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

INTRODUCTION

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

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

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

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

STATIC CASE

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 ?

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

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

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

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

One Touch v.s Super Replication

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

Straddle / Parabola

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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

A simple inequality

Dispersion Trades

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,,,

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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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)

Variance Call example

CONTINUOUS TIME

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Monetizing along the path

Static Hedging

Dynamic Hedging

Asian and Fwd Start

Linear Programming Method

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

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

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

Joint p.d.f for Asian Option Lower Bound

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

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

Super Replication of Portfolios Asian1: |x+y| LB: 2.00UB: Asian2: |0.8x+1.2y| LB: 2.027UB: S=Portfolio of (Asian1 - Asian2) LB(Asian1) - UB(Asian2) = UB(Asian1) - LB(Asian2) = LB(S)= 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)

CLAIM DECOMPOSITION

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

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 tT0 Worse scenario S tT0 SLK

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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)

European Case 1 period g a LB k

European Case Continuous Time LB a

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)+

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

Path Dependent Case

Functionals and Derivatives

Examples

Functional Itô Formula

Fragment of proof

Subfunctionals

LB and Functional Itô Formula

CONCLUSION

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