Alexander Gairat

In collaboration with IVolatility.com

Variance swaps

IntroductionThe goal of this paper is to make a reader more familiar with pricing and hedging variance swaps and to propose some practical recommendations for quoting variance swaps (see section Conclusions).

We give basic ideas of variance swap pricing and hedging (for detailed discussion see [1]) and apply this analyze to real market data. In the last section we discuss the connection with volatility swaps.

A variance swap is a forward contract on annualized variance, the square of the realized volatility.

VR = σR2 = 252 1

n − 2 „i=1

n−1ikjjjLog i

kjjjS@ti+1DS@tiD

y{zzzy{zzz2

Its payoff at expiration is equal to

PayOff = σR2 − KR2

Where

S@tiD - the closing level of stock on ti valuation date.

n - number of business days from the Trade Date up to and including the Maturity Date.

When entering the swap the strike KR is typically set at a level so that the counterparties do not have to exchange cash flows (‘fair strike’).

Variance swap and option delta-hedging.Advantage of trading variance swap rather than buying options is that it is pure play on realized volatility no path dependency is involved. Let us recall how path dependency appears in trading P&L of a delta-hedged option position.

If we used implied volatility σi on day i for hedging option then P&L at maturity is sum of daily variance spread weighted by dollar gamma.

Final P & L =1

2 ‚

i=1

n−1 Hri2 − σi2 ∆tL Γi Si2

ri = Si+1−SiSi

is stock return on day i

Γi Si2 = Γ@i, SiD Si2 - dollar gamma on day i.

So periods when dollar gamma is high are dominating in P&L.

Market example

We use here historical data for the option with one year expiration on SPX500, strike K=1150, observation period from 2004/1/2 to 2004/12/4

4ê1ê2 4ê3ê1 4ê4ê26 4ê6ê22 4ê8ê17 4ê10ê12 4ê12ê7

-0.0001

-0.00005

0.00005

5 days average of ri2−σi

2∆t

4ê1ê2 4ê3ê1 4ê4ê26 4ê6ê22 4ê8ê17 4ê10ê12 4ê12ê7

2000400060008000

10000

5 days average of dollar gamma ΓtSt2

5 days average of plain variance spread and variance spread weighted by dollar gamma

4ê1ê2 4ê3ê1 4ê4ê26 4ê6ê22 4ê8ê17 4ê10ê12 4ê12ê7

-0.008

-0.006

-0.004

-0.002

‚i=1n−1Hri

2−σi2∆tL

A∗‚i=1n−1Hri

2−σi2∆tLΓiSi

2

Where dollar gamma is normalized by factor A = 252êH⁄ Γi Si2L

2 Variance Swaps

PricingThe fair strike KR can be calculated directly from option prices (under assumptions that the underlying follows a continuous diffusion process, see Appendix 1)

(1) KR2 =2

T „

Ki≤FT

∆Ki

Ki2 r T Put@KiD +

2

T „

Ki>FT

∆Ki

Ki2 r T Call@KiD

where

FT = Hr−qL T is the forward price on the stock at expiration time T,

r risk-free interest rate to expiration,

q dividend yield.

Example: VIX

VIX CBOE volatility index it is actually fair strike KR for variance swap on S&P500 index.

VIX@TD2 =2

T „

Ki≤FT

∆Ki

Ki2 r T Put@KiD +

2

T „

Ki>FT

∆Ki

Ki2 r T Call@KiD −

1

T ikjjFTK0

− 1y{zz2

where K0 is the first strike below the forward index level FT.

The only difference with formula (1) is the correction term

−1

T ikjjFTK0

− 1y{zz2

which improves the accuracy of approximation (1).

There two problems in calculation the fair strike of variance swap in from raw market prices:

è We need quotes of European options.

They are available for indexes but not for stocks.

è Number of available option strikes can be not sufficient for accurate calculation.

The first problem can be solved by calculating prices of European options from implied volatilities of listed American options. To overcome the second problem we could use additional strikes and interpolate implied volatilities to this set.

Variance Swaps 3

Examples

In this section we calculate prices of variance swap which starts on n-th trading day counted from 1.1.2004 with the expiration on 31.12.2004. We compare two values:

The first KR is calculated based on 40 European options with standardized moneyness x

x = Log@KêFTD ë Iσè!!!!t M

x in range from -2 to 2 with step 0.1.

