ńņš. 3 |

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 diļ¬usion, 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 deļ¬ned on this process, knowledge of the transition density would suļ¬ce

at least theoretically for valuing the option. Unfortunately these equations are

often very diļ¬cult to solve explicitly.

Besides the Kolmogorov equations, we can use simple ordinary diļ¬erential

equations to arrive at some of the basic properties of a diļ¬usion. To illustrate,

consider one of the simplest possible forms of a diļ¬usion, where a(Xt , t) =

Ī±(t)+Ī²(t)Xt where the coeļ¬cients Ī±(t), Ī²(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 + Ļ(Xs , s)dWs

0 0

Rt

and by construction that last term Ļ(Xs , s)dWs is a zero-mean martingale.

0

For example its small increments Ļ(Xt , t)dWs are approximately N (0, Ļ(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 + [Ī±(s) + Ī²(s)m(s)]ds

0

and therefore m(t)solves the ordinary diļ¬erential equation

m0 (t) = Ī±(t) + Ī²(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 diļ¬usion process can be found by solving a similar ordinary

diļ¬erential equation, similar except that the diļ¬usion term has been dropped.

These are only two of many reasons to wish to solve both ordinary and

partial diļ¬erential equations in ļ¬nance. The solution to the Kolmogorov partial

diļ¬erential 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 diļ¬erential equation will allow the calculation of the expected value of

the process and this is the ļ¬rst 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 diļ¬erential equations.

However, that the information about a stochastic process obtained from a

deterministic object such as a ordinary or partial diļ¬erential equation is nec-

essarily limited. For example, while we can sometimes obtain the marginal

distribution of the process at time t it is more diļ¬cult 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 + Ļ(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 payoļ¬ function which depends only on ST , the process at time T .

Itoā™s lemma provides the ability to translate an a relation governing the

diļ¬erential dSt into a relation governing the diļ¬erential 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 ļ¬nance 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 ļ¬nancial models. We saw one version of it

at the beginning of this section and here we provide a more general version.

Itoā™s lemma.

Suppose St is a diļ¬usion process satisfying

dSt = a(St , t)dt + Ļ(St , t)dWt

and suppose V (St , t) is a smooth function of both arguments. Then V (St , t)

also satisļ¬es a diļ¬usion equation of the form

Ļ 2 (St , t) ā‚ 2 V

ā‚V ā‚V ā‚V

(2.34)

dV = [a(St , t) + + ]dt + Ļ(St , t) dWt .

ā‚S 2

ā‚S 2 ā‚t ā‚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 coeļ¬cients a and Ļ.)

1 ā‚ 2V

ā‚V ā‚V

(St+h ā’ St )2 +

V (St+h , t + h) ā V + (St+h ā’ St ) + (2.35)

h

2

ā‚S 2 ā‚S ā‚t

where we have ignored remainder terms that are o(h). Note that substituting

from 2.33 into 2.35, the increment (St+h ā’ St ) is approximately normal with

mean a(St , t ) h and variance Ļ 2 (St , t ) h. Consider the term (St+h ā’ St )2 .

Note that it is the square of the above normal random variable and has expected

value Ļ 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) ā’ V (St , t) is

approximately normally distributed with mean

Ļ 2 (St , t) ā‚ 2 V

ā‚V ā‚V

[a(St , t ) + + ]h

2

ā‚S 2 ā‚S ā‚t

ā

and standard deviation Ļ(St , t) ā‚V h justifying (but not proving!) the relation

ā‚S

2.34.

By Itoā™s lemma, provided V is smooth, it also satisļ¬es a diļ¬usion 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 ā’ 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 ļ¬nd a self-ļ¬nancing portfolio consisting only of the se-

curity and the bond which exactly replicates the option price process V (St , t)?

The self-ļ¬nancing 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-ļ¬nancing 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-ļ¬nancing, 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 Ļ(St , t)dWt

If Vt is to be exactly equal to the price V (St , t ) of an option, it follows on

ā‚V

comparing the coeļ¬cients of dt and dWt in 2.34 and 2.36, that ut = ā‚S , called

the delta corresponding to delta hedging. Consequently,

ā‚V

Vt = St + wt Bt

ā‚S

and solving for wt we obtain:

1 ā‚V

[V ā’

wt = St ].

Bt ā‚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

ā‚V

value of the option. If we own the option, then by shorting (selling) delta= ā‚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

continuous time trading commission-free trading, the perfect hedge is possible.

