Arbitrage Theory In Continuous Time
Second Edition

•

TOMAS BJORK

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with probability 1.

An arbitrage portfolio is thus basically a deterministic money making machine, and \e interpret the existence of an arbitrage portfolio as equivalent to a serious case of mispricing on the market. It is now natural to investigate when a given market model is arbitrage free, Le. when there are no arbitrage portfolios.

Proposition 2.3 The model above is free of arbitrage conditions hold: d $ (1 +R) $ u.

if and only if the following
(2.1)

Proof The condition (2.1) has an easy economic interpretation. It simply says that the return on the stock is not allowed to dominate the return on the bond and vice versa. To show that absence of arbitrage implies (2.1), we assume that (2.1) does in fact not hold, and then we show that this implies an arbitrage opportunity. Let us thus assume that one of the inequalities in (2.1) does not hold, so that ~ have, say, the inequality 8(1 + R) > su. Then we also have 8(1 + R) > sd 9J it is always more profitable to invest in the bond than in the stock. An arbitrage strategy is now formed by the portfolio h = (s, -1), Le. we sell the stock short and invest all the money in the bond. For this portfolio we obviously have Vh = 0, and as for t = 1 we have O

V1h which by assumption is positive.

= 8(1 + R) -

8Z,

8

THE BINOMIAL MODEL

Now assume that (2.1) is satisfied. To show that this implies absence of h arbitrage let us consider an arbitrary portfolio such that VO = O. We thus have x ys = 0, Le. x = -ys. Using this relation we can write the value of the portfolio at t = 1 as

+

If,h _