7%20equity%20variance%20swaps.pdf
TRANSCRIPT
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Equity Variance Swaps Trading just volatility
This presentation was prepared exclusively for instructional purposes only, it is for your information only. It
is not intended as investment research. Please refer to disclaimers at back of presentation.
Leo EvansAC
Vice President
Global Asset Allocation
J.P. Morgan Securities plc
+44(0) 20 7742 2537
April 2013
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Equity Variance Swaps Trading just volatility
Realised Volatility: Definition and characteristics
Trading volatility via straddles and delta-hedged options: path dependent P&L
Dollar gamma: How to make it constant
Variance Swaps: Mechanics, P&L, vega notional, MtM, caps, pricing, variance swap
indices (VIX, VSTOXX, VDAX)
Variance swap hedging & 2008 crisis
Volatility Swaps
Relative value
Convexity & Vol of Vol
Third generation volatility products: forward variance, conditional variance, corridor variance and gamma swaps
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We define volatility as the annualised standard deviation of the (log) daily return
of a stock (or index) price, and variance as the square of the standard-deviation
We compute the standard deviation over a fixed period of time (T days) and
then annualise it by multiplying it by the square root of the number of trading
days in a year (252) divided by the number of days in the calculation period.
We assume that the mean of the log daily return is zero in order to simplify
calculations (and because this is the measure used in the payoff of variance and
volatility swap contracts).
Realised Volatility: Definition and Characteristics (I)
2
1
2
1
2 ln252
T
i i
i
S
S
T
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Standard deviation or variance?
Standard deviation is a more meaningful measure of volatility, given that it is
measured in the same units as stock return.
However some volatility products (such as variance swaps) have payoffs in
terms of variance given that variance related-products are easier to replicate
(with plain vanilla options) and therefore to price.
Moreover, when trading vol via delta-hedged options, the P&L is a direct
function of the difference between realised and implied variance.
Variance swaps payoffs are defined in terms of realised variance. However, the
market standard is to always use volatility for communication (i.e. quoting) purposes.
Realised Volatility: Definition and Characteristics (II)
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Principal characteristics of volatility:
It grows when uncertainty increases.
It reverts to the mean.
It goes up and tends to stay up when most assets go down.
It can increase suddenly in spikes.
Realised Volatility: Definition and Characteristics (III)
Long term history of realised volatility (S&P Index)
Source: J.P. Morgan.
0%
10%
20%
30%
40%
50%
60%
70%
80%
29 34 39 44 49 54 59 64 69 74 79 84 89 94 99 04 09
SPX Index 3m realised vol. (annualised)
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Realised Volatility: Definition and Characteristics (IV)
EuroStoxx 50 (SX5E) Index Volatility: Realised vs. (BS) Implied
Source: J.P. Morgan.
Realised vol is a backward looking measure.
Implied vol (from option prices) is a forward looking measure.
0%
10%
20%
30%
40%
50%
60%
70%
80%
00 01 02 03 04 05 06 07 08 09 10 11 12
Realised 3m Vol. Implied 3m ATM Vol.
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First generation:
Plain vanilla options: gain liquidity after Black & Scholes (BS) option pricing
framework (1974).
Second generation:
Variance and volatility swaps emerge in the 90ies. Seminal papers: 1990
Neuberger (Volatility Trading), 1998 Carr & Madan (Towards a theory of
volatility trading), 1999 Derman et al. (More than you ever wanted to know
about volatility swaps).
Third generation:
Conditional variance swaps, corridor variance swaps and gamma swaps.
Volatility products
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We want to make money if realised volatility (or variance) within a future time
period is higher than a given amount.
We only want to take exposure to realised vol/variance, to nothing else.
Why would we want to do that?
How can we do that?
How can we trade volatility?
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Unlike vanilla options, variance swaps are said to provide pure exposure to
volatility, in the sense that their P&L is only a function of realised volatility:
If you buy a variance swap with notional N and expiry T, your payoff at T will be
equal to N times the difference of realised volatility up to T and a fixed (pre-
agreed) volatility strike.
In order to highlight the differences between vanilla options and variance swaps we
will first illustrate the traditional alternatives to take volatility exposure via options.
Our objective is to find a way to obtain, via options, a volatility exposure similar to
the one provided by a variance swap.
Apart from being a useful way of introducing the rationale behind variance
swaps, this will illustrate how we can replicate a variance swap via vanilla
options.
This replication strategy is the backbone of variance swaps hedging (by dealers)
as well as pricing.
Variance Swaps: How we will introduce them
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Trading volatility via straddles and delta-hedged options
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Buy an ATM call and ATM put. Using BS, its cost depends on: implied vol and time to
expiry (ignoring rates).
Implied volatility exposure: If implied vol increases, other things equal, the position
makes money. However, if the position is kept until expiry, the payoff is independent
of implied vol movements.
What is the exposure of this position to the realised volatility until expiry?
Long ATM Straddle (I)
Straddle cost and PnL at expiry Straddle instantaneous delta
Source: J.P. Morgan.
-20
-10
0
10
20
30
40
50
60
50 70 90 110 130 150
Cost today PnL at ex piry
-100%
-50%
0%
50%
100%
50 70 90 110 130 150
Delta
X-axis: stock price.
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What is the exposure of this position to the realised volatility until expiry?
Imagine realised volatility is very large, but the stock price at expiry is equal to
the strike of the straddle.
We lose money.
This isnt what we wanted.
Initially, the delta of our position is zero, but once the stock moves away from
the strike price, the delta is not zero anymore and we have exposure to the
underlying price of the stock.
This isnt what we wanted.
We wanted exposure to realised vol, to nothing else.
Long ATM Straddle (II)
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Lets analyse the P&L of buying an option and delta hedging it during a small time
interval (e.g. 1 day) first
In order to compute the delta of the option we need to rely on a pricing model.
BS is the most commonly used
Notation:
Option price
Stock price
Interest rate
Implied volatility
Delta
Gamma
Theta
Vega
Long delta hedged option (I)
tC
tS
)0( r
i
t
t
For simplicity, we
assume interest rate
and implied
volatility are
constant. This
allows us to ignore
rho and vega.
Moreover, we
assume interest
rates and dividends
are zero.
t
tV
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1 day goes by (in years) and the stock price moves to
We bought a call option and sold units of the underlying stock
P&L of our position:
The option price depends on the stock price, time to expiry and implied volatility. We
use an (approx.) Taylor expansion on the option price change with respect to the
stock price, time and vol:
Long delta hedged option (II)
t ttS
t
tttttttt SSCCLP &
tttt
itttttttttt
SS
VtSSSS
2
2
1
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Assuming implied volatility stays constant , the P&L can be approximated by
Under BS, there is a one-to-one relationship between theta and gamma (assuming
zero interest rates; see Hull, 6th edition, Chp. 15.7):
This leaves the daily P&L of a delta-hedged call option as:
Long delta hedged option (III)
tSSLP tttttt 2
2
1&
22
2
1it
Stt
t
S
SSSLP
it
t
ttttt
2
2
2
2
1&
)0( i
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Daily P&L of a delta-hedged option (call or put):
Thus, buying a delta hedged option we make money if the realised variance is above
the implied one. The P&L is also affected by the dollar gamma of the option.
Long delta hedged option (IV)
t
S
SSSLP
it
t
ttttt
2
2
2
2
1&
Dollar Gamma
Daily
return Implied
variance
Realised minus implied variance during the day
t
tt
t
ttt
S
S
S
SSln
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Using BS, we can derive a theoretical closed form solution for the dollar gamma. It
depends on:
where is the density function of a N(0,1), K is the strike price and T is the expiry.
See Hull, 6th edition, Chp. 15.
Dollar Gamma
tT
tTrKSd
tTS
d
S
i
it
it
t
t t
)()2/()/ln(
where,
2
1
1
2
)(
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Call and put options (same strike, expiry and implied vol) have the same gamma;
thus, the P&L of buying (and delta-hedging) a call or a put option is the same.