In the real world, it is said to exist only in a Japanese garden. The equation we

obtained by equating both coeļ¬cients in 2.34 and 2.36 is;

Ļ 2 (St , t) ā‚ 2 V

ā‚V ā‚V

ā’rt V + rt St (2.37)

+ + = 0.

ā‚S 2

ā‚S ā‚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 ā’ St ā‚V ) where V = Vt = V (St , t). Our return over the next time

ā‚S

increment dt if the portfolio were liquidated and the identical amount invested

in a risk-free bond would be rt (Vt ā’ St ā‚V )dt. On the other hand if we keep this

ā‚S

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

ā‚V ā‚V

d(V ā’ St ) = dV ā’ ( )dS

ā‚S ā‚S

Ļ2 ā‚ 2V

ā‚V ā‚V ā‚V

=( + +a )dt + Ļ dWt

2 ā‚S 2

ā‚t ā‚S ā‚S

ā‚V

ā’ [adt + ĻdWt ]

ā‚S

Ļ2 ā‚ 2V

ā‚V

=( + )dt

2 ā‚S 2

ā‚t

Therefore

Ļ 2 (St , t) ā‚ 2 V

ā‚V ā‚V

rt (V ā’ St )= + .

ā‚S 2

ā‚S ā‚t 2

The left side rt (V ā’ St ā‚V ) represents the amount made by the portion of our

ā‚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 satisļ¬ed by any option price process where the underlying

security satisļ¬es a diļ¬usion 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-

eļ¬cient 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 eļ¬ect 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) = ĀµSt and Ļ(St , t) = ĻSt for

constants Āµ, Ļ. 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

Ļ2 S 2 ā‚ 2 V

ā‚V ā‚V

ā’rV + + (r ā’ D0 )S (2.38)

+ = 0.

2 ā‚S 2

ā‚t ā‚S

Note that we have not used any of the properties of the particular derivative

product yet, nor does this diļ¬erential equation involve the drift coeļ¬cient Āµ.

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 diļ¬erential equations,

so-called because like the equation of a parabola, they are ļ¬rst 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 diļ¬erential

equations, the heat or diļ¬usion equation, which models the diļ¬usion of heat

along a rod. This equation takes the form

ā‚2

ā‚

(2.39)

u = k 2u

ā‚t ā‚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,

ā

when can we hope to ļ¬nd a solution of 2.39 of the form u(x, t) = g(x/ t).

By diļ¬erentiating and substituting above, we obtain an ordinary diļ¬erential

equation of the form

ā

1

g 00 (Ļ) + Ļg 0 (Ļ) = 0, Ļ = 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

ā‚

g(Ļ) = C2 exp{ā’w2 /4k} = C2 exp{ā’x2 /4kt} (2.41)

ā‚w

showing that a constant plus a constant multiple of the Normal (0, 2kt) cumu-

lative distribution function or

Z x

1

exp{ā’z 2 /4kt}dz

u(x, t) = C1 + C2 ā (2.42)

2 Ļkt ā’ā

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

ā‚u

For example if we wish a solution for the half real x > 0 with initial condition

ā‚x .

u(x, 0) = 0, u(0, t) = 1 all t > 1, we may use

Z ā

1

exp{ā’z 2 /4kt}dz, t > 0, x ā„ 0.

u(x, t) = 2P (N (0, 2kt) > x) = ā

Ļkt x

Let us consider a basic solution to 2.39:

1

exp{ā’x2 /4kt}

u(x, t) = ā (2.43)

2 Ļkt

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);

[FIGURE 2.8 ABOUT HERE]

As t ā’ 0, the function approaches a spike at x = 0, usually referred to as

the āDirac delta functionā (although it is no function at all) and symbolically

representing the derivative of the āHeaviside functionā. The Heaviside function

is deļ¬ned as H(x) = 1, x ā„ 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 ļ¬xed t of the form Ī½t (A) = A u(x, t)dx.

Note that the initial condition compatible with the above solution 2.42 can be

described somewhat clumsily as āu(x, 0) corresponds to a measure placing all

mass at x = x0 = 0 ā.In fact as t ā’ 0, we have in some sense the following

convergence u(x, t) ā’ Ī“(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 ā’ x0 ) for arbitrary x0 and the solution takes the form

1

exp{ā’(x ā’ x0 )2 /4kt}.

u(x, t) = ā (1.22)

2 Ļkt

Indeed sums of such solutions over diļ¬erent 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 ā’ x)

u0 (x) =

Note that the function

Z ā

1

exp{ā’(z ā’ x)2 /4kt}u0 (z)dz

u(x, t) = ā (1.22)

2 Ļkt ā’ā

solves 2.39 subject to the required boundary condition.

