The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 426 Wednesday, February 4, 2004 1:08 PM

426 The Mathematics of Financial Modeling and Investment Management

( –

y tk–

PSt = ukdtkS 0 ) = pk( 1 – p)

k

Next introduce a risk-free security. In the setting of a binomial

model, a risk-free security is simply a security such that d = u = 1 + r

where r > 0 is the positive risk-free rate. To avoid arbitrage it is clearly

necessary that d < 1 + r < u. In fact, if the interest rate is inferior to both

the up and down returns, one can make a sure profit by buying the risky

asset and shorting the risk-free asset. If the interest rate is superior to

both the up and down returns, one can make a sure profit by shorting

the risky asset and buying the risk-free asset. Denote by bt the price of

the risk-free asset at time t. From the definition of price movement in

the binomial model we can write: bt = (1 + r)tb 0.

Risk-Neutral Probabilities for the Binomial Model

Let’s now compute the risk-neutral probabilities. In the setting of bino-

mial models, the computation of risk-neutral probabilities is simple. In

fact we have to impose the condition:

q Q

t = Et [qt + 1 ]

which we can explicitly write as follows:

quSt + ( 1 – q)dSt

St = --------------------------------------------

1 + r

1 + r = qu + d – qd

1 + rd–

q = ---------------------

ud–

u – 1 – r

1 – q = ---------------------

ud–

As we have assumed 0 < d < 1 + r < u, the condition 0 <<q 1 holds.

Therefore we can state that the unique risk-neutral probabilities are

1 + rd–

q = ---------------------

ud–