15-ArbPric-ContState/Time Page 469 Wednesday, February 4, 2004 1:08 PM
Arbitrage Pricing: Continuous-State, Continuous-Time Models 469■ The Black-Scholes option pricing formula is obtained solving the par-
tial differential equation implied by the equality of the replicating self-
financing trading strategy and the option price process.
■ A deflator is any strictly positive Itô process; a state-price deflator is a
deflator with the property that the deflated price process is a martin-
gale.
■ If there is a (regular) state-price deflator then there is no arbitrage; the
converse is true only under a number of technical conditions.
■ Two probability measures are said to be equivalent if they assign prob-
ability zero to the same event.
■ Given a process X on a probability space with probability measure P,
the probability measure Q is said to be an equivalent martingale mea-
sure if it is equivalent to P and X is a martingale with respect to Q
(plus other conditions).
■ If there is a regular deflator such that the deflated price process admits
an equivalent martingale measure, then there is no arbitrage.
■ Under the equivalent martingale measure, all Itô price processes have
the same drift.
■ In the absence of arbitrage, a market is complete if and only if there is a
unique equivalent martingale measure.