New Developments in Vol and Var Products Christoph Burgard Outline Simple Variance Products via Gamma rent Variance swap dynamics Discrete model: VS and Skew Dynamics Continuous model: LSV Multi-asset stoch vol models Summary New Developments in Vol and Var Products 15 th CFE Workshop, Columbia University Christoph Burgard Quantitative Analytics, Barclays Capital 5th December 2008 Copyright c 2008 Barclays Capital - Quantitative Analytics, London
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NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
New Developments in Vol and Var Products15th CFE Workshop, Columbia University
Joint dynamics for VS and spotI At t = ti know VS for periond ti to ti + ∆
VSi = ξi (ti )
I Still have freedom to specify skew
I Want cheap simulation for marginal spot distributionI control skewI be consistent with VS generated (without costly calibration)I want closed form formula for log-contracts / variance swaps
I Achieve this by using Merton’s model to generate marginal spotdistributions
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Review: Merton’s jump-diffusion model
Merton’s jump diffusion model:
dSt
St= µdt + σdW + (eα+δε − 1)Sdq
with constant hazard rate λ and ε a normal variable
I Log contract is worth
(−λκ− σ2
2 + αλ)∆
with κ = eα+0.5∗δ2 − 1
I replicate VS with log-contracts
I depending on Merton parameters can generate different skewsconsistent with VS value VSi on the path at time ti
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Restrict degrees of freedomI Restrict degrees of freedom to smaller set to get fast calibration
I There are many ways to do this, e.g.I Specify reference skew with α0, δ0, σ0, qv0
I e.g. calibrated to initial skew
I Define two extra parameters linking α, σ and δ to referenceparameters
I Control the skew behaviour wrt variance level
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Linking the log contract to the Merton parametersI Define
Vri =σ2
VSi
I contribution of Merton volatility paramter to VS value
I Start from reference parameters VS0, κ0, δ0, σ0.
I Define 2 model inputsI Vr volatility-jump-ratio-amplitudeI Ab amplitude-blend
I Allow Vri to move (as driven by VS) within [Vr−,Vr+], with
Vr− = max[0.01,Vr0(1− Vra)]
andVr+ = min[1,Vr0(1 + Vra)]
where Vri is calculated according to
Vri = max
[Vr−,min
[Vr+,Vr0 + (VSi − VS0) ∗ Vr+ − Vr−
(Ab)VS0
]]
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Linking the log contract to the Merton parameters
Given VSi on MC path, compute all the Merton parameters
σ =√
Vr · VSi
κ = max[−0.5, κ0
σσ0
]δ = δ0
κκ0
α = log(1 + κ)− 12δ
2
λ = VSi−σ2
2(κ−α)
We cap λ at 10 times λ0 and adjust σ accordingly.
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Controlling the skewI Skew can be controlled wrt variance
I Can be constant
I Can steepen if variance decreases
Figure: Controlling the skew through VRA parameter
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Forward Smile Implied by the modelI Forward variances are log normally distributed
I To get forward smile, integrate over the forward starting pricesfor each variance level (Monte Carlo or quadrature)
Figure: Forward smile generated by the model without vol of var
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Forward Smile Implied by the model
Figure: Forward smile generated by the model with vol of var switched on
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Prices : impact of the vol of variance
Figure: Price of a reverse cliquet and a Napoleon
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Part 4:
Forward Vol Modelling 2:
Continuous model: LSV - local stochasticvolatility
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Local Stochastic Volatility Model (LSV)
I originates from FX (Blacher (2001))
I tight calibration to vanillas with more realistic dynamics thanlocal vol
I spot volatility has both local and exogenous component
I assume exogenous part is Heston (zero drift):
dSt = σ(St , t)√
vt dW(1)t
dvt = λ(v − vt) dt + η√
vt(ρ dW(1)t +
√1− ρ2 dW
(2)t )
I two step calibration to vanillasI step 1: determine stoch. vol parameters (calibrate to vanillas,
forward-starts, or something else)I step 2: determine σ(·) so that vanillas are matched
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
LSV calibrationI second step by forward induction based on Fokker-Planck PDE
for p(s, v , t) (e.g. Ren, Madan, and Qian Qian (2007))
I the density p(·) is used for calculating σ(·) via
σ2(K ,T ) =σ2
D(K ,T )
E[vT |ST = K ]
where σ2D(K ,T ) is the Dupire local vol.
