, dt, and
dt2 are but what are (dXt)2 and dXtdt?
(dX t ) 2  ( t dt   t dBt ) 2
             t2 dt 2   t2 dBt2  2 t  t dBt dt
            0   t2 dt  0   t2 dt

dX t dt  ( t dt   t dBt )dt  0  0  0

Then collecting terms from the Taylor’s series expansion
dYt  f ( X t  dt , t  dt )  f ( X t , t )  f ( X t  dX t , t  dt )  f ( X t , t )
      [ f x ( X t , t ) t  f t ( X t , t )  1 2  f xx ( X t , t ) t2 ]dt  f x ( X t , t ) t dBt
Note that the residual includes only higher order terms and the multiplication rules
therefore imply that it indeed vanishes. This finishes a heuristic proof of the lemma.

Note that in ordinary calculus dX is small enough so that dX2 vanishes. In stochastic
calculus dX is a random variable so dX2 does not vanish (instead it converges to  t2 dt )
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 compute fx, fxx, and ft in order to use the
lemma.

            t  0  t  1 and                   X t  Bt       so dX t  dBt
           Yt  f ( X t , t )  X     t
                                       2
                                              and      f :   [0, T ]    X t  
then
        f x ( X t , t)  2 X t             f xx ( X t , t )  2 and       ft ( X t , t)  0
and we arrive at



                                                                                                          10
dYt  [ f x ( X t , t ) t  f t ( X t , t )  1 2 f xx ( X t , t ) t2 ]dt  f x ( X t , t ) t dBt
       [2 X t  0  0  1 2  2  12 ]dt  2 X t  1  dBt
     1  dt  2 Bt dBt since X t  Bt
So Yt is an Ito process with a drift () of 1 and a diffusion () of 2Bt.


(2) Yt  3  t  exp( Bt )
 t  0  t  1 X t  Bt dX t  dBt f ( X t , t )  3  t  exp( X t )
 f x  exp( X t ), f xx  exp( X t ), f t  1
dYt  [1  1 2 exp( Bt )]dt  exp( Bt )dBt


(3) dX t  dt  dBt an arithmetic Brownian motion, can also be written X t  t  Bt
(since  is a constant). Consider a process defined by S t  S 0 exp( X t ) with S0 > 0.

St = f(Xt, t) where the function f:[0, T]   is defined by f(Xt, t) = S0exp(Xt).

            f x  S 0 exp( X t )        f xx  S 0 exp( X t )        ft  0
and
           dS t  [ S 0 exp( X t )    1 2  S 0 exp( X t ) 2 ]dt  S 0 exp( X t )    dBt
             [(  1 2   2 ) S t ]dt    S t  dBt
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) + t and
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 V
guarantees a flow of cash payments at the rate $1dt every instant of time forever. This is
a continuous-time equivalent of a risk-free perpetuity paying $1 each discrete period. If
the 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 Ito’s lemma. Here it is simple, because V does
not depend on any stochastic variable: V = V(t). Thus dV = Vtdt.

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.


                                                                                                       11
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 to
solve the equation for V some how. Here just guess that V cannot depend on t since it is
the value of a constant, risk-free perpetuity. Thus Vt = 0 and the solution is V = 1/r. This
is equivalent to the present value of a risk-free perpetuity in discrete time but here r is the
instantaneous risk free rate.

Now, suppose X follows a geometric Brownian motion: dX t  X t dt  X t dBt . A
security with value V collects Xdt continuously forever. V represents the current value
of a perpetuity whose continuous cash flow Xdt grows 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 is
the value of this security?

V = V(Xt, t) – however, just as above, since it’s perpetual (and so looks the same today as
it does in 2 years) V can’t actually depend on t only on X, so V = V(Xt). What is V(X)?

Using Ito’s lemma:
dV  [Vx    X  1 2  Vxx   2  X 2 ]  dt  Vx    X  dBt

The expected capital gain on this security is E[dV]
E[dV ]  [  X  Vx  1 2   2  X 2  Vxx ]  dt

The expected cash flow is Xdt by the construction of the security.

The total expected return on the security is the sum of these components:
Expected Total Return  [  X  Vx  1 2   2  X 2  Vxx  X ]  dt

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

rV  [  X  Vx  1 2   2  X 2  Vxx  X ]

To solve the differential equation, guess that doubling the current level of X (current cash
flow and so the expected future cash flow) will double V. Then V = X for some constant
. This implies that Vx = , and Vxx = 0. Substituting these into the differential equation
we get:
                                              1
        r    X      X  X or              and so
                                           (r   )
                 X
        V
              (r   )


                                                                                                 12
This is the equation for the present value of a perpetuity growing at a constant rate of 
when the riskless rate is r.


