(BNP Paribas) Volatility Investing Handbook
(BNP Paribas) Volatility Investing Handbook
(BNP Paribas) Volatility Investing Handbook
45%
40%
35%
Nicolas Mougeot
nicolas.mougeot@bnpparibas.com
+33 (0) 1 55 77 53 89 30%
25%
20%
15%
10%
Mar-01 Sep-01 Mar-02 Sep-02 Mar-03 Sep-03 Mar-04 Sep-04 Mar-05
€-Stoxx 50 var swap €-Stoxx 50 gamma swap S&P 500 var swap
S&P 500 gamma swap Nikkei 225 var swap Nikkei 225 gamma swap
www.eqd.bnpparibas.com This research report has been published in accordance with BNP Paribas
conflict management policy. Disclosures relating to BNP Paribas potential
conflicts of interest are available at https://eqd.bnpparibas.com
Equity Derivatives Technical Study 28 September 2005
Table of contents
Executive summary 3
Conclusion 34
Appendix 35
Hedging options when volatility is unknown 35
Valuing gamma swaps 36
The Derman et al. replication of variance swaps and its extension to gamma
swaps 38
Executive summary
The goal of this paper is to introduce investors to sophisticated derivatives
instruments that allow them to take views on volatility. Despite the fact that
volatility-sensitive derivatives have been traded for years if not centuries, it is
only recently that products giving pure exposure to volatility have appeared.
Variance swaps provide a pure view on volatility since they pay the difference
between future realized variance and a pre-defined strike price. The rapid
development of variance swaps reflects the simplicity with which they can be
valued: under certain non-restrictive assumptions, the variance swap strike
price can be shown to be equal to the value of an options portfolio that uses a
continuum of strike prices and is inversely weighted by the square of the
options’ strike prices.
Variance swaps can be used in many ways, ranging from arbitraging realized
vs. implied volatility and dispersion trading, to hedging structured products or
hedge fund strategies.
We also describe other volatility products that have been developed in recent
years. Gamma swaps are similar to variance swaps but with a notional that is a
function of the asset price. They have several advantages over variance swaps
since they do not require any caps and are a more efficient tool for dispersion
trading. Finally, we present derivatives instruments such as conditional variance
swaps or corridor variance swaps that allow investors to make asymmetric bets
on volatility and take positions on the skew and the smile.
While the emergence of variance swaps has allowed investors to take a pure
position on volatility without taking other risks, a third generation of volatility
products, including gamma corridor and conditional variance swaps, has
appeared. These products provide investors with tools for taking positions on
the skew/smile and efficiently trading dispersion.
Volatility products have historically
emerged through the development The goal of this paper is therefore to explain the methods of trading volatility,
of mathematical solutions for from straddles to variance swaps and third-generation products.
efficient pricing and hedging.
2
See “A Chronology of Derivatives” by Don M. Chance for an extensive chronology from 1700
B.C. to 1995.
Defining volatility
First of all, we need to define how volatility is measured. Throughout this report,
we define volatility as the annualized standard deviation – noted σ – of the log of
the daily return of the stock (or index) price, and variance as the square of the
standard-deviation3.
252 T 252 T
σ2 = ∑ (ln(S i S i−1 )) and σ = ∑ (ln(S i S i −1 ))
2 2
T i =1 T i =1
Both are good measures of stock variability. However, as we shall see below,
Volatility is usually defined as the standard deviation is a more meaningful measure of volatility, given that it is
sum of squared asset returns. measured in the same units as stock return. However, most of the volatility
products we present here are priced in terms of variance, reflecting the fact that
variance-related products are generally easier both to value and replicate (with
the help of vanilla options). We also explain why they are also more useful to
traders.
Straddle Strangle
60
70.00
50 60.00
50.00
40
40.00
30
30.00
∆C/∆S>0
20
20.00
∆C/∆S=0
10 10.00
∆C/∆S>0
∆C/∆S=0 -
0 50 60 70 80 90 100 110 120 130 140 150
50 60 70 80 90 100 110 120 130 140 150
at maturity 3-month, vol=20% 3-month, vol=40% 3-month, vol=20% at maturity 3-month, vol=40%
Investors who buy straddles are taking a bet on the stock price moving up or
down by a wide margin. Since straddles are composed of a call and a put, their
3
Note that standard deviation is calculated without the drift term
value rises with volatility and as such, they provide a means for investors to take
positions on future realized volatility and on changes in implied volatility. Since
the two options are at-the-money, straddles are initially quite expensive. In view
of this, one way to decrease the premium paid upfront is to buy strangles since
Straddles and strangles are the they combine two cheaper out-of-the-money options. However, an investor
easiest way to invest in volatility… buying a strangle would require the stock price to move more in order to make
money.
However, straddles and strangles do not provide pure exposure to volatility. For
example, let us assume that Stock A has an initial price of $100 and that the
investor buys a 3-month straddle. Even if the stock price moves sharply during
…but certainly not the most efficient the 3 months but ends up at $100 at maturity, the straddle will be worthless at
one.
maturity. Of course, had the implied volatility also risen, buying the straddle
would have been profitable if it had been sold before maturity.
Furthermore, once the stock price has moved away from its initial position (here
$100) the straddle delta is no longer null, as shown by the above graph. The
straddle then becomes sensitive to price movements and the same holds true
for the strangle.
One step beyond: delta-hedging In order to eliminate the sensitivity to stock prices and obtain pure exposure to
options. volatility risk, the amount of stocks held, ∆, is periodically re-adjusted to ensure
the portfolio’s sensitivity to the stock price remains null. At time t, ∆ is thus equal
to:
∂
∆t = V (St ,0;σ )
∂S
We assume that the option is delta-hedged at a constant implied volatility σ h .
