ch2 - Chapter 2 Options, Arbitrage, Martingales The aim of...

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Chapter 2 Options, Arbitrage, Martingales The aim of this chapter is to appreciate the key connection between arbitrage- free markets and the existence of equivalent measures under which the dis- counted stock price becomes a martingale. 2.1 The Arbitrage-Martingale Connection You have already met the concept of arbitrage in PAS367/6051. We now make this more mathematically precise. Defnition A self-Fnancing strategy φ is said to be an arbitrage oppor- tunity if V φ (0) = 0 ,V φ ( T ) 0 and P ( V φ ( T ) > 0) > 0 . A market is said to be arbitrage-free if there are no arbitrage opportunities. So if arbitrage opportunities exist, there is a non-zero probability that your portfolio can create wealth with no investment ! We are now going to explore the remarkable relationship between arbitrage- free markets and martingales. ±irst recall that two probability measures P and P are equivalent if they have the same sets of measure zero (i.e. A ∈F and P ( A ) = 0 if and only if P ( A ) = 0 - see Chapter 1 of PAS401/6052 and Problem 9). In the case we are considering of a Fnite market model we imposed the condition in Chapter 1 that P ( A ) > 0 for all non-empty A . Conse- quently in this context P is equivalent to P if and only if P ( A ) > 0 for all non-empty A . 9
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Defnition A probability measure P on (Ω , F ) is said to be a martingale measure for ( ± S ( n ) ,n ∈T )i f 1. P is equivalent to P . 2. ( ± S ( n ) )isa P -martingale. 1 In the sequel, all expectations of random variables computed using P will be denoted as E . The next result is of fundamental importance. Indeed it is usually called the frst Fundamental theorem oF asset-pricing . It gives a direct link between arbitrage-free markets on the one hand and existence of martingale measures on the other. Theorem 2.1.1 The market is arbitrage-Free iF and only iF there exists at least one martingale measure. ProoF. The full proof is difficult and uses concepts which lie outside the scope of this course. If you want to learn about these - go to section 4.2 in Bingham and Kiesel. We’ll just give the easy part of the proof, i.e. we show that if a martingale measure exists, then the market is arbitrage-free. If a martingale measure exists then the discounted wealth process ( ± V φ ( n ) ) is a P -martingale (see Problem 10) and so if φ is a self-Fnancing portfolio and if ± V φ (0) = 0, then E ( ± V φ ( T )) = E ( ± V φ (0)) = 0. Now suppose that ψ is an arbitrage opportunity, then by Problem 8, V ψ (0) = 0 ,V ψ ( T ) 0 and E ( V ψ ( T )) > 0 . Let v 1 ,...,v M be the values of the random variable V ψ ( T ). Since V ψ ( T ) 0wemu s thave v i 0 for all 1 i M. Let the probability distribution of V ψ ( T )be p i = P ( V ψ ( T )= v i ) for 1 i M . Since E ( V ψ ( T )) = M ² i =1 v i p i > 0 , we must have v i > 0 for at least one value of i . Let i 0 be any such value of i . Recall that each p i > 0 (since it is the probability of an event in F and we are assuming that these are all positive). As P is equivalent to P we have each p i = P ( V ψ ( T v i ) > 0. Hence E ( V ψ ( T )) v i 0 p i 0 > 0 . Since β ( T ) > 0 we deduce that E ( ± V ψ ( T ))
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ch2 - Chapter 2 Options, Arbitrage, Martingales The aim of...

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