The Multi-Variate Case:

Now introduce a second Ito process Y to the system so we have:
dX t   ( X t , Yt , t )dt   ( X t , Yt , t )dBt1
dYt   ( X t , Yt , t )dt   ( X t , Yt , t )dBt2

Define E[dB1dB2] = dt as the correlation between B1 and B2. We can also show that
E[(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 Ito’s lemma has the
following Ito differential:

dWt  f x dX t  f y dYt  f t dt  1 2 [ f xx dX t  2 f xy dX t dYt  f yy dYt ]
                                                      2                        2




terms of higher order than dX2 or dY2 vanish as before.
Now we have to expand our list of multiplication rules:
(dB1 ) 2  (dB 2 ) 2  dt dB1dB 2  dt dB1dt  dB 2 dt  0 dt 2  0

Then
dX 2  (dt  dB 1 ) 2   2 dt
dY 2  ( dt  dB 2 ) 2   2 dt
dXdY  (dt  dB 1 )( dt  dB 2 )     dt

Making these substitutions into the Ito differential we have:
dW  [f x  f y  f t  1 2  2 f xx  f xy  1 2 2 f yy ]dt  f x dB1  f y dB 2

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

Application:
Let X and Y be two real valued Ito processes with:
dX  dt  dB 1
dY  dt  dB 2

Show that Z = XY is an Ito process, that is show that it can be written:
dZ  dt  dB


                                                                                             13
First note Z  f ( X , Y , t )  XY so f x  Y , f y  X , f t  0, f xx  0, f yy  0, and , f xy  1 .

Using Ito’s lemma
dZ  [f x  f y  f t  1 2  2 f xx   f xy  1 2 2 f yy ]dt  f x dB 1  f y dB 2
         [Y  X   ]dt  YdB1  XdB 2
    Y (dt  dB 1 )  X ( dt  dB 2 )   dt
    YdX  XdY   dt
And the result follows by simple substitution.


Stochastic Integrals and Trading Gains
Trading Strategies:
A trading strategy  is a stochastic process that specifies at each state and each date the
holdings of each of the available securities.

Formally: t :   N is a measurable function with respect to Ft (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  be
a trading strategy.

The gains of trading strategy  are given by the stochastic integral of  with respect to B:
T

  dB
0
    t     t   where B may be an N dimensional Brownian motion

This integral is stochastic since B is stochastic.


Let’s consider some simple examples:
(1) Suppose that    where  is a constant for all t, then:
T

  dB
0
    t     t     ( BT  B0 )


(2) Consider a piecewise constant strategy  on [0, T] so for some 0 = t0 < t1 < … < tk = T
t = (tm-1)  t  [tm-1, tm)  m = 1, …, k




                                                                                                       14
             t




                                                            t
                             0   t1 t2   t3            T




Then  is called a “simple” trading strategy and
T                 k

  t dBt   (t m1 )( Bt  Bt )
0             m 1
                                   m      m 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
        T                          
 Lim E   [ m (t )   (t )] 2 dt   0
 m 
        0                          

The stochastic integral   t dBt is then defined by

       T
      
                         T
                                      
                                      2
                                        
Lim E    m (t )dBt    (t )dBt    0
m 
       0
                        0            

Informally we can think of this as

T                        T

  dB
0
    t   t    " Lim   m (t )dBt "
                  m 
                         0




Trading Gains for Ito Prices
Let S be an Ito process (security price process) defined by:
S0  S0
dS t   t dt   t dBt


                                                                                           15
Then the gains of trading strategy  are given by the stochastic integral of  with respect
to S:
T           T           T

  dS     dt     dB
0
    t   t
            0
                t   t
                        0
                            t   t   t   an integral with one nonstochastic and one stochastic term.



