2 Putting the Black–Scholes equation into historical perspective
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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.
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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 −
2τ
,
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 ).
dτ
dτ
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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.
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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.
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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
dη
= 12 σ 2 τ α−2β
d 2f
,
dη2
(7.4)
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