Gamma: Call & Put
Call
-20
0
20
40
60
50 70 90 110 130 150
Cost today
PnL at expiry
-120%
-100%
-80%
-60%
-40%
-20%
0%
50 70 90 110 130 150
Delta
0%
1%
2%
3%
4%
50 70 90 110 130 150
Gamma
Put
-20
0
20
40
60
50 70 90 110 130 150
Cost today
PnL at expiry
0%
20%
40%
60%
80%
100%
120%
50 70 90 110 130 150
Delta
0%
1%
2%
3%
4%
50 70 90 110 130 150
Gamma
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Dollar Gamma is not constant
Dollar Gamma & Stock Price Dollar Gamma & Time to Expiry
Example used
Source: J.P. Morgan.
Dollar gamma is larger the closer the
stock price is to the strike and the
closer we are to the options expiry
date.
0%
1%
1%
2%
2%
3%
50 70 90 110 130 150
0
50
100
150
200
250Gamma
Dollar Gamma (RHS)
0
100
200
300
400
500
600
700
800
50 70 90 110 130 150
1y to ex piry
6m to ex piry
1m to ex piry
X-axis: stock price. X-axis: stock price.
Strike 100
Ivol 20%
Int. rate 0%
Days to expiry 252
Strike K Strike K
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Total P&L of a (dynamically) delta-hedged option (held to expiry) can be
approximated by:
The total P&L of the trade is a function of the difference between realised and
implied variance. However, this is polluted by the dependence of dollar gamma on
the time to expiry and stock price.
This causes the P&L to be path dependent and, as a consequence, delta hedged
options are said to provide an impure exposure to volatility.
Anyone can think about examples?
Long delta hedged option (V)
t t
tttt t
S
SSSLP
it
2
2
2
2
1&
Realised minus implied variance during each time interval (e.g. day)
Path dependent
For each day
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Consider a long option position on a 6-month ATM call, delta-hedged everyday to
expiry. Implied volatility of the option is set at 30% and we simulate the underlying
stock price evolution based on a realised volatility of 30% (over the 6m holding
period). This simulation is repeated 1,000 times to allow for different possible
evolutions of the underlying price.
Total P&L - Delta-Hedged Option (I)
Source: J.P. Morgan.
If implied and
realised vols are 30%
the expected
(average) P&L is zero.
However, there is a
variability of P&Ls
around the zero
average.
The return distribution
varies with the hedging
frequency. The more
frequent the re-hedging
the less variable the
returns. However, the
costs of hedging will
increase and so reduce
overall returns.
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The previous example illustrates that dynamically delta-hedging an option in an
environment where realised volatility is equal to implied volatility can generate a
P&L different from zero. Equivalently, it can also be shown that, under certain
scenarios, the P&L of the trade can be negative even if realised volatility is above
implied volatility.
The contribution to the total P&L of (realised minus implied) variance on a given
day depends on the dollar gamma for that day, which is very sensitive to the time to
expiry and the stock price.
For example, if the stock price is close to the strike during the last part of the
options life, whatever happens during that period has a very large impact on the total P&L.
If we had bought the option an the stock realises very low volatility during that
period (much lower than the implied), this will have a very negative impact on
the total P&L.
The final P&L can be negative even if the realised volatility since inception to
expiry was very large (making the total realised volatility higher than the implied
one).
Total P&L - Delta-Hedged Option (II)
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Example: an option trader sells a 1-year call struck at 110% of the initial price on a
notional of $10,000,000 for an implied volatility of 30%, and delta-hedges his position
daily.
Total P&L - Delta-Hedged Option (III)
The realized
volatility (over
the options
life) is 27.50%,
yet his final
trading P&L is
down $150k.
Why?
Source: J.P. Morgan.
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The stock oscillated around the strike in the final months, triggering the dollar gamma
to soar. This would be good news if the volatility of the underlying had remained below
the 30% implied vol, but unfortunately this period coincided with a change in the (50
days realised) volatility regime from 20% to 40%.
Total P&L - Delta-Hedged Option (IV)
Negative
total P&L
even though
the realised
volatility
over the year
was below
30%!
Source: J.P. Morgan.
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Total P&L of a delta-hedged call option:
Notice that every single day counts:
For each t, what matters is the combination of (i) dollar gamma for that day
and (ii) difference between stock price % change (squared) and implied vol.
Although realised variance over the life of the option may be higher than realised ...
There will likely be many days where is negative.
If those days coincide with a very large dollar gamma, they can have a large
impact on the final P&L.
Especially if, for the days where is positive, the dollar
gamma happens to be very low.
t t
tttt t
S
SSSLP
it
2
2
2
2
1&
Long delta hedged option (V)
tS
SSi
t
ttt 2
2
tS
SSi
t
ttt 2
2
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Example Nov-01/Nov-02; 1y EuroStoxx options.
Index was initially at 3500 (with ATM implied volatility at 28.5%) and up until May
2002 remained in the range 3500-3800, realising around 20% volatility. After May,
the index fell rapidly to around the 2500 level, realising high (around 50%)
volatility on the way. Over the whole year, realised volatility was 36%.
Compare the performance of (buying and dynamically) delta-hedging a 2500 and
a 4000-strike option respectively.
Total P&L - Delta-Hedged Option (VI)
Source: J.P. Morgan.
2500-strike option 4000-strike option
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Suppose a market-maker buys and delta-hedges a vanilla option. If realised volatility is
constant and the option is delta-hedged over infinitesimally small time intervals. Then
the market-maker will profit if and only if realised volatility exceeds the level of
volatility at which the option was purchased.
However, the magnitude of the P&L will depend not only on the difference
between implied and realised volatility, but where that volatility is realised, in
relation to the option strike. If the underlying trades near the strike, especially
close to expiry (high gamma) the absolute value (either positive or negative) of
the P&L will be larger.
If volatility is not constant, where and when the volatility is realised is crucial. The
differences between implied and realised volatility will count more when the
underlying is close to the strike, especially close to expiry.
For non-constant volatility, it is perfectly possible to buy (and delta-hedge) an
option at an implied volatility below that subsequently realised, and still lose
from the delta-hedging.
For a clear recap of options path dependent volatility exposure: J.P. Morgan, Variance
Swaps, 2006, Sections 4.1-4.3.
Long delta hedged option (VII) - Recap
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Dollar Gamma: How to make it constant
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Objective: Constant Dollar Gamma (i.e. constant vol exposure)
Total P&L of a delta-hedged call option:
Our objective is to create a position which provides pure (i.e. not path dependent)
exposure to realised variance/vol. We have seen that a single (delta-hedged) option
doesnt do the work.
Can we create a position, via delta-hedged options, which provides a non-path
dependent volatility exposure?
In other words, Is there a way of building a portfolio of options such that its
dollar gamma is constant with respect to the stock price?
t t
tttt t
S
SSSLP
it
2
2
2
2
1&
Realised minus implied variance
for each time interval (e.g. day) Path
dependent
Remember: A call and a put option with the same
strike have the same gamma (and dollar
gamma). Thus, we can use one or the other.
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Objective: Constant Dollar Gamma (i.e. constant vol exposure)
Lets look first at the dollar gamma of different options. We assume a 20% implied vol,
0% interest rates and 1y to expiry.
Dollar Gamma: 50 and 150 strike options Dollar Gamma across strikes
0
50
100
150
200
250
300
350
0 50 100 150 200 250
50 150
0
50
100
150
200
250
300
350
400
0 50 100 150 200 250
25 50 75 100
125 150 175
The dollar gamma of an option has a higher peak and a higher width as the strike
increases.
Is there a way of combining a set of options to generate a constant dollar gamma?
Source: J.P. Morgan.
X-axis: stock price. X-axis: stock price.
