A Probability Path.pdf

# 0 corollary 10161 suppose the market is viable and p

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0 Corollary 10.16.1 Suppose the market is viable and P* is an equivalent martin- gale measure making {(Sn, Bn), 0::: n ::: N} a P*-martingale. Then is a P* -martingale for any self-financing strategy ¢. Proof. If¢ is self-financing, then by (10.65) n Vn(¢) = Vo(¢) + L(cPj• dj) j=l so that { (V n ( ¢), Bn), 0 ::: n ::: N} is a P *-martingale transform and hence a martingale. 0

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10.16 Fundamental Theorems of Mathematical Finance 425 10.16.4 Complete Markets A contingent claim or European option of maturity N is a non-negative BN = B-measurable random variable X. We think of the contingent claim X paying X (w) 2:: 0 at time N if the state of nature is w. An investor may choose to buy or sell contingent claims . The seller has to pay the buyer of the option X (w) dollars at time N. In this section and the next we wlfl see how an investor who sells a contingent claim can perfectly protect himself or hedge his move by selling the option at the correct price . Some examples of contingent claims include the following. European call with strike price K. A call on (for example) the first asset with strike price K is a contingent claim or random variable of the form X = - K)+ . If the market moves so that > K, then the holder of the claim receives the difference. In reality, the call gives the holder the right (but not the obligation) to buy the asset at price K which can then be sold for profit K)+. European put with strike price K. A put on (for example) the first asset with strike price K is a contingent claim or random variable of the form X = (K- The buyer of the option makes a profit if < K . The holder of the option has the right to sell the asset at price K even if the market price is lower . There are many other types of options, some with exotic names: Asian options, Russian options, American options and passport options are some examples . A contingent claim X is attainable if there exists an admissible strategy </J such that The market is complete if every contingent claim is attainable. Completeness will be shown to yield a simple theory of contingent claim pricing and hedging. Remark 10.16.1 Suppose the market is viable. Then if X is a contingent claim such that for a self-financing strategy cpwe have X= VN(</J), then <Pis admissible and hence X is attainable. In a viable market , if a contingent claim is attainable with a self-financing strat- egy, this strategy is automatically admissible. To see this, suppose P* is an equiva- lent martingale measure making { (Sn, Bn), 0 ;::: n ;::: N} a vector martingale . Then if <Pis self-financing, Lemma 10 . 16.1 implies that { (V n(</J ), Bn ), 0 ;::: n ;::: N} is a P* -martingale. By the martingale property However , 0 :::: X = V N ( <P) and hence V N ( <P) 2:: 0 so that V n ( </J) > 0 for 0 :::: n :::: N, and hence <P is admissible. o
426 10 . Martingales Theorem 10.16.2 Suppose the market is viable so an equivalent martingale mea- sure P* exists. The market is also complete iff there is a unique equivalent mar- tingale measure.

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