Bruno DUPIRE Bloomberg Quantitative Research

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

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

3. Model free properties Dominance

4. European profilesnecessary & sufficient conditions on call prices K K1 K2 K1 K2 K3 0 K Dominance

5. A conundrum Call prices as function of strike are positive decreasing: they converge to a positive value a. Do we necessary have ? • 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 a > 0. Dominance

6. Arbitrage with Infinite trading N Dominance

7. Quiz Strong smile 80 90 S0 = 100 Put (80) = 10, Put (90) = 11.Arbitrageable? Dominance

8. 90 80 90 Answer • 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. Dominance

9. European pay-off f(S). What are the non-arbitrageable prices for f? Answer: intersection of convex hull with vertical line f UB arbitrage hedge LB 0 initial cost { arbitrage >0 pay-off arbitrageable price S0 Bounds for European claims If market price < LB : buy f, sell the hedge for LB: S Dominance

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

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

12. Lookback 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. Dominance

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

14. Normal model with no interest rates Dominance

15. Digitals 1 American Digital = 2 European Digitals Reflected path From reflection principle, Proba (Max0-T > K) = 2 Proba (ST > K) K Brownian 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. Dominance

16. Down & out call DOC (K, L) = C (K) - P (2L - K) 60.00 50.00 40.00 30.00 20.00 10.00 2L-K K 0.00 50 60 70 80 90 100 110 120 130 140 150 160 170 -10.00 L -20.00 -30.00 -40.00 The hedge meets boundary conditions.If S reaches L, unwind at 0 cost. Dominance

17. 20.00 15.00 10.00 5.00 0.00 80 90 100 110 120 130 140 150 160 -5.00 -10.00 -15.00 -20.00 Up & out call 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. Dominance

18. General Pay-off 20.00 15.00 10.00 5.00 0.00 80 90 100 110 120 130 140 150 160 170 180 -5.00 -10.00 -15.00 -20.00 The hedge must meet boundary conditions, i.e. allow unwind at 0 cost. Dominance

19. Double knock-out digital 2 symmetry points: infinite reflections 0.02 0.9 0.02 0.01 0.4 0.01 0.00 -0.1 80 90 100 110 120 130 -0.01 -0.01 -0.6 -0.02 -1.1 -0.02 Price & Hedge: infinite series of digitals Dominance

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

21. Extensions Dominance

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

23. Extension: double KO 0 K 0 L Dominance

24. Martingale inequalities Dominance

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

26. Tchebitchev • Property: • In financial terms: S0 - a S0 S0 + a Dominance

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

28. Applications X Dominance

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

30. A sight of Cauchy-Schwarz Dominance

31. S0 Cauchy-Schwarz (2) • Call dominated by parabola: • In financial terms: • Hedge: • Short ATM straddle. • Buy a Par + b. Dominance

32. DOOB • Property: • Hedge at date t with current spot x and current max a: • If x < a do nothing. • If x = a -> a + da sell 4da stocks • total short position: 4 (a + da) stocks. Dominance

33. Up Crossings • Product: pays U(a,b) number of times the spot crosses the band [a,b] upward. • Dominance: 2 3 1 • 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. Dominance

34. Lookback squared • Property: (£ if S not continuous) • In financial terms: (Parabola centered on S0) • Zero cost strategy: when a new minimum is lowered by dm, buy 2 dm stocks. • At maturity: long 2 (S0-min) stocks paid in average (1/2) (S0+min). • Final wealth: Dominance

35. A simple inequality Dominance

36. Quadratic variation Strategy: be long 2xi stock at time ti In continuous time: Dominance

37. Quadratic variation: application Volatility swap: to lock (historical volatility)2 ~ QV (normal convention) 1) Buy calls and puts of all strikes to create the profile ST2 2) Delta hedge (independently of any volatility assumption) by holding at any time -2St stocks Dominance

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. Dominance

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

40. H B Sketch of proof 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 a such that: Dominance

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

42. Equality case 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: Dominance

43. Bounds for derivatives 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) Dominance

44. European pay-off f(S). What are the non-arbitrageable prices for f? Answer: intersection of convex hull with vertical line f UB arbitrage hedge LB 0 initial cost { arbitrage >0 pay-off arbitrageable price S0 Bounds for European claims1 date If market price < LB : buy f, sell the hedge for LB: S Dominance

45. Arbitrage bound for C100 - C200 ( S0=100, ST>0) Example: Call spread 100 50 100 200 ST Dominance

46. Bounds for n dates 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. Dominance

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

48. Bounds for path dependent claimscontinuous time • 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 Dominance

49. Conclusion • 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. Dominance