[university of london, jacquier] volatility seminar - some notes on variance swaps and volatility...

7/31/2019 [University of London, Jacquier] Volatility Seminar - Some Notes on Variance Swaps and Volatility Derivatives

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Volatility Seminar, Imperial College

Some notes on Variance Swaps and Volatility

derivatives

Antoine Jacquier∗

Birkbeck College, University of London

Zeliade Systems

May 2007

Abstract

We review here the theoretical approaches as well as the practical useof volatility derivatives. A clear focus will be made on variance swaps,

for historical and logical - the most liquid of any volatility instruments -

reasons. Instead of going into all the details of the calculations, we will

try to point out the useful papers on different subtopics.

Contents

1 Variance and volatility swaps 1

1.1 Replication via Vanilla options . . . . . . . . . . . . . . . . . . . 21.2 Comments on the replication result . . . . . . . . . . . . . . . . . 3

2 ”Exotic” Variance swaps 7

2.1 Forward Variance Swap . . . . . . . . . . . . . . . . . . . . . . . 72.2 Gamma Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Entropy Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Corridor and Conditional Variance Swaps . . . . . . . . . . . . . 82.5 Generalised Variance Swaps . . . . . . . . . . . . . . . . . . . . . 82.6 Moment Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7 Dispersion and correlation trading . . . . . . . . . . . . . . . . . 9

3 Volatility derivatives 9

3.1 Options on realised variance . . . . . . . . . . . . . . . . . . . . . 93.2 VIX and options on the VIX . . . . . . . . . . . . . . . . . . . . 93.3 Forward-Started options and Cliquets . . . . . . . . . . . . . . . 103.4 Advanced Volatility Derivatives . . . . . . . . . . . . . . . . . . . 11

∗[email protected].

1



1 Variance and volatility swaps

Variance swaps are products with the following payoff at maturity :

Π = N

1

T

T 0

σ2t dt− K

Where N is a notional, K the strike, expressed in variance (squared volatility)points, and T is the maturity of the option and σ2

t represents the squared returnover the infinitesimal period [t, t + dt]. As any swap, its price is such that theoption has zero value at inception, i.e. :

K = E∗

1

T

T 0

σ2t dt/F 0

1.1 Replication via Vanilla options

The idea of replication relies upon the formula by Breeden &Litzenberger (1978)

: p (S T , T ; S t, t) = ∂ 2C (S t,K,t,T )∂K 2 K =S T

= ∂ 2P (S t,K,t,T )∂K 2 K =S T

, where C (S t, K, t, T )

(resp. P ) stands for the undiscounted call option price at time t written on thestock S with maturity T and strike K . Then the value of any claim with payoff g reads (for any F ≥ 0)

E [g (S T ) /F t] =

∞

0

p (K, T ; S t, t) g (K ) dK

=

F 0

∂ 2P

∂K 2g (K ) dK +

∞

F

∂ 2C

∂K 2g (K ) dK

Integrating twice by parts and using the Put-Call parity relationship (we referto formula (11.1) for details), we obtain :

E [g (S T ) /F t] = g (F ) +

F 0

P (K ) g (K ) dK +

∞

F

C (K ) g (K ) dK

Now, consider the following stock price process : dS tS t

= rdt + σtdW t as well asthe traditional Black-Scholes assumptions, except for the volatility, which needsnot be constant. Using Ito’s lemma, we get :

log

S T S 0

=

T 0

d log(S t) =

T 0

dS tS t

− 1

2

T 0

σ2t dt

Hence, the realised variance over the period [0, T ] reads :

1

T T

0σ2t dt =

2

T T

0

dS t

S t −2

T logS T

S 0

2



Taking expectations on both sides, and noting that EdS tS t

= rdt, as well as

the following identities, for any S ∗ > 0 :

− log

S T S 0

= −S T − S ∗

S ∗

Short 1/S ∗

fwd contracts

+

S ∗0

(K − S T )+

dK

K 2+

∞

S ∗

(S T −K )+

dK

K 2

Long position in Calls and Puts struck at K

− log

S ∗S 0

we eventually get

1

T E

T 0

σ2t dt

=

2

T

rT −

S 0S ∗

erT − 1

− log

S ∗S 0

+ erT

S ∗0

dK

K 2P (K ) +

∞

S ∗

dK

K 2C (K )

1.2 Comments on the replication result

• the term S ∗ can be chosen arbitrarily. For practical reasons, it is usu-ally taken as the at-the-money forward, which corresponds to a liquiditythreshold : above this strike, the calls are more liquid, below, the puts aremore liquid.

