15-ArbPric-ContState/Time Page 454 Wednesday, February 4, 2004 1:08 PM
454 The Mathematics of Financial Modeling and Investment Management∂CS( t, t) ∂CS( t, t) 1 ∂^2 CS( t, t)
dYt = ----------------------+ ----------------------μ(St, t) + ----------------------------σ^2 (St, t) dt∂t ∂St (^2) ∂S
t
2
∂CS( t, t)
- ----------------------σ(St, t)dBt
∂St
The second approach yields
1
(
2
dYt = [θtrSt, t)Vt + θt μ(St, t)]dt + θt
2
σ(St, t)dBt
Equating coefficients in dt, Db we obtain the trading strategy
θ^11 ( ∂CS( t, t)
t = ------CSt, t) – ----------------------St
Vt ∂St
θ^2 ∂CS( t, t)
t = ----------------------
∂St
and the PDE
( , ( , ( ,
( , ( , ( ,
∂Cx t) ∂Cx t) 1 ∂^2 Cx t)
- rx t)Cx t) + rx t)---------------------x + ---------------------+ --------------------------σ ,
2
(xt) = 0
∂x ∂t (^2) ∂x^2
with the boundary conditions C(ST,T) = g(ST). Solving this equation we
obtain a candidate option pricing function. In each specific case, one
can then verify that the option pricing function effectively solves the
option pricing problem.
STATE-PRICE DEFLATORS
We now extend the concepts of state prices and equivalent martingale
measures to a continuous-state, continuous-time setting. As in the previ-
ous sections, the economy is represented by a probability space (Ω, ℑ, P)
where Ω is the set of possible states, ℑ is the σ-algebra of events, and P
is a probability measure. Time is a continuous variable in the interval
[0,T]. The propagation of information is represented by a filtration ℑt.