The P&L resulting from delta-hedging an option is by definition equal to the final
cost of the option minus its initial cost minus the cost of delta-hedging the
position. We show in the appendix that this P&L can be broken up into three
components:
⎡ 1T ⎤ T
14
(
4244 3
) ( ) ( )
P & L = σ i2 − σˆ 2 Tg 0 + σˆ 2 − σ h2 T ⎢ g 0 − ∫ g t dt ⎥ + ∫ σˆ 2 − σ t2 g t dt
T 0
1 14444⎣2444 43⎦ 1 0
442443
2 3
e r (T − t ) ∂
gt = V (St , T − t ;σ h )
2σ h ∂σ
Options delta-hedging implies three II : A “Vega risk” factor, which stems from the fact that the option is hedged at
different sources of risk: the implied volatility σ h instead of at the realized volatility σˆ . This term is indeed
- Variance risk null if the trader is able to hedge at the realized – but unknown – volatility.
- Vega risk
- Volatility path dependency risk III: A “Volatility path dependency risk” or “model risk” factor that depends on the
or model risk. historical behavior of realized volatility. Under the Black & Scholes assumption,
the instantaneous volatility σ t is constant and thus equal to the realized volatility
between time t and T. However, should the volatility vary over time, (σˆ 2 − σ t2 ) will
no longer be zero. As gt is a decreasing function of time to maturity, this term
will be positive if instantaneous – or intraday – volatility rises during the life of
the option. The term also depends on the true distribution of stock returns.
Based on realistic simulations, Blanc also shows that variance risk only
represents 52% of the total P&L resulting from delta-hedging an option. As a
result, delta-hedging options does not provide pure exposure to volatility, given
that the P&L not only depends on variance risk but also on a vega risk which
itself results from the fact that risk cannot be hedged at the - unknown future -
realized volatility and that volatility may not be constant over time.
Indeed, delta-hedging options yields further risk sources not indicated above.
The above analysis was done without taking into account dividends and by
assuming a constant interest rate. In practice, traders face the risk of unknown
dividends being paid during the life of the option, while with interest rates liable
to vary over time, the option vega may change if the interest rate changes.
These issues paved the way for the introduction of new derivatives instruments
that enable investors to take a view on volatility without bearing any other risks.
4
For a discrete time version of the demonstration, see N. Blanc, Index Variance Arbitrage :
Arbitraging Component Correlation, BNP Paribas technical studies, 2004
simple payoffs
2
A simple pay-off: σ *N
( )
Pay − off = σ 2 − KVar * N
Investor :
variance swap
BNP
variance
Paribas :
swap
buyer seller
K *N
Var
252 T
∑
σ2 = (ln(S i S i −1 ))2
T i =1
Where Si is the closing price of the asset, T the number of days in the
observation period and ln the natural logarithm. Also, variance swaps are
usually quoted in terms of squared volatility, e.g. (20%)2 since volatility is more
economically meaningful than variance.
Note that there is no correction for dividend payment in the above formula.
Should a stock pay a dividend, the final payout will not be adjusted for the jump
implied by the dividend. However, variance swaps can be structured in such a
way as to adjust the above formula for dividends. One should also bear in mind
that variance swaps are not exempt from counterparty risks. Since the product
is OTC, both counterparties face the risk of the other going bankrupt.
60%
50%
20%
10%
0%
maximum loss/gain of a short position
-10%
%
%
0%
4%
8%
12
16
20
24
28
32
36
40
44
48
52
56
60
Re a lize d va ria nce
m is the multiplier setting the cap. The graph below compares the payoffs from
capped and uncapped variance swaps. The maximum gain (and thus loss for
the counterparty) is limited and thereby facilitates the hedging of the instrument.
As we will demonstrate in the next section, a variance swap can in theory be
replicated by a portfolio of options with a continuum of strike prices. In practice,
however, there is no liquidity for options whose strike prices are far from ATM.
Traders are therefore unable to completely hedge variance swaps, whereas
capped variance swaps do not require a continuum of strike prices.
120%
100%
60%
40%
20%
0%
-20%
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
Realized volatility
As the graph below shows, the variance vega declines as the stock price moves
The strike price may be determined away from the strike price and is also an increasing function of the strike price.
by a static portfolio of options The goal is therefore to create an options portfolio with a constant variance
inversely weighted by the square of vega.
their strike price.
Variance vega of 3-month options with different strike prices
The addition of options with strike prices away from ATM flattens variance vega
and thus makes the portfolio sensitive to variance but invariant to stock price.
40 50 60 70 80 90 100 110 120 130 140 150 160 40 50 60 70 80 90 100 110 120 130 140 150 160
Firstly, Ito’s Lemma states that any smooth function f(Ft) can be rewritten as:
T T
1
f (FT ) = f (F0 ) + ∫ f ' (Ft )dFt + Ft 2 f ' ' (Ft )σ t2 dt
2 ∫0
(1)
0
Let us consider the following function:
Ft
f (Ft ) = ln(F0 Ft ) + −1
F0
Where Ft stands for the futures price. This function has a slope and a value
equal to zero when Ft = F0.
Therefore, we have:
1
T
F T
⎛ 1 1⎞
∫ σ t2 dt = ln (F0 FT ) + T − 1 − ∫ ⎜⎜ − ⎟⎟dFt (2)
20 F0 0⎝ 0
F Ft ⎠
Carr and Madan further assume that a market exists for futures options of all
strikes. In this case, they show that any payoff f(FT) of the futures price FT can
be broken up into:
2
64444 4744444 8
[ ]
} 1
κ ∞ (3)
+ ∫ f ' ' (κ )(K − FT ) dK + ∫ f ' ' (κ )(FT − K ) dK
+ +
0
1 44424443 κ144424443
3 4
Where κ is an arbitrary number. As Carr and Madan point out, the above terms
can be interpreted as:
In the absence of arbitrage, the above breakdown must prevail for initial values.