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Objective: Constant Dollar Gamma (i.e. constant vol exposure)
Lets use options with strikes 75, 100, , 250, 275. Lets buy each one with a notional
equal to 1/strike (1/K).
Dollar Gamma of each option (1/K) Total dollar gamma
Not quite. Each option, weighted by 1/K, has a similar (peak) dollar gamma, but the
portfolio dollar gamma is not constant with respect to the stock price.
Any other idea?
Source: J.P. Morgan.
0.0
0.5
1.0
1.5
2.0
2.5
50 100 150 200 250
0
1
2
3
4
5
6
7
8
9
50 100 150 200 250
X-axis: stock price. X-axis: stock price.
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Objective: Constant Dollar Gamma (i.e. constant vol exposure)
Lets use options with strikes 75, 100, , 250, 275. Lets buy each one with a notional
equal to 1/K2.
Dollar Gamma of each option (1/K2) Total dollar gamma
Weighting each option by (1/K2) generates a constant dollar gamma exposure.
The area where the dollar gamma of the portfolio is constant depends on the number of
options used; the more the better.
Source: J.P. Morgan.
0.000
0.005
0.010
0.015
0.020
0.025
0.030
50 100 150 200 250
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
50 100 150 200 250
Constant
X-axis: stock price. X-axis: stock price.
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Objective: Constant Dollar Gamma (i.e. constant vol exposure)
To achieve a constant dollar gamma across strikes what kind of portfolio is needed?
One important observation is that (peak) dollar gamma increases linearly with strike
(top-left figure next page).
It may be thought that weighting the options in the portfolio (across all strikes) by the
inverse of the strike will achieve a constant dollar gamma. It does have the property
that each option in the portfolio has an equal peak dollar gamma (top-right figure next
page).
However, the dollar-gammas of the higher strike options spread out more, and the
effect of summing these 1/K-weighted options across all strikes still leads to a dollar-
gamma exposure which still increases with the underlying (bottom-right figure next
page).
In fact, in can be shown that this increase is linear, and therefore weighting each
option by the inverse of the strike-squared will achieve a portfolio with constant dollar
gamma (bottom figures next page).
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Dollar Gamma of each option (1/K2) Total dollar gamma
Dollar Gamma of each option (1) Dollar Gamma of each option (1/K)
Source: J.P. Morgan.
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Objective: Constant Dollar Gamma (i.e. constant vol exposure)
The area where the dollar gamma of the portfolio is constant depends on the number of
options used; the more the better.
In the limit a portfolio of options with a continuum of strikes (from 0 to
infinity) will generate a constant dollar gamma.
This is not possible in practice; it would be very costly even if it was possible.
Using a subset of options will generate a dollar gamma which is fairly constant
on a local area.
We can always increase/reduce the number of options as well as the strike
area to suit our purposes.
Lets look at a couple of examples.
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1/Strike2 Using options to get a constant dollar gamma
We move from a portfolio of 75 to 225 strike options (25 apart) to a portfolio of 125 to 325
options (50 apart). 1y expiry, 20% vol, 0% rates.
75 to 225 strikes, 25 apart 125 to 325 strikes; 50 apart
The second portfolio generates a lower dollar gamma, in absolute level, so we will have
to do more notional of each option (or use a finer grid).
As time approaches expiry, the dollar gamma profile of the portfolio also changes.
Source: J.P. Morgan.
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
50 100 150 200 250
0
0.005
0.01
0.015
0.02
0.025
50 100 150 200 250
Constant Constant
X-axis: stock price. X-axis: stock price.
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1/Strike2 Using options to get a constant dollar gamma
Dollar gamma of a portfolio of 75 to 250 strike options (25 apart). 20% vol, 0% rates.
Dollar gamma as a function of time to expiry
As expiry approaches, we will likely need to increase the number of options in our
portfolio to maintain the constant dollar gamma exposure (i.e. use a finer grid of
strikes).
Source: J.P. Morgan.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
50 70 90 110 130 150 170 190 210 230 250
1y 6m 3m 1m
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Objective: Constant Dollar Gamma (i.e. constant vol exposure)
Remember the total P&L of a dynamically delta-hedged call/put option:
Using a portfolio of options, appropriately weighted, we can generate a constant
dollar gamma exposure.
Thus, delta-hedging this portfolio of options will generate a position with a P&L
directly dependent of realised volatility; which is what we were looking for.
t t
tttt t
S
SSSLP
it
2
2
2
2
1&
Realised minus implied
variance for each time
interval (e.g. day)
Can make this
constant!!
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Objective: Constant Dollar Gamma (i.e. constant vol exposure)
If we could buy the entire strike continuum of options, we wouldnt need to modify
the amount of options.
I.e. static hedge (on the options side; well always have to delta-hedge).
If we assume an initial flat implied volatility skew, the P&L will be just a
function of realised and implied volatility.
When the initial implied volatility is different across strikes, i.e. no flat skew, this will
have an impact given that we buy options with different strikes.
Thus, the price of a variance swap will be a function of the volatility skew.
t t
ttt tS
SSXLP
i
2
2
2
1&
Flat skew:
implied vol is the
same for all
strikes.
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Objective: Constant Dollar Gamma (i.e. constant vol exposure)
Assume we put together a portfolio of (delta-hedged) options with constant dollar
gamma. If is one business day, i.e. 1/252 years, and we run the trade from day 0
to day T
varImplied varRealised2522
ln252
2522
252ln
2252
1ln
2
2
1&
2
1
2
1
2
1
2
11
2
1
2
1
1
2
2
1
1
TX
S
S
T
TX
T
S
SX
S
SX
tS
SSXLP
i
ii
i
T
t t
t
T
t t
tT
t
T
t t
t
T
t t
tt
t
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Objective: Constant Dollar Gamma (i.e. constant vol exposure)
A portfolio of options (calls and/or puts) where each option is weighted by 1/strike-
squared, has constant dollar-gamma;
Delta-hedging this portfolio provides constant exposure to the difference between
implied and realised variance regardless of where and when the volatility is realised;
Hence the P&L from delta-hedging this portfolio is proportional to difference between
realised and implied variance.
This is the idea behind variance swaps: payoff, pricing and hedging.
Source: J.P. Morgan.
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Variance Swaps
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A variance swap offers straightforward and direct exposure to the variance (and
indirectly volatility) of an underlying stock or index.
It is a swap contract where the parties agree to exchange a pre-agreed variance
level (the implied variance, or strike) for the actual amount of variance
realised by the stock or index (the realised variance) over a specified period.
Cash settled at expiry of the swap; no other cash flows.
Variance swaps offer investors a means of achieving direct exposure to realised
variance without the path-dependency issues associated with delta-hedging options.
Variance swap mechanics ref.: J.P. Morgan, Variance swaps, 2006, Section 1.
What is a Variance Swap?
Variance
Seller Variance
Buyer
Realised
variance
Implied (agreed) variance
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Mechanics
The strike of a variance swap, not to be confused with the strike of an option,
represents the level of volatility bought of sold and is set at trade inception.
The strike is set according to prevailing market conditions so that the swap
initially has zero value.
If the subsequent realised volatility is above the level set by the strike, the
buyer of a variance swap will make a profit; and if realised volatility is below,
the buyer will make a loss. A buyer of a variance swap is therefore long
volatility.
Similarly, a seller of a variance swap is short volatility and profits if the level of
variance sold (the variance swap strike) exceeds that realised.
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P&L (I)
The P&L, at expiry, of a (long) variance swap is given by:
where is the variance swap strike (expressed in volatility terms), is the realised
variance and is the variance notional.
22& KNLP rVarVar K 2
rVarN
Example 1:
An investor wishes to gain exposure to the volatility of an underlying asset (e.g. Euro Stoxx 50) over the next year.
The investor buys a 1-year variance swap, and will be delivered the difference between the realised variance over
the next year and the current level of implied variance, multiplied by the variance notional.
Suppose the trade size is 2,500 variance notional, representing a P&L of 2,500 per point difference between realised and implied variance.
