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2 Putting the Black–Scholes equation into historical perspective

2 Putting the Black–Scholes equation into historical perspective

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102



Part One mathematical and financial foundations



• chemical reactions, such as the Belousov–Zhabotinsky reaction which exhibits fascinating

wave structure

• electrical activity in the membranes of living organisms, such as the Hodgkin–Huxley model

• dispersion of populations, such as individuals moving both randomly and to avoid overcrowding

• pursuit and evasion in predator-prey systems

• pattern formation in animal coats, such as the formation of zebra stripes

• dispersion of pollutants in a running stream

In most of these cases the resulting equations are more complicated than the Black–Scholes

equation.

The simplest heat equation for the temperature in a bar is usually written in the form

∂u

∂ 2u

= 2

∂t

∂x



(6.1)



where u is the temperature, x is a spatial coordinate and t is time. This equation comes from

a heat balance. Consider the flow into and out of a small section of the bar. The flow of heat

along the bar is proportional to the spatial gradient of the temperature

∂u

∂x

and thus the derivative of this, the second derivative of the temperature, ∂ 2u/∂x 2 , is the heat

retained by the small section. This retained heat is seen as a rise in the temperature, represented

mathematically by

∂u

.

∂t

The balance of the second x-derivative and the first t-derivative results in the heat equation,

Equation (6.1). (There would be a coefficient in the equation, depending on the properties of

the bar, but I have set this to one.)



6.3 THE MEANING OF THE TERMS IN THE BLACK–SCHOLES

EQUATION

The Black–Scholes equation can be accurately interpreted as a reaction-convection-diffusion

equation.

The basic diffusion equation is a balance of a first-order t derivative and a second-order S

derivative:

∂V

∂ 2V

+ 12 σ 2 S 2 2 .

∂t

∂S

If these were the only terms in the Black–Scholes equation it would still exhibit the smoothing-out effect, that any discontinuities in the payoff would be instantly diffused away. The



partial differential equations Chapter 6



only difference between these terms and the terms as they appear in the basic heat or diffusion

equation, is that the diffusion coefficient is a function of one of the variables S. Thus we really

have diffusion in a nonhomogeneous medium.

The first-order S-derivative term

rS



∂V

∂S



can be thought of as a convection term. If this equation represented some physical system, such

as the diffusion of smoke particles in the atmosphere, then the convective term would be due

to a breeze, blowing the smoke in a preferred direction.

The final term

−rV

is a reaction term. Balancing this term and the time derivative would give a model for decay

of a radioactive body, with the half-life being related to r.

Figure 6.1 shows how a call option payoff would evolve in three cases: (i) diffusion only,

(ii) convection only and (iii) reaction only.

Putting these terms together we get a reaction-convection-diffusion equation. An almost

identical equation would be arrived at for the dispersion of pollutant along a flowing river with

absorption by the sand. In this, the dispersion is the diffusion, the flow is the convection, and

the absorption is the reaction.



140

120

Diffusion



100



Convection



Value



Reaction

80

60

40

20

0

0



50



100



150



200



Stock



Figure 6.1 Call option value before expiration: (i) diffusion only, (ii) convection only and

(iii) reaction only.



103



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Part One mathematical and financial foundations



6.4 BOUNDARY AND INITIAL/FINAL CONDITIONS

To specify a problem uniquely we must prescribe boundary conditions and an initial or final

condition. Boundary conditions tell us how the solution must behave for all time at certain

values of the asset. In financial problems we usually specify the behavior of the solution at S = 0

and as S → ∞. We must also tell the problem how the solution begins. The Black–Scholes

equation is a backward equation, meaning that the signs of the t derivative and the second S

derivative in the equation are the same when written on the same side of the equals sign. We

therefore have to impose a final condition. This is usually the payoff function at expiry.

The Black–Scholes equation in its basic form is linear and homogeneous, and therefore

satisfies the superposition principle; add together two solutions of the equation and you will

get a third. This is not true of non-linear equations. Linear diffusion equations have some very

nice properties. Even if we start out with a discontinuity in the final data, due to a discontinuity

in the payoff, this immediately gets smoothed out; this is due to the diffusive nature of the

equation. Another nice property is the uniqueness of the solution. Provided that the solution

is not allowed to grow too fast as S tends to infinity the solution will be unique. This precise

definition of ‘too fast’ need not worry us, as we will not have to worry about uniqueness for

any problems we encounter.



6.5 SOME SOLUTION METHODS

We are not going to spend much time on the exact solution of the Black–Scholes equation.

Such a solution is important, but current market practice is such that models have features

which preclude the exact solution. The few explicit, closed-form solutions that are used by

practitioners will be covered in the next two chapters.