Solution of the Diļ¬usion Equation.

We now consider the general solution to the diļ¬usion equation of the form 2.37,

rewritten as

Ļ 2 (St , t) ā‚ 2 V

ā‚V ā‚V

= rt V ā’ rt St ā’ (2.44)

ā‚S 2

ā‚t ā‚S 2

where St is an asset price driven by a diļ¬usion equation

(2.45)

dSt = a(St , t)dt + Ļ(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 diļ¬erent measure). Recall from the

86 CHAPTER 2. SOME BASIC THEORY OF FINANCE

backwards Kolmogorov equation 2.27 that if a related process Xt satisļ¬es the

stochastic diļ¬erential equation

(2.47)

dXt = r(Xt , t)Xt dt + Ļ(Xt , t)dWt

ā‚

Ā· z|Xt = s] satisļ¬es a partial

then its transition kernel p(t, s, T, z) = ā‚z P [XT

diļ¬erential equation similar to 2.44;

ā‚p Ļ 2 (s, t) ā‚ 2 p

ā‚p

= ā’r(s, t)s ā’ (2.48)

ā‚s2

ā‚t ā‚s 2

For a given process Xt this determines one solution. For simplicity, consider

the case (natural in ļ¬nance 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 satisļ¬ed, 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{ā’ 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

PDEā™s, the ālucky guessā 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 deļ¬nition 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

ā‚V ā‚

= p(t, s, T, z)q(t, T )w(z)dz

ā‚t ā‚t

Z

ā‚p Ļ 2 (St , t) ā‚ 2 p

ā’

= [ā’r(St , t)St 2]q(t, T )w(z)dz

ā‚s 2 ā‚s

Z

p(t, St , T, z)q(t, T )w(z)dz by 2.48

+ r(St , t)

Ļ 2 (St , t) ā‚ 2 V

ā‚V

= ā’r(St , t)St ā’ + r(St , t)V (St , t)

ā‚S 2

ā‚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]

satisļ¬es both the partial diļ¬erential 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 payoļ¬ (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

Duļ¬e, appendix E) are satisļ¬ed. 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{ā’ 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 diļ¬usion, the equivalent martingale measure shares the

same diļ¬usion coeļ¬cient 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 Ļ(t) are deterministic functions

of time only. Then we can solve the stochastic diļ¬erential equation (2.22) to

obtain

Z T

Xt Ļ(u)

(2.54)

XT = + dWu

q(t, T ) q(u, T )

t

The ļ¬rst 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 qĻ(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 qĻ(u,T ) du.

Xt

2

The special case of 2.53 of most common usage is the Black-Scholes model:

suppose that Ļ(S, t) = SĻ(t) for Ļ(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 Āµ and variance Ļ 2 . Then we

say that the distribution of X = eZ is lognormal with mean Ī· = exp{Āµ + Ļ 2 /2}

and volatility parameter Ļ. The lognormal probability density function with

mean Ī· > 0 and volatility parameter Ļ > 0 is given by the probability density

function

1

ā exp{ā’(log x ā’ log Ī· ā’ Ļ2 /2)2 /2Ļ 2 }. (2.55)

g(x|Ī·, Ļ) =

xĻ 2Ļ

The solution to (2.18) with non-random functions Ļ(t), r(t) is now

ZT ZT

(r(u) ā’ Ļ 2 (u)/2)du + Ļ(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) ā’ Ļ 2 (u)/2)du and variance t Ļ 2 (u)du. It follows that the

conditional distribution is lognormal with mean Ī· = Xt q(t, T ) and volatility

qR

T2

parameter Ļ (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 payoļ¬ function is V0 (ST ) =

max(ST ā’ E, 0). Now it is helpful to use the fact that for a standard normal

random variable Z and arbitrary Ļ > 0, ā’ā < Āµ < ā we have the expected

value of max(eĻZ+Āµ , 0) is

Āµ Āµ

2

eĀµ+Ļ /2

+ Ļ) ā’ Ī¦( ) (2.57)