If the variance swap strike is 20 (implied variance is 400) and the subsequent variance realised over the course of
the year is 15%2 = 0.0225 (quoted as 225), the investor will make a loss because realised variance is below the level
bought.
Overall loss to the long = 437,500 = 2,500 x (400 225) . The short will profit by the same amount.
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Quotation
Variance swap strikes are quoted in terms of volatility, not variance; but their
payoff is based on the difference between the level of variance implied by the strike
(in fact the strike squared given that the strike is expressed in vol terms) and the
subsequent realised variance.
When quoting and computing the payoff of a variance swap, we wont use
volatility in % terms; well quote 15% volatility as 15.
Example: you buy a variance swap with a variance notional of 100 and 15 strike. At
expiry, realised volatility is 20% during the period
Your payoff will be 100 x (202 - 152 ) = 100 x (400 - 225) = 17,500
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P&L (II)
Definition of realised variance for variance swap payoff:
where is the stock price and is the number of days.
We express variance in annualised terms.
T
i i
ir
S
S
T 1
2
1
2 ln252
iS T
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Variance Swaps - Recap
Variance swap pays the difference between fixed (implied) and realised variance
Payout = variance amount x (realised variance - strike2)
A variance swap is a pure play on volatility
Example: Buy 2,500 notional of 6-month variance swap @ 30 strike (variance = 900)
if realised vol = 25 (var = 625) loss = (625 - 900) x variance amount
= 275 x variance amount
= 687,500
if realised vol = 35 (var = 1225) profit = (1225 - 900) x variance amount
= 325 x variance amount
= 812,500
Realised variance is calculated using the formula :
variance
swap
seller
variance
swap
buyer Fixed Payment
(implied variance = strike2)
Realised variance
i i
i
S
S
T 1
2ln252
Source: J.P. Morgan.
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Vega Notional (I)
Since volatility is a more familiar concept than variance and that most variance swap
investors also have option positions, it is useful to express the notional of a variance
swap in terms of Vega Notional (rather than Variance Notional).
Vega notional is defined as an approximate P&L on a variance swap for a 1% change in volatility.
Taking the first derivative of the P&L of a var. swap w.r.t realised volatility we
get , which depends on the final realised volatility.
Given that final realised volatility is expected to be equal to the swap strike , a
good approximation to the P&L of the swap for a 1% change in volatility is
Thus, it is market convention to define vega notional as , which
makes the final P&L equal to
rrVarN 2
KNN VarVega 2
K
KNVar 2
22222
& KK
NKNLP r
Vega
rVarVar
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Vega Notional (II)
Using variance or vega notional is irrelevant for the P&L of the swap. However,
market participants will speak in terms of vega notional given that it is related to
volatility, which is the standard measure used in options.
The P&L of a variance swap is often expressed in terms of vega notional.
Example 2:
Suppose a 1-year variance swap is struck at 20 with a vega notional of 100,000.
If the index realises 25% volatility over the next year, the long will receive 562,500 = 100,000 x (252 202) / (2 x 20). However if the index only realises 15%, the long will pay 437,500 = 100,000 x (152202) / (2 x 20). Therefore the average exposure for a realised volatility being 5% away from the strike is 500,000 or 5 times the vega notional, as expected.
Note that the variance notional is 100,000 / (2 x 20) = 2,500, giving the same calculation as that used in Example 1.
The P&L of a variance swap is often expressed in terms of vega notional.
In Example 2, a gain of 562,500 is expressed as a profit of 5.625 vegas (i.e. 5.625 times the vega notional). Similarly a loss of 437,500 represents a loss of 4.365 vegas. The average exposure to the 5% move in realised volatility is therefore 5 vegas, or 5 times the vega notional.
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Variance Swaps are Convex on Realised Volatility
Although variance swap payoffs are linear with variance they are convex with realised
volatility.
The vega notional represents only the average P&L for a 1% change in volatility.
A long variance swap position will always profit more from an increase in
volatility than it will lose for a corresponding decrease in volatility (see Recap
example).
This difference between the magnitude of the gain and the loss increases with
the change in volatility. This is the convexity of the variance swap.
If we differentiate the variance swap final P&L w.r.t realised volatility we obtain:
Thus, the sensitivity of the variance swap P&L to volatility is not constant: it is higher
the higher the volatility realised.
r
Vega
rVar
r
Var
K
NN
LP
2
&
Var
VegaN
K
N
2
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Volatility Swaps are Linear on Realised Volatility
A volatility swap will have a linear P&L w.r.t. realised volatility, i.e.:
In a vol swap the vega notional is not an approximation to the average P&L if
volatility changes 1%, it is an exact (and constant) amount.
If we differentiate the volatility swap final P&L w.r.t realised volatility we obtain:
Thus, the sensitivity of the volatility swap P&L to volatility is constant and
independent of the level of volatility realised.
KNLP rVegaSwapVol &
Vega
r
SwapVolN
LP
&
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Variance Swaps vs. Volatility Swaps
The P&L of a variance swap is linear w.r.t. variance and positively convex w.r.t.
volatility.
The P&L of a volatility swap is linear w.r.t. volatility and negatively convex w.r.t.
variance.
P&Ls vs. Realised Vol P&Ls vs. Realised Variance
Source: J.P. Morgan.
-60
-40
-20
0
20
40
60
0% 20% 40% 60% 80% 100%
Var. Sw ap P&L Vol. Sw ap P&L
Final realised volatility
-60
-40
-20
0
20
40
60
0% 20% 40% 60% 80% 100%
Var. Sw ap P&L Vol. Sw ap P&L
Final realised variance
Both with 50% strike (in vol terms) ................................ 50% x 50% = 25% in var terms
50 strike, 1 vega notional on both var & vol swaps
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Variance is additive
In annualised terms, the realised variance between 0 and T is the weighted
average of the realised variances between 0 and t and 0 and T:
0 t days T
2
,0 t2
,TtRealised variance from 0 to t (annualised) Realised variance from t to T (annualised)
ln252
0
2
1
t
i i
i
S
S
t ln
2522
1
T
ti i
i
S
S
tT
2
,
2
,0
2
,0 TttTT
tT
T
t
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Mark-to-Market - Exercise I
At time 0, you buy a variance swap with:
Notional
Expiry
Strike
Right after you open the trade, the quoted strike for the variance swap moves to ,
which we assume to be higher than
Questions:
What (offsetting) trade would you have to do in order to lock-in a sure positive
payoff at T?
What is that (sure) payoff at T?
In order to compute the MtM of your original trade at time 0, you would just need to
discount (risk-free) the (sure) payoff at T that you could achieve by doing the offsetting
trade.
TK ,0
VarN
T
New
TK ,0TK ,0
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Mark-to-Market - Exercise I (cont.)
0 T
Trade 1: Buy variance notional , at time 0, strike , expiry .
Payoff at expiry T =
Trade 2: Sell variance notional , at time 0, strike , expiry .
Payoff at expiry T =
Total (net) payoff at expiry T adding both trades is known with certainty at time 0
Thus, trade 1 MtM at time 0:
TK ,0VarN T
Realised variance from 0 to T (annualised)
2
,0 T
][ 2,02
,0 TTVar KN New
TK
,0VarN T
])[( 2,02
,0 T
New
TVar KN
])[( 2,02
,0 T
New
TVar KKN
TT
New
TVar DFKKN ,02
,0
2
,0 ])[(
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Mark-to-Market - Exercise II
0 t T2
,0 t
At time 0, you buy a variance swap with:
Notional
Expiry
Strike
You keep your trade open and, at time
The (annualised) realised variance from 0 to has been
The quoted strike for a variance swap starting at t and expiring at is
TK ,0
VarN
T
TtK ,T
t
t2
,0 t
Realised variance from 0 to t (annualised)
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Mark-to-Market - Exercise II (cont.)
What is your MtM at time ?