Therefore, the initial value of the payoff is equal to:
κ ∞
V f0 = f (κ )B0 + f ' (κ )[C 0 (κ ) − P0 (κ )] + ∫ f ' ' (κ )P0 (κ )dK + ∫ f ' ' (κ )C 0 (κ )dK
0 κ
Carr and Madan thus prove that an arbitrary pay-off can be obtained from bond
and option prices without making strong assumptions about the stochastic
process driving the stock price5.
F ∞
FT 0
1 1
ln(F0 FT ) + − 1 = ∫ 2 (K − FT ) dK + ∫ 2 (FT − K ) dK
+ +
F0 0 K F0 K
T
Therefore, in order to receive σ 2 dt at time T, a trader should buy a continuum
∫ t 0
of puts with strike prices ranging from 0 to F0 and calls with strikes prices
ranging from F0 to infinity. The initial cost is equal to:
F0 ∞
2 2
∫0 K 2 0
P ( K )dK + ∫F K 2 C0 (K )dK (A)
0
5
More complete proof of the result can be found in Peter Carr and Dilip Madan, 1998,
‘Towards a Theory of Volatility Trading’, in Volatility: New Estimation Techniques for Pricing
Derivatives, pp. 417-427.
Since the initial cost of achieving this strategy is given by (A), the fair forward
value of the variance at time 0 should be equal to:
⎡ F0 1 ∞ ⎤
[ ]
V f σ 02,T =
2e rT
T
1
⎢ ∫ 2 P0 (K )dK + ∫ 2 C0 (K )dK ⎥
⎢⎣ 0 K F0
K ⎥⎦
Furthermore, let us recall that the P&L of a delta-hedged option is given by:
⎡ T
⎤ T
( ) ( ) 1
( )
P & L = σ i2 − σˆ 2 Tg 0 + σˆ 2 − σ h2 T ⎢ g 0 − ∫ gt dt ⎥ + ∫ σˆ 2 − σ t2 g t dt
T0
⎣ ⎦ 0
Options traders are therefore linearly exposed to variance and are thus likely to
be more interested in variance swaps. The same reasoning holds true for an
investor wishing to hedge the volatility exposure of his options portfolio.
However, the strike price of a variance swap is often defined in terms of volatility
as it is more economically meaningful. For example, a variance swap contract
will specify that the volatility strike is equal to 20% which means that the actual
variance strike price is equal to (20%)2.
σ − KVOL ≈
1
2 KVOL
(σ 2 − KVAR )
Where KVOL is the strike price measured in terms of volatility (i.e. the square root
of the variance strike price).
Characteristics of volatility
The specific dynamics of volatility are of particular interest and these may be
summarized in the following four points:
The following graphs display 6-week historical volatility for the VIX6 and the S&P
500. The graph on the left shows that volatility may move from around 20% to
north of 100% when the market experiences a serious downturn such as in
October 1987. Furthermore, as opposed to stock returns, volatility tends to
revert back towards its mean and usually remains within a high or a low regime
for a long period of time. S&P 500 volatility, for example, was in a low regime
between 1992 and 1996 and then moved into a high regime until 2003.
140 45
40
120
35
100
30
80 25
60 20
15
40
10
20
5
0 0
Jan-86
Jan-88
Jan-90
Jan-92
Jan-94
Jan-96
Jan-98
Jan-00
Jan-02
Jan-04
Dec-88
Dec-89
Dec-90
Dec-91
Dec-92
Dec-93
Dec-94
Dec-95
Dec-96
Dec-97
Dec-98
Dec-99
Dec-00
Dec-01
Dec-02
Dec-03
Dec-04
VIX S&P 500 6-week historical volatility VIX S&P 500 6-week historical volatility
6
The Chicago Board Options Exchange SPX Volatility, or VIX, Index reflects a market estimate
of future volatility, based on the weighted average of the implied volatilities for a wide range of
strikes. 1st & 2nd month expirations are used until 8 days from expiration, then the 2nd and 3rd
are used.
contradicts the Black & Scholes assumption of lognormal prices and highlights
the fact that the distribution of stock returns is skewed.
6-week correlation between changes in VIX and S&P 500 daily returns
60%
40%
20%
0%
-20%
-40%
-100%
Jan-86
Jan-88
Jan-90
Jan-92
Jan-94
Jan-96
Jan-98
Jan-00
Jan-02
Jan-04
6-week correlation between changes in VIX and daily S&P 500 returns
First, let us compare the 6-month variance swap strike price with 6-month ATM
implied volatility for European options on the €-Stoxx 50 index. Variance swaps
would have traded an average of 1.11 volatility points above ATM implied
volatility between August 2001 and August 2005, a difference that appears to be
statistically significant.
Variance swap strike prices are ATM implied Skew Var swap - Var swap -
Var swap
vol (90% - 100%) ATM implied vol realized vol
higher than realized volatility on
average. minimum 13.17% 12.43% 1.75% 0.31% -19.86%
maximum 40.04% 39.52% 3.32% 2.74% 16.39%
average 26.28% 25.17% 2.40% 1.11% 1.11%
Source – BNP Paribas
45%
50%
40%
40%
35%
30%
30%
20% 25%
20%
10%
15%
0%
10%
-10%
5%
-20% 0%
Aug-01 Feb-02 Aug-02 Feb-03 Aug-03 Feb-04 Aug-04 Feb-05 Aug-01 Feb-02 Aug-02 Feb-03 Aug-03 Feb-04 Aug-04 Feb-05
The following table plots the regression of the €-Stoxx 50’s 1-year historical
standard-deviation against its 1-year variance strike price (expressed in terms of
volatility) and against 1-year ATM implied volatility. Since regression β is far
from one in the case of variance swap strike prices, it does not provide an
unbiased proxy for future realized volatility. The same conclusion holds for ATM
implied volatility despite the fact that the constant and the slope are closer to
zero and one, respectively.