• This formula shows that the realised variance, i.e. the strike of a varianceswap, can be replicated by a portfolio of options, each of them weightedby the squared strikes.

• It can be shown that such a weighting scheme makes the portfolio’s vegabe independent of the stock price. This is a very convenient feature, asthe variance swap can hence be considered as a pure volatility product.For a proof of this statement, see [14].

• The whole replication above has been done within the Black-Scholes frame-work, which means in particular, no jumps and flat skew. Let us start bythe problem of the skew, i.e. the influence of the skew on the price of thevariance swap. We are here mainly inspired from [14]. We state here afew results and refer to [14] for details on the derivations.

– Skew linear in strike

Let suppose that the implied volatility has the following parameter-isation :

σimp (K ) = σ0 − β K − S

S

Where σ is the fair volatility in the flat world. The higher the β , thesteeper the slope of the skew. Then, the fair value of the varianceswap is modified as

K Var = σ20

1 + 3T β 2 + . . .

3



– Gatheral approach

Interestingly, Gatheral (see in the book, page 139) points out twothings: first, that the fair value of the variance swap can indeed beexpressed as a weighted average of implied variance ; then he presentsa different results for the impact of the skew : according to him, theslope of the skew has no impact, whereas the curvature of the smile

does affect (in the same direction) the price of the variance swap. Heindeed gets the following formula :

E

1

T

T 0

σ2t dt

=

R

dzN (z) σ2BS (z)

Where N is the cumulative distribution function of the Gaussian andz is a level of log-moneyness : z = log

S K

.

• In the replication above, one of the main assumptions was the existenceof a continuum of options for all strikes, which is of course not observedon the market. So one is only able to trade in a limited range of strikes.It is easily shown that this limited replication has a lower value than thefull replication. See [14] (section 5) for details.

• Here, we only consider the pricing of a variance swap at the inceptionof the contract. Thanks to the additive properties of the variance, it isfairly easy to decompose the variance swap at an intermediate time into arealised variance and a future variance. For more details on this, see [10].

• In the derivation of the replicated portfolio, we assumed the continuityof the stock price. Suppose that a jump occurs and that the stock price

jumps downwards (J < 0) or upwards (J > 0) from S to S (1 + J ). Then,because of the additivity of variance, we have, in discrete time :

1

T

∆S

S

2

=1

T

∆S

S

2

no jump

+1

T

∆S

S

2

jump

And 1T ∆S

S 2

jump = J 2

T

Now, the replication strategy reads

RS =2

T

N i=1

∆S iS i−1

− log

S i

S i−1

So the influence on the jump on the replicating strategy is worth 2T (J − log (1 + J )).

And so, the P&L due to this jump is worth

P &LJump =2

T (J − log (1 + J ))− J 2

T

Taylor expanding the log function around J (assuming J is small enough),we get :

P &LJump ≈ −23

J 3T

4



This means that a big move upwards (J > 0) leads to a loss for the varianceswap seller, and vice versa. For more details on this, see [14] (Section 5).

• Convexity adjustment for volatility swapsVolatility swaps look like variance swap. The payoff of a volatility swapis N vol(

√ V − K vol). At first, we can use the following approximation :

E√ x ≈ E(x). Therefore, we can replicate a volatility swap (K vol,N vol)using a variance swap (K var ,N var), where N vol is noted in units of currencyper volatility point, and N var in units of currency per variance point. Wemust take care that :

K var = K 2vol

N var = N vol2K vol

The second equality is explained as follows :

√ V −K vol =

V −K 2vol√ V + K vol

≈ 1

2K vol

V −K 2vol

But, in fact, there is a convexity bias between the volatility swap and the

variance swap due to the non linearity of the square root function.As explained in Brockhaus and Long [1], we can have a much better ap-proximation of a volatility swap using the results found with a varianceswap. Given the square-root function F : x → √

x, the second orderTaylor expansion around x0 gives us

F (x) ≈ F (x0) + F (x0)(x − x0) +1

2F (x0)(x − x0)2

F (x) ≈ √ x0 +

x− x0

2√

x0− 1

8

(x− x0)2 x3

0

F (x) ≈ x + x0

2√

x0− (x − x0)2

8 x3

0

Applying this formula with x = v and x0 = E[v] and taking expectations,we have

E√

v ≈

E[v] − V[v]

8 E3[v]

This formula gives us the approximate value of the convexity bias, which

is thus equal to V[v]

8√ E3[v]

.

