burgard christoph dec 08

NewDevelopments in

Vol and VarProducts

ChristophBurgard

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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

Christoph Burgard

Quantitative Analytics, Barclays Capital

5th December 2008

Copyright c© 2008 Barclays Capital - Quantitative Analytics, London

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ChristophBurgard

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Overview

Simple Variance Products via Gamma rent

Variance swap dynamics

Discrete model: VS and Skew Dynamics

Continuous model: LSV

Multi-asset stoch vol models

Summary

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Part 1:

Replication of simple variance productsvia gamma rent

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Simple Variance Products

Applicable to e.g. Variance Swaps, Corridor Variance Swaps, Gammaswaps.

Basic ideaI Delta hedging a ”replication option” under 0-vol asumption

I see Carr/Madan (2002)

I Gamma rent ”tracking error” generates var product payout

I Price is cost of setting up ”replication options”I Cost of replication option in real worldI Minus cost of replication option in 0-vol world

Here:I Apply to payouts on spot (rather than forward) - e.g. CVS

I maturity dependent replication option

I Comparison to standard approach

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Gamma rent

Gamma rent ”tracking error”I Delta hedge option V asumming zero vol σ0 = 0

I Gamma rent ”tracking error” if realised vol is σ

Γerr =

∫1

2(σ2 − σ2

0)F 2 ∂2Vσ0(F , t)

∂F 2dt

=

∫1

2σ2F 2 ∂

2Vσ0(F , t)

∂F 2dt

I Choose V such that F 2 ∂2Vσ0

(F ,t)

∂F 2 produces required factor.

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Examples

Example payouts (on F(t, T ), 0 strike, cont. diff. limit)

Var Swap: VScont =∫σ2(t,Ft,T )dt

Corridor VS: CVScont =∫σ2(t,Ft,T )1{A<Ft,T<B}dt

Replication payouts

Var Swap: Vσ0 = −2 ln Ft,T

Corridor VS: Vσ0 = −2(ln(F A,Bt,T )− Ft,T

FA,Bt,T

)

F A,Bt,T = max(A,min(B,Ft,T )))

Figure: Replication payout for CVS

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CVS in spot

In practise payouts on St rather than Ft,T

I For CVS:I const corridors in St mean t-dependent corridors in Ft,T

I gamma rent approach allows for efficient definition of replicationoption and hedging

Figure: Replication of CVS for flat barriers in spot

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Comparison to ”standard approach”

I Standard approachI Ito on FA,B

t,T and discretise in timeI Gamma rent approach

I performance significantly better in backtesting (for same numberof instruments)

I easy over/underhedgingI better hedging of discrete dividends

Figure: Backtesting of CVS replication approaches

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Part 2:

Variance Swap dynamics and options on realisedvariance

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Summary

Mean reverting models long T limitI Commonly accepted and desirable property of volatility driver is

mean reversion

dSt/St = · · ·+ f (vt) dWt , vt mean reverting

I Typically this implies v has stationary distribution µ and isergodic (Heston, Scott-Chesney, Schobel-Zhu)

I As a consequence

limT→∞

1

T

∫ T

0

g(vt) dt =

∫R

g dµ ∀g

I Underlying of the options on variance approaches constant forlarge T

I Options on variance have BS vol approaching zero as T →∞I For single factor model this limit can be reached too quickly

I Better to use 2-factor model (can move one factor out to longmaturities)

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Modelling Expected Instantaneous Variance

I Define ξT (t) = E(σ2

T |Ft

)I Variance Swap price (undiscounted):

V T1,T2t =

1

T2 − T1

∫ T2

T1

ξτ (t) dτ

I From initial VS curve

ξT (0) =∂

∂T(TV 0,T

0 )

I Realised variance at T

RV0,T0 =

1

T

∫ T

0

ξτ (τ) dτ

I Free to impose any martingale dynamics on ξT (t)

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Example Heston type model (for comparison)

dSt

St= rt dt +

√vt dW S

t

dvt = −λ(vt − v) dt + η√

vt dW vt

Take expectation of variance SDE to find

ξT (t) = E (vT |Ft) = v + (vt − v) exp(−λ(T − t))

dξT (t) = η√

eλ(T−t)(ξT (t)− v) + vdW vt

I Analytic VS price

I Analytic approximation of VS dynamics

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Heston Type Model Calibration

Figure: Calibration on the historical volatilities of VS

I With 1 factor can’t calibrate short and long end simultaneously

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Two-factor model

I Specify dynamics of ξT (t) directly (after Bergomi (2005)):

dξTt

ξTt

= ω[exp(−k1(T − t)) dW 1

t + θ exp(−k2(T − t)) dW 2t

]I Match intial VS curve by construction

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Summary

Modelling Expected Instantaneous Variance

VS dynamics with 2 factor model

Vt,t+τt =

1

τ

∫ t+τ

t

ξs(t) ds

Evolving with t gives:

dVt,t+τt = (. . . )dt + A1(τ)dW 1

t + θA2(τ)dW 2t

where Ai (τ) =ω

τeki t

∫ t+τ

t

ξs(t)e−ki sds

VS equivalent volatility

σt,t+τt =

√A2

1 + θ2A22 + 2ρθA1A2

Vt,t+τt

I Can calibrate on historical dynamics

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Two-Factor Model Calibration

Figure: Calibration on the historical volatilities.

I Better flexibility due to the short and long term parameters

I Can calibrate short end and long end simultaneously.

