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Super-Replication Bruno Dupire Bloomberg LP AIMS Day 3 Cape Town, February 19, 2011

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INTRODUCTION

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Background The crisis has revived 2 important concerns: Model risk Stress testing Super-replication is at the crossroad of both (model free properties, valid under all circumstances) It is VaR 100%, a worst case policy (“la politique du pire”) The link : Liquid => Illiquid Can be established by - Calibration - Super-replication

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Claim Pricing Complete Markets perfect replication unique price Incomplete Markets imperfect replication range of prices Minimum variance hedge Indifference pricing (utility based) Super-replication

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Bad press for Super-replication Common belief: + In an uncertain world, robustness is most valuable - A hedge that works for every possible scenario is prohibitively expensive. Example: cheapest super-hedge of call with stocks and bonds S CallStock Payoff Overpay a lot at time 0 to may be get something at maturity T

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However … It is not the case because 1. A wide variety of instruments can help to super-replicate 1. There are strategies to cash-in along the way 1. Of benefit of portfolio diversification

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STATIC CASE

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Static hedge in Stocks and Bonds ST Stock ST Linear ST Bond 1 Constant ST Portfolio a.ST + b Affine ST g (ST) S0 Claim gives g(ST) at T Arbitrage free prices for the claim at 0 ?

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Arbitrage free prices for g 1. Take convex hull of (S, g(S)) in the plane 1. Intersect with the vertical line at S0 S0 LB UB

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Price Range 1. If α > UB Sell Claim for α Buy super- replication for UB 1. Prices in (LB, UB) are possible Binomial model with states = contact points S0 UB α S0

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Examples FVIX2(tradable) LB UB VIX Futures FVIX S0K LB UB Call S0K1K1 LB UB K2K2 Call Spread

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Click to edit Master text styles Second level Third level Fourth level Fifth level

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Using Vanilla options K Listed strikes KL Non-listed Strike Variance Swap One Touch Option

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One Touch v.s Super Replication

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Cauchy - Schwarz Click to edit Master text styles Second level Third level Fourth level Fifth level

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Straddle / Parabola

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Click to edit the outline text format Second Outline Level Third Outline Level Fourth Outline Level Fifth Outline Level Sixth Outline Level Seventh Outline Level Eighth Outline Level Ninth Outline LevelClick to edit Master text styles Second level Third level Fourth level Fifth level Click to edit the outline text format Second Outline Level Third Outline Level Fourth Outline Level Fifth Outline Level Sixth Outline Level Seventh Outline Level Eighth Outline Level Ninth Outline LevelClick to edit Master text styles Second level Third level Fourth level Fifth level Forward Start Options is replicable 2. where α = price at T0 of T2 OptionsT1 OptionsΔ hedge at T1

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

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A simple inequality

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Dispersion Trades

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Static: The cheapest super replication is Its price at 0 is Let, Static position in options and delta hedge in x and y to capture QV of,,,

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Dynamic option position: Noting that We get At t, the market value of the initial upper bound B is Earn on QV of price spread and QV of volatility spread. Volatility spread moves may be excessive.

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Click to edit the outline text format Second Outline Level Third Outline Level Fourth Outline Level Fifth Outline Level Sixth Outline Level Seventh Outline Level Eighth Outline Level Ninth Outline LevelClick to edit Master text styles Second level Third level Fourth level Fifth level Click to edit the outline text format Second Outline Level Third Outline Level Fourth Outline Level Fifth Outline Level Sixth Outline Level Seventh Outline Level Eighth Outline Level Ninth Outline LevelClick to edit Master text styles Second level Third level Fourth level Fifth level Theory of static case Delbaen-Schachermayer/ El Karoui (1994): Duality Result Price decomposition With static position in options and dynamic position in stocks where Ci 0 is the price of option Ci at 0

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Use of T-maturity options Skew of maturity T: Risk Neutral density φ(ST) Mapping with Skorohod embedding problem (Hobson, Rogers, Obloj) gives optimal super- hedge for - lookbacks, barriers (Azema-Yor solution) - variance options (Root/Rost solution)

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Variance Call example

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CONTINUOUS TIME

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Click to edit the outline text format Second Outline Level Third Outline Level Fourth Outline Level Fifth Outline Level Sixth Outline Level Seventh Outline Level Eighth Outline Level Ninth Outline LevelClick to edit Master text styles Second level Third level Fourth level Fifth level Results El Karoui - Quenez (1995): LB is a submartingale Dynamic version of Delbaen - Schachermayer (1994) Kramkov (1996): Optimal decomposition theorem where k is a non decreasing process Interpretation: The increasing process k comes from improving LB whenever you can In other words, if you prepare for the worst, you can only have good surprises