T-stat
constant β T-stat (β) R-square
Variance swap strike prices are not (constant)
a good proxy for future realized var swap 0.10 5.27 0.59 8.56 8%
volatility. ATM implied volatility 0.06 3.75 0.76 11.85 13%
Source – BNP Paribas
One of the reasons why variance-swap strike prices do not provide a fair proxy
of future realized volatility is again the presence of a variance risk premium.
7
β is the sensitivity of the variance-swap strike price (or ATM implied volatility) to realized
volatility. The T-stat measures whether this parameter shows a statistically significant
difference from zero: it needs to be below -1.96 or above 1.96 to be significant at the 95%
confidence level. The R-square measures the overall explanatory power of the equation. An R-
square of 40% means that realized volatility explains 40% of the variability of ATM implied
volatility.
To check whether a variance risk premium exists, we follow Carr and Wu’s
suggestion: in the absence of a variance risk premium, the average variance
swap strike price should be equal to the realized variance. The log ratio of the
two - ln(Var Swap/realized variance)- should thus be equal to zero.
The table below reports the t-statistics for ln(var swap/realized variance) for the
€-Stoxx 50 and its 10 largest constituents at present using data from March
2001 to September 2005. The t-stat is significant and positive both for the €-
Stoxx 50 and for 9 out of its top 10 constituents, a situation denoting the
presence of a significant variance risk premium at both the index and single
stocks levels.
8
Carr and Wu (2005) provide a thorough analysis of variance risk premium for several US
stocks and indices. See also Driessen et al. (2005) for a comparison of variance risk premia for
single stocks and indices.
20%
y = 13.11x - 0.30
15% R2 = 21.67%
10%
0%
-5%
-10%
-15%
-20%
-25%
1.50% 1.70% 1.90% 2.10% 2.30% 2.50% 2.70% 2.90% 3.10% 3.30% 3.50%
Skew (90% - 100%)
T-stat
constant β T-stat (β) R-square
(constant)
var swap - realized vol -0.30 -15.11 13.03 15.81 21.67%
Source – BNP Paribas
Variance swaps can thus be used for different strategies such as:
One way to play an expected rise in volatility term structure is to enter into two
different variance swaps with two different maturities. Let us assume for
Variance swaps are an efficient tool example that the variance strike price for a 1-year variance swap is currently
for investing in volatility and equal to (20%)2 , but that one expects the 6-month variance swap to be priced
arbitraging discrepancies between at (30%)2 in 6 months with the realized volatility remaining constant over the
implied and realized volatilities. next 12 months. An investor can then take advantage of this expected
steepening of volatility term structure by:
If the investor is right, the payoff will be equal to N*(10%)2. This type of strategy
may suitably be executed via the use of forward-start variance swaps9 which we
describe in more detail in the next section.
9
The next section looks into forward variance swaps in greater detail.
45%
40%
35%
30%
25%
20%
15%
10%
5%
0%
3M 6M 1Y 2Y
Today In 6 months (expected)
Our implied volatility valuation model can also provide some ideas for implied
volatility pairs trading. This model helps calculate fair values for the implied
volatility of single stocks according to their beta, 5-year CDS, size and stock
returns10.
10
For more details on this model, see “Predicting uncertainty: A new method for valuing implied
volatility”, BNP Paribas Technical Studies, published on October 14, 2004.
10 30
5 20
0 10
-5 0
Sep-03 Nov-03 Jan-04 Mar-04 May-04 Jul-04 Sep-04 Nov-04 Jan-05 Mar-05 May-05 Jul-05
Dispersion trading
Dispersion trading consists of buying the volatility of an index and selling the
volatility of its constituents according to their index weights. It is defined as11:
N
Dispersiont = ∑α i ,tσ i2,t − σ I2,t
Pairs trading can be extended to
dispersion trading, i.e. arbitraging i =1
between the volatility of an index
Where α i,t is the weight of stock i in the index. By definition, it changes over
and the volatility of its constituents.
time as stock prices change and is equal to:
α i ,t = ni S i ,t I t
It thus depends on:
o the correlation
11
Dispersion can also be expressed as the square root of the mentioned formula
5%
4%
3%
2%
1%
0%
Sep-03 Dec-03 Mar-04 Jun-04 Sep-04 Dec-04 Mar-05 Jun-05 Sep-05
Realised dispersion Implied dispersion
On the other hand, option-based products such as ODBs, are usually positively
related to volatility. On this basis, variance swaps could help structured-product
managers to hedge their volatility risks efficiently.
Correlation between CSFB/Tremont hedge fund indices and S&P 500 and €-Stoxx 50 return and volatility
CSFB/Tremont
Convertible Dedicated Emerging Event Global Long/Short Multi-
Hedge Fund Distressed
Arbitrage Short Markets Driven Macro Equity Strategy
Index
S&P 500 volatility -31% 21% 21% -29% -45% -41% -9% -32% -4%
€-Stoxx 50 volatility -35% 13% 17% -24% -47% -41% -3% -26% -6%
S&P 500 return 41% 7% -75% 63% 53% 44% 11% 44% 30%
€-Stoxx 50 return 45% 4% -63% 57% 49% 36% 17% 49% 28%
Source: BNP Paribas, CSFB/Tremont
Gamma swaps
As discussed previously, the gamma exposure of variance swaps is insensitive
to the level of the underlying asset. In the event the stock price rises or declines,
the gamma exposure depends solely on the initial value of the portfolio.
Variance swaps are thus said to have constant “cash” gamma exposure.
