valuation of volatility derivativesfaculty.baruch.cuny.edu/jgatheral/paris2005.pdfvaluation of...

Valuation of Volatility Derivatives

Jim GatheralGlobal Derivatives & Risk Management 2005ParisMay 24, 2005

Jim Gatheral, Merrill Lynch, May-2005

The opinions expressed in this presentation are those of the author alone, and do not necessarily reflect the views of of Merrill Lynch, its subsidiaries or affiliates.


Outline of this talkValuing variance swaps under compound Poisson assumptionsThe impact (or lack thereof) of jumps on the valuation of variance swapsFinding the risk neutral distribution of quadratic variationOptions on quadratic variationHow to value VIX futuresEstimating volatility of volatilityVolatility of volatility as a traded parameter


Quadratic variation for a compound Poisson processLet denote the return of a compound Poisson process so that

TN

T ii

X Y=∑with iid and a Poisson process with mean .Define the quadratic variation as

TX

iY TN Tλ

[ ]

2

2 2 ( )

TN

iTi

T iT

X Y

X N Y T y y dyλ µ

=

⎡ ⎤⎡ ⎤ = =⎣ ⎦ ⎣ ⎦

∑

∫E E E

Also, [ ]

( )22 2 2

( )

( ) ( ) ( )

T

T

X T y y dy

X T y y dy T y y dy

λ µ

λ µ λ µ

=

⎡ ⎤ = +⎣ ⎦

∫∫ ∫

E

E

So . Expected QV = Variance of terminal distribution for compound Poisson processes! Obviously not true in general (e.g. Heston).

[ ]var TTX X⎡ ⎤ =⎣ ⎦E


Examples of compound Poisson processesMerton jump-diffusion model (constant volatility lognormal plus independent jumps).AVG (Asymmetric variance gamma)CGMY (More complicated version of AVG)NIG (Normal inverse Gaussian)

List does not include time-changed models such as VG-CIR


Valuing variance swapsWe can express the first two moments of the final distribution in terms of strips of European options as follows:

So, if we know European option prices, we may compute expected quadratic variation – i.e. we may value variance swaps as

[ ] [ ]0

00

02 2

00

ln( / ) ( ) ( )

ln( / ) 2 ( ) 2 ( )

T T

T T

X S S dk p k dk c k

X S S dk k p k dk k c k

∞

−∞

∞

−∞

= = − −

⎡ ⎤ ⎡ ⎤= = − −⎣ ⎦ ⎣ ⎦

∫ ∫

∫ ∫

E E

E E

[ ] 22var TT T TX X X X⎡ ⎤⎡ ⎤ ⎡ ⎤= = −⎣ ⎦ ⎣ ⎦⎣ ⎦E E E


Valuing variance swaps under diffusion assumptionsIf the underlying process is a diffusion, expected quadratic variation may be expressed in terms of an infinite strip of European options (the log-strip):

2( ) ( )BSTX dz N z zσ

∞

−∞

′⎡ ⎤ =⎣ ⎦ ∫E

with

2( , )( )

2( , )BS

BS

k T Tkz d kk T T

σσ

−≡ = −

0

0

2 ( ) 2 ( )TX dk p k dk c k∞

−∞

⎡ ⎤ = +⎣ ⎦ ∫ ∫E

Equivalently, we can compute expected quadratic variation directly from implied volatilities without computing intermediate option prices using the formula


What is the impact of jumps?In summary, if the underlying process is compound Poisson, we have the above formula to value a variance swap in terms of a strip of European options and if the underlying process is a diffusion, we have the usual well-known formula.

In reality, we don’t know the underlying process but we do know the prices of European options. Suppose we were to assume a diffusion but the underlying process really had jumps. What would the practical valuation impact be?


The jump correction to variance swap valuationOnce again, if the underlying process is a diffusion, we can value a variance swap in terms of the log-strip:

Also,

[ ]2 TTX X⎡ ⎤ = −⎣ ⎦E E

[ ] 00( )TiuX

T u u T uuX i e i uφ

==⎡ ⎤= − ∂ = − ∂⎣ ⎦E E

where is the characteristic function. Our assumptions that jumps are independent of the diffusion leads to factorization of the characteristic function into a diffusion piece and a pure jump piece.

From the Lévy-Khintchine representation, we arrive at

( )T uφ

( ) ( ) ( )C JT T Tu u uφ φ φ=

0( ) (1 ) ( )J y

u T ui u T y e y dyφ λ µ

=⎡ ⎤− ∂ = + −⎣ ⎦∫


The jump correction continuedOn the other hand, we already showed that

where the superscript J refers to the jump component of the process.It follows that the difference between the fair value of a variance swap and the value of the log-strip is given by

2var ( )J JTT

X X T y y dyλ µ⎡ ⎤ ⎡ ⎤⎡ ⎤= =⎣ ⎦ ⎣ ⎦⎣ ⎦ ∫E

Example:• Lognormally distributed jumps with mean and standard deviation

[ ] 202 ln( / ) 2 (1 / 2 ) ( )y

TTX S S T y y e y dyλ µ⎡ ⎤⎡ ⎤ + = + + −⎣ ⎦ ⎣ ⎦∫E E

α δ

[ ]

( )

2 22 2

/ 20

2 2 4

2 ln( / ) 2 12

3 ( )3

TTX S S T e

T O

α δα δλ α

λ α α δ α

+⎡ ⎤+⎡ ⎤ + = + + −⎢ ⎥⎣ ⎦⎣ ⎦

= − + +

E E

• With and (from BCC) we get a correction of 0.00122427 per year which at 20% vol. corresponds to 0.30% in volatility terms.

0.09, 0.14α δ= − = 0.61λ =


RemarksJumps have to be extreme to make any practical difference to thevaluation of variance swaps.The standard diffusion-style valuation of variance swaps using the log-strip works well in practice for indices

• From the perspective of the dealer hedging a variance swap using the log-strip, the statistical measure is the relevant one.

– How often do jumps occur in practice and how big are they?• Jumps in the risk-neutral measure are driven by the short-dated smile.

However, the model may be mis-specified. Jumps may not be the main reason that the short-dated skew is so steep.

Single stocks may be another story – jumps tend to be frequent even in the statistical measure.


A simple lognormal modelDefine the quadratic variation

Assume that is normally distributed with mean and variance .

2

0

: ( , )T

TX s dsσ ω= ∫

2 2/ 2 2 2;s sT T

X e X eµ µ+ +⎡ ⎤ ⎡ ⎤= =⎣ ⎦⎣ ⎦E E

( )logT

X µ2s

Then is also normally distributed with mean and variance .Volatility and variance swap values are given by respectively

( )logT

X 2µ24s

Solving for and givesµ 2s2

2 2 log ; log TT

T T

XXs

X Xµ

⎛ ⎞⎛ ⎞ ⎡ ⎤⎡ ⎤ ⎜ ⎟⎣ ⎦ ⎣ ⎦⎜ ⎟= = ⎜ ⎟⎜ ⎟⎡ ⎤ ⎡ ⎤⎜ ⎟ ⎜ ⎟⎣ ⎦⎣ ⎦⎝ ⎠ ⎝ ⎠

EE

E E


A simple lognormal model continuedNote that under this lognormal assumption, the convexity adjustment (between volatility and variance swaps) is given by

( )2 / 2 1sT T T

X X e X⎡ ⎤ ⎡ ⎤⎡ ⎤ − = −⎣ ⎦ ⎣ ⎦ ⎣ ⎦E E E

In the lognormal model, given volatility and variance swap prices, the entire distribution is specified and we may price any claim on quadratic variation!Moreover, the lognormal assumption is reasonable and widely assumed by practitioners.


VIX from 1/1/1990 to 4/25/2005

0

50

100

150

200

250

300

-3 -2.5 -2 -1.5 -1 -0.5 0

ln(VIX)

Freq

uenc

y

Unconditional distribution of VIX vs lognormal

Consistent with lognormal volatility dynamics!

1.67; 0.31µ σ= − =


Vol term structure and skew under stochastic volatilityAll stochastic volatility models generate volatility surfaces with approximately the same shapeThe Heston model has an implied volatility term structure that looks to leading order like

It’s easy to see that this shape should not depend very much on the particular choice of model.Also, Gatheral (2004) shows that the term structure of the at-the-money volatility skew has the following approximate behavior for all stochastic volatility models of the form

So we can estimate by regressing volatility skew against volatility level.

