15-ArbPric-ContState/Time Page 451 Wednesday, February 4, 2004 1:08 PM
Arbitrage Pricing: Continuous-State, Continuous-Time Models 451∂CS( t,t) ∂CS( t,t) 1 ∂^2 CS( t,t) 2
σ
2- rC S( t,t)+ r----------------------St + ----------------------+ ----------------------------St = 0
∂St ∂t (^2) ∂S
t
2
If the function C(St,t) satisfies this relationship, then the coefficients
in dt match. The above relationship is a partial differential equation
(PDE). In Chapter 9 we discussed how to solve this equation with suit-
able boundary conditions. Boundary conditions are provided by the
payoff of the option at the expiry date:
YT = CS( T,T)= max(ST – K, 0 )
The closed-form solution of the above PDE with the above boundary
conditions was derived by Fischer Black and Myron Scholes^3 and
referred to as the Black-Scholes option pricing formula:
CS( t,t)= xΦ()z – e –rT ( – t)KΦ(z – σ Tt– )
with
log (S ^12 –
t ⁄K)+ r + ---σ
(Tt)
2
z = --------------------------------------------------------------------------
σ Tt–
and where Φis the cumulative normal distribution.
Let’s stop for a moment and review the logical steps we have fol-
lowed thus far. First, we defined a market made by a stock whose price
process follows a geometric Brownian motion and a bond whose price
process is a deterministic exponential. We introduced into this market a
European call option. We then made two assumptions: (1) The option’s
price process is a deterministic function of the stock price process; and
(2) the option’s price process can be replicated by a self-financing trad-
ing strategy.
If the above assumptions are true, we can write a stochastic differ-
ential equation for the option’s price process in two different ways: (1)
Using Itô’s lemma, we can write the option price stochastic process as a
function of the stock stochastic process; and (2) using the assumption
that there is a replicating trading strategy, we can write the option price
(^3) Fischer Black and Myron Scholes, “The Pricing of Options and Corporate Liabili-
ties,” Journal of Political Economy 81 (1973), pp. 637–654.