15-ArbPric-ContState/Time Page 449 Wednesday, February 4, 2004 1:08 PM
Arbitrage Pricing: Continuous-State, Continuous-Time Models 449θ^12
t Vt + θt St < YtAn investor could then sell the option for Yt, make an investment
1 2
θt Vt + θt St in the trading strategy, and at time T liquidate both the
1 2
option and the trading strategy. As θTVT + θt ST = YT the final liquida-
tion has value zero in every state of the world, so that the initial profit
1 2
Yt – θt VT + θt ST is a risk-free profit. A similar reasoning could be
applied if, at any time t < T, the strategy (θt
1 2
, θt ) had a value higher
than the option. Therefore, we can conclude that if there is a self-financ-
ing trading strategy that replicates the option’s payoff, the value of the
strategy must coincide with the option’s price at every instant prior to
the expiry date.
Observe that the above reasoning is an instance of the law of one
price that we discussed in the previous chapter. If two portfolios have
the same payoffs at every moment and in every state of the world, their
price must be the same. In particular, if a trading strategy has the same
payoffs of an asset, its value must coincide with the price of that asset.The Black-Scholes Option Pricing Formula
Let’s now see how the price of the option can be computed. Assume that
the price of the option is a function of time and of the price of the
underlying stock: Yt = C(St,t). This assumption is reasonable but needs
to be justified; for the moment it is only a hint as to how to proceed
with the calculations. It will be justified later by verifying that the pric-
ing formula produces the correct final payoff.
As St is assumed to be an Itô process, in particular a geometric
Brownian motion, Yt = C(St,t)—which is a function of St—is an Itô pro-
cess as well. Therefore, using Itô’s formula, we can write down the sto-
chastic equation that Yt must satisfy. Recall from Chapter 8 that Itô’s
formula prescribes that:∂CS( t, t) ∂CS( t, t) 1 ∂ ( ∂CS( t, t)
2
CSt, t) 2
dYt = ----------------------+ ----------------------Stμ + ----------------------------St σ^2 dt + ----------------------σStdB∂t ∂St (^2) ∂S
t
(^2) ∂St
1
Suppose now that there is a self-financing trading strategy Yt = θt Vt
2
- θt St. We can write this equation as
t t t
∫dYt = θt^1 ∫ dVt + θt^2 ∫ dSt
0 0 0