6.5.1 Transformation to Constant Coefficient Diffusion Equation



It can sometimes be useful to transform the basic Black–Scholes equation into something a

little bit simpler by a change of variables. If we write

V (S, t) = eαx+βτ U (x, τ ),

where

α = − 12



2r

− 1 , β = − 14

σ2



2r

+1

σ2



2



, S = ex and t = T −





,

σ2



then U (x, τ ) satisfies the basic diffusion equation

∂U

∂ 2U

.

=

∂τ

∂x 2



(6.2)



This simpler equation is easier to handle than the Black–Scholes equation. Sometimes that

can be important, for example when seeking closed-form solutions, or in some simple numerical

schemes. We shall not pursue this any further.



partial differential equations Chapter 6



6.5.2



Green’s Functions



One solution of the Black–Scholes equation is

V (S, t) =



e−r(T −t)





e

σ S 2π (T − t)



2

1

log(S/S )+(r− 2 σ 2 )(T −t) /2σ 2 (T −t)



(6.3)



for any S . (You can verify this by substituting back into the equation, but we’ll also be seeing

it derived in the next chapter.) This solution is special because as t → T it becomes zero

everywhere, except at S = S . In this limit the function becomes what is known as a Dirac

delta function. Think of this as a function that is zero everywhere except at one point where

it is infinite, in such a way that its integral is one. How is this be of help to us?

Expression (6.3) is a solution of the Black–Scholes equation for any S . Because of the

superposition principle we can multiply (6.3) by any constant, and we get another solution. But

then we can also get another solution by adding together expressions of the form (6.3) but with

different values for S . Putting this together, and thinking of an integral as just a way of adding

together many solutions, we find that

e−r(T −t)



σ 2π (T − t)







e



2

1

− log(S/S )+(r− 2 σ 2 )(T −t) /2σ 2 (T −t)

f (S



0



)



dS

S



is also a solution of the Black–Scholes equation for any function f (S ). (If you don’t believe

me, substitute it into the Black–Scholes equation.)

Because of the nature of the integrand as t → T (i.e. that it is zero everywhere except at S

and has integral one), if we choose the arbitrary function f (S ) to be the payoff function then

this expression becomes the solution of the problem:

V (S, t) =



e−r(T −t)



σ 2π(T − t)







e



2

1

− log(S/S )+(r− 2 σ 2 )(T −t) /2σ 2 (T −t)



0



Payoff(S )



dS

.

S



The function V (S, t) given by (6.3) is called the Green’s function.

6.5.3



Series Solution



Sometimes we have boundary conditions at two finite (and non-zero) values of S, Su and Sd ,

say (we see examples in Chapter 23). For this type of problem, we postulate that the required

solution of the Black–Scholes equation can be written as an infinite sum of special functions.

First of all, transform to the nicer basic diffusion equation in x and τ . Now write the solution as





ai (τ ) sin(iωx) + bi (τ ) cos(iωx),



eαx+βτ

i=0



for some ω and some functions a and b to be found. The linearity of the equation suggests that

a sum of solutions might be appropriate. If this is to satisfy the Black–Scholes equation then

we must have

dai

dbi

= −i 2 ω2 ai (τ ) and

= −i 2 ω2 bi (τ ).







105



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Part One mathematical and financial foundations



You can easily show this by substitution. The solutions are thus

ai (τ ) = Ai e−i



2 ω2 τ



and bi (τ ) = Bi e−i



2 ω2 τ



.



The solution of the Black–Scholes equation is therefore





e



e−i



αx+βτ



2 ω2 τ



(Ai sin(iωx) + Bi cos(iωx)) .



(6.4)



i=0



We have solved the equation; all that we need to do now is to satisfy boundary and initial

conditions.

Consider the example where the payoff at time τ = 0 is f (x) (although it would be expressed

in the original variables, of course) but the contract becomes worthless if ever x = xd or

x = xu .1

Rewrite the term in large brackets in (6.4) as

Ci sin iω



x − xd

xu − xd



+ Di cos iω



x − xd

xu − xd



.



To ensure that the option is worthless on these two x values, choose Di = 0 and ω = π. The

boundary conditions are thereby satisfied. All that remains is to choose the Ci to satisfy the

final condition:





eαx



Ci sin iω

i=0



x − xd

xu − xd



= f (x).



This also is simple. Multiplying both sides by

sin j ω



x − xd

xu − xd



,



and integrating between xd and xu we find that

Cj =



2

xu − xd



xu



f (x)e−αx sin j ω



xd



x − xd

xu − xd



dx.



This technique, which can be generalized, is the Fourier series method. There are some

problems with the method if you are trying to represent a discontinuous function with a sum

of trigonometrical functions. The oscillatory nature of an approximate solution with a finite

number of terms is known as Gibbs phenomenon.



