<< стр. 3(всего 3)СОДЕРЖАНИЕ
в€‚t в€‚x

Note that if we were able to solve these equations, this would provide the
transition density function p, giving the conditional distribution of the process.
It does not immediately provide other characteristics of the diп¬Ђusion, such as
the distribution of the maximum or the minimum, important for valuing various
exotic options such as look-back and barrier options. However for a European
76 CHAPTER 2. SOME BASIC THEORY OF FINANCE

option deп¬Ѓned on this process, knowledge of the transition density would suп¬ѓce
at least theoretically for valuing the option. Unfortunately these equations are
often very diп¬ѓcult to solve explicitly.
Besides the Kolmogorov equations, we can use simple ordinary diп¬Ђerential
equations to arrive at some of the basic properties of a diп¬Ђusion. To illustrate,
consider one of the simplest possible forms of a diп¬Ђusion, where a(Xt , t) =
О±(t)+ОІ(t)Xt where the coeп¬ѓcients О±(t), ОІ(t) are deterministic (i.e. non-random)
functions of time. Note that the integral analogue of 2.26 is
Z Z
t t
(2.29)
Xt = X0 + a(Xs , s)ds + Пѓ(Xs , s)dWs
0 0
Rt
and by construction that last term Пѓ(Xs , s)dWs is a zero-mean martingale.
0

For example its small increments Пѓ(Xt , t)dWs are approximately N (0, Пѓ(Xt , t)dt).
Therefore, taking expectations on both sides conditional on the value of X0 , and
letting m(t) = E(Xt ), we obtain:
Z t
(2.30)
m(t) = X0 + [О±(s) + ОІ(s)m(s)]ds
0

and therefore m(t)solves the ordinary diп¬Ђerential equation

m0 (t) = О±(t) + ОІ(t)m(t). (2.31)

(2.32)
m(0) = X0

Thus, in the case that the drift term a is a linear function of Xt , the mean or
expected value of a diп¬Ђusion process can be found by solving a similar ordinary
diп¬Ђerential equation, similar except that the diп¬Ђusion term has been dropped.
These are only two of many reasons to wish to solve both ordinary and
partial diп¬Ђerential equations in п¬Ѓnance. The solution to the Kolmogorov partial
diп¬Ђerential equations provides the conditional distribution of the increments of
a process. And when the drift term a(Xt , t ) is linear in Xt , the solution of an
ordinary diп¬Ђerential equation will allow the calculation of the expected value of
the process and this is the п¬Ѓrst and most basic description of its behaviour. The
MODELS IN CONTINUOUS TIME 77

appendix provides an elementary review of techniques for solving partial and
ordinary diп¬Ђerential equations.
However, that the information about a stochastic process obtained from a
deterministic object such as a ordinary or partial diп¬Ђerential equation is nec-
essarily limited. For example, while we can sometimes obtain the marginal
distribution of the process at time t it is more diп¬ѓcult to obtain quantities
such as the joint distribution of variables which depending on the path of the
process, and these are important in valuing certain types of exotic options such
as lookback and barrier options. For such problems, we often use Monte Carlo
methods.

The Black-Scholes Formula

Before discussing methods of solution in general, we develop the Black-Scholes
equation in a general context. Suppose that a security price is an Ito process
satisfying the equation

(2.33)
dS t = a(St , t ) dt + Пѓ(St , t) dW t

Assumed the market allows investment in the stock as well as a risk-free bond
whose price at time t is Bt . It is necessary to make various other assumptions
as well and strictly speaking all fail in the real world, but they are a reasonable
approximation to a real, highly liquid and nearly frictionless market:

1. partial shares may be purchased

2. there are no dividends paid on the stock

3. There are no commissions paid on purchase or sale of the stock or bond

4. There is no possibility of default for the bond

5. Investors can borrow at the risk free rate governing the bond.

6. All investments are liquid- they can be bought or sold instantaneously.
78 CHAPTER 2. SOME BASIC THEORY OF FINANCE

Since bonds are assumed risk-free, they satisfy an equation

dBt = rt Bt dt

where rt is the risk-free (spot) interest rate at time t.
We wish to determine V (St , t), the value of an option on this security when
the security price is St , at time t. Suppose the option has expiry date T and
a general payoп¬Ђ function which depends only on ST , the process at time T .
ItoвЂ™s lemma provides the ability to translate an a relation governing the
diп¬Ђerential dSt into a relation governing the diп¬Ђerential of the process dV (St , t).
In this sense it is the stochastic calculus analogue of the chain rule in ordinary
calculus. It is one of the most important single results of the twentieth century
in п¬Ѓnance and in science. The stochastic calculus and this mathematical result
concerning it underlies the research leading to 1997 Nobel Prize to Merton and
Black for their work on hedging in п¬Ѓnancial models. We saw one version of it
at the beginning of this section and here we provide a more general version.