The convexity bias needs either a specified model for the dynamics of the volatility or an assumption on the level of the forward volatility andvariance. In terms of replication, there is no static hedging. But, as in thetradional Black-Scholes framework, we can produce a dynamic hedgingfor these volatility swaps, using variance swaps all along the life time of

the swaps so that it becomes instantaneously independent on the moves

5



in the volatility. Mathematically, we want to find (α, β ) minimizing thefunction Φ defined as :

Φ (α, β ) = E

σt − ασ2

t − β 2

When differentiating the above function, we get

∂Φ∂α (α, β ) = E

2ασ4

t − 2σ3t + 2αβσ2

t

∂Φ∂β (α, β ) = E

2β − 2σt + 2ασ2

t

which finally gives

E [σt] = αE

σ2t

+ β

E

σ3t

= αE

σ4t

+ β E

σ2t

This two-equalities result hence needs the forward levels of volatility andvariance. This explains why variance swaps are nowadays much moreliquid than volatility swaps. Furthermore, despite the fact that we can’t

calculate a mark-to-market price for volatility swap without a specifiedstochastic volatility model, Morokoff, Akesson and Zhou established in[18] the existence of lower and upper bounds for such products, usingarbitrage arguments. Carr and Lee [5] also found an interesting result: the volatility swap admits a robust supereplication using a varianceswap and has a lower bound equal to the at-the-money implied volatility(ATMIV). They also provide an approximation using this ATMIV and thetotal variance which is

E0σ̄T ≈ a04 +

a20

+σ̄20

2 T

4 + a20T

where a0 represents the at-the-money implied volatility at time 0.

In a similar way to variance swaps, the hedge for volatility swaps isn’tstatic but has to be dynamically rebalanced using the log-contracts. How-ever, the large number of options needed to synthetically create this hedgemakes the transaction costs too much.For practical as well as technical details on the convexity adjustment, see[21].

• P&L and ”Cash” Gamma

Let us consider an option V t (for example a Vanilla Call option or a Vari-ance Swap). Using Black-Scholes method for constructing a delta-hedgedportfolio, one can easily show that the P&L of such a portfolio on theperiod [t, t + dt] reads

P &L[t,t+dt] =1

2 ΓS 2t dS t

S t2

− σ2t dt6



Proof : We briefly sketch the proof We have

P &L[t,t+dt] = Delta P&L+Gamma P&L+Theta P&L = ∆ (∆S )+1

2Γ (∆S )

2+Θdt

But Θ ≈ − 12 ΓS 2σ2

imp, where σ2imp stands for the implied volatility. Hence,

when delta-hedging the portfolio,

P &L[t,t+dt] =1

2Γ

(∆S )2 − σ2

impS 2dt

=1

2S 2Γ

∆S

S

2

− σ2impdt

The first-term in the bracket corresponds to the realised variance, and thesecond term to the implied variance. For more insights on this, see [4](part 8)Now, the Gamma of the Variance Swap reads (see [17] for the greeks of aVariance Swap)

ΓVarSwap =2

S 2t T

Plugging this into the P&L formula, we find

P &L[t,t+dt] =1

T

∆S

S

2

− σ2impdt

This means that the P&L of a Variance Swap over a small period of timedoes not depend on the level of the stock price. This is why it is called aconstant ”Cash” Gamma. In fact, the P&L, as we can see it, only dependson the difference between implied and realised variance.

• A finite-difference approach for thepricing of Variance Swaps has also beendevelopped, see [16].

2 ”Exotic” Variance swaps

2.1 Forward Variance Swap

2.2 Gamma Swaps

We saw that the vega of the variance swap does not depend on the level of thestock price, and also that the Variance Swap had a constant ”Cash” Gamma.Suppose the stock price jumps downwards ; the Gamma of the variance swapdoes not change, nor its Vega. If one uses variance swaps for hedging purposes,this could be a problem. Hence, it would be quite convenient ot be get a Vegawhich adjusts itself automatically with the level of the stck price, so as to reducethe potential exposure to volatility. The Gamma swap is an answer to this andits payoff reads

V = N 1

T T

0σ2t

S t

S 0 dt− K 7



With this definition, Gamma Swaps will be replicated exactly like VarianceSwaps, but with a weighting scheme equal to 1

K instead of 1K 2 . Practically

speaking, Gamma Swaps have a constant ”Share” Gamma , and, as they areweighted by the level of the stock, they do take into account the possibility of

jumps.