Ī¦(

Ļ Ļ

where Ī¦(.) denotes the standard normal cumulative distribution function. As

a result, in the special case that r and Ļ are constants, (2.53) results in the

famous Black-Scholes formula which can be written in the form

V (S, t) = SĪ¦(d1 ) ā’ Eeā’r(T ā’t) Ī¦(d2 ) (2.58)

where

ā

log(S/E) + (r + Ļ 2 /2)(T ā’ t)

ā , d2 = d1 ā’ Ļ T ā’ t

d1 =

Ļ T ā’t

90 CHAPTER 2. SOME BASIC THEORY OF FINANCE

are the values Ā±Ļ2 (T ā’ t)/2 standardized by adding log(S/E) + r(T ā’ t) and

ā

dividing by Ļ T ā’ 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 Ī· = S(t)er(T ā’t) .

The mean of the log-normal in the risk neutral world S(t)er(T ā’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 payoļ¬

function given by V0 (ST ) is the expected value of this function against this

lognormal probability density function, then discounted to present value

Zā

ā

ā’r(T ā’t)

V0 (x)g(x|S(t)er(T ā’t) , Ļ T ā’ t)dx. (2.59)

e

0

Notice that the Black-Scholes derivation covers any diļ¬usion process govern-

ing the underlying asset which is driven by a stochastic diļ¬erential equation of

the form

(2.60)

dS = a(S)dt + ĻSdWt

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 ā’ 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 ā’ t in years,

the annual volatility Ļ 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 ā’ C = Eeā’r(T ā’t) or in this

case 1432.25 + 14.625 ā’ 23 = 1425eā’r(7/252) . We might solve this relation to

obtain the spot interest rate r. In order to conļ¬rm that a diļ¬erent 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 ā’ 152 = 1400eā’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 inļ¬‚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 eļ¬ect 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 diļ¬erence 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 Ī£ is a positive deļ¬nite covariance matrix and Ī· a column vector.

Show that the set of all possible pairs of standard deviation and mean

ā P

return ( wT Ī£w, Ī· 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 oļ¬ered 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 oļ¬er 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 ļ¬ne 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 diļ¬erential 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 = Āµdt + ĻdWt .

Ī²

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

number Ī².

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

dSt = ĀµSt dt + ĻSt dWt

n

Find the diļ¬usion equation satisļ¬ed 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 Āµ. How does this allow constructing

estimators of Ļ that do not require knowledge of the value of Āµ?

9. Consider an Ito process of the form

dSt = a(St )dt + Ļ(St )dWt

Is it possible to ļ¬nd 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 + Ļ(St )dWt

Is it possible to ļ¬nd a function f (St ) which has constant diļ¬usion term?

RT P

g(t)dWt ā

11. Consider approximating an integral of the form g(t){W (t+

0

h) ā’ 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 ā’ 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 ļ¬nd 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 = ĀµYt dt + ĻYt dWt .

Derive a stochastic diļ¬erential 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 = ĀµYt dt + ĻYt dWt

(1) (2)

and Wt , Wt are correlated Brownian motion processes with correlation

Ļ.

13. Prove the Shannon inequality that

X qi

)ā„0

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) = Āµ. Show

that the solution, if it exists, is given by

exp(Ī·f (i))

qi = pi

m(Ī·)

PROBLEMS 95

P m0 (Ī·)

where m(Ī·) = i pi exp(Ī·f (i))] and Ī· is chosen so that = Āµ. (This

m(Ī·)

shows that the closest distribution to P which satisļ¬es the constraint is

dQ

obtained by a simple āexponential tiltā or Esscher transform so that dP (x)

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

15. Let Qā— 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, Qā— ) + H(Qā— , P ).

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

(2.62)

EQ gi (X) = ci

ā—

and suppose we deļ¬ne Pn as the solution of

max H(P )

P

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

ā— ā— ā— ā— ā— ā— ā— ā—

H(Pn , P1 ) = H(Pn , Pnā’1 ) + H(Pnā’1 , Pnā’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 ā’ 1.

Then show that the value of this bond at time t is

exp{ā’(r + k)} ā’ exp{ā’(r + k){T ā’ t)}

Pt = d

1 ā’ exp{ā’(r + k)}

+ pF exp{ā’r(T ā’ t)} + (1 ā’ p)F exp{ā’(r + k)(T ā’ 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 ) Ā· H(pX ).

(b) Show that if X has any discrete distribution over n values, then its

entropy is Ā· log(n).

ńņš. 3 |