ItoвЂ™s lemma.

Suppose St is a diп¬Ђusion process satisfying

dSt = a(St , t)dt + Пѓ(St , t)dWt

and suppose V (St , t) is a smooth function of both arguments. Then V (St , t)
also satisп¬Ѓes a diп¬Ђusion equation of the form

Пѓ 2 (St , t) в€‚ 2 V
в€‚V в€‚V в€‚V
(2.34)
dV = [a(St , t) + + ]dt + Пѓ(St , t) dWt .
в€‚S 2
в€‚S 2 в€‚t в€‚S

Proof. The proof of this result is technical but the ideas behind it are

simple. Suppose we expand an increment of the process V (St , t) ( we write V
MODELS IN CONTINUOUS TIME 79

in place of V (St , t) omitting the arguments of the function and its derivatives.
We will sometimes do the same with the coeп¬ѓcients a and Пѓ.)

1 в€‚ 2V
в€‚V в€‚V
(St+h в€’ St )2 +
V (St+h , t + h) в‰€ V + (St+h в€’ St ) + (2.35)
h
2
в€‚S 2 в€‚S в€‚t

where we have ignored remainder terms that are o(h). Note that substituting
from 2.33 into 2.35, the increment (St+h в€’ St ) is approximately normal with
mean a(St , t ) h and variance Пѓ 2 (St , t ) h. Consider the term (St+h в€’ St )2 .
Note that it is the square of the above normal random variable and has expected
value Пѓ 2 (St , t)h + a2 (St , t)h2 . The variance of this random variable is O(h2 ) so
if we ignore all terms of order o(h) the increment V (St+h , t + h) в€’ V (St , t) is
approximately normally distributed with mean

Пѓ 2 (St , t) в€‚ 2 V
в€‚V в€‚V
[a(St , t ) + + ]h
2
в€‚S 2 в€‚S в€‚t
в€љ
and standard deviation Пѓ(St , t) в€‚V h justifying (but not proving!) the relation
в€‚S

2.34.

By ItoвЂ™s lemma, provided V is smooth, it also satisп¬Ѓes a diп¬Ђusion equation of
the form 2.34. We should note that when V represents the price of an option,
some lack of smoothness in the function V is inevitable. For example for
a European call option with exercise price K, V (ST , T ) = max(ST в€’ K, 0)
does not have a derivative with respect to ST at ST = K, the exercise price.
Fortunately, such exceptional points can be worked around in the argument,
since the derivative does exist at values of t < T.
The basic question in building a replicating portfolio is: for hedging pur-
poses, is it possible to п¬Ѓnd a self-п¬Ѓnancing portfolio consisting only of the se-
curity and the bond which exactly replicates the option price process V (St , t)?
The self-п¬Ѓnancing requirement is the analogue of the requirement that the net
cost of a portfolio is zero that we employed when we introduced the notion of
80 CHAPTER 2. SOME BASIC THEORY OF FINANCE

arbitrage. The portfolio is such that no funds are needed to be added to (or re-
moved from) the portfolio during its life, so for example any additional amounts
required to purchase equity is obtained by borrowing at the risk free rate. Sup-
pose the self-п¬Ѓnancing portfolio has value at time t equal to Vt = ut St + wt Bt
where the (predictable) functions ut , wt represent the number of shares of stock
and bonds respectively owned at time t. Since the portfolio is assumed to be
self-п¬Ѓnancing, all returns obtain from the changes in the value of the securities
and bonds held, i.e. it is assumed that dVt = ut dSt + wt dBt . Substituting from
2.33,

(2.36)
dVt = ut dSt + wt dBt = [ut a(St , t) + wt rt Bt ]dt + ut Пѓ(St , t)dWt

If Vt is to be exactly equal to the price V (St , t ) of an option, it follows on
в€‚V
comparing the coeп¬ѓcients of dt and dWt in 2.34 and 2.36, that ut = в€‚S , called
the delta corresponding to delta hedging. Consequently,
в€‚V
Vt = St + wt Bt
в€‚S
and solving for wt we obtain:

1 в€‚V
[V в€’
wt = St ].
Bt в€‚S
The conclusion is that it is possible to dynamically choose a trading strategy, i.e.
the weights wt , ut so that our portfolio of stocks and bonds perfectly replicates the
в€‚V
value of the option. If we own the option, then by shorting (selling) delta= в€‚S

units of stock, we are perfectly hedged in the sense that our portfolio replicates
a risk-free bond. Surprisingly, in this ideal word of continuous processes and
In the real world, it is said to exist only in a Japanese garden. The equation we
obtained by equating both coeп¬ѓcients in 2.34 and 2.36 is;
Пѓ 2 (St , t) в€‚ 2 V
в€‚V в€‚V
в€’rt V + rt St (2.37)
+ + = 0.
в€‚S 2
в€‚S в€‚t 2
MODELS IN CONTINUOUS TIME 81

Rewriting this allows an interpretation in terms of our hedged portfolio. If we
own an option and are short delta units of stock our net investment at time t
is given by (V в€’ St в€‚V ) where V = Vt = V (St , t). Our return over the next time
в€‚S

increment dt if the portfolio were liquidated and the identical amount invested
in a risk-free bond would be rt (Vt в€’ St в€‚V )dt. On the other hand if we keep this
в€‚S

hedged portfolio, the return over an increment of time dt is

в€‚V в€‚V
d(V в€’ St ) = dV в€’ ( )dS
в€‚S в€‚S
Пѓ2 в€‚ 2V
в€‚V в€‚V в€‚V
=( + +a )dt + Пѓ dWt
2 в€‚S 2
в€‚t в€‚S в€‚S
в€‚V
в€‚S
Пѓ2 в€‚ 2V
в€‚V
=( + )dt
2 в€‚S 2
в€‚t

Therefore
Пѓ 2 (St , t) в€‚ 2 V
в€‚V в€‚V
rt (V в€’ St )= + .
в€‚S 2
в€‚S в€‚t 2
The left side rt (V в€’ St в€‚V ) represents the amount made by the portion of our
в€‚S

portfolio devoted to risk-free bonds. The right hand side represents the return
on a hedged portfolio long one option and short delta stocks. Since these
investments are at least in theory identical, so is their return. This fundamental
equation is evidently satisп¬Ѓed by any option price process where the underlying
security satisп¬Ѓes a diп¬Ђusion equation and the option value at expiry depends
only on the value of the security at that time. The type of option determines
the terminal conditions and usually uniquely determines the solution.
It is extraordinary that this equation in no way depends on the drift co-
eп¬ѓcient a(St , t). This is a remarkable feature of the arbitrage pricing theory.
Essentially, no matter what the drift term for the particular security is, in order
to avoid arbitrage, all securities and their derivatives are priced as if they had
as drift the spot interest rate. This is the eп¬Ђect of calculating the expected values
under the martingale measure Q.
This PDE governs most derivative products, European call options, puts,
82 CHAPTER 2. SOME BASIC THEORY OF FINANCE

futures or forwards. However, the boundary conditions and hence the solution
depends on the particular derivative. The solution to such an equation is possi-
ble analytically in a few cases, while in many others, numerical techniques are
necessary. One special case of this equation deserves particular attention. In
the case of geometric Brownian motion, a(St , t) = ВµSt and Пѓ(St , t) = ПѓSt for
constants Вµ, Пѓ. Assume that the spot interest rate is a constant rand that a
constant rate of dividends D0 is paid on the stock. In this case, the equation
specializes to

Пѓ2 S 2 в€‚ 2 V
в€‚V в€‚V
в€’rV + + (r в€’ D0 )S (2.38)
+ = 0.
2 в€‚S 2
в€‚t в€‚S
Note that we have not used any of the properties of the particular derivative
product yet, nor does this diп¬Ђerential equation involve the drift coeп¬ѓcient Вµ.
The assumption that there are no transaction costs is essential to this analysis,
as we have assumed that the portfolio is continually rebalanced.
We have now seen two derivations of parabolic partial diп¬Ђerential equations,
so-called because like the equation of a parabola, they are п¬Ѓrst order (derivatives)
in one variable (t) and second order in the other (x). Usually the solution of such
an equation requires reducing it to one of the most common partial diп¬Ђerential
equations, the heat or diп¬Ђusion equation, which models the diп¬Ђusion of heat
along a rod. This equation takes the form

в€‚2
в€‚
(2.39)
u = k 2u
в€‚t в€‚x
A solution of 2.39 with appropriate boundary conditions can sometime be found
by the separation of variables. We will later discuss in more detail the solution
of parabolic equations, both by analytic and numerical means. First, however,
в€љ
when can we hope to п¬Ѓnd a solution of 2.39 of the form u(x, t) = g(x/ t).
By diп¬Ђerentiating and substituting above, we obtain an ordinary diп¬Ђerential
equation of the form
в€љ
1
g 00 (П‰) + П‰g 0 (П‰) = 0, П‰ = x/ t (2.40)
2k
MODELS IN CONTINUOUS TIME 83