Which (new) trade would you do (at time ) if you wanted to lock-in that MtM for sure?
Lets start by working this out:
At time you enter into a new trade (keeping your initial one):
Sell a variance swap with expiry ; notional ; strike .
Compute the payoff of both trades, and the net payoff, at time .
Does that payoff depend on something which you dont know for sure at time
? If it does, then you havent locked-in your MtM.
Which notional should you trade at time to lock-in your MtM for sure (i.e. to
have a payoff at which is known with certainty at )?.
The discounted value of such payoff will be the MtM at on your original
trade.
t
t
t
T VarN TtK ,
T
t
ttT
t
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Mark-to-Market - Exercise II (Solution a)
Trade 1: Buy variance notional , at time 0, strike , expiry .
Payoff at expiry T =
Trade 2: Sell variance notional , at time t, strike , expiry .
Payoff at expiry T =
Total (net) payoff at expiry T adding both trades:
TK ,0VarN T
][ 2,02
,0 TTVar KN
TtK ,VarN T
][ 2,2
, TtTtVar KN
][][ 2,2
,
2
,0
2
,0 TtTtVarTTVar KNKN
0 t T2
,0 tRealised variance from 0 to t (annualised)
2
,TtRealised variance from t to T (annualised)
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Mark-to-Market - Exercise II (Solution b)
Total payoff at expiry T adding both trades:
2
,
2
,
2
,0
2
,
2
,0
2
,0
2
,
2
,
2
,0
2
,
2
,0
2
,
2
,
2
,0
2
,0
2
,
2
,
2
,0
2
,0 ][][
TtTtTTtTtVar
TtTtTTttVar
TtTtTTVar
TtTtVarTTVar
KT
tKK
T
tTK
T
tN
T
tT
T
tK
T
tT
T
tK
T
tT
T
tN
KKN
KNKN
1 1
Known at time t Realised variance between t and T,
unknown at time t
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Mark-to-Market - Exercise II (Solution c)
Selling variance with a notional at time t doesnt generate a sure payoff at T.
Rather than selling variance with a notional at t, try this:
Sell variance notional , at time t, strike , expiry .
VarN
VarN
TtK ,T
tTNVar
T
Trade 1: Buy variance notional , at time 0, strike , expiry .
Payoff at expiry T =
Trade 2: Sell variance notional , at time t, strike , expiry .
Payoff at expiry T =
TK ,0VarN T
][ 2,02
,0 TTVar KN
TtK , T
][ 2,2
, TtTtVar KT
tTN
T
tTNVar
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Mark-to-Market - Exercise II (Solution d)
Total payoff at expiry T adding both trades:
2
,0
2
,
2
,0
2
,0
2
,
2
,
2
,0
2
,
2
,0
2
,
2
,
2
,0
2
,0
2
,
2
,
2
,0
2
,0 ][][
TTtTtVar
TtTtTTttVar
TtTtTTVar
TtTtVarTTVar
KKT
tTK
T
tN
KT
tT
T
tT
T
tT
T
tK
T
tT
T
tN
KT
tT
T
tTKN
KT
tTNKN
1
Known at time t
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Mark-to-Market (I)
Marking to market of variance swaps is easy: variance is additive. At an intermediate
point in the lifetime of a variance swap, the expected variance at maturity is simply the
time-weighted sum of the variance realised over the time elapsed, and the implied
variance (i.e. new var swap strike) over the remaining time to maturity.
All that is needed to compute the mark-to-market of a variance swap is:
The realised variance since the start of the swap; and
the implied variance (variance strike) from the present time until expiry.
Since the variance swap is cash settled at maturity, a discount factor between the
present time and expiry is also required
TtTTtTtVart DFKKT
tTK
T
tNMtM ,
2
,0
2
,
2
,0
2
,0
where inception is time 0, t is today, T is expiry, DF is discount factor, is the
annualised realised variance from 0 to t, was the original (i.e. at time 0) strike, and
is the current strike.
2
,0 t
TK ,0
TtK ,
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Variance Amount x ( [Strike2 - realised2] x elapsed time + [Strike2 - Newstrike2] x remaining time )
= 2,500 x [302 - 252] x 3/12 + [302 - 272] x 9/12
= 197 x 2,500
= 492,500
3.3 vegas = ( 492,500 / 150,000)
Example: We are short a 12-month variance swap on a stock
Strike 30%
Variance notional 2,500
Vega notional 150,000 ( = 2 x 2,500 x 30)
Assume that over the next 3 months the stock has a realised volatility of 25% and the
variance swap for the remaining 9 months is quoted at 27.
If we then buy a 9 months var swap with strike 27 and var notional 1,875 ( = 2,500 x 9
/ 12), the P&L would be calculated as :
Mark-to-Market (II)
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Mark-to-Market (III)
Example 3: Suppose a 1-year variance swap is stuck at 20 with a vega notional of 100,000 (variance notional of 2,500). If the volatility realised over the first 3 months is 15%, but the volatility realised over the following 9 months is 25%,
then, since variance is additive, the variance realised over the year is:
Variance = [ x 152 ] + [ x 252 ] = 525 (22.9 volatility). At expiry the P&L would be 2,500 x (22.92 202) = 312,500.
Now, suppose again that realised volatility was 15% over the first 3 months. In order to value the variance swap MtM after
3 months we need to know both the (accrued) realised volatility to date (15%) and the fair value of the expected variance
between now and maturity. This is simply the prevailing strike of a 9-month variance swap. If this is currently trading at
25, then the same calculation as above gives a fair value at maturity for the 1-year variance swap of 312,500.
Although the fair value at maturity (now 9 months in the future) is 312,500, we wish to realise this p/l now (after 3-months). It is therefore necessary to apply an appropriate interest rate discount factor.
If, after 3-months, the discount factor is 0.97, the MtM would be equal to about 303,400.
Source: J.P. Morgan.
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Caps (I)
Variance swaps, especially on single-stocks, are usually sold with caps.
These are often set at 2.5 times the strike of the swap capping realised
volatility at this level.
P&L with caps:
Variance swap caps are useful for short variance positions, where investors are then
able to quantify their maximum possible loss.
22,Min& KKCapNLP rVar
Capped vs. Uncapped P&L
Source: J.P. Morgan.
Capped vs. Uncapped P&L
-20
0
20
40
60
80
100
120
0% 10% 20% 30% 40% 50% 60% 70%
Var. Swap ( strike K = 20% ) P&L
Capped Var. Swap 2.5x P&L
Final realised volatility
-20
0
20
40
60
80
100
0% 5% 10% 15% 20% 25% 30% 35% 40%
Var. Swap ( strike K = 20% ) P&L
Capped Var. Swap 2.5x P&L
Final realised variance
20 strike, 1 vega notional on both swaps
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Caps (II)
In practice caps are rarely hit especially on index underlyings and on longer-
dated variance swaps.
When caps are hit, it is often due to a single large move e.g. due to an M&A event or major earning surprise on an individual name, or possibly from a
dramatic sell-off on an index.
Single-day moves needed to cause a variance swap cap to be hit are large and
increase with maturity.
A 1-month variance swap struck at 20 and realising 20% (annualised) on all
days except for one day which has a one-off 14% move, will hit its cap.
A similar 3-month maturity swap would need a 1-day 24% move to hit the
cap
The required 1-day move on a 1-year swap would be 46%.
For lower strikes the required moves are also lower.
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Caps (III)
A 1-y variance swap struck at 20 and realising 20% (annualised) on all days except for
one day which has a one-off 46% move, will hit its cap.
To hit the cap we need:
where T = 252 and K = 20, i.e.
We assume that, for all days except one (i.e. 251 days), the stock realises 20% (annualised), i.e.
How much does the stock need to change in the other day m, i.e. ln(Sm+1/Sm) as a proxy for (Sm+1-Sm)/Sm , for the final realised variance to be equal to the cap?
