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