I Computational challenges for η2 > 2λv (usual in equities):
I uniqueness does not hold for the Fokker-Planck PDE, even forplain CIR (Feller (1951))
I boundary conditions are part of the solution ⇒ exact boundaryconditions are needed
I typical consequence of incorrect boundary conditions – ”massleakage”’ from the computed density, calibration fails
I solution is unbounded in the vicinity of v = 0
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
LSV calibrationI correct boundary conditions: obtain from generalization of
Feller’s “zero flux across boundary” at v = 0 (Lucic (2008)):(η2
2
∂
∂v(vp) + λp(v − v) + vρη
∂ (σp)
∂x
)∣∣∣∣v=0
= 0
I in addition, for good accuracy, change of variable isrecommended to deal with unbounded sol’n
I for efficient MC simulation, (modification of) Andersen steppingpreferable
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Options on variance with LSVI Example: opt on var - effect of Heston params
I VSD price with ω = 2.5; vol-of-fwd-var stationary limit: 60.8%I Napoleon 4.6%I Reverse Cliquet 13.95%
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Propagated Forward SkewI forward skew in LSV does not flatten due to stochastic volatility
component tends to Heston forward skew
I Once we hit the stationary regime, the skew follows the termstructure of the vol
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Vol of Var and Forward SkewI LSV pricing, λ = 1.5, ρ = −70%, v0 = v = 50%
I The vol of var increases the slope and curvature of the skew.
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Mean Reversion Rate and Forward SkewI LSV pricing, η = 150%, ρ = −70%, v0 = v = 50%
I Opposite effect to vol of var. A high mean reversion rateconstrains the vol convexity.
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Correlation and Forward SkewI LSV pricing, λ = 1.5, η = 150%, v0 = v = 50%
I The correlation dictates the slope of the forward skewirrespective of the shape of the skew seen from today.
I Positive correlation reverses the skew at the stationary regime.
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
In summary, have
I Fast discrete time modelI linked to realistic dynamics of VSI some control over skew dynamicsI calibrated to VS but not necessarily vanillas expiring on later legs
I Continuous time model (LSV)I matches all vanillasI decent speedI Decent control over dynamics. The local vol captures the
vanillas today and the stochastic vol parameters allow to controlthe forward skew.
I continuous time - can price all payoffs consistently
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Part 5:
Multi-asset stochastic vol model
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Multidimensional Stochastic Volatility ExperimentI Price (.STOXX50E, .SPX) basket in multi-asset LSV and LV
I stoch vol in LSV has ”decorrelation effect” on spot trajectories
I for same ATMF basket prices need to increase spot/spot corr inLSV compared to LV
I impact on other exotic follows similar decorrelation logic
I challenge in applications: come up with nice way ofparameterizing correlation
I typically assume spot-spot and spot-vol diagonal correlations areknown ⇒ matrix completion problem
I Kahl (2007) proposes a method equivalent to maximumdeterminant completion
I complete starting from (spoti , spotj ) and (spoti , voli )
I this approach allows no other degrees of freedomI ideas based on minimum relative entropy are explored
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Multidimensional Stochastic Volatility ExperimentI prices of the OTMF options are relatively close (puts below
forward, calls above)
I some skew effect
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Multidimensional Stochastic Volatility ExperimentI more significant effects on best-of call (2Y)
Effect of the stochastic vol parametersI η (0.5,1,1.5,2) λ(0.5,1,2,3) ρ(-.9,-.8,-.7,-.5). Spot/Spot correl =
86% and Vol/Vol correl = 70%
I Increasing the vol convexity increases the decorrelation effect.
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
Summary
Looked at a number of topics of variance and vol product modelling
I Replication of simple variance products via tracking error ongamma rent
I VS dynamics and options on realised variance
I Link of that to a skew dynamics in discrete time model
I Some notes on LSV
I Decorrelation effects in multi-asset stochastic vol models
Acknowledgements
I would like to acknowledge significant contributions from presentand former members of the quantitative analytics team at BarCap, inparticular Tom Hulme, Abdessamad Khaled, Vladimir Lucic, GabrielManceau, Vladimir Piterbarg, Olaf Torne and Franck Viollet.
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
L. Bergomi, Smile Dynamics II, Risk Magazine, Oct 2005
Peter Carr, Dilip Madan, Towards a theory of volatility trading,Working paper, 2002
B. Dupire, Arbitrage Pricing with Stochastic Volatility, BanqueParibas, May 1993
Gatheral (2005): Valuation of volatility derivatives, ICBI GlobalDerivatives, May 2005
Ren, Madan and Qian Qian (2007): Calibrating and pricing withembedded local volatility models, Risk Magazine, Sept 2007
G. Blacher (2001): A new approach for designing and calibratingstochastic volatility models for optimal delta-vega hedging ofexotics, Global Derivatives 2001
NewDevelopments in
Vol and VarProducts
ChristophBurgard
Outline
Simple VarianceProducts viaGamma rent
Variance swapdynamics
Discrete model:VS and SkewDynamics
Continuousmodel: LSV
Multi-asset stochvol models
Summary
C. Kahl (2007): Modelling and simulation of stochastic volatilityin finance, Doctoral disertation, Univeristy of Wuppertal 2007
W. Feller (1951): Two singular diffusion problems , The Annalsof Mathematics, July 1951.
V. Lucic (2008): Boundary conditions for computing densities inhybrid models via PDE methods, SSRN, 2008.