15-ArbPric-ContState/Time Page 450 Wednesday, February 4, 2004 1:08 PM

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450 The Mathematics of Financial Modeling and Investment Management

or, in differential form, as

dY^12 12

t = θt dVt + θt dSt = (θt rVt + θt μSt)dt + θt

(^2) σS
tdBt
If the trading strategy replicates the option price process, the two
expressions for dYt—the one obtained through Itô’s lemma and the
other obtained through the assumption that there is a replicating self-
financing trading strategy—must be equal:
(θ^12
t rVt + θt μSt)dt + θt
(^2) σS
tdBt
∂CS( t, t) ∂CS( t, t) 1 ∂^2 CS( t, t) 2 ∂CS( t, t)
= ----------------------+ ----------------------Stμ + ----------------------------St σ^2 dt + ----------------------σStdBt
∂t ∂St (^2) ∂S
t
(^2) ∂St
The equality of these two expressions implies the equality of the
coefficients in dt and dB respectively. Equating the coefficients in dB
yields,
θt
2

∂CS( t, t)
∂St

As Yt = CS( t, t) = θt^1 Vt + θt^2 St, substituting, we obtain

θt

1

=

1

Vt

• CS( t, t) –

∂CS( t, t)

∂St

• St

We have now obtained the self-financing trading strategy in function of

the stock and option prices. Substituting and equating the coefficients of

dt yields,

1 ( (

(

∂CSt, t) ∂CSt, t)

------ CSt, t) – ----------------------St rVt + ----------------------μSt

Vt ∂St ∂St

∂CSt, t) ∂CSt, t) 1 ∂

(^2) CS
= ----------------------( + ----------------------( S ( t, t)^2 σ^2
tμ + ----------------------------St
∂t ∂St^2 ∂S
t
2
Simplifying and eliminating common terms, we obtain