Gamma swaps have a constant Pay − off Gamma swap = (Gamma − K Gamma ) * N
“share” gamma exposure which
T
⎡ Si ⎤ ,
makes them more efficient tools for Where Gamma = 252 ∑ ⎢(ln(S i S i −1 ))
2
⎥ KGamma is the strike and N the
options hedging. T i =1 ⎣ S0 ⎦
notional amount. In continuous-time, the gamma swap payoff is equal to:
T
1
T ∫0
Γ0,T = σ t2 S t S 0 dt
40 50 60 70 80 90 100 110 120 130 140 150 160 40 50 60 70 80 90 100 110 120 130 140 150 160
2e 2 rT ⎡ 0 1 ⎤
F ∞
V f [Γ0,T ] =
1
Gamma-swap strike prices may be
⎢ ∫ P0 (K )dK + ∫ C0 (K )dK ⎥
replicated by an options portfolio TS 0 ⎣⎢ 0 K F0
K ⎦⎥
with a continuum of strike prices
The gamma-swap’s strike price may thus be replicated by a continuum of puts
inversely weighted by their strike
and calls inversely weighted by their strike prices.
prices.
By virtue of their payoff and insofar as squared returns (ln (S i S i −1 ))2 are
weighted by the performance of the stock S i S 0 , gamma swaps underweight big
downward index move relative to variance swaps. This means that if the
distribution of stock returns is skewed to the left, gamma swaps minimize the
effect of a crash, thereby making it easier for the trader to hedge. In this case,
hedging does not require additional caps, unlike variance swaps which need to
be capped.
6-month gamma and variance swaps’ strike prices on the €-Stoxx 50 index
50% 22%
45%
20%
40%
35%
18%
30%
25% 16%
20%
14%
15%
10%
12%
5%
0% 10%
Aug-01 Feb-02 Aug-02 Feb-03 Aug-03 Feb-04 Aug-04 Feb-05 Aug-04 Oct-04 Dec-04 Feb-05 Apr-05 Jun-05
Gamma swap strike prices should thus be lower than variance swap strike
prices. The following graphs show the strike prices of 6-month variance and
gamma swaps on the €-Stoxx 50. The gamma swap strike price would
systematically have been slightly lower than the variance swap strike price by
1.03 volatility points on average.
N
Dispersiont = ∑ α i ,t σ i2,t − σ I2,t
i =1
Gamma swaps are a means to Where α i,t is the weight of stock i in the index. By definition, it changes over
optimally trade dispersion. time as stock prices change and is equal to:
α i ,t = ni S i ,t I t
If an investor trades dispersion using variance swaps weighted by the initial
weights of the stocks in the index, he faces the risk of a possible change in
weights over time until maturity of the variance swap. Gamma swaps, however,
offer a more efficient way to trade dispersion. If one sells a gamma swap on the
index and buys ni Si,0 /I0 – or α i , 0 – gamma swaps on each stock i, the payoff
at maturity should be equal to:
N ni S i , 0 1 T 2 1
T
P & LT = ∑ ∫ σ S
i ,t i , t S i ,0 dt − ∫ σ I2,t I t I 0 dt
i =1 I0 T 0 T 0
Rearranging the terms, it gives:
T T
1 ⎛ N
2 ⎞ 1
P & LT = ∫ ⎜ I t I 0 ∑ α i ,t σ i ,t ⎟dt − ∫ σ I2,t I t I 0 dt
T 0⎝ i =1 ⎠ T 0
T
1 It
T ∫0 I 0
= Dispersiont dt
The payoff is thus equal to the average dispersion over the period [0,T]
weighted by index performance.
T'
252
σ2 = ∑ (ln(S Si −1 ))
2
i
T '−T i = T +1
Or in continuous-time:
T'
1
T − T ' T∫
σˆ T2,T ' = σ t2 dt
Forward-start variance swaps
enable positions to be taken in Carr and Madan again show that forward-start variance swaps may also be
variance between two future dates. priced as the difference between a variance swap maturing at T’ and a variance
swap maturing at T. The value of a forward-start variance swap maturing at T’
and starting on T is thus equal to:
⎡ F0 1 ∞ ⎤
[
V f σ T2,T ' = ] 2e rT '
T'
1
⎢ ∫ 2 P0 (K , T ')dK + ∫ 2 C0 (K , T ')dK ⎥
⎢⎣ 0 K F0
K ⎥⎦
2e rT ⎡ 0 1 ⎤
F ∞
1
− ⎢∫ 2 0P ( K , T )dK + ∫ 2
C0 (K , T )dK ⎥
T ⎢⎣ 0 K F0
K ⎥⎦
As a result, forward-start variance swaps enable investors to take positions on
future volatility without having to enter two different variance swaps.
252 T
2
K Corr = ∑ (ln(Si Si−1 ))21St −1∈[κ −∆;κ +∆ ]
T i =1
Hence, squared returns are counted in if the stock price lies within a pre-
specified range [κ −∆ ;κ + ∆ ] .
Corridor variance swaps therefore enable bets to be taken on the pattern of the
2
stock. If the stock move sideways and stays within the defined range, K Corr will
be high. If the stock moves sharply upward or downward and leaves the range
2
quickly, K Corr will be low.