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Pricing

Two-factor model

Evolve dynamic of ξT (t) for each T .

I Discretization: realised variance is 1n

∑ni=1 ξ

ti (ti )∆i with∆i = ti − ti−1

I Run analytic approximation (moment matching, see below) orMC

I Can use other points on the variance swap curve for options onimplied variance (options on VIX, etc)

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Compare skew Heston-type / 2f model

Figure: Skew on realised var skew between Heston and 2f models

I The two models produce different skewI The 2f model has a more realistic one (upwards sloping)I Heston one downward sloping (as it’s a sqrt process)

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Pricing approximations for 2f model

Moment matching methods for options on realised varianceI Calculate moments of realised variance distribution

I pth moment:

E(

(RV0,T0 )p

)=

1

np

n∑i1=1

· · ·n∑

ip=1

E(ξti1 (ti1)∆i1 . . . ξ

tip (tip )∆ip

)Group and compute every terms O(np)

I Match and price with easily tractable distributions

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2 moment matching for 2 factor model

Example:Variance of 2 factor model

Var(RV0,T

0

)=

1

n2

( n∑i=1

(ξi (0))2(

exp(ω

2 1 − e−2k1δi

2k1

+ω2θ

2 1 − e−2k2δi

2k2

+2ω2θρ

1 − e(k1+k2)δi

k1 + k2

)−1

)

+2

n−1∑i=1

n∑j=i+1

(ξi (0)ξj (0)

)(exp(ω

2e−k1δ(j−i) 1 − e−2k1δi

2k1

+ ω2θ

2e−k2δ(j−i) 1 − e−2k2δi

2k2

+

ω2θρ

1 − e(k1+k2)δi

k1 + k2

(e−k1δ(j−i) + e−k2δ(j−i)))− 1

))

Equivalent log-normal volatility

Σ0,T0 =

√√√√√√ 1

Tln

Var(RV0,T

0

)(Fwd0,T

0

)2 + 1

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Comparison market prices, MC and momentmatching

Figure: Options on variance skew with moment matching methodscompare to Monte Carlo and market bid offer (EuroStoxx, Jan 08)

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Part 3:

Forward Vol Modelling 1:

Discrete model: VSD - linking VS dynamics withskew process

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Foward volatility modelling

For forward vols would want to have a model that

I has realistic vol dynamics

I has controllable vol dynamics

I is easy calibrated to vanillas, var swaps and forward voldependent products

I is preferably continuous in time

I is fast

In general, something of above needs to give.

Discuss two models with different sets of above properties.

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Smile Dynamics RequirementsI Extend 2f model from above to include dynamics for skew

I Independently specifyI dynamics of VS volatilitiesI level of the short-term forward skewI link between VS and skew

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Large steps for variance simulation

Can express 2f model for forward variance (discretised) in terms oftwo OU processes Xt and Yt :

ξi (t) = ξi (0) exp (ωe−k1(Ti−t)Xt + ωθe−k2(Ti−t)Yt

−ω2

2 [e−2k1(Ti−t)Var(Xt) + θ2e−2k2(Ti−t)Var(Yt)+2θe−(k1+k2)(Ti−t)Cov(Xt ,Yt)])

This allows large step sizes in MC.

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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

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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

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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

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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

]]

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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.

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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

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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

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Forward Smile Implied by the model

Figure: Forward smile generated by the model with vol of var switched on

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Prices : impact of the vol of variance

Figure: Price of a reverse cliquet and a Napoleon

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Part 4:

Forward Vol Modelling 2:

Continuous model: LSV - local stochasticvolatility

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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

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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

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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

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Options on variance with LSVI Example: opt on var - effect of Heston params

I LSV parameters: ρ = −70%,√

v0 =√

v = 50% withI base/low mean rev: λB = 1.5, λL = 0.5I base/high vol-of-var: ηB = 1.25, ηH = 2.0

I High vol of var - price up; low mean rev - long end up

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Local Stochastic Volatility Model (LSV)

I price vol swap, model calibrates to vanillas

I Carr-Lee approximation does not apply to LSV due to explicitdepends of local vol term on spot

I Verifies Gatheral (2005) conjecture that vol convexity is cheaperin local vol

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Pricing ExamplesI 3 yr Napoleon, coupon 8%, single payment at the back

I 3 yr reverse cliquet, monthly resets, yearly coupon 20%,pay-as-you-go

I LSV pricing, ρ = −80%, v0 = v = 17.6%

vol of var 25% 35% 45% 60% 35%rev rate 1.5 1.5 1.5 1.5 0.5

Napoleon 2.81% 3.23% 3.69% 4.53% 4.7%reverse cliquet 8.46% 10.29% 12.24% 15.59% 15.48%

I VSD price with ω = 2.5; vol-of-fwd-var stationary limit: 60.8%I Napoleon 4.6%I Reverse Cliquet 13.95%

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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

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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.

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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.

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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.

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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

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Part 5:

Multi-asset stochastic vol model

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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

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Multidimensional Stochastic Volatility ExperimentI prices of the OTMF options are relatively close (puts below

forward, calls above)

I some skew effect

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Multidimensional Stochastic Volatility ExperimentI more significant effects on best-of call (2Y)

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Multidimensional Stochastic Volatility ExperimentI ...and outperformance option (2Y)

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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.

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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.

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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

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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.

burgard christoph dec 08

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