20
2
1 5.2 ln252
KS
S
T
T
t t
t
2
2
1 20ln1
252
t
t
S
S1 day realised (annualised) variance: , which implies:
252
20ln
22
1
t
t
S
Sfor 251 days
22
1
22
1
2
1
0
2
1 205.2ln1252
20251ln1ln251 ln
m
m
m
m
t
tT
t t
t
S
S
S
S
S
S
S
S
2252
0
2
1 5.2 ln KS
S
t t
t
8.45252
20251205.2ln
221
m
m
S
S
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Possible ways to exit a variance swap:
Go back to the original counterparty to unwind
Tear-up the contract (probably after some payment How much?)
No future cash flows & legal risk.
Enter into an offsetting transaction
If you bought variance with counterparty A, you sell them with counterparty
B; keeping both positions.
Residual risks: counterparty risk if any of the two swaps is not cleared.
What are the complications introduced by the caps?
Exiting variance swap positions (before expiry)
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Offsetting capped variance swaps example
Suppose that an investor buys a 6-month variance swap with a strike of 20. This has
the standard 2.5x cap meaning the exposure to realised volatility will be capped at
50.
The very same day, the (6-month) variance swap trades at a strike of 30 leading to a
significant mark-to-market P&L.
The investor wants to lock in this profit.
With the strike now at 30, the cap on an new variance swap contract will by default
be set at 2.5 x 30 = 75.
Then if the investors sells this 30-strike variance swap in an attempt to close out his
position the difference in caps will mean he takes on a short volatility exposure if the
subsequent realised volatility is above 50% (although capped at 75%).
In effect, in the course of trying to close out his position, he will have sold a
50%/75% call spread on volatility. Whilst the price he gets for selling the variance swap will reflect this higher cap, the residual volatility exposure is presumably
unwanted, and the investor would be best either trading directly with the original
counterparty or negotiating a bespoke contract with another counterparty in order to
fully close out his outstanding contract.
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Two offsetting capped var swaps Net payoff position
Two offsetting var swaps Net payoff position
Source: J.P. Morgan.
-20
0
20
40
60
80
100
0% 5% 10% 15% 20% 25% 30% 35% 40%
Var. Swap ( strike K = 20% ) P&L
Var. Swap ( strike K = 30% ) P&L
Final realised variance
-75
-55
-35
-15
5
25
45
65
0% 5% 10% 15% 20% 25% 30% 35% 40%
Net payoff at expiry
Final realised variance
-75
-55
-35
-15
5
25
45
65
0% 10% 20% 30% 40% 50% 60% 70% 80%
Net payoff at expiry
Final realised variance
-50
-30
-10
10
30
50
70
90
110
130
150
0% 10% 20% 30% 40% 50% 60% 70% 80%
Capped Var. Swap P&L (strike K=20; 50 cap)
Capped Var. Swap P&L (strike K=30; 75 cap)
Final realised variance
Notional on all swaps: 1 vega notional (using the original strike, i.e. 20) i.e. a variance swap notional of 1 / 2 x 20 = 0.025.
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Two offsetting capped var swaps Net payoff position
Two offsetting var swaps Net payoff position
Source: J.P. Morgan.
-30
-20
-10
0
10
20
30
40
0% 5% 10% 15% 20% 25% 30% 35% 40%
Var. Swap ( strike K = 20% ) P&L
Var. Swap ( strike K = 30% ) P&L
Final realised volatility-75
-55
-35
-15
5
25
45
65
0% 5% 10% 15% 20% 25% 30% 35% 40%
Net payoff at expiry
Final realised volatility
-40
-20
0
20
40
60
80
100
120
140
0% 20% 40% 60% 80% 100%
Capped Var. Swap P&L (strike K=20; 50 cap)
Capped Var. Swap P&L (strike K=30; 75 cap)
Final realised volatility
-75
-55
-35
-15
5
25
45
65
0% 20% 40% 60% 80% 100%
Net payoff at expiry
Final realised volatility
Notional on all swaps: 1 vega notional (using the original strike, i.e. 20) i.e. a variance swap notional of 1 / 2 x 20 = 0.025.
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Variance swaps were initially developed on index underlyings.
In Europe, variance swaps on the Euro Stoxx 50 index are by far the most liquid,
but DAX and FTSE are also frequently traded.
Variance swaps are also tradable on the more liquid equity underlyings especially Euro Stoxx 50 constituents, allowing for the construction of variance
dispersion trades.
Variance swaps are tradable on a range of indices across developed markets and
increasingly also on emerging markets.
The most liquid variance swap maturities are generally from 3 months to around 2
years, although indices and more liquid stocks have variance swaps trading out to 3 or
even 5 years and beyond.
Maturities generally coincide with the quarterly options expiry dates, meaning
that they can be efficiently hedged with exchange-traded options of the same
maturity.
Variance swap market ref.: J.P. Morgan, Variance swaps, 2006, Section 2.
Variance Swap Market
No Exam
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Pricing a variance swap involves determining its strike price K, i.e. the fixed level
of volatility which will be used to settle the swap (vs. realised volatility) at expiry.
The fair value of the variance swap is determined by the cost, expressed in
volatility terms, of a replicating portfolio.
We illustrated earlier how a portfolio of options, delta-hedged and weighted by the
inverse of their squared strike, generates an exposure with a constant dollar gamma,
i.e. a constant exposure to realised variance (minus implied variance).
Our objective here is not to analytically derive how to price variance swaps. Main
references for those interested on that:
1990 Neuberger (Volatility Trading), 1998 Carr & Madan (Towards a theory of
volatility trading), 1999 Derman et al. (More than you ever wanted to know
about volatility swaps), 2006 Gatheral (The volatility surface), 2006 J.P.
Morgan (Variance Swaps).
Variance Swap Pricing (I)
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A variance swap can be replicated by a (dynamically delta-hedged) portfolio of
options with a continuum of strikes weighted by the inverse of the squared strike.
Variance Swap Pricing (II)
Dollar Gamma: Var. Swap vs. (Imperfect) Replicating options portfolio
50 70 90 110 130 150 170 190 210 230 250
Portfolio of options w eighted 1/K2 Var. Sw ap
Stock price
(Delta-hedged options w ith
strikes from 75 to 250, 25
apart.)
Pricing-wise, the variance swap price can be thought of as a weighted average
of the entire volatility skew (i.e. implied volatilities for all the option strikes).
Thus, the drivers of variance swap prices are essentially the same drivers as for
options volatilities and skews (plus particular demand-supply issues on the variance
swap market).
Source: J.P. Morgan.
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Flat and linear skews SX5E 1y Imp. Vol Skew
Source: J.P. Morgan.
10%
15%
20%
25%
30%
35%
40%
45%
40 60 80 100 120 140 160
1y Implied vol vs. Strike (% current index)
25%
27%
29%
31%
33%
35%
37%
39%
70 80 90 100 110 120
Flat skew
Flat skew (for Derman's approx.)
90/100 put skew
Strike (expressed as % of current stock price)
As of Mar-12
Skew refers to implied volatility (derived from traded option prices) across strikes.
Put skew
Implied vols for strikes below
100 (% of current stock price)
Call skew
ATM Vol
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A variance swap represents a kind of weighted average of volatilities across the
skew curve, with the closer-to-the-money volatilities higher weighted
Rules of thumb:
Given a flat skew, variance swaps should price (theoretically) at the same
level as ATM vol.
High skews will increase variance swap prices
This is the case for both put and call skews (where OTM calls have higher
volatilities than ATM).
ATM volatility will provide the greatest contribution to variance swap
prices
Variance Swap Pricing (III)
iSkewfK ATMiTheo ,,
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Whilst it is necessary to have prices available for the entire strip of (OTM) options in
order to calculate the true theoretical price of a variance swap, reasonable
approximations for variance swap prices can be made under certain assumptions
about the skew. (See J.P. Morgan 2006, Sections 2 & 4).
