1. 1 Bruno DUPIRE Bloomberg Quantitative Research Arbitrage, Symmetry and Dominance NYU Seminar New York 26 February 2004

2. Background Dominance Can we say anything about option prices and hedges when (almost) all assumptions are relaxed? REAL WORLD: anything can happen infinite number of possible prices, infinite potential loss MODEL: stringent assumption 1 possible price, 1 perfect hedge

3. 3 Dominance Model free properties

4. 4 European profiles necessary & sufficient conditions on call prices Dominance K K1K1 K2K2 K2K2 K3K3 K1K1 0K

5. 5 A conundrum Dominance Do we necessary have ? Call prices as function of strike are positive decreasing: they converge to a positive value . It depends which strategies are admissible! If all strikes can be traded simultaneously, C has to converge to 0. If not, no sure gain can be made if  > 0.

6. 6 Arbitrage with Infinite trading Dominance N

7. 7 Quiz Dominance Strong smile Put (80) = 10, Put (90) = 11. Arbitrageable? 8090S 0 = 100

8. 8 Answer Dominance At first sight: P(80) < P(90), no put spread arbitrage. At second sight: P (90) - (90/80) P (80) is a PF with final value > 0 and premium < 0. 8090

9. 9 Bounds for European claims Dominance European pay-off f(S). What are the non-arbitrageable prices for f? Answer: intersection of convex hull with vertical line S S0S0 UB LB f arbitrageable price arbitrage hedge If market price < LB : buy f, sell the hedge for LB: 0 initial cost >0 pay- off { arbitrage

10. 10 Call price monotonicity Dominance Call prices are decreasing with the strike: are they necessarily increasing with the initial spot? non NO. counter example 1: 0T 90 110 100 0T 90 100 80 120 25% 75% counter example 2: martingale 100

11. 11 Call price monotonicity Dominance If model is continuous Markov, Calls are increasing with the initial spot (Bergman et al) Take 2 independent paths  x and  y starting from x and y today. (1)  x and  y do not cross. (2)  x and  y cross. x y x y Knowing that they cross, the expectation does not depend on the initial value (Markov property).

12. 12 Lookback dominance Dominance Domination of Portfolio: Strategy: when a new maximum is reached, i.e. sell The IV of the call matches the increment of IV of the product.

13. 13 Lookback dominance (2) Dominance More generally for To minimise the price, solve thanks to Hardy-Littlewood transform (see Hobson).

14. 14 Dominance Normal model with no interest rates

15. 15 Digitals Dominance 1 American Digital = 2 European Digitals From reflection principle, Proba (Max 0-T > K) = 2 Proba (S T > K) K Brownian path Reflected path As a hedge, 2 European Digitals meet boundary conditions for the American Digital. If S reaches K, the European digital is worth 0.50.

16. 16 Down & out call Dominance DOC (K, L) = C (K) - P (2L - K) The hedge meets boundary conditions. If S reaches L, unwind at 0 cost. -40.00 -30.00 -20.00 -10.00 0.00 10.00 20.00 30.00 40.00 50.00 60.00 5060708090100110120130140150160170 K2L-K L

17. 17 Up & out call Dominance UOC (K, L) = C (K) - C (2L - K) - 2 (L - K) Dig (L) The hedge meets boundary conditions for the American Digital. If S reaches L, unwind at 0 cost. -20.00 -15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 20.00 8090100110120130140150160

18. 18 General Pay-off Dominance The hedge must meet boundary conditions, i.e. allow unwind at 0 cost. -20.00 -15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 20.00 8090100110120130140150160170180

19. 19 Double knock-out digital Dominance 2 symmetry points: infinite reflections Price & Hedge: infinite series of digitals -1.1 -0.6 -0.1 0.4 0.9 90100110130 -0.02 -0.01 0.00 0.01 0.02 80120

20. 20 Max option Dominance (Max - K) + = 2 C (K) Hedge: when current Max moves from M to M+  M sell 2 call spreads C (M) - C (M+  M), that is 2  M European Digitals strike M. Pricing: K

21. 21 Dominance Extensions

22. 22 Extension to other dynamics Dominance No interpretation in terms of hedging portfolio but gives numerical pricing method. Principle: symmetric dynamics w.r.t L antisymmetric payoff w.r.t L L K 2L-K 0

23. 23 Extension: double KO Dominance L K 0 0

24. 24 Dominance Martingale inequalities

25. 25 Cernov Dominance Property: In financial terms: Hedge: Buy C (K), sell AmDig (K+ ). If S reaches K+, short 1 stock. K K+

26. 26 Tchebitchev Dominance Property: In financial terms: S0S0 S 0 + aS 0 - a

27. 27 Jensen’s inequality Dominance E[X]X f

28. 28 Applications Dominance X

29. 29 Cauchy-Schwarz Dominance Property: Let us call: Which implies:

30. 30 A sight of Cauchy-Schwarz Dominance

31. 31 Cauchy-Schwarz (2) Dominance Call dominated by parabola: In financial terms: S0S0 Hedge: Short ATM straddle. Buy a Par + b.

