Security Markets
X
Miloslav S. Vosvrda
Theory of Capital Markets
A continous-time version of the
crown valuation
The problem solved by the Black-Scholes Option
Pricing Formula is a special case of the following
continuous-time version of the crown valuation
problem, treated in a binomial random walk setting.
We are given the riskless security defined by a
constant interest rate r and a risky security whose
price process S is described by
dS t  mS t dt  vS t dBt ,
with dividend rate   0.
t  0,
We are interested in the value of a security, say
a crown, that pays a lump sum of g  S T 
at a future time T , where is g sufficiently well
behaved to justify the fillowing calculations. (It is
certainly enough to know that g is bounded and
twice continuosly differentiable with a bounded
derivative.)
In the case of an option on the stock with exercise
price K and exercise date T , the payoff
function is defined by gST   ST  K   maxST  K,0,
which is sufficiently well behaved. We will suppose
that the value of the crown at any time t  0, T is
C St , t  ,where C is a function that is twice
continuously differentiable for t  0, T .
In particular, C ST , T   g ST .
For convenience, we use the notation
C
C
C
 C (s,t)
(s,t) 
;
 s
s
ss
t
(s,t) 

2
C (s,t)
;
2
 s
 C (s,t)
(s,t) 
.
 t
We can solve the valuation problem by explicity
determining the function C . For simplicity, we
supose that the riskless security is a discount bond
maturing after T, so that its market value  t at
rt

e
. Suppose an investor decides to
time t is
0
hold the portfolio at , bt  of stock and bond at any
time t , where a t  C s  S t , t  and
bt  C S t , t   C s S t , t S t  /  t .
This particular trading strategy has two special
properties. First, it is self-financing, meaning that it
requires an initial investment of a 0 S 0  b0  0 ,
but neither generates nor requires any further funds
after time zero. To see this fact, one must only show
t
t
that
at St  bt   a0S0  b0 0   a dS   b d .
0
0
The left hand side is the market value of the
portfolio at time t; the right hand side is the sum of
its initial value and any interim gains or losses from
trade.
This equation can be verified by an application of
Ito’s Lemma in the following form,
If f : R 2  R is twice continuously differentiable and
X defined by the stochastic differential equation
dX k   X t dt   X t dBt , t  0
,
then for any time t  0,
t
t  f ( X , )

f ( X t , t )  f ( X 0 ,0)   Df ( X , )d  
 ( X )dB ,
0
0
x
where
 f ( x, t )  f ( x, t )
1  2 f ( x, t )
2
Df ( x, t ) 

( x) 

(
x
)
.
2
t
x
2 x
The second important property of the trading
strategy at , bt  is the equality
a t S t  bt  t  C ( S t ),
which follows immediately from the definitions of
at and bt
. From Ito’s Lemma, we have

t
0
t
DC ( S  ,  )d    C s ( S  ,  )vS  dB 
0
t
t
0
0
  a dS    b d    0 .
Using
and
dS   mS  d   vS  dB
d    r  d  ,
(*)
we can collect the terms in d and dB  separately.
If (*) holds, the integrals involving d and dB 
must separately sum to zero. Collecting the terms in
d alone,

t
0
[ C t  S  ,    C s  S  ,  rS  
1 2 2
 v S  C ss  S  ,    rC  S  ,  ]d   0 .
2
for all t  0, T . But then this expresion implies
that C must satisfy the partial differential equation
1 2 2


0 Ct (S ,) Cs (S ,)rS  2v S Css(S ,) rC(S ,)dr0,
t
for s , t   0,    0, T . We have the boundary
C s , T   g s , s  0 .
condition
The partial differential equation with boundary
condition can be shown to have the solution

C s , t   E e
 r T  t 

g se
Z
,
where Z is normally distributed with mean
2
2
T  t .
v
and
variance
T  t  r  v / 2


For the case of the call option payoff function,
g s   s  K  ,

we can quickly check that C S ,0 
is precisely the Black-Scholes Option Pricing
Formula. More generally,

C s , t   E e
 r T  t 

g se
Z
,
can be solved numerically by standard Monte Carlo
simulation and variance reduction methods.