Stochastic Calculus

23 Stochastic Calculus i The last couple of chapters were heavily mathematical with not much reference to option theory.. Brownian motion was investigated in some detail and we developed

23 Stochastic Calculus

(i) The last couple of chapters were heavily mathematical with not much reference to option theory Brownian motion was investigated in some detail and we developed a form of calculus

0 a t dWt, which was constructed to be a martingale, and we derived the following three explicit results from first principles:

0

0

W T dWt = 1

2

T −1

2T ;

0

The second and third results were rather surprising and reflect the fact that the quadratic

varia-tion of Brownian movaria-tion is equal to T, and not zero as it would be for a differentiable funcvaria-tion.

(ii) The reader will be disappointed (or perhaps relieved) to learn that we cannot go very much further in deriving explicit integrals In classical calculus, virtually any continuous function can

Riemann integration can be equated to reverse differentiation, so that a large library of standard integrals has been established Differentiation with respect to time has no meaning in stochastic calculus, so this approach is not available The reader’s first reaction to this news must be to wonder whether it was worth plowing through all the stuff in the last two chapters just to derive

a calculus which is so puny that it can only manage three integrals But thanks to Ito’s lemma which is discussed next, some powerful calculation techniques do emerge

(iii) This brings us to an important definitional point: the whole motivation for these chapters

a(S t , t)δt + b(S t , t)δW t Presumably, in the limit of infinitesimal time intervals, this could

be written as the differential equation dSt = a(St , t)dt + b(S t , t)dW t The reader might have

noticed that books on stochastic theory (including this one) have sections entitled Stochastic Differential Equations, which deal with equations of this type Yet in the last chapter it was emphasized that differential calculus does not apply to Brownian motion; so what’s going on?

Let us ignore the atδt term for the moment, as this is not where the difficulty arises, and

(C) You can certainly rewrite this relationship as ST − S0=0T

d S t =0T

b t dWt We have, after all, just spent a chapter defining exactly what this integral means

0

T

b t dWt =T

0 b t dW t

dt dt.

The punch-line is that when we see the differential equation dSt = at dt + bt dWt, what is

really meant is

S T − S0=



0

T

a t dt+



0

T

b t dWt

where the first integral is a Riemann integral and the second integral is the Ito integral which was defined in Chapter 22 The differential form is mere shorthand and should immediately

be hidden if a serious mathematician drops by The justification for this shorthand is that first,

it is a simple and intuitive representation of a process and second, everybody else does it In

this spirit of imprecision, we can state that dSt = bt dWtis a martingale

(iv) The next section uses the properties of differentials extensively, so at the risk of belaboring the obvious, it is worth reviewing when differential calculus can be used and when not A

stock price St is stochastic, as is the price of the derivative of the stock f (St) But despite

the fact that they are both stochastic, f (St ) is a well-behaved, differentiable function of St In fact,∂ f (S t) /∂ S t, is just the delta of the derivative Similarly,∂ f (S t) /∂t is well defined, even

the partial derivatives of f (St ) with respect to both St and t are meaningful, dSt /dt does not

make the grade It is impossible to attach a meaning to this when we have no idea whether the

next move in St will be up or down, or by how much Similarly, d f (St)/dt is meaningless; this

seems a little surprising since the partial derivative was well behaved, but remember that the

total derivative does not hold Stconstant over the infinitesimal time interval Finally, although

d f (St , t)/dt is not allowed, a close relative defined by

A f (S t , t) = lim

δt→0

E[ f (St + δS, t + δt) | Ft]− f (St , t)

δt

does have a respectable place in stochastic calculus We revisit this in Section 23.8

(i) In general, a small increment in the price of a derivative may be given by a Taylor expansion

as follows:

δ f (St , t) = ∂ f t

∂t δt +

∂ f t

∂ S t

2

∂2f t

∂ S2

t

(δSt)2

2

∂2f t

∂t2 (δt)2+1

2

∂2f t

∂ S∂t(δSt)(δt) + · · ·

δt or δWt However, if the stock price can be written dSt = a(St , t) dt + b(S t , t) dW t, then the

in its integral form

0

A t (dWt)2= lim

δN→∞; δt→0

N

i=1

A i (Wi − Wi−1)2 This is Brownian quadratic variation, which unlike an analytic quadratic variation, does not

vanish to zero in the limit (dWt)2is just not small enough to ignore in the Taylor expansion,

260

23.3 STOCHASTIC INTEGRATION

which was explained in Section 22.2(ii) Our Taylor expansion was of course written in terms of (δSt)2, which leads to additional terms a2(dt)2and ab dSt dt, but these can be safely dropped

as they are O[(δt)2] and O[(δt)3/2].