( )1/ 2

2

0

1/ 2

(1 ), 1

with / 2

T

BSx

v ex Tx T T

v

β λ

β

ρησλ λ

λ λ ρη

′− −

=

−

⎧ ⎫∂ −≈ −⎨ ⎬′ ′∂ ⎩ ⎭

′ = −

( )dv v v dt v dZβλ η= − − +

( )dv v v dt v dZλ η= − − +

( ) ( )2 (1 ),T

BSex T v v vT

λ

σλ

−−≈ + −

β


SPX 3-month ATM volatility skew vs ATM 3m volatility

0.25 0.3 0.35ATM vol

-0.13

-0.12

-0.11

-0.09

-0.08

ATM vol skew


Interpreting the regression of skew vs volatilityRecall that if the variance satisfies the SDE

at-the-money variance skew should satisfy

and at-the-money volatility skew should satisfy

The graph shows volatility skew to be roughly independent of volatility level so again consistent with lognormal volatility dynamics.

~dv v dZβ

1/ 2

0k

v vk

β −

=

∂∝

∂

2 2

0

BSBS

kkβσ σ −

=

∂∝

∂

1β ≈


A variance call option formulaAssuming this lognormal model, we obtain a Black-Scholes style formula for calls on variance:

( ) ( )22 21 2

sT

X K e N d K N dµ+ +⎡ ⎤− = −⎣ ⎦E

with21 1

2 21 2

log 2 log;K s Kd ds sµ µ− + + − +

= =


Example: One year Heston with BCC parametersWe compute one year European option prices in the Heston model using parameters from Bakshi, Cao and Chen. Specifically

0.04; 0.39; 1.15; 0.64v v η λ ρ= = = = = −

We obtain the following volatility smile:

Obviously, the fair value of the variance swap is 0.04T

X⎡ ⎤ =⎣ ⎦E

-1 -0.75 -0.5 -0.25 0.25 0.5k

0.15

0.25

0.3

0.35

Implied Vol


Heston formula for expected volatilityBy a well-known formula

Then, taking expectations

3/ 20

1 12

yey dλ

λλπ

∞ −−= ∫

We know the Laplace transform from the CIR bond formula.So we can also compute expected volatility explicitly in terms of the Heston parameters.With the BCC parameters, we obtain

3/ 20

112

TX

T

eX d

λ

λλπ

−∞ ⎡ ⎤− ⎣ ⎦⎡ ⎤ =⎣ ⎦ ∫

EE

TXe λ−⎡ ⎤⎣ ⎦E

0.187429T

X⎡ ⎤ =⎣ ⎦E

Our lognormal approximation then has parameters 2 0.129837; -1.73928s µ= =


1 Year options on varianceNow we value one year call options on variance (quadratic variation)

• Exactly in the Heston model using BCC parameters• Using our simple Black-Scholes style formula with

Results are as follows:

Blue line is exact; dashed red line is lognormal formula

2 0.129837; -1.73928s µ= =

0.05 0.1 0.15 0.2K

0.01

0.02

0.03

0.04

Variance CallT

X K+

⎡ ⎤−⎣ ⎦E


pdf of quadratic variationEquivalently we can plot the pdf of the log of quadratic variation

• Exactly in the Heston model using BCC parameters• Using our lognormal approximation

Results are as follows:

Blue line is exact Heston pdf; dashed red line is lognormal approximation

-5 -4 -3 -2 -1ln K

0.1

0.2

0.3

0.4

0.5

density


CorollaryWe note that even when our assumptions on the volatility dynamics are quite different from lognormal (i.e. Heston), results are good for practical purposes.In practice, we believe that volatility dynamics are lognormal so results should be even better!

So, if we know the convexity adjustment or equivalently, if we have market prices of variance and volatility swaps, we can use our simple lognormal model to price any (European-style) claim on quadratic variation with reasonable results.


Are volatility option prices uniquely determined by European option prices?

We know from Carr and Lee (and then from Friz and Gatheral) that the prices of options on quadratic variation are uniquely determined if the correlation between volatility moves and moves in the underlying is zero.Moreover, we showed how to retrieve the pdf of quadratic variation from option prices under this assumption.

What happens if correlation is not zero?


