The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 416 Wednesday, February 4, 2004 1:08 PM

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The Mathematics of Financial Modeling and Investment Management

Risk-Neutral Probabilities

Probabilities computed according to the equivalent martingale measure

Q are the risk-neutral probabilities. Risk-neutral probabilities can be

explicitly computed. Here is how. Call qωthe risk-neutral probability of

state ω. Let’s write explicitly the relationship

i

S Q dj

t

i = E

t ---------

Rtj,

as follows:

q i ω i ω

ω

T d

j() qω

T d

= j()

SAkt ∑ ------------------- ∑ -------------- = ∑ ---------------------------- ∑ --------------

ω ∈AktQA(^ kt)j = t + 1 Rtj, ω ∈Akt 

 ∑ qω

 j = t + 1 Rtj



,

ω ∈Akt

The above system of equations determines the risk-neutral probabil-

ities. In fact, we can write, for each risky asset, Mt linear equations,

where Mt is the number of sets in the partition It plus the normalization

equation for probabilities. From the above equation, one can see that

the system can be written as

T di ω

j()^

∑ qω ∑ --------------– SA

i

kt =^0

ω ∈ Akt, j = t + 1 Rtj,

S

∑qω =^1

ω= 1

This system might be determined, indetermined, or impossible. The

system will be impossible if there are arbitrage opportunities. This sys-

tem will be indetermined if there is an insufficient number of securities.

In this case, there will be an infinite number of equivalent martingale

measures and the market will not be complete.

Now consider the relationship between risk-neutral probabilities and

state-price deflators. Consider a probability measure P and a nonnegative

random variable Y with expected value on the entire space equal to 1.

Define a new probability measure as Q(B) = E[1BY] for any event B and

where 1B is the indicator function of the event B. The random variable Y

is called the Radon-Nikodym derivative of Q and it is written