2.3 Entropy Swaps

The Entropy Swap has been considered by Buehler in [3]. It looks like a GammaSwap, though it is much less common. Its payoff reads T

0

S td log(S t) =

T 0

S tσ2t dtdt

We refer to [3] for details on this product. Its replication is very similar to thatof the Gamma Swap.

2.4 Corridor and Conditional Variance Swaps

Corridor and conditional Variance Swaps allow investors to take exposure onsome future level of volatility given that the underlying has traded in a speci-fied range. This conditionality makes these products less expensive than pureVariance Swaps. There are basically two types of conditional variance swaps: up-variance swaps accrue realised volatility when the underlying stock or in-dex is above a certain threshold, whereas down-variance swaps have gets higherwhen the underlying remains below a pre-specified level. The two products arevery similar, up to two small differences :

• In the Conditional Variance Swap, the accrued variance is divided by thenumber of days spent in the specied range, whereas it is divided by thetotal number of days for a Corridor Variance Swap.

• (Consequence of the first point) : the P&L of the Conditional Variance

Swap is scaled to the proportion of time spent within the range.More formally, we can write :

P &LCond =D

T

S

D− K Cond

=

S

T −K Cond

Corridor VS

+

K Corr − D

T K Cond

Range accrual

From a pricing point of view, the replication scheme is almost the same than theone used for pure Variance Swaps : the weights are identical, but the integrals

are truncated at the levels specified by the barriers(down or up).General references

8



• A.Sepp, Variance swaps under no conditions , (Risk, March 2007)

• P.Carr, K.Lewis, Corridor variance swaps , [7]

2.5 Generalised Variance Swaps

This is just a generalisation of the Vanilla Variance Swaps and the so-called ”ex-otic” Variance Swaps. In fact, instead of considering the discrete-time realisedvariance, consider the following :

S =1

T

T −1t=0

F t log2

S t+1

S t

Then, one can see that

• If F t = 1, then this represents a pure Variance Swaps.

• If F t = S tS 0

, then this is a Gamma Swaps

• if F =

1, if L ≤ S t−1 ≤ U 0 otherwise

, then we are back to a Corridor Variance

Swap. (The down-Variance Swap assumes U < ∞, and for the up-VarianceSwap, we have U = ∞).

2.6 Moment Swaps

This product has been proposed by Schoutens (see [20]). The payoff of such anoption reads

Π = N

1

T

T 0

dS tS t

n

−K

Where N represents the notional of the product, K the strike, T the maturity,and n the exponential power. Using Taylor expansion method, Schoutens showsthat a nth-order Moment Swap can be replicated by trading log-contract (i.e.

a portfolio of European options), a dynamic strategy in Futures and a series of moment swaps of order strictly smaller that n. For now, it seems like this kindof product has been purely of theoretical interest.

2.7 Dispersion and correlation trading

3 Volatility derivatives

3.1 Options on realised variance

The natural extension of Variance Swaps is an option on the realised variance,basically a call option, with payoff

1

T T

0σ2t dt −K

+

9



Carr & Lee [5] found a replication strategy for such an option under zero cor-relation between the spot price and the volatility. In particular, they were ableto replicate any payoff of the form of the exponential of the quadratic variation.See also [6] for very recent results on the replication of volatility derivatives,using variance and volatility swaps as hedging instruments.For pricing methods of options on realised variance under Levy processes, we

refer the reader to [9].

3.2 VIX and options on the VIX

The VIX is a volatility index launched by the CBOE in 1993. Its purpose wasto replicate the one-month implied volatility of the S&P 100 index. In 2003,the calculation method was changed and expanded to replicate the S&P 500.Options on the VIX are quite recent :

• March 2004 : VIX Futures listed by the CBOE.

• 2004 : Variance Futures listed by the CBOE.

•2005 : New volatility indices, VDAX, VSTOXX, VSMI.

• 2005 : Volatility Futures on thesevol indices.

• 2006 : Options on the VIX were launched.

