Peter J¨ackelSTOCHASTIC VOLATILITY MODELS:PAST, PRESENT AND FUTUREAbstractThere are many models for the uncertainty in future instantaneous volatility. Whenit comes to an actual implementation of a stochastic volatility model for the purpose ofthe management of exotic derivatives, the choice of model is rarely made to capturethe particular dynamical features relevant for the specific contract structure at hand.Instead, more often than not, the model is chosen that provides the greatest easewith respect to market calibration by virtue of (semi-)closed form solutions for theprices of plain vanilla options. In this presentation, the further implications of variousstochastic volatility models are reviewed with particular emphasis on both the dynamicreplication of exotic derivatives and on the implementation of the model. Also, a newclass of models is suggested that not only allows for the level of volatility, but also forthe observed skew to vary stochastically over time.Peter J¨ackelOverview and Introduction• Why stochastic volatility?• What stochastic volatility?• One model to rule them all?• Mathematical features of stochastic volatility models• A stochastic skew model• Monte Carlo methods and stochastic volatility models• Finite differencing methods and stochastic volatility modelsStochastic Volatility Models: Past, Present and Future 1Peter J¨ackelWhy stochastic volatility?• Realised volatility of traded assets displays significant variability. It wouldonly seem natural that any model used for the hedging of derivative con-tracts on such assets should take into account that volatility is subject tofluctuations.• More and more derivatives are explicitly sensitive to future (both impliedand instantaneous) volatility levels. Examples are cliquets, globally flooredand/or capped cliquets, and many more.• Some (apparently) comparatively straightforward exotic derivatives suchas double barrier options are being being re-examined for their sensitivityto uncertainty in volatility.• New trading ideas such as exotic volatility options and skew swaps, how-ever, give rise to the need for a new kind of stochastic volatility model: thestochastic skew model.Why stochastic volatility? 2Peter J¨ackel3500400045005000550060006500700002-Jun-2000 07-Feb-2001 04-Oct-2001 06-Jun-2002 06-Feb-2003FTSE100FTSE 100.Why stochastic volatility? 3Peter J¨ackel0%10%20%30%40%50%60%70%80%02-Jun-2000 07-Feb-2001 04-Oct-2001 06-Jun-2002 06-Feb-2003historical volatility5 day volatility10 day volatility30 day volatilityFTSE 100 realised volatility.Why stochastic volatility? 4Peter J¨ackelWhat stochastic volatility?The concept of stochastic volatility, or rather the idea of a second sourceof risk affecting the level of instantaneous volatility, should not be seen inisolation from the nature of the underlying asset or deliverable contract.For the three most developed modelling domains of equity, FX, and inter-est rate derivatives, different effects are considered to be at least partiallyresponsible for the smile or skew observed in the associated option markets.What stochastic volatility? 5Peter J¨ackelEconomic effects giving rise to an equity skew• Leverage effects [Ges77, GJ84, Rub83]. A firm’s value of equity can beseen as the net present value of all its future income plus its assets minusits debt. These constituents have very different relative volatilities whichgives rise to a leverage related skew.• Supply and demand. Equivalently, downwards risk insurance is more de-sired due to the intrinsic asymmetry of positions in equity: by their finan-cial purpose it is more natural for equity to be held long than short, whichmakes downwards protection more important.• Declining stock prices are more likely to give rise to massive portfolio re-balancing (and thus volatility) than increasing stock prices. This asymme-try arises naturally from the existence of thresholds below which positionsmust be cut unconditionally for regulatory reasons.What stochastic volatility? 6Peter J¨ackelEconomic effects giving rise to an FX skew and smile• Anticipated government intervention to stabilise FX rates.• Government changes that are expected to change policy on trade deficits,interest rates, and other economic factors that would give rise to a marketbias.• Foreign investor FX rate protection.Economic effects giving rise to an interest rate skew and smile• Elasticity of variance and/or mean reversion. In other words, interest ratesare for economic reasons linked to a certain band. Unlike equity or FX, in-terest rates cannot be split, bought back or re-valued and it is this intrinsicdifference that connects volatilities to absolute levels of interest rates.• Anticipated central bank action.What stochastic volatility? 7Peter J¨ackelNone of these effects are well described by strong correlation be-tween the asset’s own driving factor and a second factor govern-ing the uncertainty in volatility since the are all based on deter-ministic relationships.Still, most stochastic volatility models incorporate a skew byvirtue of strong correlation of volatility and stock. The strongcorrelation is usually needed to match the pronounced skew ofshort-dated plain vanilla options.In this context, one might wonder if it wouldn’t be more ap-propriate to let the stochasticity of volatility explain the market-observed features related to or associated with uncertainty involatility, and use other mechanisms to account for the skew.What stochastic volatility? 8Peter J¨ackelOne model to rule them all?An important question that must be asked when a stochastic volatility modelis considered is: what is it to be used for?• Single underlying moderate exotics with strong dependence on forwardvolatility? Forward starting options? Cliquets?• Single underlying exotics with strong dependence on forward skew? Glob-ally floored and/or capped cliquets and friends?• Single underlying exotics with strong path dependence? Barriers of allnatures (single, double, layered, range accruals).One model to rule them all? 9Peter J¨ackel• Multiple underlying moderate exotics with strong dependence on forwardvolatility? Options on baskets. Cliquets on baskets.• Multiple underlying moderate exotics with strong dependence on forwardskew? Mountain range options.• Multiple underlying moderate exotics with strong dependence on correla-tion? Mountain range options.Not all of these applications would necessarily suggest the