The Friz inversion algorithmWe recall from Hull and White that in a zero-correlation world, we may write ( ) ( ) ( , )BSc k dy g y c k y= ∫where is the total variance. In words, we can compute an option price by averaging over Black-Scholes option prices conditioned on the BS total implied variance.We assume that the law of is given by Then

2: BSy Tσ=

2( ) ( , ) izi BS

ic k p c k e=∑

By construction, the mean and variance of the approximate lognormal pdf match the mean and variance of the true pdf. We thus use

as an initial guess to the and minimize the objective

( )logT

X⎡ ⎤⎣ ⎦Eii z

i

p δ∑

2 2( ) / 2

2

12

iz siq e

sµ

π− −∝

ip2

2( , ) - ( ) ( , )izi BS j j

j ip c k e c k d p qβ

⎡ ⎤⎧ ⎫ +⎨ ⎬⎢ ⎥⎩ ⎭⎣ ⎦

∑ ∑where d(p,q) is some measure of distance such as relative entropy.


0.05 0.1 0.15 0.2K

0.01

0.02

0.03

0.04

Variance Call

A local volatility computationWe generate European option prices from the Heston model with BCC parameters and compute local volatilities.We use Monte Carlo simulation to compute the payoff of an option on quadratic variation on each path.

Local volatility (the green curve) underprices volatility options.

Note that is uniquely determined by European option prices!

TX⎡ ⎤⎣ ⎦E

-2 -1 1 2k

0.02

0.04

0.06

0.08

0.1

Total Variance


0.05 0.1 0.15 0.2K

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Variance Call

What happens if we change the correlation?Regenerate European option prices from the Heston model with BCC parameters but and recompute local volatilities.Heston exact results and the lognormal approximation are both insensitive to the change in correlation. What about the local volatility approximation?

The local volatility result (orange line) with is lower still.

0ρ =

-2 -1 1 2k

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Total Variance

0ρ =


A commentAs Dupire has pointed out, the zero correlation assumption is very strong.

• Local volatility is a diffusion process that is consistent with all given European option prices. But volatility moves and stock price moves are (locally) perfectly correlated.

Given the prices of European options of all strikes and expirations, even restricting ourselves to diffusion processes, the pricing of volatility options is not unique.

Dupire’s recent construction of upper and lower bounds on the price of an option on volatility suggests the following conjecture:


A conjectureGiven the prices of European options of all strikes and expirations, of all possible underlying diffusions consistent with these option prices, the lowest possible value of a volatility option is achieved by assuming local volatility dynamics.


Dupire’s method for valuing VIX futuresRoughly speaking, at time ,the VIX futures pays

where is quadratic variation between and .Also, as before

1 1 21,

:T T TX Y⎡ ⎤ =⎣ ⎦E

1 2 2 1,:

T T T TX X X= − 1T 2T

1T

[ ] [ ]2 21 1 1vart tY Y Y⎡ ⎤= −⎣ ⎦E E

We know in terms of the and log-strips. is estimated as the historical variance of VIX futures prices.

The fair value of the VIX future is then given by

A practical problem is that the VIX futures don’t trade enough to give accurate historical vols.

1 2

21 ,t t T T

Y X⎡ ⎤⎡ ⎤ =⎣ ⎦ ⎣ ⎦E E 1T 2T[ ]1var Y

[ ] [ ]21 1 1vart tY Y Y⎡ ⎤= −⎣ ⎦E E


Estimating volatility of volatilityWe can compute historical volatility of the VIX

Annualized 60-day volatility of VIX

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

3/29

/199

0

3/29

/199

1

3/29

/199

2

3/29

/199

3

3/29

/199

4

3/29

/199

5

3/29

/199

6

3/29

/199

7

3/29

/199

8

3/29

/199

9

3/29

/200

0

3/29

/200

1

3/29

/200

2

3/29

/200

3

3/29

/200

4

3/29

/200

5

…pretty stable at around 80-100%


Then we apply the volatility envelopeWe note that if , variance swaps should behave as

( )dv v v dt noiseλ= − − +

1: ( )T

T T

eW X v v v Tλ

λ

−−⎡ ⎤= − +⎣ ⎦ ∼E

Then assuming that changes in instantaneous volatility drive most of the changes in the volatility surface, we obtain

1 T

TeW v

λ

λ

−−∆ ∆∼

and changes in forward starting variance swaps are given by1 2

1

1 2, 2 1( )T T

TT T

e eW v e v T Tλ λ

λ

λ

− −−−

∆ ∆ ∆ −∼

We see that the volatility of should decay exponentially with maturity . We can identify changes in with changes in the VIX.We can estimate from the term structure of skew for example. Or we can use an empirically estimate volatility envelope to decay volatility as a function of time to maturity.

21Y

1T vλ


Decaying volatilities


RemarksNote that historical and implied volatilities on Bloomberg are consistent with the historical volatility of the VIX.So we have a practical way of estimating volatility of volatility.And we can compute variance swaps from the log-strips.We can then use our simple lognormal model to price any European-style claim on quadratic variation.


Trading volatility of volatility: implied vol of volWe have shown how volatility of volatility might be estimated from historical data or alternatively, from a knowledge of the variance-volatility convexity adjustment.In practice, even where the convexity adjustment

is traded, there is a bid-offer spread.So volatility of volatility becomes another implied parameter that is effectively quoted and traded with its own bid-offer spread.An example: the market for the one-year SPX variance-volatility convexity adjustment is currently around 0.80-1.50 with the variance swap at 17.3% (mid-market).

• Under our lognormal assumption,

• Then the volatility of volatility s is roughly 0.31 bid at 0.43 offered.

T TX X⎡ ⎤⎡ ⎤ −⎣ ⎦ ⎣ ⎦E E

2 2 log T

T

Xs

X

⎛ ⎞⎡ ⎤⎣ ⎦⎜ ⎟= ⎜ ⎟⎡ ⎤⎜ ⎟⎣ ⎦⎝ ⎠

E

E


0.5 1 1.5 2T

0.8

1.2

1.4

1.6

1.8

2

Comparing implied vol of vol with historical0.31 bid @ 0.43 offered is the effective market for (average) vol of volof one-year SPX implied volatility.According to our earlier analysis, the volatility of one-year volatility should be given by

21

1 eVIXλ

σλ

−−∆ ∆∼

From the Bloomberg historical VIX futures volatilities, we estimateNote that the shape of is not very different from

Then we have

1.3.λ ∼

2 21 11.66

1VIX

e λ

λ σ σ−∆ ∆ = ∆−

∼

This translates to an implied spot VIX volatility bid-offer spread of 0.52-0.73. ... not inconsistent with historical VIX implied volatility!

1.3(1 )Te−− 1/ T1/ T

1.3(1 )Tconst e−× −


SummaryWe showed how to compute expected quadratic variation for compound Poisson processes.We computed the magnitude of the jump correction to the diffusion-based valuation of variance swaps and suggested that at least for indices, jumps have little impact on the valuation.We described a simple lognormal model for estimating the value of options on volatility and related the parameters to the values of variance and volatility swaps.Under the zero correlation assumption, we showed how to recover the unique density of quadratic variation from European option prices. We investigated whether or not volatility option prices are in general uniquely determined by European option prices and showed that they are not.We described Dupire’s method for valuing VIX futures.Finally, we suggested how to estimate volatility of volatility.


ReferencesBakshi, G., Cao C. and Chen Z. (1997). Empirical performance of alternative option pricing models. Journal of Finance, 52, 2003-2049.Cont, Rama and Peter Tankov (2004), Financial Modelling with Jump Processes, Chapman and Hall.Duanmu, Zhenyu (2005). Rational pricing of options on realized volatility. Available at http://www.ieor.columbia.edu/feseminar/duanmu.pptDupire, Bruno (2005). Model free results on volatility derivatives. Available at http://www.math.nyu.edu/%7Ecarrp/mfseminar/bruno.pptFriz, Peter and Gatheral, J. (2005). Valuation of Volatility Derivatives as an Inverse Problem, Merrill Lynch and Courant Institute, NYU.Gatheral, J. (2004). Case studies in financial modeling lecture notes. http://www.math.nyu.edu/fellows_fin_math/gatheral/case_studies.htmlHeston, Steven (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies 6, 327-343.Hull, John and Alan White (1987), The pricing of options with stochastic volatilities, The Journal of Finance 19, 281-300.Lewis, Alan R. (2000), Option Valuation under Stochastic Volatility : with Mathematica Code, Finance Press.

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