(ii) Ito’s Lemma: The arguments in the last section have been couched in terms of a derivative

which is a function of a stock price The conclusions apply more generally to any function of

a Brownian motion For future reference, the results can be stated as follows

If a stochastic variable, driven by a Brownian motion, follows the process

dxt = a (xt , t) dt + b (x t , t) dW t

either differential or integral form:

d ft = ∂ f t

∂t dt+

∂ f t

∂x t

2

∂2f t

∂x2

t (dxt)2

= ∂ f t

∂t dt+

∂ f t

∂x t

2b

2

t

∂2f t

∂x2

t dt

f T − f0=

0

∂ f t

∂t + at

∂ f t

∂x t

2b

2

t

∂2f t

∂x2

t



0

b t

∂ f t

∂x t

Remember that from the definition of an Ito integral, the last term of this second equation is a martingale

f T − f0=

0

1 2

∂2f t

∂W2

t

0

∂ f t

∂W t

At the beginning of this chapter it was observed that a stochastic integral cannot be considered the reverse of a stochastic differential with respect to time, simply because the latter does not exist The result is that stochastic calculus can never build up the battery of standard integrals possessed by analytical calculus In fact, the store of standard results is so poor that any insights are gratefully received Ito’s lemma confirms in a very simple manner a couple

of the results we derived from first principles and gives us a procedure for integrating by parts

f T − f0= Wt =

0

dWt

which is the simplest integral, derived from first principles in Section 22.2(i)

261

(ii) A slightly more complex integral, derived in Section 22.2(iii), is obtained by putting ft = W2

t Again, substituting this in equation (23.3) gives

f T − f0 = W2

0

0

or

0

W t dWt =1

2

T−1

2T

equation (23.2) becomes

f T − f0= WT g(t)=

0

x t ∂g(t)

∂t dt+

0

g(t) dW t

which immediately gives a stochastic form of integration by parts

0

g(t) dW t = WT g(T )−

0

∂g(t)

(i) The simplest stochastic differential equation (SDE) of interest in option theory has constant coefficients:

which may be simply integrated to give

From this very simple expression for xT, it is clear that

(ii) Stock Price Distribution: The most frequently used SDE for a stock price movement, which

underlies Black Scholes analysis, is the following:

f T − f0− lnS T

S0 =

0

µ −1

2σ2



0

σ dW t =

µ −1

2σ2



or

S T = S0e(µ−1σ2)T +σ W T

found explicitly in Section 3.2 by plugging in the explicit normal distribution and slogging through the integral We are now able to achieve the same result with a lighter touch by using

262

23.4 STOCHASTIC DIFFERENTIAL EQUATIONS

Ito’s lemma From the last result

E



S T

S0



y T − y0=

0

1

2σ2y t dt+

0

σ y t dWt

Both of the integrals in this equation contain random variables Take the expectation at time

integral is zero (martingale property):

Y T − Y0=

0

1

2σ2Y t dt

The random variables have been eliminated from this equation by taking time zero expectations;

Substituting this solution back in equation (23.5) gives

E



S T

S0



= eµ T

Precisely the same technique, using an intermediate variable, allows us to write

E



S T

S0

2

= e2(µ−1σ2)T

E[e2σ W T]= e(2µ+σ )T

giving a variance

var



S T

S0



= E

#

S T

S0

2$

− E2



S T

S0



= e2µT(eσ2T− 1)

var[ln ST /S0]= σ2T precisely We may, however, make the approximation var[S t /S0]≈ σ2δt

for smallδt, by expanding the full expression to first order in δt.