Let us solve this using MAPLE.
eqn := diff(g(w),w,w)+(w/(2*k))*diff(g(w),w)=0;
dsolve(eqn,g(w));

and because the derivative of the solution is slightly easier (for a statistician)
to identify than the solution itself,
> diff(%,w);
giving
в€‚
g(П‰) = C2 exp{в€’w2 /4k} = C2 exp{в€’x2 /4kt} (2.41)
в€‚w
showing that a constant plus a constant multiple of the Normal (0, 2kt) cumu-
lative distribution function or
Z x
1
exp{в€’z 2 /4kt}dz
u(x, t) = C1 + C2 в€љ (2.42)
2 ПЂkt в€’в€ћ

is a solution of this, the heat equation for t > 0. The role of the two constants is
simple. Clearly if a solution to 2.39 is found, then we may add a constant and/or
multiply by a constant to obtain another solution. The constant in general is
determined by initial and boundary conditions. Similarly the integral can be
removed with a change in the initial condition for if u solves 2.39 then so does
в€‚u
For example if we wish a solution for the half real x > 0 with initial condition
в€‚x .

u(x, 0) = 0, u(0, t) = 1 all t > 1, we may use
Z в€ћ
1
exp{в€’z 2 /4kt}dz, t > 0, x в‰Ґ 0.
u(x, t) = 2P (N (0, 2kt) > x) = в€љ
ПЂkt x

Let us consider a basic solution to 2.39:

1
exp{в€’x2 /4kt}
u(x, t) = в€љ (2.43)
2 ПЂkt

This connection between the heat equation and the normal distributions is fun-
damental and the wealth of solutions depending on the initial and boundary
conditions is considerable. We plot a fundamental solution of the equation as
follows with the plot in Figure 2.8:
84 CHAPTER 2. SOME BASIC THEORY OF FINANCE

Figure 2.8: Fundamental solution of the heat equation

>u(x,t) := (.5/sqrt(Pi*t))*exp(-x^2/(4*t));

>plot3d(u(x,t),x=-4..4,t=.02..4,axes=boxed);

As t в†’ 0, the function approaches a spike at x = 0, usually referred to as
the вЂњDirac delta functionвЂќ (although it is no function at all) and symbolically
representing the derivative of the вЂњHeaviside functionвЂќ. The Heaviside function
is deп¬Ѓned as H(x) = 1, x в‰Ґ 0 and is otherwise 0 and is the cumulative distrib-
ution function of a point mass at 0. Suppose we are given an initial condition
of the form u(x, 0) = u0 (x). To this end, it is helpful to look at the solu-
tion u(x, t) and the initial condition u0 (x) as a distribution or measure (in this
case described by a density) over the space variable x. For example the density
R
u(x, t) corresponds to a measure for п¬Ѓxed t of the form ОЅt (A) = A u(x, t)dx.
Note that the initial condition compatible with the above solution 2.42 can be
described somewhat clumsily as вЂњu(x, 0) corresponds to a measure placing all
mass at x = x0 = 0 вЂќ.In fact as t в†’ 0, we have in some sense the following
convergence u(x, t) в†’ Оґ(x) = dH(x), the Dirac delta function. We could just as
easily construct solve the heat equation with a more general initial condition of
MODELS IN CONTINUOUS TIME 85

the form u(x, 0) = dH(x в€’ x0 ) for arbitrary x0 and the solution takes the form

1
exp{в€’(x в€’ x0 )2 /4kt}.
u(x, t) = в€љ (1.22)
2 ПЂkt

Indeed sums of such solutions over diп¬Ђerent values of x0 , or weighted sums, or
their limits, integrals will continue to be solutions to 2.39. In order to achieve
the initial condition u0 (x) we need only pick a suitable weight function. Note
that
Z
u0 (z)dH(z в€’ x)
u0 (x) =

Note that the function
Z в€ћ
1
exp{в€’(z в€’ x)2 /4kt}u0 (z)dz
u(x, t) = в€љ (1.22)
2 ПЂkt в€’в€ћ

solves 2.39 subject to the required boundary condition.

Solution of the Diп¬Ђusion Equation.

We now consider the general solution to the diп¬Ђusion equation of the form 2.37,
rewritten as
Пѓ 2 (St , t) в€‚ 2 V
в€‚V в€‚V
= rt V в€’ rt St в€’ (2.44)
в€‚S 2
в€‚t в€‚S 2
where St is an asset price driven by a diп¬Ђusion equation

(2.45)
dSt = a(St , t)dt + Пѓ(St , t)dWt ,

V (St , t) is the price of an option on that asset at time t, and rt = r(t) is the
spot interest rate at time t. We assume that the price of the option at expiry
T is a known function of the asset price

(2.46)
V (ST , T ) = V0 (ST ).

