14-Arbitrage Page 439 Wednesday, February 4, 2004 1:08 PM
Arbitrage Pricing: Finite-State Models 439SUMMARY
■ The law of one price states that a given asset must have the same price
regardless of the means by which one goes about creating that asset.
■ Arbitrage is the simultaneous buying and selling of an asset at two dif-
ferent prices in two different markets.
■ A finite-state one-period market is represented by a vector of prices and
a matrix of payoffs.
■ A state-price vector is a strictly positive vector such that prices are the
product of the state-price vector and the payoff matrix.
■ There is no arbitrage if and only if there is a state-price vector.
■ A market is complete if an arbitrary payoff can be replicated by a port-
folio.
■ A finite-state one-period market is complete if there are as many lin-
early independent assets as states.
■ A multiperiod finite-state economy is represented by a probability
space plus an information structure.
■ In a multiperiod finite-state market each security is represented by a
payoff process and a price process.
■ An arbitrage is a trading strategy whose payoff process is nonnegative
and not always zero.
■ A market is complete if any nonnegative payoff process can be repli-
cated with a trading strategy.
■ A state-price deflator is a strictly positive process such that prices are
random variables equal to the conditional expectation of discounted
payoffs.
■ A martingale is a process such that at any time t its conditional expec-
tation at time s, s > t coincides with its present value.
■ In absence of arbitrage there is an artificial probability measure in
which all price processes, appropriately discounted, become martin-
gales.
■ Given a probability measure P, the probability measure Q is said to be
equivalent to P if both assign probability zero to the same events.
■ The binomial model assumes that there are two positive numbers, d,
and u, such that 0 < d < u and such that at each time step the price S of
the risky asset changes to dS or to uS.
■ The distribution of prices of a binomial model is a binomial distribu-
tion.
■ The binomial model is complete.
■ The Arbitrage Pricing Theory (APT) asserts that each asset’s return is
equal to the risk-free rate plus a linear combination of factors.
■ The APT can be tested with maximum likelihood methods.