252 T
∑ (ln(S i S i−1 )) 1St −1 >B
2
2
K Upcorr =
T i =1
Hence, squared returns are counted in if and only if the stock price lies above a
predefined level denoted B. Indeed, one can also define a down corridor
variance swap whose payoff would be defined by:
252 T
2
K downcorr = ∑ (ln(Si Si−1 ))21St −1<B
T i =1
Aggregating a down corridor variance swap and an up corridor variance swap
yields the classic variance swap. This particular payoff has several advantages:
Comparison of variance swaps and down corridor variance swaps on the S&P 500 index
40 1800 30 1600
1600 1500
35
25
1400
1400
30
1300
1200 20
25
1200
1000
20 15 1100
800
1000
15
600 10
900
10
400 800
5
5 200 700
- 0 - 600
May-99
May-00
May-01
May-02
May-03
May-04
May-05
Aug-76
Aug-78
Aug-80
Aug-82
Aug-84
Aug-86
Aug-88
Aug-90
Aug-92
Aug-94
Aug-96
Aug-98
Aug-00
Aug-02
Aug-04
Jan-99
Sep-99
Jan-00
Sep-00
Jan-01
Sep-01
Jan-02
Sep-02
Jan-03
Sep-03
Jan-04
Sep-04
Jan-05
variance swap down corridor variance swap S&P 500 index variance swap down corridor variance swap S&P 500 index
The ATM down corridor variance swap systematically yields a lower payoff and
should thus be cheaper than the variance swap. Furthermore, the graph shows
that the down corridor variance swap’s payoff is negatively correlated with the
market trend. With squared returns not counted in when the market rises, the
down corridor variance swap offers a lower strike price. This is confirmed by the
statistics reported in the following table. As a result, up and down corridor
variance swaps enable a combined view to be taken on volatility, correlation
and market direction.
Comparison of 1-year variance swaps with 1-year ATM up and down corridor variance swaps (in vol terms)
( )∑ 1
T
Payoff Upcond = K 2 − K Upcond
2
S i −1 > B
i =1
T
252
∑ (ln(S S i −1 )) 1S t −1 > B
2
2
K Upcond = T i
∑1
i =1
S i −1 > B
i =1
( )∑1
T
Payoff Downcond = K 2 − KUpcond
2
Si −1 < B
i =1
T
252
∑ (ln(S S i −1 )) 1St −1 < B
2
2
K Downcond = T i
∑1
i =1
S i −1 > B
i =1
The conditional variance swap payoff is such that if the stock price never trades
above the threshold B, it will be null whereas the up corridor variance swap
2
payoff would be equal to K . It also enables bets to be taken on very specific
volatility behavior. Take the case of an index whose initial value is 100. If the
index stands above 100 for a few days with a high volatility, say 40%
annualized, and then drops below 100, an ATM up conditional variance swap
will yield a much higher payoff than an ATM up corridor variance swap, given
that the payoff’s floating leg is divided by the number of days the index stays
above the threshold (above 100 in this case). As a result, timing is less of an
issue for an investor who buys a conditional variance swap rather than a
corridor variance swap.
Optimally, one would need to buy or sell corridor variance swaps whose
range [κ −∆ ;κ + ∆ ] would be such that ∆ tends to zero. In this case, the conditional
variance swap would pay when the index is around κ. Although theoretical
models exist for pricing such products, they are not traded as yet12.
The following graph displays the smile for three different (90%-110%) skews,
but which display the same average implied volatility of 22.5%. The table plots
the prices of 1-year variance swaps and gamma swaps for the three different
skews. One can see that the spread between the two increases when the skew
rises. Indeed, variance swaps and gamma swaps yield different vega
exposures. By being long a variance swap and short a gamma swap, an
investor does not get a vega-neutral position. As a result, trading the smile by
this method further requires to delta-hedge the position in order to cancel the
vega sensitivity of the strategy. As delta-hedging necessitates an option model,
investors further face model risk.
Note that gamma swaps generally yield lower payoffs than variance swaps, but
that trading variance swaps against gamma swaps is a simple technique for
trading the smile or the skew.
12
See again Peter Carr and Dilip Madan, 1998, ‘Towards a Theory of Volatility Trading’, in
Volatility: New Estimation Techniques for Pricing Derivatives, pp. 417-427, for the pricing of
conditional variance swaps along a specific strike.
45
40
35
30
25
20
15
10
0
40% 60% 80% 100% 120% 140% 160%
Correlation trading
Correlation measures how closely stocks move together or in opposite
directions. The correlation between two stocks is denoted by ρi,j and calculated
as:
cov(Ri , R j ) ∑R i ,t R j ,t
ρ i, j = = t =1
σ iσ j T T
∑ Ri2,t
t =1
∑R
t =1
2
j ,t
The higher the correlation the higher the index volatility, since both the index
weights α i and the stock volatilities σ i are positive. Average correlation
provides a good measure of overall correlation. Average correlation is
calculated by assuming the same pairwise correlation for all stocks:
n n −1 n
σ I2 = ∑ α i2σ i2 + ∑∑ α iα j σ iσ j ρ
i =1 i =1 j =i
n
σ I2 − ∑ α i2σ i2
ρ= n −1 n
i =1
∑∑α α σ σ
i =1 j =i
i j i j
The same breakdown holds for both implied and realized volatilities, thus
meaning that option prices can be used to extract an average implied
correlation for a given maturity.
There are several interesting features with correlation:
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Feb-03 Aug-03 Feb-04 Aug-04 Feb-05 Aug-05
Realised correlation Implied correlation
The fact that stocks generally all tend to head down when the bears are out
means that correlation usually rises when markets decline.
Trading correlation can take different forms. One can approximate future
correlation using variance or gamma swaps but the replication will never be
perfect13. However, the simplest way to trade correlation is to take a position in
a correlation swap.
A correlation swap has a payoff similar to the variance swap and one which is
equal to:
13
see N. Blanc, Index Variance Arbitrage: Arbitraging Component Correlation, BNP Paribas
technical studies, 2004, for a discussion regarding several methods for optimizing correlation
trades.
ρ realized is the average of all correlations ρi,j. ρi,j is the correlation of the log daily
returns of the stocks prices between stock i and j:
n −1 n
2
ρ Re alised = ∑∑ ρ i, j where n is the number of stocks in the index
n × (n − 1) ) i =1 j =i
or basket.
Conclusion
The combination of greater liquidity in the options market and new quantitative
tools has now made trading volatility almost as easy as trading stocks or bonds.
Variance swaps are now traded every business day on both indices and single
stocks.
These financial instruments are not just arbitrage tools, but also efficient
hedging tools. As we have shown in this report, they can be used to hedge
structured product books or to hedge volatility up to a certain level.