The second value KR HRawL is calculated based on the raw market prices (either they are European or American). We use only the strikes of options with valid implied volatilities IV, i.e. IV is in range from 0 to 1. These strikes we call valid strikes. The number of valid strikes we denote by N[K].

SPX

The differences here are quite small less than 1 volatility point. It is natural to expect small difference for SPX because in both cases we used European option and, what is more important, the number of valid strikes is rather large more than 40.

50 100 150 200

12.5

17.5

20

22.5

25

KR KRHRawL

4 Variance Swaps

50 100 150 200

-0.8

-0.6

-0.4

-0.2

KR−KRHRawL

50 100 150 200

40

45

50

N@KD

EBAY INC

This example is different. Only American options are available. But still deference are not too large. And we see that diver-gence happens due to the small number of strikes available for calculation of KR HRawL. Still the number of valid strikes is relatively large close to 20.

50 100 150 200 250

27.5

30

32.5

35

37.5

40

42.5

KR KRHRawL

Variance Swaps 5

50 100 150 200 250

-1

-0.5

0.5

1

1.5

2

2.5

KR−KRHRawL

50 100 150 200 250101214161820N@KD

In the following two examples we will see that the quality of approximation (1) with raw market prices drops dramatically due to small number of valid strikes.

ADVANCED MICRO DEVICES

50 100 150 200 250

45

50

55

60

65

70

KR KRHRawL

6 Variance Swaps

50 100 150 200 250

-20

-15

-10

-5

5

10

15

KR−KRHRawL

50 100 150 200 250

8

10

12

14N@KD

Time Warner INC

50 100 150 200 250

20

25

30

KR KRHRawL

Variance Swaps 7

50 100 150 200 250

-5

5

10

KR−KRHRawL

50 100 150 200 250

4

6

8

10N@KD

HedgingReplication strategy for variance V[T] follows from the relation (for details see Appendix)

1

T V@TD ≈

1

T „i=1

n−1ikjjjLog i

kjjjS@ti+1DS@tiD

y{zzzy{zzz2

≈2

T ikjjjjj‚i=1

n−1 S@ti+1D − S@tiDS@tiD

− Log@ST êS0Dy{zzzzz

The first term in the brackets

‚i=1

n−1 S@ti+1D − S@tiDS@tiD

can be thought as P&L of continuous rebalancing a stock position so that it is always long 1êSt shares of the stock.

The second term

−Log@ST êS0D

represent static short position in a contract which pays the logarithm of the total return.

Example

As an example we take the stock prices of PFIZER INC, for one year period from 1/1/2004 to 1/1/2005.

8 Variance Swaps

50 100 150 200 250Trading days from 1ê1ê2004

26

28

30

32

34

36

38

St PFIZER INC

For this stock prices we calculate realized variance Vt and its replication Πt for each trading day

Πt = 2 ‚ti<t

HS@ti+1D − S@tiDLêS@tiD − 2 Log@St êS0D

Difference of realized variance and replication in volatility points

50 100 150 200 250

-0.1

-0.05

0.05

0.1

100Hè!!!!!!!!!!!!Vt êt −è!!!!!!!!!!!!Πt êt L

The typical differences in this plot are less than 0.1 volatility point. Large differences in the first 10 days are explained by large ratio of expiration period to time step - 1 day. Large difference around 240-th trading day appears due to the jump of the stock price.

Variance Swaps 9

We see that combination of dynamic trading on stock and log contract can efficiently replicate variance swap.

The payoff of log contract can be replicated by linear combination of puts and calls payoffs (see Appendix)

−Log@ST êS0D ≈ −ST − S0S0

+ „Ki≤S0

∆Ki

Ki2 HKi − STL+ + „

Ki>S0

∆Ki

Ki2 HST − KiL+

Hence log contract can be replicated by standard market instruments:

è short position in 1/S forward contracts strike at S0

è long position in ∆Ki êKi2 put options strike at K, for all strikes Ki from 0 to S0,

è long position in ∆Ki êKi2 call options strike at K, for all strikes Ki > S0

So this portfolio (with doubled positions) and dynamic trading on stock replicates variance swap. The replication also gives the fair strike value of volatility swap at time t

(2) KR@tD2 ≈2

T i

kjjjjjj ‚ti+1<t

S@ti+1D − S@tiDS@tiD

+ Log@FT@tDêStDy

{zzzzzz

Hdynamic tradingL

+2

T ikjjjjS0 − FT@tD

S0+ „

Ki≤S0

∆Ki

Ki2 r HT−tL Put@St, Ki, T − tD +

„Ki>S0

∆Ki

Ki2 r HT−tL Call@St, Ki, T − tDy

{zzzz Hstatic replicationL

where r is interest rate.

