The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 446 Wednesday, February 4, 2004 1:08 PM

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446 The Mathematics of Financial Modeling and Investment Management

1. Prices Sti are equal to the normalized conditional expectation of pay-
   offs deflated with state prices under the real probabilities:

T

1

St

i

= -----Et ∑ πjdj

i

πt j = t + 1

2. Prices Sti are equal to the conditional expectation of discounted payoffs
   under the risk-neutral probabilities

T

Q dj

i

Sti = Et ∑

Rtj

j = t + 1 ,

State-price deflators and risk-neutral probabilities can be computed solv-

ing systems of linear equations for a kernel of basic assets. The above

relationships are algebraic linear equations that fix all price processes.

In a continuous-state, continuous-time setting, the principle of arbi-

trage pricing is the same. In the absence of arbitrage, given a number of

basic price and payoff stochastic processes, other processes are fixed.

The latter are called redundant securities as they are not necessary to fix

prices. If markets are complete, every price process can be fixed in this

way. In order to make computations feasible, some additional assump-

tions are made, in particular all payoff-rate and price processes are

assumed to be Itô processes.

The theory of arbitrage pricing in a continuous-state, continuous-

time setting uses the same tools as in a discrete-state, discrete-time set-

ting. Under an equivalent martingale measure, all price processes

become martingales. Therefore prices can be determined as discounted

present value relationships. Equivalent martingale measures are the

same concept as state-price deflators: After appropriate deflation, all

processes become martingales. The key point of arbitrage pricing theory

is that both equivalent martingale measures and state-price deflators can

be determined from a subset of the market. All other processes are

redundant.

In the following sections we will develop the theory of arbitrage

pricing in steps. First, we will illustrate the principles of arbitrage pric-

ing in the case of options, arriving at the Black-Scholes option pricing

formula. We will then extend this theory to more general derivative

securities. Subsequently, we will state arbitrage pricing theory in the

context of equivalent martingale measures and of state-price deflators.