0 corollary 10161 suppose the market is viable and p

Info icon This preview shows pages 436–439. Sign up to view the full content.

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
Image of page 436

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

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
Image of page 437
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.
Image of page 438

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Image of page 439
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern