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Unformatted text preview: EC3070 FINANCIAL DERIVATIVES BLACK–SCHOLES OPTION PRICING
The Diﬀerential Equation The BlackScholes model of option pricing assumes that the price St of the underlying asset has a geometric Brownian
motion, which is to say that
dS = µS dt + σ S dw, (1) where µ and σ are constant parameters.
Let f = f (S, t) be the price of a derivative, which might be a call option
contingent on the price S of the underlying asset. Then, according to Ito’s
lemma, there is
Ω
æ
2
∂f
∂f
∂f
2 21 ∂ f
df = µS
+
+σ S
dt + σ S
dw.
(2)
2
∂S
∂t
2 ∂S
∂S Because the forcing function in both equations (1) and (2) is the same
Wiener process, it is possible to construct a portfolio that eliminates risk, which
is to say that the Wiener increment dw can be eliminated from the equation
expressing the value of the portfolio.
The relevant portfolio comprises ∂ f /∂ S units of the asset and 1 unit of
the derivative or option as a liability. The value of the portfolio at time t is
given by
∂f
Vt =
St − ft ;
(3)
∂S
and the change in its value is
∂f
dVt =
dSt − dft .
(4)
∂S
Substituting (1) and (2) into the latter and cancelling various terms gives
Ω
æ
2
∂f
2 21 ∂ f
dVt = −
+σ S
dt,
(5)
∂t
2 ∂S2
which does not involve the stochastic increment dw.
On the instant, this portfolio must earn the same as a sum Vt invested in
a riskless asset at a rate of return of r, which is to say that
dVt = rVt dt.
By putting (5) into the LHS of this and (3) into the RHS, we get
Ω
æ
Ω
æ
2
∂f
∂f
2 21 ∂ f
+σ S
dt = r ft −
St dt
∂t
2 ∂S2
∂S (6) (7) whence ∂f
1 ∂2f
∂f
+ σ2 S 2
+r
dSt = rft .
2
∂t
2 ∂S
∂S
This it the BlackScholes diﬀerential equation for option pricing.
1 (8) ...
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This note was uploaded on 03/02/2012 for the course EC 3070 taught by Professor D.s.g.pollock during the Spring '12 term at Queen Mary, University of London.
 Spring '12
 D.S.G.Pollock

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