Market examplesIn this section we test replication performance on a set of market data.

For each day we calculate price of volatility swap è!!!!!!!!!!!!!!!!!!!!Vt@TDêT started on 1.1.2004 with expiration on 1.1.2005 and the

price of replication strategy KR@tD.

At time t the volatility swap consists of the realized variance and the expected future variance

1

T Vt@TD =

1

T „

ti+1<t

ikjjjLog i

kjjjS@ti+1DS@tiD

y{zzzy{zzz2

´̈ ¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨̈̈ ¨¨¨¨¨¨̈ ¨̈≠ ƨ¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨̈̈ ¨¨̈ ¨¨¨̈¨realized variance

+

2

T r HT−tL

ikjjjj„

Ki≤FT

∆Ki

Ki2 Put@St, Ki, T − tD + „

Ki>FT

∆Ki

Ki2 Call@St, Ki, T − tDy

{zzzz

´̈ ¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨̈ ¨̈ ¨≠ ƨ¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨̈ ¨̈expected variance

10 Variance Swaps

For replication we use 40 European options with strikes K such that the standardized moneyness x

x = Log@KêFT@0DD ë Iσè!!!!T M

x is in n range from -2 to 2 with step 0.1.

The implied volatilities for these strikes we get by linear interpolation of the market implied volatilities. Then we calculate option prices and get the fair strike value KR@tD from Eq.(2).

To estimate the performance of replication we define the replication error

Error@tD = 100∗i

k

jjjjjjj$%%%%%%%%%%%%%%%%%%%1

T Vt@TD − KR@tD

y

{

zzzzzzz

We also plot the number of valid market strikes N[K] (strikes with well defined implied volatilities).

Variance Swaps 11

Replication Error

SPX

50 100 150 200

40

45

50

N@KD

50 100 150 200-0.2-0.1

0.10.20.30.4

Error

50 100 150 200

14

16

18

KR

12 Variance Swaps

PFIZER INC

50 100 150 200 250

68

101214

N@KD

50 100 150 200 250

-0.6

-0.4

-0.2

0.2Error

50 100 150 200 250

21

22

23

24

25

26

KR

Variance Swaps 13

EBAY INC

50 100 150 200 250

10

12

14

16

18

20

N@KD

50 100 150 200 250

-1.5-1.25

-1-0.75

-0.5-0.25

Error

50 100 150 200 250

34

35

36

37

38

KR

14 Variance Swaps

ADVANCED MICRO DEVICES

50 100 150 200 250

8

10

12

14

N@KD

50 100 150 200 250

-1.5

-1

-0.5

0.5

Error

50 100 150 200 250

47.5

52.5

55

57.5

60

KR

Variance Swaps 15

GENERAL MOTORS CORP

50 100 150 200 250

6

8

10

12

N@KD

50 100 150 200 250

0.2

0.4

0.6

Error

50 100 150 200 250

24

26

28

30

32

KR

16 Variance Swaps

Time Warner Inc.