32. 32 DOOB Dominance Property: Hedge at date t with current spot x and current max  : If x <  do nothing. If x =  ->  sell 4  stocks total short position: 4 (  ) stocks.

33. 33 Up Crossings Dominance Product: pays U(a,b) number of times the spot crosses the band [a,b] upward. Dominance: Hedge: Buy 1/(b-a) calls strike a. First time b is reached, short 1/(b-a) stocks. Then first time a is reached, buy 1/(b-a) stocks. etc. 1 2 3

34. 34 Lookback squared Dominance Property: (  if S not continuous) In financial terms: (Parabola centered on S 0 ) Zero cost strategy: when a new minimum is lowered by  m, buy 2  m stocks. At maturity: long 2 (S 0 -min) stocks paid in average (1/2) (S 0 +min). Final wealth:

35. 35 A simple inequality Dominance

36. 36 Quadratic variation Dominance Strategy: be long 2x i stock at time t i In continuous time:

37. 37 Quadratic variation: application Dominance Volatility swap: to lock (historical volatility) 2 ~ QV (normal convention) 1)Buy calls and puts of all strikes to create the profile S T 2 2)Delta hedge (independently of any volatility assumption) by holding at any time -2S t stocks

38. 38 Dominance We have quite a few examples of the situation for any martingale measure, which can be interpreted financially as a portfolio dominance result. Is it a general result? ; i.e. if you sell A, can you cover yourself whatever happens by buying B and delta-hedge? The answer is YES.

39. 39 General result: “Realise your expectations” Dominance Theorem: If for any martingale measure Q Then there exists an adapted process H (the delta-hedge) such as for any path  : That is: any product with a positive expected value whatever the martingale model (even incomplete) provides a positive pay-off after hedge.

40. 40 Sketch of proof Dominance Lemma: If any linear functional positive on B is positive on f, then f is in B Proof: B is convex so if by Hahn-Banach Theorem, there is a separating tangent hyperplane H, a linear functional and a real  such that: B H

41. 41 Sketch of proof (2) Dominance The lemma tells us: If for any martingale measure Q, then stoch. int.positive Which concludes the theorem.

42. 42 Equality case Dominance Corollary of theorem: If for any martingale measure Q, Then there exists H adapted such that Proof: apply Theorem to f and -f: Adding up:

43. 43 Bounds for derivatives Dominance The theorem does not give a constructive procedure: In incomplete markets, some claims do not have a unique price. What are the admissible prices, under the mere assumption of 0 rates (martingale assumption)

44. 44 Bounds for European claims 1 date Dominance European pay-off f(S). What are the non-arbitrageable prices for f? Answer: intersection of convex hull with vertical line S S0S0 UB LB f arbitrageable price arbitrage hedge If market price < LB : buy f, sell the hedge for LB: 0 initial cost >0 pay- off { arbitrage

45. 45 Example: Call spread Dominance 100 50 200 Arbitrage bound for C 100 - C 200 ( S 0 =100, S T >0) 100 STST

46. 46 Bounds for n dates Dominance Natural idea: intersection of convex hull of g with (0,…,0) vertical line This corresponds to a time deterministic hedge: decide today the hedge at each date independently from spot.

47. 47 Bounds n dates (2) Dominance Lower bound: Apply recursively the operator A used in the one dimensional case, i.e. define gives the lower bound

48. 48 Bounds for path dependent claims continuous time Dominance Brownian case: El Karoui-Quenez (95) Analogous to American option pricing American: sup on stopping times Upper bound: sup on martingale measures In both cases, dynamic programming For upper bound: Bellman equation

49. 49 Conclusion Dominance It is possible to obtain financial proofs / interpretation of many mathematical results If claim A has a lesser price than claim B under any martingale model, then there is a hedge which allows B to dominate A for each scenario If a mathematical relationship is violated by the market, there is an arbitrage opportunity.