UVA-F-0945
DERIVATION OF THE BLACK-SCHOLES OPTION-PRICING MODEL
Assume that we have one asset, W, whose price at time t, W(t), depends on the
price of another asset at time t, S(t). In this case, we can write the price of W(t) as a
function of the price of S at time t, or W(t) = F[S(t), t]. The simplest way of thinking of
this is to think of the price of something depending on something else. For example, the
prices of bonds depend on interest rates. In the discussion that follows, we focus on call
options, where the price of the call option depends on the stock price. The basic
definitions we will use are
W(t) = price of options at t, and
W(t) = F[S(t),t], where S(t) is stock price at time t.
Assume dS(t) = asSdt + σsSdz. This stochastic differential equation assumes that
the change in the stock price over time, dS(t), follows a log-normal distribution.
Additional definitions we will use are
N1(t) = number of shares of S,
N2(t) = number of options,
Q(t) = money in Rf, and
P(t) = value of the portfolio.
The value of a portfolio with holdings in the stock, option, and risk-free asset at time t is
P(t)= N 1 (t)S(t) + N 2 (t)W(t) + Q(t), 1
where N1(t)S(t) is the value of the position in the stock, N2(t)W(t) is the value of the
position in the options, and Q(t) is the dollar amount in the risk-free asset at time t.
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UVA-F-0945
The change in the value of P(t) over time can be defined as
dP = d( N 1 S) + d( N 2 W) + dQ,
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which we can rewrite as
dP =
d( N 1 S)
d( N 2 W)
dQ
 N1 S +
 N 2W +
 Q.
Q
N1 S
N 2W
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Then d(N1S) and d(N2W) can be decomposed into two parts as follows:
d( N 1 S) dN 1 dS
=
+
S
N1 S
N1
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d( N 2 W) dN 2 dW
=
+
.
W
N 2W
N2
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and
Assuming
dN 1
dN 2
 0 and
0 ,
N1
N2
then
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d( N 1 S) dS
d( N 2 W) dW
=
and
=
.
S
W
dN 1 S
N 2W
The assumption that the percentage change in N1 and N2 is almost zero implies
that any adjustments are very small per unit of time. Because the dollars invested in the
risk-free asset earn the risk-free rate, Rf, dQ = QRfdt, with dt as the time measure.
time:
With these assumptions, we can write the change in the portfolio value over
Because
dQ = QR f dt,
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dP =
dS
dW
( N 1 S) +
( N 2 W) + R f Qdt.
S
W
Now let
dW
= a w dt +  w dZ.
W
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UVA-F-0945
This stochastic differential equation describes how the price of the option, W(t), changes
over time.
Using this equation and the definition for the stochastic differential equation for
the stock price, dS(t), we can do the following:
Substituting for
dS
dW
and
,
S
W
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dP = ( as dt + s dZ) N 1 S + ( aw dt + w dZ) N 2 W + R f Qdt.
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Dividing by P yields
dP
N S
N W Rf Q
= ( as dt +  s dZ) 1 + ( aw dt +  w dZ) 2 +
dt
P
P
P
P
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The next step is to define weights for the portfolio. If P = N1S + N2W + Q, then
(N1S)/P is the percentage of the value of P invested in the stock. The same would be
true for the options and the risk-free asset. Define
1= N1
and substitute to yield
or, collecting terms,
P
S
W
Q
3= ,
P
P
2= N2
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dP
= ( a s dt +  s dZ)  1 + ( a w dt +  w dZ)  2 + R f  3 dt,
P
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dP
= ( a s  1 + a w  2 + R f  3 )dt + (  s  1 +  w  2 )dZ.
P
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At this point, we have a stochastic differential equation for the portfolio. If we
would like to make the portfolio risk free, we must eliminate the stochastic part of the
equation or the dZ part. We can do this as follows. Set
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(  s  1 + w 2 )= 0
or
 
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1= - w w ,
s
dP
and the result is that
is nonstochastic.
P
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UVA-F-0945
For this portfolio, the expected return is
 dP 
E   = ( a s  1 + a w  2 + R f  3 ) dt,
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 P 
but if the return is nonstochastic, it must be fixed per time unit, and the resulting
portfolio should earn the risk-free rate of return, or
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( as  1 + aw  2 + R f  3 )dt = R f dt.
At this point, we have three structural equations relating to the risk-free portfolio
described previously:
 *1 +  *2 +  *3 = 1
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  *1 +  w  *2 = 0
and
as  1 + aw  2 + R f (1 -  1 -  2 ) = R f .
*
*
*
*
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Solving yields
as - r
s
=
aw - r
w
.
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The result is an arbitrage condition that must hold at all times.
We can now use this result to solve the differential equation for the option price.
The key is to use the no-arbitrage result and Ito's Lemma to derive values for aw and
σw. By Ito's Lemma, if y(t) = y(x,t), where dx = f(x,t)dt + σ(x,t)dZ, then
1
dy = ( yt + y x  f(x,t) +  y xx   2 (x,t)) dt + y x   (x,t) dZ.
2
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For W(t)=F(x,t), where dS = asSdt + σsSdZ, then
1


dW =  F t + a s  S  F s +  F xx   2s  S 2  dt + F s  S   s  dZ.
2


Thus, if W(t) = F(S,t) = awWdt + σwWdZ, then aw and σw can be written as
1
2
2
a w  W = F t + a s  S  F s +  F ss    S
2
and
 w W =  s  S  F s .
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From the no-arbitrage condition, we know that
as - R f aw - R f
=
,
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s
w
so we can use the definitions of aw and σw we found to substitute into the no-arbitrage
condition. The result is
1 2 2
F
t + a s S F s +  S F ss - R f W
as R f
2
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=
.
s
 s S Fs
Noting that W = F, we rearrange this equation to
1
2
a s  S  F s - R f  S  F s = F t + a s  S  F s +  F ss    S - R f  F. 28
2
Canceling terms and bringing everything to the right-hand side of the equation yields
the basic Black-Scholes stochastic differential equation:
1
0 = R f  S  F s + F t +  F ss   2s  S - R f  F.
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In order to derive the exact option-pricing formula, we need to add the boundary conditions that
apply to a call option, which are W(T)=Max [0, ST - X] and W(t)  0 for all t, where T is
the maturity date of the call option and X is the strike price. The solution to the
stochastic differential equation turns out to be well known as the heat-transfer equation.
The solution in this case is the Black-Scholes option-pricing model:
Call price = S  N( d 1 ) - X  eR f  (T-t)  N( d 2 ),
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where S= current stock price, X= strike price, (T-t)= time to maturity, Rf= risk-free rate
of interest, N(.) is the cumulative standard normal, and
S 
2
ln   +  R f +   (T - t)
2 
X 
d1=
  T -t
S 
2
ln   +  R f -   (T - t)
2 
X 
= d1 -   T - t .
d2=
  T -t
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