50 100 150 200 250

4

6

8

10

N@KD

50 100 150 200 250-0.1

0.10.20.30.40.50.6

Error

50 100 150 200 250

22

24

26

28

30

KR

Variance Swaps 17

Citigroup Inc

50 100 150 200 250

456789

10

N@KD

50 100 150 200 250

-0.2-0.1

0.10.20.30.40.5

Error

50 100 150 200 250

18

20

22

24

KR

18 Variance Swaps

EXXON MOBIL CORP

50 100 150 200 250

468

1012

N@KD

50 100 150 200 250

-0.5

-0.4

-0.3

-0.2

-0.1

Error

50 100 150 200 250

18

19

20

21

KR

Variance Swaps 19

MORGAN STANLEY DEAN WITTER CORP

50 100 150 200 250

6

8

10

N@KD

50 100 150 200 250

-0.4

-0.2

0.2

Error

50 100 150 200 250

26

28

30

32

KR

20 Variance Swaps

FORD MOTOR CORP

50 100 150 200 250

4

6

8

10

N@KD

50 100 150 200 250

-0.3

-0.2

-0.1

0.1

Error

50 100 150 200 250

30

32

34

36

38

40

KR

Variance Swaps 21

GUIDANT CORP

50 100 150 200 250

4

6

8

10

12

N@KD

50 100 150 200 250

-0.3-0.2-0.1

0.10.2

Error

50 100 150 200 250

30

32

34

KR

22 Variance Swaps

TENET HEALTHCARE CORP

50 100 150 200 250

468

1012

N@KD

50 100 150 200 250-0.5

0.51

1.52

2.53

Error

50 100 150 200 250

50

55

60

65

70

KR

Variance Swaps 23

MICRON TECHNOLOGY INC

50 100 150 200 250

468

101214

N@KD

50 100 150 200 250

-1-0.75

-0.5-0.25

0.250.5

Error

50 100 150 200 250

38

40

42

44

46

48

50

KR

24 Variance Swaps

ConclusionIn section Pricing we tested two methods for estimation variance swap price. One method is based on the raw option prices. Another is based on a virtual set of European options with implied volatilities extracted from the market data. We saw that the both methods give very close results in case of large number of valid market strikes. But if this number is small (less than 15-10) the second method gives more regular historical prices. Under condition that implied volatilities are available this method is simple, fast and can be recommended for quoting variance swaps.

In section Hedging we tested replication strategy for hedging variance swaps. The results confirm that described replication is quite robust, the error of replication is typically less than one volatility point. However practical attractiveness of this method can be restricted by small number of listed options on underlying. In this case trader needs to use OTC market to construct replication portfolio.

Variance Swaps 25

Variance and Volatility swapsIt is interesting to compare fair strike level KR for variance swap with ATM Implied Volatility (IVATM). This topic is exten-sively discussed in [2] (see also [3] ) with examples of VIX and VXO. This comparison is interesting if we think of IVATM as an estimate of future volatility σR. In this case it is natural to expect that there is a positive spread between KR and IVATM. It follows from convexity adjustment argument (see Appendix). Let us recall it.

Suppose the future variance V@TD has mean V¯ and variance W (under risk-neutral measure)

V¯

= E V@TD,W = E HV@TD − WL.

Then

KR =è!!!!!!!!!!!!!!EV@TD =

"####V¯

And

KR − σR ≈W

8 KR3

We plot here 5 days moving average of fair strike level KR calculated for the period (t,T) calculated by Eq. (1) and ATM implied volatility of the same period. Indeed we see quite stable positive spread between KR and ATM implied volatility. If interpretation of ATM implied volatility as expected future volatility really holds then from this spread we can estimate new parameter W variance of variance. This parameter can be used for price estimates of volatility or variance derivatives.