Flat Skew:
In the hypothetical case where the skew surface is flat (i.e. all strikes trade at
identical implied volatilities) the variance swap theoretical level will be the
(constant) implied volatility level.
Linear skew:
If the skew is assumed to be linear, at least for strikes relatively close to the
money, then Dermans approximation can be used.
Other approximations: long-linear skew, Gatherals formula.
Different (more flexible) assumptions regarding the skew.
Variance Swap Pricing: Approximations (I)
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Linear skew: Dermans approximation.
is the implied volatility for the forward strike, is the years to expiry and
is generally taken to be the 90/100 put skew.
In practice, this approximation tends to work best for short-dated index variance (up
to about 1-year), where put skews are often relatively linear and call skews
relatively flat, at least close to the money.
As maturity increases and the OTM strikes have a greater effect on the variance
swap price (given the higher prob on ending ITM), the contribution of the skew
becomes more important, but the inability of the approximation to account for the
skew convexity can make it less accurate.
Similarly, for single stocks, where the skew convexity can be much more significant,
even at shorter dates, the approximation can be less successful.
Ref.: Derman et al., More than you ever wanted to know about volatility swaps. 1999.
Variance Swap Pricing: Approximations (II)
2
, 31 SkewTK ATMiTheo
ATMi , T
Skew
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Variance Swap Pricing: Approximations (III)
Source: J.P. Morgan.
No Exam
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The dollar gamma of low strike options is higher at the peak but falls much more
aggressively than the dollar gamma of high strike options.
As a consequence, the dollar gamma of a replicating portfolio (with equally spaced
options) generally tends to fall aggressively as the stock price falls.
50 75 100 125 150 175 200
Portfolio of options w eighted 1/K2 Var. Sw ap
Stock price
(Delta-hedged options w ith strikes
from 75 to 200, 25 apart.)
Dollar Gamma: Var. Swap vs.
replicating options portfolio
Variance Swap Pricing Insights (I)
Dollar Gamma of two options divided by
their squared strike
50 100 150 200 250
75 150
Source: J.P. Morgan.
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What does this mean?
In practice, low strike options are generally more important than high strike
options to hedge a variance swap: if one is forced to use only a few options, it
is less risky to use options with lower strikes (or at least to use more of
them).
Generally, put options are more liquid for low strikes.
Variance Swap Pricing Insights (II)
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A long capped variance swap can be thought of as a standard variance swap plus a
short call on variance, stuck at the cap level.
A standard cap of 2.5x current implied variance strike is relatively far out-of-the-
money, assuming that the volatility of volatility is not too large. This means that the
value of the cap should be relatively small compared to the variance swap strike and
should not have a major effect on pricing.
Variance Swap Pricing: Capped Swaps
Capped vs. Uncapped P&L However, for a long position, a
variance swap with a cap will always
be worth less than an uncapped
variance swap of the same strike.
Therefore capped variance swaps must
trade with strikes slightly below their
uncapped equivalents the difference,
in theory, representing the current
value of the call on variance. -20
0
20
40
60
80
100
120
0% 10% 20% 30% 40% 50% 60% 70%
Var. Sw ap ( strike K = 20% ) P&L
Capped Var. Sw ap 2.5x P&L
Final realised volatility
Source: J.P. Morgan.
1 vega notional on all swaps
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SX5E 1y Var Swap vs. Implied ATM Vol SX5E 1y Var Swap vs. Implied ATM Vol
SX5E Variance Swaps SX5E Variance Swaps Term Structure
Source: J.P. Morgan.
0%
10%
20%
30%
40%
50%
60%
70%
Jan-07 Jan-08 Jan-09 Jan-10 Jan-11 Jan-12
6m Variance Swap 1y Variance Swap
-10%
-8%
-6%
-4%
-2%
0%
2%
4%
Jan-07 Jan-08 Jan-09 Jan-10 Jan-11 Jan-12
1y minus 6m Var Swaps
0%
10%
20%
30%
40%
50%
60%
Jan-07 Jan-08 Jan-09 Jan-10 Jan-11 Jan-12
Implied 1y ATM Vol. 1y Variance Swap
-5%
0%
5%
10%
15%
20%
25%
Jan-07 Jan-08 Jan-09 Jan-10 Jan-11 Jan-12
1y Var Swap minus Implied ATM Vol.
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SX5E 1y Var Swap vs. Implied ATM Vol SX5E 1y Imp. Vol Skew
SX5E 1y 90/100 Implied Vol Skew Variance generally prices above ATM
vol.
This is due, among other things, to the
existence of the volatility skew (given
that the variance swap price can be
thought of as a weighted average of the
entire volatility skew).
-5%
0%
5%
10%
15%
20%
25%
Jan-07 Jan-08 Jan-09 Jan-10 Jan-11 Jan-12
1y Var Swap minus Implied ATM Vol.
As of Dec-10
10%
15%
20%
25%
30%
35%
40%
45%
40 60 80 100 120 140 160
1y Implied vol vs. Strike (% current index)
As of Mar-12
2.0%
2.2%
2.4%
2.6%
2.8%
3.0%
3.2%
3.4%
3.6%
3.8%
Jan-07 Jan-08 Jan-09 Jan-10 Jan-11 Jan-12
1y 90/100 Skew
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SX5E and S&P Var Swap Term Structure
SX5E Term Structure:
Var Swap vs. ATM Implied Vol
As of March 2012
10%
15%
20%
25%
30%
35%
0m 3m 6m 9m 12m 15m 18m 21m 24m
SXE5 S&P
10%
15%
20%
25%
30%
35%
0m 3m 6m 9m 12m 15m 18m 21m 24m
Var Swap ATM Implied Vol.
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Variance Swap Indices (I)
The VIX, VSTOXX and VDAX indices represent the theoretical prices of 1-month
variance swaps on the S&P500, Euro Stoxx and DAX indices respectively, and are
calculated by the exchanges from listed option prices, interpolating to get 1-
month maturity.
Widely used as benchmark measures of equity market risk, even though they
are only short-dated measures and are not directly tradable.
The short-dated nature of these variance swaps indices means the
principal driver of the volatility index level is recent realised volatility.
In reality, longer dated (e.g. 1-year) variance, spreads of implied to
realised variance, skew levels or even ratios of put to call open-interest
would perhaps be a better proxy for the level of risk-aversion present in
the market.
The design of these indices is based on the square root of implied variance and
incorporates the volatility skew by incorporating OTM puts and calls in the
calculation. A rolling index of 30 days to expiration is derived via linear interpolation
of the two nearest option expiries.
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Variance Swap Indices (II)
VIX vs. Implied Vol VIX minus Implied Vol
Source: J.P. Morgan.
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
Mar-02 Mar-04 Mar-06 Mar-08 Mar-10
VIX S&P 1m ATM Implied Vol.
0%
2%
4%
6%
8%
10%
12%
14%
Mar-02 Mar-04 Mar-06 Mar-08 Mar-10
VIX minus 1m ATM Implied Vol
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Variance Swap Indices (III)
VIX and VSTOXX serve as underlying for listed future and option contracts (Eurex
and CBOE)
Futures
Trading forward variance.
These futures do not expire on the normal index (futures) expiry dates,
but 30 calendar days beforehand. This expiry is chosen because on that
date, the listed options have exactly 30 calendar days remaining maturity
and the VSTOXX calculation does not need to interpolate from any other
maturities.
Reference: J.P. Morgan VDAX, VSTOXX and VSMI Futures, 2005.
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Variance Swap Indices (IV)
VIX and VSTOXX serve as underlying for listed future and option contracts (Eurex
and CBOE)
Options:
Trading vol of vol.
In April 2005, options on the VIX index were launched. These represented
the first available exchange traded options on variance. As for the futures,
these expire 30 days before an index expiry and are listed to expire 30
days before the corresponding quarterly options expiry dates for the
underlying.
Reference: J.P. Morgan Options on implied volatility, 2010.