The Black-Scholes Model
Review:
Binomial option pricing formula – single period
2 states of nature
3 securities
        1 – risky stock
        2 – riskless bond
        3 – call option on the stock
the idea is that you replicate the call option using the stock and the bond
a simple portfolio of the 2 securities spans the 2 states of nature

Binomial option pricing formula – multi (T) period
2T states of nature
3 securities
        1 – risky stock
        2 – riskless bond
        3 – call option on the stock
the idea is that you replicate the option with the stock and the bond at each date
2 long-lived securities and a dynamic trading strategy span the 2T states of nature


Black-Scholes Model (continuous time)
Infinite number of states of nature
3 securities
        1 – risky stock
        2 – riskless bond
        3 – call option on the stock
the idea is that you replicate the option with the stock and the bond using a continuous
trading strategy
2 long-lived securities and a continuous trading strategy span an infinite number of states
of nature

The Available Securities
Stock – the price of the risky stock is assumed to follow a geometric Brownian motion
S0  S0  0
dS t  S t dt  S t dBt




                                                                                                16
Bond – the price of the riskless bond follows the process
0  0  0
d t  r t dt
where r is the (assumed constant) instantaneous riskless rate of return.

We can derive this price process as follows. By definition, for any t,  t   0  e r t .
Differentiate this with respect to t:

d t
       0 e r t r  r t (constant exponential growth) then d t  r t dt
 dt

Option – a European call option that matures at time T and has an exercise price equal to
k. The price process of the European call option is denoted C, where C ( S t , 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 t in 0 < t < T. Let at be the holdings of the stock at time t and bt be the
holdings of the riskless bond. Then the trading strategy (a, b) is self financing if
                 t            t
a 0 S 0  b0  0   a dS   b d   at S t  bt  t for any time t in 0 < t < T.
                 0            0


Assume that C(St, t), the function relating the price of the call option to the stock price (or
the call price process), is twice continuously differentiable. Then, using Ito’s lemma

(1)     dCt  [Cs (St , t )St  Ct (St , t )  1 2 Css (St , t ) 2 St2 ]dt  Cs (St , t )St dBt


Claim
Suppose that there exists a self financing strategy (a, b) with
C(ST , T )  aT ST  bT  T (so the time T payoffs on the strategy and the call are the same)
then in the absence of arbitrage opportunities the current (t = 0) price of the European call
option on the stock is given by C(S 0 ,0)  a0 S 0  b0  0 .

Proof
Assume for contradiction that
C(S 0 ,0)  a0 S 0  b0  0 .
Then the trading strategy (a, b, -1) (long the strategy and short the call option) has a t = 0
payoff (cost) of C(S 0 ,0)  a0 S 0  b0  0  0 and the time T payoff of this position is zero
hence it is an arbitrage opportunity.

If we instead assume that
C(S 0 ,0)  a0 S 0  b0  0 .


                                                                                                     17
Then the trading strategy (-a, -b, 1) (short the strategy and long the call option) is an
arbitrage. So the replicating portfolio’s (trading strategy’s) initial cost must be the initial
option price if the strategy is self financing.

The same argument applied at each date t in the interval [0, T] implies that
(2)    C ( S t , t )  at S t  bt  t t  [0, T ]

It follows that
       dC t  at dS t  bt d t
                at ( S t dt  S t dBt )  bt r t dt
(3)         (at S t  bt r t )dt  at S t dBt
so Ct follows an Ito process as defined in (3)

Using (1) and (3) and matching coefficients in dBt we see it must be true that
C s ( S t , t )S t  at S t
or
 at  C s ( S t , t ) (the amount of the stock held at each time t is the option’s delta at that
time)

Now using (2)
     C ( S t , t )  at S t
bt 
               t
or
                      C (S t , t )  Cs (S t , t )S t
(4)            bt 
                                    t
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

Cs (St , t )St  Ct (St , t )  1 2 Css (St , t ) 2 St2  at St  bt r t

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

bt r t  Ct (S t , t )  1 2 Css (S t , t ) 2 S t2

and using (4) we find

 rC(St , t )  rCs (St , t )St  Ct (St , t )  1 2 Css (St , t ) 2 St2  0

This is the Black-Scholes partial differential equation. The boundary condition is of
course:
C ( ST , T )  ( ST  k )   max( ST  k ,0)


                                                                                                   18
This PDE can be solved in several ways – none of which we will pursue. By direct
calculation of the derivatives you can verify that the Black-Scholes formula satisfies the
PDE.

C ( S t , t )  S t N (d1 )  k exp( r (T  t )) N (d 2 )
where

    Ln
      
            St
                                  
                    r   2 (T  t )
                 k
d1                        2
                    T t
d 2  d1   T  t

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, the
call price is the stock price times the option delta less the discounted value of the exercise
price 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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