S T = S0exp

0

µ(S t , t) − 1

2σ2(St , t)



0

σ(S t , t) dW t



(iv) An Interesting Martingale: As a further exercise and because we need the result in the next

chapter, consider the process

2φ2

t dt − φt dWt

Defineξ t = ex t and use Ito’s lemma to give

ξ T − ξ0= −

0

φ t ξ t dWt

263

or in differential shorthand

dξ t

ξ t

clearly,ξ t is a martingale

(v) Ornstein–Uhlenbeck Process: This process, which is of interest in the study of interest rates,

has the following SDE:

The stochastic term is the same as before, but the drift term is more interesting: the negative

sign and the proportionality to xtmeans that the larger this term becomes, the larger the effect

While it is observed in finance that a stock price is usually well described by Brownian motion, interest rates usually move within a fairly narrow band We don’t often come across interest rates of 50% (at least in markets where we want to do derivatives), but we often see stock prices that start at $10 and after a few years have reached $100 Interest rates are

assumed to display mean reversion They do not of course mean revert to zero (as implied by

the Ornstein–Uhlenbeck process), but we stick with this most basic process for simplicity of exposition

Let’s try out the function ft = xteat Ito’s lemma then gives

f T − f0= xTeaT − x0 =

0

σ e at dWt

or

x T = x0e−aT+ e−aT σ

0

eat dWt

We are not able to solve the integral explicitly, but we can nevertheless obtain some useful results The integral is a martingale, so taking expectations of the last equation and of its square gives

E



x T

x0



E



x T

x0

2

= e−2aT+e−2aT σ2

x2 0

E

T

0

eat dWt

2$

The cross term in this last equation has disappeared on taking expectations Substituting for the squared integral from equation (22.7) gives

E

#

x T

x0

2$

x2 0

0

e2at dt



var



x T

x0



= E

#

x T

x0

2$

− E2



x T

x0



2ax02{1 − e−2aT}

264

23.5 PARTIAL DIFFERENTIAL EQUATIONS

By now, the reader has probably thought to himself that this stochastic calculus is all very well, but there is not much in the way of concrete answers (i.e numbers) to real problems One of the main bridges between the rather abstract theory and “answers” is the relationship between stochastic differential equations and certain non-stochastic partial differential equa-tions (PDEs) Partial differential equaequa-tions may be hard to solve analytically, but they can be forced to yield tangible results using numerical methods

(i) Feynman–Kac Theorem: The basic trick in deriving the PDEs relies very simply on Ito’s

equation is dxt = a(xt , t) dt + b(x t , t) dW t Ito’s formula [equation (23.2)] gives the process

for Mt, and this is a martingale if and only if the drift term (the integral with respect to t) is zero This implies that

∂ M t

∂ M t

∂x t

2b(x t , t) ∂2M t

∂x2

t

The PDE approach consists of setting up functions which are martingales and then using Ito’s lemma to obtain PDEs for these functions

(ii) The first and most obvious choice for a martingale on which to try out this method is the

discounted derivative price f t∗ = B−1

t f t Substituting f t∗for Mt in equation (23.6) gives the

following PDE for ft:

∂ f t

∂t + a(xt , t)

∂ f t

∂x t

2b(x t , t) ∂2f t

∂x2

t

t

∂ B t

∂t f t Take the Black Scholes case where xt = St , a(x t , t) = r S t , b(x t , t) = σ S t and Bt = er t The last equation then simply becomes the Black Scholes partial differential equation which was

∂ f t

∂ f t

∂ S t

2σ2S t2∂2f t

∂ S2

t

= r ft

deferred until the next chapter; but all this will be obvious to anyone who is familiar with the risk-neutrality arguments of Chapter 4 This very slick derivation of the Black Scholes equation is shown here in order to demonstrate the power of the PDE approach to martingales

Scholes equation in the last section, where should we go for the next martingale? It turns out that there exists a machine for cranking out martingales on demand

We are used to making the distinction between a

random variable xtand its expected value Ext , which

is not a random variable But expectations can be

constructed in such a way that they are also random

variables: suppose xt is a stochastic process which we are anticipating at time 0 Then E [ xt | F0]

and E [ xT | F0] are clearly not random variables; but E [ xT | F t] definitely is a random variable

265

Define the function f (xt , t) = E[φ(x T)| Ft] where φ is a well-behaved function of x T From this definition and the tower property we have