The exact calculation of the VIX index can be found at:
http://www.cboe.com/micro/vix/vixwhite.pdf
See also J.P. Morgan Cross-asset hedging with VIX, 2012.
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Variance Swaps: Hedging and 2008 Crisis
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A variance swap can be statically hedged with a portfolio of (European-style) options,
weighted according to the inverse squares of their strikes.
This makes it easy, in theory, to perfectly hedge a variance swap with options,
assuming option prices are available across the entire range of strikes.
In practice, traded strikes are not continuous, although for major liquid indices they
are closely spaced (0.4% notional apart for the S&P, 1% for the FTSE, 1.4% for the Euro
Stoxx).
A more serious limitation is the lack of liquidity in OTM strikes, especially for puts,
as these provide a relatively large component of the variance swap price in the
presence of steep put skews.
S&P options are listed down to a strike of 600, FTSE to 3525 and Euro Stoxx
down to 600, although in reality, liquidity does not even reach this far.
Variance Swap Hedging in Practice (I)
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In practice, market-makers will not attempt to hedge with the entire strip of options
but typically will use only a few.
One problem with this kind of approach is that the partial hedge is no longer static,
and must be dynamically managed.
The constant dollar gamma would be maintained by a combination of holding a
portfolio which has roughly constant dollar gamma if the underlying does not
move too much, and re-hedging by trading more options if the underlying does
move significantly.
Thus, market makers are unable to buy the complete theoretical hedge, and instead
have to use a portfolio comprised of a limited number of options. The resulting
portfolio hedges the variance swap well within a range of asset levels near the spot at
inception, but not outside this range
See J.P. Morgan, Variance Swaps, 2006, Section 4.8for an explanation of how to construct a replicating
portfolio, i.e. absolute amount of each option traded to generate the variance notional of the variance swap.
Variance Swap Hedging in Practice (II)
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Assume the client goes long variance and the dealer sets up an (imperfect) replicating
portfolio In the event that the market falls significantly and realised volatility is higher
than the variance swap strike, the overall hedge will lose money (if its not rebalanced).
50 75 100 125 150 175 200
Portfolio of options w eighted 1/K2 Var. Sw ap
Stock price
(Delta-hedged options w ith strikes
from 75 to 200, 25 apart.)
Dollar Gamma: Var. Swap vs.
replicating options portfolio
Variance Swap Hedging in Practice (III)
Client (long
variance) gets
this P&L
The replicating
hedge gives the
dealer this P&L
Source: J.P. Morgan.
One can imagine what happened in 2008/2009 market crash For a detailed
explanation, see J.P. Morgan, Volatility Swaps, 2010, Section 4.
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2008: Market makers books were generally short single stock variance swaps
Why? Due to investors selling index correlation to capture the implied
correlation premium.
We will review these trades in a later lecture, but essentially, if an investor
wants to short index correlation he sells index vol (via var. swaps) and buys
single stocks vol (via var. swaps). This leaves dealers long index variance swaps
and short single name variance swaps.
In order to hedge their variance swap books, market makers were holding portfolios of
single stock options and delta-hedging daily.
We saw in the previous slide what can go wrong if a dealer has sold a variance swap
and hedges it with a partial hedge.
The 2008 crisis led to large drops in single stock prices, and many market makers
found themselves unhedged in the new trading range
2008 Crisis & Variance Swap Hedging (I)
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By selling single stock variance swaps, traders had committed to deliver the P/L of a constant
dollar gamma portfolio, irrespective of the spot level, but their replicating portfolio did not
have sufficient dollar gamma at the new spot levels. Market makers were therefore forced to
buy low strike options at the post-crash volatility level, which was much higher than the one
prevailing when they sold the variance swap and therefore incurred heavy losses.
2008 Crisis & Variance Swap Hedging (II)
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Not re-hedging the gamma risk was not a possibility, as this would have left the books
exposed to potentially catastrophic losses if the stock prices declined further, and
volatility continued to increase.
This situation led to large losses for many market-makers in the single stock variance
swap markets. In turn this led banks to re-assess the risk of making markets in these
instruments and to a substantial reduction of the liquidity in the single stock variance
swap market.
Index variance swap markets did not experience a similar disruption and were actively
traded throughout the crisis, despite a widening of their bid-ask spreads. Index
variance swaps continued to trade because of the high liquidity and depth of the
index options markets. A wider range of OTM strikes are listed for index options
compared to single stock options. Additionally, the 'gap risk' of a sudden large decline
is significantly lower for indices than for single stocks.
2008 Crisis & Variance Swap Hedging (III)
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Vol Swaps
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Following the de facto shutdown of the single stock variance swap market in the
aftermath of the 2008 credit crisis, volatility swaps gained liquidity as an instrument
for providing direct exposure to volatility for single stock underliers.
Why?
Although pricing and hedging volatility swaps is more complex than variance
swaps, when hedging volatility swaps with options traders are a lot less
exposed to tail risks (i.e. extreme moves in the stock price and volatility).
There is not a static hedge for volatility swaps, thus hedging them requires
dynamicaly trading options.
Volatility Swaps (I)
Reference: J.P. Morgan, Volatility Swaps, 2010.
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A common complaint about variance swaps is that they pay-off based on realised
variance (volatility squared) and not simply realised volatility.
Remember that the strike of variance swaps is actually quoted in terms of
volatility, and the notional of variance swaps is generally measured with
respect to the (average) sensitivity of the swap to volatility (vega notional).
Why dont we then just trade volatility swaps directly? I.e. a product with a
payoff linear in volatility, not in variance?
Volatility Swaps (II)
The P&L for a (long) volatility swap is given by:
where is the volatility swap strike, is the realised volatility and is
the vega notional (i.e. P&L for each realised volatility point).
KNLP rVega &
K r VegaN
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Whilst volatility can be seen as more of an intuitive measure (being a standard
deviation it is measured in the same units as the underlying), variance is in some
sense more fundamental especially because it is additive.
The exposure of delta-hedged options to volatility, after accounting for the dollar
gamma, is actually an exposure to the difference between implied and realised
volatility squared. In this sense, a variance swap mirrors a kind of ideal delta-hedged
option whose dollar gamma remains constant. Furthermore, variance swaps are
relatively easy to replicate. Once the replicating portfolio of options has been put in
place, only delta-hedging is required; no further buying or selling of options is
necessary.
The main theoretical difficulty with volatility swaps is that they cannot be
statically replicated through options.
Volatility Swaps (III)
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Delta-hedging options leads to a P&L linked to the variance of returns rather than
volatility. To achieve the linear exposure to volatility (which volatility swaps
provide) it is therefore necessary to dynamically trade in portfolios of options,
which would otherwise provide an exposure to the square of volatility.
There doesnt exist a neat and simple hedging strategy for volatility swaps as it
does for variance swaps (using delta-hedged options with a notional of 1 / strike
squared).
A volatility swap can be replicated using a delta-hedged portfolio of options,
where the portfolio of options is dynamically rebalanced (on the option side,
not only on the delta side) to replicate the vega and gamma profile of the
volatility swap across the range of spot prices.
Volatility Swaps (IV)
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Why does the hedging of volatility swaps expose traders to lower risks in extreme
price movements?
Volatility Swaps (V)
Dollar Gamma of a Volatility Swap & a Strangle
80-120% 1.9x 1 Ratio
strangle consists on selling
1.9 80% strike puts and
selling one 120% call.
The dollar gamma of
volatility swaps decreases as
the stock price moves away
from par, as opposed to the
dollar gamma of variance
swaps, which is constant for
all stock prices.
The dollar gamma of
volatility swaps is similar
to the dollar gamma of
option strangles.
No Exam
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For traders proxy hedging volatility and variance swaps, volatility swaps appear to
be easier to manange than variance swaps as the dollar gamma decreases following
large moves in the spot.
Volatility Swaps (VI)
50 75 100 125 150 175 200
Portfolio of options w ei