Somewhat strangely, the option is priced using a related but not identical process
(or, equivalently, the same process under a diп¬Ђerent measure). Recall from the
86 CHAPTER 2. SOME BASIC THEORY OF FINANCE

backwards Kolmogorov equation 2.27 that if a related process Xt satisп¬Ѓes the
stochastic diп¬Ђerential equation

(2.47)
dXt = r(Xt , t)Xt dt + Пѓ(Xt , t)dWt

в€‚
В· z|Xt = s] satisп¬Ѓes a partial
then its transition kernel p(t, s, T, z) = в€‚z P [XT

diп¬Ђerential equation similar to 2.44;

в€‚p Пѓ 2 (s, t) в€‚ 2 p
в€‚p
= в€’r(s, t)s в€’ (2.48)
в€‚s2
в€‚t в€‚s 2

For a given process Xt this determines one solution. For simplicity, consider
the case (natural in п¬Ѓnance applications) when the spot interest rate is a function
of time, not of the asset price; r(s, t) = r(t). To obtain the solution so that
terminal conditions is satisп¬Ѓed, consider a product

(2.49)
f (t, s, T, z) = p(t, s, T, z)q(t, T )

where
Z T
q(t, T ) = exp{в€’ r(v)dv}
t

is the discount function or the price of a zero-coupon bond at time t which pays
1\$ at maturity.
Let us try an application of one of the most common methods in solving
PDEвЂ™s, the вЂњlucky guessвЂќ method. Consider a linear combination of terms of
the form 2.49 with weight function w(z). i.e. try a solution of the form
Z
(2.50)
V (s, t) = p(t, s, T, z)q(t, T )w(z)dz

for suitable weight function w(z). In view of the deп¬Ѓnition of pas a transition
probability density, this integral can be rewritten as a conditional expectation:

(2.51)
V (t, s) = E[w(XT )q(t, T )|Xt = s]

the discounted conditional expectation of the random variable w(XT ) given the
current state of the process, where the process is assumed to follow (2.18). Note
MODELS IN CONTINUOUS TIME 87

that in order to satisfy the terminal condition 2.46, we choose w(x) = V0 (x).
Now
Z
в€‚V в€‚
= p(t, s, T, z)q(t, T )w(z)dz
в€‚t в€‚t
Z
в€‚p Пѓ 2 (St , t) в€‚ 2 p
в€’
= [в€’r(St , t)St 2]q(t, T )w(z)dz
в€‚s 2 в€‚s
Z
p(t, St , T, z)q(t, T )w(z)dz by 2.48
+ r(St , t)

Пѓ 2 (St , t) в€‚ 2 V
в€‚V
= в€’r(St , t)St в€’ + r(St , t)V (St , t)
в€‚S 2
в€‚S 2

where we have assumed that we can pass the derivatives under the integral
sign. Thus the process

(2.52)
V (t, s) = E[V0 (XT )q(t, T )|Xt = s]

satisп¬Ѓes both the partial diп¬Ђerential equation 2.44 and the terminal conditions
2.46 and is hence the solution. Indeed it is the unique solution satisfying certain
regularity conditions. The result asserts that the value of any European option
is simply the conditional expected value of the discounted payoп¬Ђ (discounted to
the present) assuming that the distribution is that of the process 2.47. This
result is a special case when the spot interest rates are functions only of time of
the following more general theorem.

Theorem 13 ( Feynman-Kac)

Suppose the conditions for a unique solution to (2.44,2.46) (see for example
Duп¬ѓe, appendix E) are satisп¬Ѓed. Then the general solution to (2.15) under the
terminal condition 2.46 is given by
Z T
(2.53)
V (S, t) = E[V0 (XT )exp{в€’ r(Xv , v)dv}| Xt = S]
t
88 CHAPTER 2. SOME BASIC THEORY OF FINANCE

This represents the discounted return from the option under the distribution
of the process Xt . The distribution induced by the process Xt is referred to
as the equivalent martingale measure or risk neutral measure. Notice that when
the original process is a diп¬Ђusion, the equivalent martingale measure shares the
same diп¬Ђusion coeп¬ѓcient but has the drift replaced by r(Xt , t)Xt . The option
is priced as if the drift were the same as that of a risk-free bond i.e. as if the
instantaneous rate of return from the security if identical to that of bond. Of
course, in practice, it is not. A risk premium must be paid to the stock-holder
to compensate for the greater risk associated with the stock.

There are some cases in which the conditional expectation 2.53 can be deter-
mined explicitly. In general, these require that the process or a simple function
of the process is Gaussian.

