Open issues in equity derivatives modelling

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1 Open issues in equity derivatives modelling Lorenzo Bergomi Equity Derivatives Quantitative Research ociété Générale

2 al Outline Equity derivatives at G A brief history of equity derivative products Prehistory 997 History Modern times 003 Modelling issues, algorithmic issues Ris measurement and management Conclusion Lorenzo Bergomi

3 Equity derivatives at G G regarded by industry participants as No in equity derivatives Lorenzo Bergomi

4 A brief history of equity derivative products Prehistory 997 Products Riss Barrier options / Digitals ew: level / dynamics (little) Max / Min options K same Asian options mile max t ( ) t K t N i Baset options Correlation (level) N i K Volatility swaps ln σ ˆ mile, VolOfVol imple cliquets Forward smile K K Models / algos - Blac choles / local vol - PDE / straight Monte Carlo Lorenzo Bergomi 3

5 A brief history of equity derivative products History Capital-guaranteed products distributed by retail networs Everest years / stocs 00% min j j 0 Emerald years / 0 stocs Every year, the stoc whose perfomance since t = 0 is the largest gets frozen and removed from the baset, and its level is floored at 00% ot its initial value. 00% maximum perormance of yearly baset values since t = 0, floored at 0. and many, many, many, other variations trying to find closed-form formulas for specific exotic payoffs now irrelevant and useless Lorenzo Bergomi 4

6 Lorenzo Bergomi 5

7 A brief history of equity derivative products History Variance waps 3 months 5 years Pays realized variance usually measured using daily returns stocs / indices Napoleon 5 years / index Every year, pays coupon reduced by worst of monthly performances of the index. Accumulator 3 years / index At maturity pays the sum if it is positive of the monthly performances, capped and and floored. Lorenzo Bergomi min C %,,% min max σ ˆ ln 6

8 Modern times Corridor variance swaps Daily variance only counted when underlying is inside given interval [ ] L, H ln σ ˆ t [ L, H ], [ L, ], [ 0, H ] Correlation swaps Pays realized correlation over 3 years by stocs of an index Gap notes Maturity = year, a series of daily puts on daily returns of an index with stries 85%, 90% Options on realized variance On indices, maturities: 3 months to years τ ln σ ˆ K τ τ imer options Vanilla payoff, paid when realized variance Q t reaches set level: Q τ = τ ln = i i Hybrids Equities / Rates / Forex / Commodities Arbitrary payoffs Lorenzo Bergomi 7 i

9 Modelling issues Why not just delta-hedge? Variance of residual P&L too large use other options Options are hedged with options Once we start using options as hedging instruments Less sensitivity to historical parameters, more sensistivity to implied parameters Model the dynamics of implied parameters Example of simple cliquet t 008 ˆσ P ( σ,, L ) ˆ r mile 3 mois K = 95 - K= 05 /5/00 /5/00 /5/003 /5/004 /4/005 /4/006 /4/007 /4/008 Lorenzo Bergomi 8

10 Modelling issues How should calibration be done? Do we really need to calibrate? Not compulsory: charge a hedging cost. We hedge parameter p by trading instrument O so that sensitivity to p vanishes: dp = dp λ do dp Model price P is adjusted so as to include hedging cost: Price P ( p ˆ ) λ ( O ( p Maret ) O ( p )) ( p = p ) = P ˆ Maret hen what is the point in calibrating? Ensures price factors in hedging costs incurred at t = 0 not future costs! Necessary to calibrate model on relevant set of hedging instruments Useless if one is unable to specify how to hedge the exotic with the hedge instruments Lorenzo Bergomi 9

11 Modelling issues 3 Volatility ris models Eurostoxx 50 - mat an 5 mile original mile = 05% mile = 95% 0 «Old models» 5 Local volatility Heston ABR Models based on process of instantaneous variance: Jump / Lévy d = K dt V dw dv = K dt ( ) dw V Challenge: Build models that give control on joint dynamics of implied volatilities and spot: First step: model dynamics of curve of forward variances Next step: model dynamics of the implied volatility surface Direct modelling of dynamics of implied volatilities is a dead end Low-dimensional Marov representation desirable How much freedom are we allowed? Lorenzo Bergomi 0

12 Modelling issues 4 Hybrids Equities Interest rates Forex Commodities Hybrid models are not built by simply glueing together models for each asset class Passive hybrids: payoff involves one asset class only Long-dated equity, Forex options Credit / Equity: convertible bonds Active hybrids : payoff involves all asset classes Require state-of-the art models for each asset class Even local vol calibration for equity smiles not easy when interest rates are stochastic Lorenzo Bergomi

13 Modelling issues 5 Correlation how de we put together correlation matrices? How do we build the large correlation matrices needed in hybrid modelling? impler question: imagine a -factor stoch. vol model and a payoff involving securities How do we set the cross-correlations??? Even simpler question how do we measure correlations? Example of European / Japanese stocs no overlap Europe Japan o C o C o C o C o C o C o C o C Correlation how de we measure correlation ris? Corrrelation how to model correlation smile? Lorenzo Bergomi

14 Algorithmic issues Monte Carlo How can we speed up pricing? Quasi-random numbers Discretization of DEs? Callable / putable options Computing sensitivies to Initial conditions Parameters of dynamics (volatilities / correlations, etc..) Lorenzo Bergomi 3

15 Conclusion hese are exciting times for doing quantitative finance Lots of new instruments / product / algorithmic issues Rich mathematical toolbox from which to pic Lorenzo Bergomi 4

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