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# Arbitrage

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### Arbitrage

1. 1. ARBITRAGE María José González
2. 2. Problem: Given a finite matrix of prices for traded European options on a financial asset, is there an arbitrage opportunity? Example : Call Options - 2 3 months 3 - 1 month 120 100 T/K - - 21 6 months 1 - 3 months - 12 - 1 month 120 100 80 T/K
3. 3. Fundamental Theorem of Asset Pricing : The existence of a martingale measure for the (discounted) price process of the stock is “essentially” equivalent to the absence of arbitrage opportunities. ARBITRAGE
4. 4. If is a martingale: . C(k,T) is non-increasing in K . C(K,T) is convex in K . C(K,T) is non-decreasing in T, since is a submartingale.
5. 5. Consequently, if this martingale were consistent with the given call quotes, these quotes would have to satisfy the same constrains. x>3 and x<2 arbitrage x<21, y >12 and (y-x)< (1-y), i.e. y<(1+x)/2, arbitrage - 2 3 months 3 x 1 month 120 100 T/K - - 21 6 months 1 y x 3 months - 12 - 1 month 120 100 80 T/K
6. 6. <ul><li>Collection of strikes: </li></ul><ul><li>Collection of maturities: </li></ul><ul><li>Data : </li></ul><ul><li>NOTE: By assumption the collection of strikes is the same for all maturities, (might fail to hold in practice) </li></ul><ul><li>Alternative PROBLEM: How to interpolate the missing data?. </li></ul>Simple setting: Carr-Madan (2005)
7. 7. <ul><li>Theorem : The rectangular grid of European call quotes are free of arbitrage if and only if: </li></ul><ul><li>Vertical spreads : </li></ul><ul><li>Butterfly spreads: </li></ul><ul><li>Calendar spreads: </li></ul><ul><li>This result has been extended by Hobson & Davis , and by Cousot to allow different strike prices at the different maturities. </li></ul>
8. 8. Results due to Hardy-Littlewood-Polya (convex ordering of measures) or later by Kellerer and Sherman (fitting martingales to given marginals),imply: There exists a martingale measure matching the given marginals, i.e. the provided call quotes are arbitrage free .