For example, suppose that both r(t) and Пѓ(t) are deterministic functions
of time only. Then we can solve the stochastic diп¬Ђerential equation (2.22) to
obtain

Z T
Xt Пѓ(u)
(2.54)
XT = + dWu
q(t, T ) q(u, T )
t

The п¬Ѓrst term above is the conditional expected value of XT given Xt . The
second is the random component, and since it is a weighted sum of the normally
distributed increments of a Brownian motion with weights that are non-random,
it is also a normal random variable. The mean is 0 and the (conditional) vari-
R T 2 (u)
ance is t qПѓ(u,T ) du. Thus the conditional distribution of XT given Xt is normal
2
R T 2 (u)
with conditional expectation q(t,T ) and conditional variance t qПѓ(u,T ) du.
Xt
2

The special case of 2.53 of most common usage is the Black-Scholes model:
suppose that Пѓ(S, t) = SПѓ(t) for Пѓ(t) some deterministic function of t. Then
the distribution of Xt is not Gaussian, but fortunately, its logarithm is. In this
case we say that the distribution of Xt is lognormal.
MODELS IN CONTINUOUS TIME 89

Lognormal Distribution

Suppose Z is a normal random variable with mean Вµ and variance Пѓ 2 . Then we
say that the distribution of X = eZ is lognormal with mean О· = exp{Вµ + Пѓ 2 /2}
and volatility parameter Пѓ. The lognormal probability density function with
mean О· > 0 and volatility parameter Пѓ > 0 is given by the probability density
function

1
в€љ exp{в€’(log x в€’ log О· в€’ Пѓ2 /2)2 /2Пѓ 2 }. (2.55)
g(x|О·, Пѓ) =
xПѓ 2ПЂ
The solution to (2.18) with non-random functions Пѓ(t), r(t) is now
ZT ZT
(r(u) в€’ Пѓ 2 (u)/2)du + Пѓ(u)dWu }. (2.56)
XT = Xt exp{
t t

Since the exponent is normal, the distribution of XT is lognormal with mean
RT RT
log(Xt ) + t (r(u) в€’ Пѓ 2 (u)/2)du and variance t Пѓ 2 (u)du. It follows that the
conditional distribution is lognormal with mean О· = Xt q(t, T ) and volatility
qR
T2
parameter Пѓ (u)du.
t

We now derive the well-known Black-Scholes formula as a special case of
2.53. For a call option with exercise price E, the payoп¬Ђ function is V0 (ST ) =
max(ST в€’ E, 0). Now it is helpful to use the fact that for a standard normal
random variable Z and arbitrary Пѓ > 0, в€’в€ћ < Вµ < в€ћ we have the expected
value of max(eПѓZ+Вµ , 0) is

Вµ Вµ
2
eВµ+Пѓ /2
+ Пѓ) в€’ О¦( ) (2.57)
О¦(
Пѓ Пѓ

where О¦(.) denotes the standard normal cumulative distribution function. As
a result, in the special case that r and Пѓ are constants, (2.53) results in the
famous Black-Scholes formula which can be written in the form

V (S, t) = SО¦(d1 ) в€’ Eeв€’r(T в€’t) О¦(d2 ) (2.58)

where
в€љ
log(S/E) + (r + Пѓ 2 /2)(T в€’ t)
в€љ , d2 = d1 в€’ Пѓ T в€’ t
d1 =
Пѓ T в€’t
90 CHAPTER 2. SOME BASIC THEORY OF FINANCE

are the values В±Пѓ2 (T в€’ t)/2 standardized by adding log(S/E) + r(T в€’ t) and
в€љ
dividing by Пѓ T в€’ t. This may be derived by the following device; Assume (i.e.
pretend) that, given current information, the distribution of S(T ) at expiry is
lognormally distributed with the mean О· = S(t)er(T в€’t) .
The mean of the log-normal in the risk neutral world S(t)er(T в€’t) is exactly
the future value of our current stocks S(t) if we were to sell the stock and invest
the cash in a bank deposit. Then the future value of an option with payoп¬Ђ
function given by V0 (ST ) is the expected value of this function against this
lognormal probability density function, then discounted to present value
Zв€ћ
в€љ
в€’r(T в€’t)
V0 (x)g(x|S(t)er(T в€’t) , Пѓ T в€’ t)dx. (2.59)
e
0

Notice that the Black-Scholes derivation covers any diп¬Ђusion process govern-
ing the underlying asset which is driven by a stochastic diп¬Ђerential equation of
the form
(2.60)
dS = a(S)dt + ПѓSdWt

regardless of the nature of the drift term a(S). For example a non-linear function
a(S) can lead to distributions that are not lognormal and yet the option price
is determined as if it were.

Example: Pricing Call and Put options.

Consider pricing an index option on the S&P 500 index an January 11, 2000 (the
index SPX closed at 1432.25 on this day). The option SXZ AE-A is a January
call option with strike price 1425. The option matures (as do equity options in
general) on the third Friday of the month or January 21, a total of 7 trading
days later. Suppose we wish to price such an option using the Black-Scholes
model. In this case, T в€’ t measured in years is 7/252 = 0.027778. The annual
volatility of the Standard and Poor 500 index is around 19.5 percent or 0.195
and assume the very short term interest rates approximately 3%. In Matlab we
can value this option using
MODELS IN CONTINUOUS TIME 91

[CALL,PUT] = BLSPRICE(1432.25,1425,0.03,7/252,0.195,0)
CALL = 23.0381
PUT = 14.6011
Arguments of the function BLSPRICE are, in order, the current equity price,
the strike price, the annual interest rate r, the time to maturity T в€’ t in years,
the annual volatility Пѓ and the last argument is the dividend yield in percent
which we assumed 0. Thus the Black-Scholes price for a call option on SPX
is around 23.03. Indeed this call option did sell on Jan 11 for \$23.00. and
the put option for \$14 5/8. From the put call parity relation (see for example
Wilmott, Howison, Dewynne, page 41) S + P в€’ C = Eeв€’r(T в€’t) or in this
case 1432.25 + 14.625 в€’ 23 = 1425eв€’r(7/252) . We might solve this relation to
obtain the spot interest rate r. In order to conп¬Ѓrm that a diп¬Ђerent interest rate
might apply over a longer term, we consider the September call and put options
(SXZ) on the same day with exercise price 1400 which sold for \$152 and 71\$
respectively. In this case there are171 trading days to expiry and so we need to
solve 1432.25 + 71 в€’ 152 = 1400eв€’r(171/252) , whose solution is r = 0.0522 .
This is close to the six month interest rates at the time, but 3% is low for the
very short term rates. The discrepancy with the actual interest rates is one of
several modest failures of the Black-Scholes model to be discussed further later.
The low implied interest rate is inп¬‚uenced by the cost of handling and executing
an option, which are non-negligible fractions of the option prices, particularly
with short term options such as this one. An analogous function to the Matlab
function above which provides the Black-Scholes price in Splus or R is given
below:
blsprice=function(So,strike,r,T,sigma,div){
d1<-(log(So/strike)+(r-div+(sigma^2)/2)*T)/(sigma*sqrt(T))
d2<-d1-sigma*sqrt(T)
call<-So*exp(-div*T)*pnorm(d1)-exp(-r*T)*strike*pnorm(d2)
put=call-So+strike*exp(-r*T)
92 CHAPTER 2. SOME BASIC THEORY OF FINANCE

c(call,put)}

Problems

1. It is common for a stock whose price has reached a high level to split or
issue shares on a two-for-one or three-for-one basis. What is the eп¬Ђect of
a stock split on the price of an option?

2. If a stock issues a dividend of exactly D (known in advance) on a certain
date, provide a no-arbitrage argument for the change in price of the stock
at this date. Is there a diп¬Ђerence between deterministic D and the case
when D is a random variable with known distribution but whose value is
declared on the dividend date?

3. Suppose ОЈ is a positive deп¬Ѓnite covariance matrix and О· a column vector.
Show that the set of all possible pairs of standard deviation and mean
в€љ P
return ( wT ОЈw, О· T w) for weight vector w such that i wi = 1 is a
convex region with a hyperbolic boundary.

4. The current rate of interest is 5% per annum and you are oп¬Ђered a random
bond which pays either \$210 or \$0 in one year. You believe that the
probability of the bond paying \$210 is one half. How much would you
pay now for such a bond? Suppose this bond is publicly traded and a
large fraction of the population is risk averse so that it is selling now for
\$80. Does your price oп¬Ђer an arbitrage to another trader? What is the
risk-neutral measure for this bond?

5. Which would you prefer, a gift of \$100 or a 50-50 chance of making \$200?
A п¬Ѓne of \$100 or a 50-50 chance of losing \$200? Are your preferences
self-consistent and consistent with the principle that individuals are risk-
averse?
PROBLEMS 93

6. Compute the stochastic diп¬Ђerential dXt (assuming Wt is a Wiener process)
when

(a) Xt = exp(rt)
Rt
(b) Xt = 0 h(t)dWt

(c) Xt = X0 exp{at + bWt }

(d) Xt = exp(Yt ) where dYt = Вµdt + ПѓdWt .