6.6 SIMILARITY REDUCTIONS

Apart from the Green’s function, we’re not going to use any of the above techniques in this

book; rarely will we even find explicit solutions. But one technique that we will find useful is

the similarity reduction. I will demonstrate the idea using the simple diffusion equation; we

will later use it in many other, more complicated problems.

1



This is an example of a double knock-out option, see Chapter 23.



partial differential equations Chapter 6



The basic diffusion equation

∂u

∂ 2u

= 2

∂t

∂x



(6.5)



is an equation for the function u which depends on the two variables x and t. Sometimes, in

very, very special cases we can write the solution as a function of just one variable. Let me

give an example.

The function

x/t 1/2



u(x, t) =



1 2



e− 4 ξ dξ



0



satisfies (6.5). This is easy to verify. But in this function x and t only appear in the combination

x

t 1/2



.



Thus, in a sense, u is a function of only one variable.

A slight generalization, but also demonstrating the idea of similarity solutions, is to look for

a solution of the form

u = t −1/2 f (ξ )



(6.6)



where

ξ=



x

t 1/2



.



Substitute (6.6) into (6.5) to find that a solution for f is

1 2



f = e− 4 ξ ,

so that

t −1/2 e−(1/4)(x



2 /t)



is also a special solution of the diffusion equation.

Be warned, though. You can’t always find similarity solutions; not only must the equation

have a particularly nice structure but also the similarity form must be consistent with any initial

condition or boundary conditions.



6.7



OTHER ANALYTICAL TECHNIQUES



The other two main solution techniques for linear partial differential equations are Fourier and

Laplace transforms. These are such large and highly technical subjects that I really cannot begin

to give an idea of how they work, space is far too short. Recently these techniques have become

useful in some higher dimensional models such as those incorporating stochastic volatility.



107



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Part One mathematical and financial foundations



6.8 NUMERICAL SOLUTION

Even though there are several techniques that we can use for finding solutions, in the vast

majority of cases we must solve the Black–Scholes equation numerically. But we are lucky.

Parabolic differential equations are just about the easiest equations to solve numerically. Obviously, there are any number of really sophisticated techniques, but if you stick with the simplest

then you can’t go far wrong. In Chapters 15, 77 and 78 we discuss these methods in detail. I

want to stress that I am going to derive many partial differential equations from now on, and I

am going to assume you trust me that we will at the end of the book see how to solve them.



6.9 SUMMARY

This short chapter is only intended as a primer on partial differential equations. If you want to

study this subject in depth, see the books and articles mentioned below.



FURTHER READING

• Grindrod (1991) is all about reaction-diffusion equations, where they come from and their

analysis. The book includes many of the physical models described above.

• Murray (1989) also contains a great deal on reaction-diffusion equations, but concentrates

on models of biological systems.

• Wilmott & Wilmott (1990) describe the diffusion of pollutant along a river with convection

and absorption by the river bed.

• The classical reference works for diffusion equations are Crank (1989) and Carslaw &

Jaeger (1989). But also see the book on partial differential equations by Sneddon (1957)

and the book on general applied mathematical methods by Strang (1986).



CHAPTER 7



the Black–Scholes

formulae and the

‘greeks’

In this Chapter. . .











7.1



the derivation of the Black–Scholes formulae for calls, puts and simple digitals

the meaning and importance of the ‘greeks,’ delta, gamma, theta, vega and rho

the difference between differentiation with respect to variables and to parameters

formulae for the greeks for calls, puts and simple digitals

INTRODUCTION



The Black–Scholes equation has simple solutions for calls, puts and some other contracts. In

this chapter I’m going to walk you through the derivation of these formulae step by step. This

is one of the few places in the book where I do derive formulae. The reason that I don’t

often derive formulae is that the majority of contracts do not have explicit solutions for their

theoretical value. Instead much of my emphasis will be placed on finding numerical solutions

of the Black–Scholes equation.

We’ve seen how the quantity ‘delta,’ the first derivative of the option value with respect to

the underlying, occurs as an important quantity in the derivation of the Black–Scholes equation.

In this chapter I describe the importance of other derivatives of the option price, with respect

to the variables (the underlying asset and time) and with respect to some of the parameters.

These derivatives are important in the hedging of an option position, playing key roles in risk

management. It can be argued that it is more important to get the hedging correct than to be

precise in the pricing of a contract. The reason for this is that if you are accurate in your hedging

you will have reduced or eliminated future uncertainty. This leaves you with a profit (or loss)

that is set the moment that you buy or sell the contract. But if your hedging is inaccurate, then it

doesn’t matter, within reason, what you sold the contract for initially; future uncertainty could

easily dominate any initial profit. Of course, life is not so simple, in reality we are exposed

to model error, which can make a mockery of anything we do. However, this illustrates the

importance of good hedging, and that’s where the ‘greeks’ come in.