The mathematical definition of the VIX is as follows

σ2 =2

T

n

∆K iK 2i

Q (K i)− 1

T

F

K 0− 1

2

Where

σ = V IX100

F : forward index level derived from index option prices

K i : Strike of the ith optionQ (K i) : Midpoint of the bid-ask spread for an option with strike K i

Now, let us consider a stock price with spot volatility σt. Then one can derivethe following formula :

V IX 2t = EQt

1

τ

t+τ

t

σ2sds

Where τ represents the one-month period. Using this result, closed-form orsemi-closed form formula can be found for options (in particular Futures) onthe VIX, if one assumes a stochastic volatility model.For details on the pricing of VIX Futures, see [22], [15] for closed-form solutionsunder various stochastic volatility models or directly on the VIX website forpractical details : http://www.cboe.com/micro/vix/vixoptions.aspx

10



More advanced results on lower and upper bounds for VIX Futures have beenobtained in [8] and [11]. In particular, the bounds obtained by Carr and Wu in[8] can be expressed in terms of a portfolio of European options (for the upperbound) and of a Forward-tart ATM call option (for the lower bound), which aredirectly observable.

3.3 Forward-Started options and Cliquets

We don’t detail anything here, as these products have already been studied ina previous talk. Just to mention that, if one considers a Forward-Start Call-Spread, with payoff (S T /S t − K 1)+ − (S T /S t −K 2)+, then one can see that itsdelta is theoretically zero (hence pure volatility product), and that it purelydepends on the skew between the two strikes K 1 and K 2. They are in fact theunderlying blocks of cliquet options, which heaviley rely on the forward level of variance. See [2] for some details on this.

3.4 Advanced Volatility Derivatives

References

[1] O. Brockhaus, D. Long. Volatility swaps made simple , Risk Magazine, Jan-uary, 2000.

[2] H. Buehler. Stochastic volatility models and products , Risk Training Course,Hong Kong, 2004

[3] H. Buehler. Volatility Markets, Consistent modeling, hedging and practical

implementation , PhD Thesis, 2006

[4] P. Carr. FAQ’s in option pricing theory , Courant Institute, NYU, 2002

[5] P. Carr, R. Lee. Robust replication of volatility derivatives , Courant Insti-tute, NYU, Stanford University, 2003

[6] P. Carr, R. Lee. Realized Volatility and Variance: Options via Swaps , Risk,2007

[7] P. Carr, R. Lee. Hedging variance options on continuous semimartingales ,Courant Institute, NYU, Stanford University, 2006

[7] P. Carr, K. Lewis. Corridor variance swaps , Risk, February 20004, 2006

[8] P. Carr, L. Wu. A tale of two indices , Journal of Derivatives, Spring 2006.

[9] P. Carr, H. Geman, D. Madan, M. Yor. Pricing options on realised variance ,Finance and Stochastics, 2005, issue 4

[10] N. Chriss, W. Morokoff. Market Risk for Volatility and Variance Swaps ,Risk, July 1999

11



[11] B. Dupire. Model free results on volatility derivatives , Bloomberg, 2006

[12] Chicago Board Options Exchange. VIX-CBOE Volatility Index , 2003,http://www.cboe.com/micro/vix/vixwhite.pdf

[13] P. Friz, J. Gatheral. Valuation of volatility derivatives as an inverse prob-

lem , Quantitative Finance, 2005

[14] K. Demeterfi, E. Derman, M. Kamal, J. Zou. More than you ever wanted

to know about variance swaps , Goldman Sachs, Quantitative Strategies Re-search Notes, 1999.

[15] Y. Lin. Pricing VIX Futures on Affine Stochastic Volatility Models with Si-

multaneous Sate-Dependent Jumps both in the S&P 00 Price and Variance

Processes , Working paper, 2006.

[16] T. Little, V. Pant. A finite difference method for variance swap, Workingpaper, 2001.

[17] D.E. Kuenzi. Variance swaps and non constant Vega , Risk, October 2005.

[18] W. Morokoff, F. Akesson, Y. Zhou. Risk management of volatility and vari-ance swaps , Risk Quantitative Modeling Notes, Goldman Sachs, 1999.

[19] A. Neuberger. The log contract , Journal of Portfolio Management, 20(2):74-80, 1994.

[20] W. Schoutens. Moment swaps , Working paper, 2005.

[21] C. Youssfi. Convexity adjustment for volatility swaps , Merril Lynch presen-tation, 2005.

[22] Y. Zhu, J.E. Zhang. Variance term structure and VIX Futures pricing ,International Journal of Theoretical and Applied Finance, Vol. 10, No. 1(2007) 111-127.

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[university of london, jacquier] volatility seminar - some notes on variance swaps and volatility...

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