ОІ
7. Show that if Xt is a geometric Brownian motion, so is Xt for any real
number ОІ.

8. Suppose a stock price follows a geometric Brownian motion process

dSt = ВµSt dt + ПѓSt dWt

n
Find the diп¬Ђusion equation satisп¬Ѓed by the processes (a) f (St ) = St ,(b)
log(St ), (c) 1/St . Find a combination of the processes St and 1/St that
does not depend on the drift parameter Вµ. How does this allow constructing
estimators of Пѓ that do not require knowledge of the value of Вµ?

9. Consider an Ito process of the form

dSt = a(St )dt + Пѓ(St )dWt

Is it possible to п¬Ѓnd a function f (St ) which is also an Ito process but with
zero drift?

10. Consider an Ito process of the form

dSt = a(St )dt + Пѓ(St )dWt

Is it possible to п¬Ѓnd a function f (St ) which has constant diп¬Ђusion term?
RT P
g(t)dWt в‰€
11. Consider approximating an integral of the form g(t){W (t+
0

h) в€’ W (t)} where g(t) is a non-random function and the sum is over val-
ues of t = nh, n = 0, 1, 2, ...T /h в€’ 1. Show by considering the distribution
94 CHAPTER 2. SOME BASIC THEORY OF FINANCE

RT
of the sum and taking limits that the random variable g(t)dWt has a
0

normal distribution and п¬Ѓnd its mean and variance.

12. Consider two geometric Brownian motion processes Xt and Yt both driven
by the same Wiener process

dXt = aXt dt + bXt dWt

dYt = ВµYt dt + ПѓYt dWt .

Derive a stochastic diп¬Ђerential equation for the ratio Zt = Xt /Yt . Suppose
for example that Xt models the price of a commodity in \$C and Yt is the
exchange rate (\$C/\$U S) at time t. Then what is the process Zt ? Repeat
in the more realistic situation in which

(1)
dXt = aXt dt + bXt dWt
(2)
dYt = ВµYt dt + ПѓYt dWt

(1) (2)
and Wt , Wt are correlated Brownian motion processes with correlation
ПЃ.

13. Prove the Shannon inequality that
X qi
)в‰Ґ0
H(Q, P ) = qi log(
pi
for any probability distributions P and Q with equality if and only if
all pi = qi .

14. Consider solving the problem
X qi
min H(Q, P ) = qi log( )
pi
q
P P
subject to the constraints i qi = 1 and EQ f (X) = qi f (i) = Вµ. Show
that the solution, if it exists, is given by
exp(О·f (i))
qi = pi
m(О·)
PROBLEMS 95

P m0 (О·)
where m(О·) = i pi exp(О·f (i))] and О· is chosen so that = Вµ. (This
m(О·)

shows that the closest distribution to P which satisп¬Ѓes the constraint is
dQ
obtained by a simple вЂњexponential tiltвЂќ or Esscher transform so that dP (x)

is proportional to exp(О·f (x)) for a suitable parameter О·).

15. Let Qв€— minimize H(Q, P ) subject to a constraint

(2.61)
EQ g(X) = c.

Let Q be some other probability distribution satisfying the same con-
straint. Then prove that

H(Q, P ) = H(Q, Qв€— ) + H(Qв€— , P ).

16. Let I1 , I2 ,... be a set of constraints of the form

(2.62)
EQ gi (X) = ci

в€—
and suppose we deп¬Ѓne Pn as the solution of

max H(P )
P

subject to the constraints I1 в€© I2 в€© ...In . Then prove that

в€— в€— в€— в€— в€— в€— в€— в€—
H(Pn , P1 ) = H(Pn , Pnв€’1 ) + H(Pnв€’1 , Pnв€’2 ) + ... + H(P2 , P1 ).

17. Consider a defaultable bond which pays a fraction of its face value F p
on maturity in the event of default. Suppose the risk free interest rate
continuously compounded is r so that Bs = exp(sr). Suppose also that a
constant coupon \$d is paid at the end of every period s = t + 1, ..., T в€’ 1.
Then show that the value of this bond at time t is

exp{в€’(r + k)} в€’ exp{в€’(r + k){T в€’ t)}
Pt = d
1 в€’ exp{в€’(r + k)}
+ pF exp{в€’r(T в€’ t)} + (1 в€’ p)F exp{в€’(r + k)(T в€’ t)}
96 CHAPTER 2. SOME BASIC THEORY OF FINANCE

18. (a) Show that entropy is always positive and if Y = g(X) is a function
of X then Y has smaller entropy than X, i.e. H(pY ) В· H(pX ).

(b) Show that if X has any discrete distribution over n values, then its
entropy is В· log(n).

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