Appendix
dS t
= µ t dt + σ t dWt
St
Where µ t is the stock return, σ t its unknown instantaneous volatility and Wt a
Brownian motion. Let us denote the initial implied volatility by σ i and assume
that the trader chooses to hedge his option position by applying the Black-
Scholes formula and using the hedge volatility σ h Under these assumptions,
Carr shows that the P&L of an option V should be equal to:
T
∂2
( ) S2
2
P & L = [V (S0 ,0; σ i ) − V (S0 ,0; σ h )]erT + ∫ er (T −t ) σ h2 − σ t2 t
V (St , t; σ h )dt
0
∂S 2
Let us now follow Blanc’s suggestion14 to linearize the first two terms in σ2
around σ h . The P&L can be rewritten as:
T
∂2 S 2 ∂2
( ) S2 ( )
2
P & L = Te rT σ i2 − σ h2 0
V (S 0 ,0; σ h ) + ∫ e r (T −t ) σ h2 − σ t2 t V (S t , t ; σ h )dt
∂S 2
0
2 ∂S 2
Let us denote the volatility over the period 0 to T by σˆ . σˆ is defined by:
T
1
T ∫0
σˆ 2 = σ t2 dt
P&L =
S 02 ∂ 2
(σ i
2
)
− σˆ 2 Te rT
2 ∂S 2
V (S 0 , T ; σ h )
⎡ S 2 ∂2 T
S 2 ∂2 ⎤
( )
+ σˆ 2 − σ h2 T ⎢e rT 0
2 ∂S 2
1
V (S 0 , T ;σ h ) − ∫ e r (T −t ) t
T 2 ∂S 2
V (S t , T − t ;σ h )dt ⎥
⎣ 0 ⎦
T
S t2 ∂ 2
( )
+ ∫ σˆ 2 − σ t2 e r (T −t )
2 ∂S 2
V (S t , T − t ;σ h )dt
0
Defining gt as:
St2 ∂ 2 e r (T − t ) ∂
g t = e r (T − t ) V (S t , T − t ; σ h ) = V (St , T − t ;σ h )
2 ∂S 2
2σ h ∂σ
The P&L is thus equal to:
⎡ T
⎤ T
( ) ( 1
)
P & L = σ i2 − σˆ 2 Tg 0 + σˆ 2 − σ h2 T ⎢ g 0 − ∫ g t dt ⎥ + ∫ σˆ 2 − σ t2 g t dt
T0
( )
⎣ ⎦ 0
14
See N. Blanc, Index Variance Arbitrage: Arbitraging Component Correlation, BNP Paribas
technical studies, 2004.
https://eqd.bnpparibas.com/research/getdocument.asp?doc_id=126089&lang_id=1
T
Γ0,T = ∫ σ t2 S t S 0 dt
0
If f(Ft,,t) is a function of both Ft, and t, then Ito’s Lemma states that:
T T T
Ft
f (FT ) = f (F0 ) + ∫ f ' F (Ft )dFt + ∫ f t ' (Ft )dFt + ∫ f F ' ' (Ft )σ t2 dt
0 0 0
2
Let us consider the following function:
T T
1
S tσ t2 dt = e rT [FT ln (FT F0 ) − FT + F0 ] − r ∫ e rt ((1 + FT ) ln (FT F0 ) − FT + F0 )dFt
2 ∫0 0
Note that Carr and Madan show that if a market exists for futures options of all
strikes, any payoff f(FT) of the futures price FT can be broken down as:
[
f (FT ) = f (κ ) + f ' (κ )(FT − κ ) − (κ − FT )
+ +
]
κ ∞ (A)
+ ∫ f ' ' (κ )(K − FT ) dK + ∫ f ' ' (κ )(FT − K ) dK
+ +
0 κ
In the absence of arbitrage, the above breakdown must prevail among initial
values. Therefore, the initial value of the payoff is equal to:
κ ∞
V f0 = f (κ )B0 + f ' (κ )[C 0 (κ ) − P0 (κ )] + ∫ f ' ' (κ )P0 (κ )dK + ∫ f ' ' (κ )C 0 (κ )dK
0 κ
0
T
Therefore, in order to receive S σ 2 dt at time T, a trader should buy a
continuum of puts with strike prices
∫0 t ranging
t
from 0 to F0 and calls with strikes
ranging from F0 to infinity, with everything weighted by the price of a discount
bond erT. The initial cost is equal to:
⎡ F0 2 ∞
2 ⎤
e rT ⎢ ∫ P0 (K )dK + ∫ C0 (K )dK ⎥ (B)
⎢⎣ 0 K F0
K ⎥⎦
The trader also needs to roll a futures position, holding at t:
F0
2
∞
2
T
⎛ ⎛F ⎞ ⎞
∫ K (K − F ) ∫ K (F − K ) dK − 2 ∫ re rt ⎜⎜ (1 + Ft ) ln⎜⎜ t ⎟⎟ − Ft + F0 ⎟dFt
+ +
e rT T dK + T ⎟
0 F0 0 ⎝ ⎝ F0 ⎠ ⎠
⎛ ⎛F ⎞ ⎞ T ⎛ ⎛F ⎞ ⎞
= 2e rT ⎜⎜ FT ln⎜⎜ T ⎟⎟ − FT + F0 ⎟⎟ − 2 ∫ re rt ⎜⎜ (1 + Ft ) ln⎜⎜ t ⎟⎟ − Ft + F0 ⎟dFt
⎟
⎝ ⎝ F0 ⎠ ⎠ 0 ⎝ ⎝ F0 ⎠ ⎠
T
= ∫ S t σ t2 dt
0
Since the initial cost of executing this strategy is given by (B), the fair forward
value of the gamma swap at time 0 should be equal to:
2e 2 rT ⎡ F0 1 ∞ ⎤
V f [Γ0,T ] =
1
⎢ ∫ P0 (K )dK + ∫ C0 (K )dK ⎥
TS0 ⎢⎣ 0 K F0
K ⎥⎦
2⎡ FT ⎤
f (FT ) = ⎢ln (F0 FT ) + − 1⎥
T⎣ F0 ⎦
plus a dynamic position on a futures contract. We also showed that the initial
value of this log contract should be equal to:
F0 ∞
2 1 1
∫K 2
P0 (K )dK + ∫ 2
C 0 (K )dK
T 0 F0 K
The issue for valuing the above continuum of options is that in practice, only a
limited number of options are available. Therefore, Derman et al. propose a
piecewise linear approximation of f() as described by the following graph.