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Click to edit the outline text format Second Outline Level Third Outline Level Fourth Outline Level Fifth Outline Level Sixth Outline Level Seventh Outline Level Eighth Outline Level Ninth Outline LevelClick to edit Master text styles Second level Third level Fourth level Fifth level Click to edit the outline text format Second Outline Level Third Outline Level Fourth Outline Level Fifth Outline Level Sixth Outline Level Seventh Outline Level Eighth Outline Level Ninth Outline LevelClick to edit Master text styles Second level Third level Fourth level Fifth level Example A super-replication A put in place today will still work tomorrow. If a cheaper one, B is available tomorrow, roll from A to B to collect the improvement If at tn+1 is cheaper than Sell A(tn) and buy B(tn+1): you collect the difference and still super- replicate Optimal strike at tn: minimize A KnKn L A Kn+ 1 B Optimal strike at tn+1 KnKn L

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Monetizing along the path

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Static Hedging

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Dynamic Hedging

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Asian and Fwd Start

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Linear Programming Method

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Basket LB (0.414)UB(2.414) Bachelier (2.236) Asian LB (2)UB (2.403) Asian/Basket (x+y)

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Spread LB (0.414)UB(2.414) Forward Start LB (0.455)UB (1.245) Fwd Start/Spread (x-y) Bachelier (1)

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Joint p.d.f for Asian Option Upper Bound:

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Joint p.d.f for Asian Option Lower Bound

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Joint p.d.f for Forward Start Option Upper Bound

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Joint p.d.f for Forward Start Option Lower Bound

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Super Replication of Portfolios Asian1: |x+y| LB: 2.00UB: 2.403 Asian2: |0.8x+1.2y| LB: 2.027UB: 2.487 S=Portfolio of (Asian1 - Asian2) LB(Asian1) - UB(Asian2) = -0.487 UB(Asian1) - LB(Asian2) = 0.376 LB(S)=-0.165 UB(S)=0.007 Super Rep. of individual Asians LB (-0.487)UB(0.376) Super Rep. of Asian portfolio LB (-0.165)UB (0.007)

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CLAIM DECOMPOSITION

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How to exploit a positive claim? S0 = 100 You receive C120,T for free You may sit on it and wait for T Are there any other ways to monetize it for sure? ST 120100

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Possible choices What NOT to do: Delta hedging in a model (say BS 30%) You may incur a loss Instead, convert a Call to Put and vice versa when you are In-the-money The Kramkov strategy consists of doing it in continuous time, which collects the local time at the strike Bad scenario S 120 100 tT0 Worse scenario S 120 100 tT0 SLK

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Click to edit the outline text format Second Outline Level Third Outline Level Fourth Outline Level Fifth Outline Level Sixth Outline Level Seventh Outline Level Eighth Outline Level Ninth Outline LevelClick to edit Master text styles Second level Third level Fourth level Fifth level Click to edit the outline text format Second Outline Level Third Outline Level Fourth Outline Level Fifth Outline Level Sixth Outline Level Seventh Outline Level Eighth Outline Level Ninth Outline LevelClick to edit Master text styles Second level Third level Fourth level Fifth level Parabola Example If you delta hedge at stopping times τi, you collect the discrete quadratic variation No hedge Hedge at τi Continuous hedge (quadratic variation of S) STS0 (ST – S0)2 (mechanics of Variance Swaps)

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Which strategy? Some τi may be better than other ones: i. τi time based: Frequencygram (daily/weekly historical volatility) i. τi move based (better estimate of volatility in view of delta hedging)

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European Case 1 period g a LB k

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European Case Continuous Time LB a

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Basket Option: k2 Let the underlying of the Basket option be X and Y 1. Payoff = (X + Y – K)+ For simplicity, K = 0: (X+Y)+ ≤ X+ + Y+ So, X+ + Y+ - (X+Y)+

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Basket Parabola: k1 Let the underlying of the Basket option be X and Y 2.Payoff = (X + Y – K)2 For simplicity, K = 0: (X+Y)2 ≤ 2(X2 + Y2) So, 2(X2 + Y2) - (X+Y)2 = (X-Y)2

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Path Dependent Case

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Functionals and Derivatives

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Examples

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Functional Itô Formula

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Fragment of proof

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Subfunctionals

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LB and Functional Itô Formula

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CONCLUSION

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Conclusion Super-replication is a potent approach when - other options can be used - super-hedge is rolled down to monetize before maturity - applied to a portfolio (Path dependent) Claims can be decomposed in a canonical way. It refines the Kramkov result and splits the increasing process in 2 Super-replication is a powerful approach: - always robust - often accurate - gives insight on the hedge

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