KR, σR and KR − σR

SPX

50 100 150 200

5

10

15

20

KR−IVATM

IVATM

KR

26 Variance Swaps

PFIZER INC

50 100 150 200 250

5

10

15

20

25

30

KR−IVATM

IVATM

KR

EBAY INC

50 100 150 200 250

10

20

30

40

KR−IVATM

IVATM

KR

ADVANCED MICRO DEVICES

50 100 150 200 250

10

20

30

40

50

60

KR−IVATM

IVATM

KR

GENERAL MOTORS CORP

50 100 150 200 250

5

10

15

20

25

30

35

KR−IVATM

IVATM

KR

Variance Swaps 27

Time Warner INC

50 100 150 200 250

5

10

15

20

25

30

KR−IVATM

IVATM

KR

Citigroup INC

50 100 150 200 250

5

10

15

20

25

KR−IVATM

IVATM

KR

EXXON MOBIL CORP

50 100 150 200 250

5

10

15

20

KR−IVATM

IVATM

KR

MORGAN STANLEY DEAN WITTER

50 100 150 200 250

5

10

15

20

25

30

KR−IVATM

IVATM

KR

28 Variance Swaps

FORD MOTOR

50 100 150 200 250

10

20

30

40

KR−IVATM

IVATM

KR

GUIDANT CORP

50 100 150 200 250

5

10

15

20

25

30

35

KR−IVATM

IVATM

KR

TENET HEALTHCARE CORP

50 100 150 200 250

10

20

30

40

50

60

70

KR−IVATM

IVATM

KR

MICRON TECHNOLOGY INC

50 100 150 200 250

10

20

30

40

50

KR−IVATM

IVATM

KR

Variance Swaps 29

References[1] K. Demeterfi, E. Derman, M. Kamal, J. Zou More Than You Ever Wanted To Know About Volatility Swaps. Quantita-tive Strategies: Research Notes, Goldman Sachs 1999

[2] P.Carr and L.Wu A Tale of Two Indices http://www.math.nyu.edu/research/carrp/papers/pdf/vixov_florida3.pdf

[3] P.Carr and R. Lee Robust replication of volatility derivatives http://www.math.nyu.edu/research/carrp/papers

AppendixVariance Replication

If stock price is follows continuous diffusion with volatility σt

StSt

= µt t + σt Wt

Then by Ito's lemma

HLog@StDL = µt t + σt Wt −1

2 σt

2 t

Or

1

2 σt

2 t =StSt

− Log@StD

Hence for total variance from 0 to T we get

(A1.1) V@TD = ‡0

T

σt2 t = 2 ‡0

T StSt

− 2 Log@ST êS0D

Log payoff decomposition

It is easy to check the following identity for any S>0

(A1.2) Log@ST êSD =1

S HST − SL − ‡

0

S1

K2 HK − STL+ K − ‡

S

∞1

K2 HST − KL+ K

The fair strike value of variance swap van be calculated by taking expectation of future variance under risk-neutral measure at time t

KR@tD2 =1

T E V@TD =

2

T ikjjjE ‡

0

T Sτ

Sτ

− E Log@ST êS0Dy{zzz =

2

T ikjjjj‡

0

t Sτ

Sτ

+ Hr − qL HT − tLy{zzzz −

2

T E Log@ST êS0D

30 Variance Swaps

Using (A1.2) with S = S0

E Log@ST êS0D =

1

S0 HFT@tD − S0L − ‡

0

S01

K2 r HT−tL Put@K, T − tD K − ‡

S0

∞1

K2 r HT−tL Put@K, T − tD K

where

FT@tD = Hr−qL HT−tL.

Finally we have

KR@tD2 =2

T ikjjjj‡

0

t Sτ

Sτ

+ Hr − qL HT − tLy{zzzz +

2

T i

k

jjjjjjjj1

S0 HFT@tD − S0L + ‡

0

S01

K2 r HT−tL Put@K, T − tD K + ‡

S0

∞1

K2 r HT−tL Put@K, T − tD K

y

{

zzzzzzzz

Pricing at initial moment, t=0

Expression for the fair strike KR = KR@0D at time-0 can be simplified in the following way

KR2 =1

T E V@TD =

2

T ikjjjE ‡

0

T StSt

− E Log@ST êS0Dy{zzz =

2

T HHr − qL T − E Log@ST êS0DL

= −2

T E Log@ST êFTD

Using (A1.2) with S = FT

−E Log@ST êFTD = ‡0

FT1

K2 rT Put@KD K + ‡

FT

∞1

K2 rT Call@KD K

Hence

KR2 =rT

T i

k

jjjjjjjj‡0

FT1

K2 Put@KD K + ‡

FT

∞1

K2 rT Call@KD K

y

{

zzzzzzzz

Convexity Adjustment

Variance Swaps 31

V

σ

KR

sRconvexityadjustment

The expected value of future volatility σR is equal to expectation of square root of future variance

σ@TD =è!!!!!!!!!!!V@TD

σR = Eσ@TD = E è!!!!!!!!!!!V@TD

Taking second order approximation of square root in V¯

è!!!!!!!!!!!V@TD ≈

"####V¯

+V@TD − V

¯

2 V¯1ê2 −

HV@TD − V¯L2

8 V¯3ê2

And hence

σR = E è!!!!!!!!!!!V@TD ≈

"####V¯

−W

8 V¯3ê2

32 Variance Swaps