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Arbitrage Theory In Continuous Time
Second Edition


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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 ofarbitrage conditions hold: d $ (1 +R) $ u.

if and only if the following

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 arbitrageopportunity. 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 obviouslyhave Vh = 0, and as for t = 1 we have O

which by assumption is positive.

= 8(1 + R) -




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 _{YS [u - (1 + R)], if Z = u. I ys [d - (1 + R)], if Z = d.
Assl,lme nO'Y.that Y > O. Then h is an arbitrage strategy if and only if we have the mequahtles

u> 1 +R, d> 1 +R,
but this is impossible because of the condition (2.1). The case y similarly.


0 is treated


At first glance this result is perhaps only moderately exciting, but we may write it in a more suggestive form. To saythat (2.1) holds is equivalent to saying that 1 + R is a convex combination of u and d, Le.

1 + R .= qu . U + qd . d,
where q" qd ~ 0 and qu qd = 1. In particular we see that the weights qu and qd can be interpreted as probabilities for a new probability measure Q with the property Q(Z = u) = q" Q(Z = d) = qd. Denoting expectation w.r.t. this measure by Eq we now have the following easycalculation


We thus have the relation

which to an economist is a well-known relation. It is in fact a risk neutral valuation formula, in the sense that it gives today's stock price as the discounted expected value of tomorrow's stock price. Of course we do not assume that the agents in our market are risk neutral-what we have shown is only that if we use the Q-probabi1itiesinstead of theobjective probabilities then we have in fact a risk neutral valuation of the stock (given absence of arbitrage). A probability measure with this property is called a risk neutral measure, or alternatively a risk adjusted measure or a martingale measure. Martingale measures will playa dominant role in the sequel so we give a formal definition.



Definition 2.4 Aprobability measure the following condition holds:

Q is called a martingale measure if

\e may now state the condition of no arbitrage in the following way.

Proposition 2.5 The market model is arbitrage free a martingale measure Q.

if and only if there exists

Rr the binomial model it is easy to calculate the martingale probabilities. The rroof is left to the reader.
Proposition 2.6 For...
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