In other words, your best guess of what your best guess will be in the future has to be the same

as your best guess now Rather obvious perhaps, but it does generate more candidates for the partial differential equation of the last paragraph! An important application of this principle is given next

(iv) Kolmogorov Backward Equation: The general process from which the PDE was constructed

was dxt = a(xt , t) dt + b(x t , t) dW t In terms of the probability distributions of classical

statis-tics, the conditional expectations of the last subsection may be written

f (x t , t) = Eφ(x T , T ) | F t =



all x T

φ(x T , T )F(x T , x t; t) dxT

where F(xT , x t ; t) is the probability density function Since F(xT , x t ; t) is the only part of the

∂ f (x t , t)

∂x t

=



φ ∂ F

∂x t dxt; ∂2f (x t , t)

∂x2

t

=



φ ∂2F

∂x2

t dxt; ∂ f (x t , t)

∂t =



φ ∂ F

∂τ dxt

Substituting this back into equation (23.6) immediately gives the backward equation, which was derived using other techniques in Appendix A.3:

∂ F(x t , t)

∂ F(x t , t)

∂x t

2b(x t , t)2∂2F (x t , t)

∂x2

t

The material in this section is used to analyze the stop-go paradox in Section 25.3 and may be omitted until then

stretching things rather, as one of the preconditions of Ito’s lemma is that the function should

second differentials of this function can be defined in terms of Heaviside functions and Dirac delta functions, as shown in equations A.7(ii) and (iii) of the Appendix The simplified form

of Ito’s lemma [equation (23.3)] is

f T − f0=

0

∂ f t

∂W t

2

0

∂2f t

∂W2

t

∂W t

= 1[X <W t <∞]; ∂2f t

∂W2

t

= lim

ε→0

1

2ε1[X−ε<W t <X+ε] = δ(Wt − X)

(ii) The first integral in equation (23.9) appears at first sight to be an adequate representation of the left-hand side of the equation Does this mean that the second integral, which comes from

266

23.6 LOCAL TIME

the quadratic variation term of Ito’s lemma, is identically equal to zero? Let us write

L T (X , ε) =

0

1

2ε1[X−ε<W t <X+ε] dt (23.10)

X + e

t 1

W t

t

X − e

X

t 2

t 4

t 3

During those periods when X − ε < Wt < X + ε, the integrand is just equal to unity; outside

this range, it is equal to zero The effect of the integration is therefore to add up all those time

Asε → 0 we expect each of the time periods τ ito shrink to zero On the face of it, we might

Brownian motion which we described in Section 21.1: as soon as a Brownian path achieves

shrinks to zero, there are an infinite number of them It may be formally shown that in the limit

ε → 0, L T (X , ε) is well defined, unique and non-zero, although the proof goes a bit beyond

the scope of this chapter

0

1[X−dX/2<W t <X+dX/2] dt

spent by a Brownian path between a and b as

a

0

1[a <W t <b] dt

vicinity of X It is called the local time of the Brownian motion It might save the reader some

time in the future if he notes that about half the literature uses the notation local time= LT (X ),

while the other half uses local time= 2L T (X ).

267

(iv) Using the Dirac delta function representation above, local time may alternatively be written as

L T (X )=

0

If h(X ) is any reasonable function of X, we can write

−∞ h(X )L T (X )dX =

−∞ h(X )

0

=

0

where we have made the heroic, but as it happens perfectly valid, assumption that we can switch the order of integration

equa-tion (23.9) becomes

0

1[X <W t <∞] dWt+1

2L T (X )

0

1[−∞<W t <X] dWt+1

2L T (X ) Adding the last two equations together gives the result

0

where

The literature rather loosely refers to any of the last three equations as Tanaka’s formula (vi) The local time results derived for simple Brownian motion can be generalized to the

and unsurprisingly yields

0

1[X < x t < ∞] dxt+ lim

ε→0

1 2

0

b2t 1

2ε1[X −ε < x t < X+ε] dt

The last term again results from the quadratic variation term of Ito’s lemma and is interpreted

as a generalized local time It is written as12 T (X ) and is subject to the same lack of notational

2 T (X ) for the same function.

(vii) Using the Dirac delta function notation

 T (X )=

0

268