110



Part One mathematical and financial foundations



7.2 DERIVATION OF THE FORMULAE FOR CALLS, PUTS

AND SIMPLE DIGITALS

The Black–Scholes equation is

∂V

∂ 2V

∂V

+ 12 σ 2 S 2 2 + rS

− rV = 0.

∂t

∂S

∂S



(7.1)



This equation must be solved with final condition depending on the payoff; each contract will

have a different functional form prescribed at expiry t = T , depending on whether it is a call,

a put or something more fancy. This is the final condition that must be imposed to make

the solution unique. We’ll worry about final conditions later, for the moment concentrate on

manipulating (7.1) into something we can easily solve.

The first step in the manipulation is to change from present value to future value terms.

Recalling that the payoff is received at time T but that we are valuing the option at time t this

suggests that we write

V (S, t) = e−r(T −t) U (S, t).

This takes our differential equation to

∂U

∂U

∂ 2U

+ 12 σ 2 S 2 2 + rS

= 0.

∂t

∂S

∂S

The second step is really trivial. Because we are solving a backward equation, discussed in

Chapter 6, we’ll write

τ = T − t.

This now takes our equation to

∂U

∂U

∂ 2U

= 12 σ 2 S 2 2 + rS .

∂τ

∂S

∂S

When we first started modeling equity prices we used intuition about the asset price return

to build up the stochastic differential equation model. Let’s go back to examine the return and

write

ξ = log S.

With this as the new variable, we find that





= e−ξ

∂S

∂ξ



and



2



∂2

−2ξ ∂

=

e

− e−2ξ .

∂S 2

∂ξ

∂ξ 2



Now the Black–Scholes equation becomes

∂ 2U

∂U

∂U

= 12 σ 2 2 + r − 12 σ 2

.

∂τ

∂ξ

∂ξ

What has this done for us? It has taken the problem defined for 0 ≤ S < ∞ to one defined

for −∞ < ξ < ∞. But more importantly, the coefficients in the equation are now all constant,



the Black–Scholes formulae and the ‘greeks’ Chapter 7



independent of the underlying. This is a big step forward, made possible by the lognormality

of the underlying asset. We are nearly there.

The last step is simple, but the motivation is not so obvious. Write

x = ξ + r − 12 σ 2 τ ,

and U = W (x, τ ). This is just a ‘translation’ of the coordinate system. It’s a bit like using the

forward price of the asset instead of the spot price as a variable. After this change of variables

the Black–Scholes becomes the simpler

∂ 2W

∂W

= 12 σ 2 2 .

∂τ

∂x



(7.2)



To summarize,

V (S, t) = e−r(T −t) U (S, t) = e−rτ U (S, T − τ ) = e−rτ U (eξ , T − τ )

= e−rτ U e



1

x− r− 2 σ 2 τ

,T



−τ



= e−rτ W (x, τ ).



To those of you who already know the Black–Scholes formulae for calls and puts the variable

x will ring a bell:

x = ξ + r − 12 σ 2 τ = log S + r − 12 σ 2 (T − t).

Having turned the original Black–Scholes equation into something much simpler, let’s take

a break for a moment while I explain where we are headed.

I’m going to derive an expression for the value of any option whose payoff is a known

function of the asset price at expiry. This includes calls, puts and digitals. This expression will

be in the form of an integral. For special cases, I’ll show how to rewrite this integral in terms

of the cumulative distribution function for the Normal distribution. This is particularly useful

since the function can be found on spreadsheets, calculators and in the backs of books. But

there are two steps before I can write down this integral.

The first step is to find a special solution of (7.2), called the fundamental solution. This

solution has useful properties. The second step is to use the linearity of the equation and the

useful properties of the special solution to find the general solution of the equation. Here we go.

I’m going to look for a special solution of (7.2) of the following form

W (x, τ ) = τ α f



(x − x )

,

τβ



(7.3)



where x is an arbitrary constant. And I’ll call this special solution Wf (x, τ ; x ). Note that the

unknown function depends on only one variable (x − x )/τ β . As well as finding the function f

we must find the constant parameters α and β. We can expect that if this approach works, the

equation for f will be an ordinary differential equation since the function only has one variable.

This reduction of dimension is an example of a similarity reduction, discussed in Chapter 6.

Substituting expression (7.3) into (7.2) we get

τ α−1 αf − βη



df





= 12 σ 2 τ α−2β



d 2f

,

dη2



(7.4)



111



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