Log payoff and its discrete approximation in the case of variance swaps
K 2P
K 2C
K 1P
K0 K 1C
20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220
The segment between K0 and K1C resembles the payoff of a call with strike K0
and the number of options one needs to buy is equal to the slope of the
segment:
f (K1C ) − f (K 0C )
wc (K 0 ) =
K 1C − K 0C
The segment [K1C, K2C] resembles a combination of calls with strike prices K0
and K1C , bearing in mind that we already own wc calls with strike price K0. The
number of options with strike price K1 should thus be equal to:
f (K 2 C ) − f (K 2 C )
wc (K1 ) = − wc (K 0 )
K 2C − K 1C
As a result, this method may be used to calculate all the weights for the calls
position as well as the number of puts needed to be bought in order to replicate
the left-hand side of the log payoff.
Indeed, this method can be also applied to the valuation of gamma swaps, the
only difference being the function f() we use. When valuing gamma swaps, the
function f is equal to:
2e rT
f (FT ) = (FT ln(FT F0 ) − FT + F0 )
T
CURRENCY USD
VALUATION DATES Each Exchange Business Day from and including the Effective
Date to and including the Final Valuation Date, regardless of
the occurrence of a Market Disruption Event.
INITIAL STOCK LEVEL (P1) Closing Level of the Stock on Trade Date
REALIZED VOLATILITY 2
⎡ ⎛ ( Pi +1) + ( Di +1) ⎞⎤
n −1
∑ ⎢ln⎜
i =1 ⎣ ⎝ Pi
⎟⎥
⎠⎦
100 X X Business Days Per Year
n −1
where:
n = number of expected Valuation Dates, known as of trade
date
CASH SETTLEMENT Three Currency Business Days following the Final Valuation
PAYMENT DATE Date
Carr, Peter and Dilip Madan, 1998, Towards a Theory of Volatility Trading, in
Volatility: New Estimation Techniques for Pricing Derivatives, p. 417-427
Carr, Peter and Liuren Wu, 2005, Variance Risk Premia, Courant Institute
working paper
Derman, Emanuel, Michael Kamal, Joseph Zou and Kresimir Demeterfi, 1999, A
Guide to Volatility and Variance Swaps, The Journal of Derivatives, Summer
p.1-32
In accordance with US regulations, the analyst(s) named in this report certifies that 1) all of the views expressed herein accurately reflect the
personal views of the analyst(s) with regard to any and all of the content, securities and issuers and 2) no part of the compensation of the
analyst(s) was, is, or will be, directly or indirectly, related to the specific recommendation or views expressed by the analyst(s) in this report.
It is a BNP Paribas policy that analysts are prohibited from trading in any of the stocks in the sector that they cover.
THIS PUBLICATION IS NOT INTENDED FOR PRIVATE CUSTOMERS AS DEFINED IN THE FSA RULES AND SHOULD NOT BE PASSED
ON TO ANY SUCH PERSON.
The material in this report was produced by BNP Paribas SA, a limited company, whose head office is in Paris. BNP Paribas SA is registered
as a bank with the “Comité des Etablissements de Crédit et des Entreprises d’Investissement (“CECEI”) and is regulated by the “Autorité des
Marchés Financiers (“AMF”) for the conduct of its designated investment business in France.
The information and opinions contained in this report have been obtained from public sources believed to be reliable, but no representation or
warranty, express or implied, is made that such information is accurate or complete and it should not be relied upon as such. Information and
opinions contained in the report are published for the assistance of recipients, but are not to be relied upon as authoritative or taken in
substitution for the exercise of judgement by any recipient, and are subject to change without notice. This report is not, and should not be
construed as, an offer document or an offer or solicitation to buy or sell any investments. Any reference to past performance should not be
taken as an indication of future performance. No BNP Paribas Group Company accepts any liability whatsoever for any direct or consequential
loss arising from any use of material contained in this report. This report is confidential and is submitted to selected recipients only. It may not
be reproduced (in whole or in part) to any other person. A BNP Paribas Group Company and/or persons connected with it may effect or have
effected a transaction for their own account in the investments referred to in the material contained in this report or any related investment
before the material is published to any BNP Paribas Group Company's customers. On the date of this report a BNP Paribas Group Company,
persons connected with it and their respective directors and/or representatives and/or employees may have a long or short position in any of
the investments mentioned in this report and may purchase and/or sell the investments at any time in the open market or otherwise, in each
case either as principal or as agent. Additionally, a BNP Paribas Group Company within the previous twelve months may have acted as an
investment banker or may have provided significant advice or investment services to the companies or in relation to the investment(s)
mentioned in this report.
This report is prepared for professional investors and is not intended for Private Customers, and should not be passed on to any such persons.
For the purpose of distribution in the United States this report is only intended for persons which can be defined as 'Major Institutional Investors'
under U.S. regulations. Any U.S. person receiving this report and wishing to effect a transaction in any security discussed herein, must do so
through a U.S. registered broker dealer. BNP Paribas Securities Corp. is a U.S. registered broker dealer.
By accepting this document you agree to be bound by the foregoing limitations.
© BNP Paribas (2004). All rights reserved.