Overview
In this note we introduce the properties of variance swaps, and give details on the hedging and
valuation of these instruments.
Section 1 gives quick facts about variance swaps and their applications.
Section 2 is written for traders and market professionals who have some degree of
familiarity with the theory of vanilla option pricing and hedging, and explains in ‘intuitive’
mathematical terms how variance swaps are hedged and priced.
Section 3 is written for quantitative traders, researchers and financial engineers, and gives
theoretical insights into hedging strategies, impact of dividends and jumps.
Appendix A is a review of the concepts of historical and implied volatility.
Appendices B and C cover technical results used in the note.
We thank Cyril Levy-Marchal, Jeremy Weiller, Manos Venardos, Peter Allen, Simone Russo for
their help or comments in the preparation of this note.
These analyses are provided for information purposes only and are intended solely for your use. The analyses have been derived
from published models, reasonable mathematical approximations, and reasonable estimates about hypothetical market
conditions. Analyses based on other models or different assumptions may yield different results. JPMorgan expressly disclaims
any responsibility for (i) the accuracy of the models, approximations or estimates used in deriving the analyses, (ii) any errors or
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Table of Contents
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D T O KN OW A B O U T VARIANCE SWAPS
Overview……………………………………………………………………………….. 1
Table of Contents ……………………………………………………………………… 2
1. Variance Swaps …………………………………………………………………… 3
1.1. Payoff 3
Convexity 4
Rules of thumb 5
1.2. Applications 5
Volatility Trading 5
Forward volatility trading 5
Spreads on indices 6
Correlation trading: Dispersion trades 7
1.3. Mark-to-market and Sensitivities 8
Mark-to-market 8
Vega sensitivity 9
Skew sensitivity 9
Dividend sensitivity 9
2. Valuation and Hedging in Practice ………………………………………………11
2.1. Vanilla Options: Delta-Hedging and P&L Path-Dependency 11
Delta-Hedging 11
P&L path-dependency 12
2.2. Static Replication of Variance Swaps 14
Interpretation 16
2.3. Valuation 16
3. Theoretical Insights ………………………………………………………………18
3.1. Idealized Definition of Variance 18
3.2. Hedging Strategies & Pricing 18
Self-financing strategy 19
Pricing 19
Representation as a sum of puts and calls 20
3.3. Impact of Dividends 20
Continuous Monitoring 21
Discrete Monitoring 21
3.4. Impact of Jumps 23
Appendix A — A Review of Historical and Implied Volatility ……………………..24
Appendix B — Relationship between Theta and Gamma………………………….27
Appendix C — Peak Dollar Gamma………………………………………………….28
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References & Bibliography……………………………………………………………29
1. Variance Swaps
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D T O KN OW A B O U T VARIANCE SWAPS
1.1. Payoff
A variance swap is an instrument which allows investors to trade future realized (or historical)
volatility against current implied volatility. As explained later in this document, only variance
—the squared volatility— can be replicated with a static hedge. [See Sections 2.2 and 3.2 for
more details.]
Sample terms are given in Exhibit 1.1.1 below.
Exhibit 1.1.1 — Variance Swap on S&P 500 : sample terms and conditions
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VARIANCE SWAP ON S&P500
SPX INDICATIVE TERMS AND CONDITIONS
Instrument: Swap
Trade Date: TBD
Observation Start Date: TBD
Observation End Date: TBD
Variance Buyer: TBD (e.g. JPMorganChase)
Variance Seller: TBD (e.g. Investor)
Denominated Currency: USD (“USD”)
Vega Amount: 100,000
Variance Amount: 3,125 ( determined as Vega Amount/(Strike*2) )
Underlying: S&P500 (Bloomberg Ticker: SPX Index)
Strike Price: 16
Currency: USD
Equity Amount: T+3 after the Observation End Date, the Equity Amount will be calculated and paid in
accordance with the following formula:
Final Equity payment = Variance Amount * (Final Realized Volatility
2
– Strike
Price2
)
If the Equity Amount is positive the Variance Seller will pay the Variance Buyer the
Equity Amount.
If the Equity Amount is negative the Variance Buyer will pay the Variance Seller an
amount equal to the absolute value of the Equity Amount.
where
Final Realised Volatility = 100
Expected _ N
252 t N
t 1
2
t 1
t
P
P ln
×
×∑
=
=
−
Expected_N = [number of days], being the number of days which, as of the Trade Date, are
expected to be Scheduled Trading Days in the Observation Period
P0 = The Official Closing of the underlying at the Observation Start Date
Pt
= Either the Official Closing of the underlying in any observation date t or, at
Observation End Date, the Official Settlement Price of the Exchange-Traded
Contract
Calculation Agent: JP Morgan Securities Ltd.
Documentation: ISDA
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D T O KN OW A B O U T VARIANCE SWAPS
Note:
Returns are computed on a logarithmic basis:
t−1
t
P
P ln .
The mean return, which normally appears in statistics textbooks, is dropped. This is
because its impact on the price is negligible (the expected average daily return is 1/252nd
of the money-market rate), while its omission has the benefit of making the payoff
perfectly additive (3-month variance + 9-month variance in 3 months = 1-year variance.)
It is a market practice to define the variance notional in volatility terms:
Strike
Vega Notional Variance Notional
× = 2
With this adjustment, if the realized volatility is 1 ‘vega’ (volatility point) above the strike
at maturity, the payoff is approximately equal to the Vega Notional.
Convexity
The payoff of a variance swap is convex in volatility, as illustrated in Exhibit 1.1.2. This
means that an investor who is long a variance swap (i.e. receiving realized variance and paying
strike at maturity) will benefit from boosted gains and discounted losses. This bias has a cost
reflected in a slightly higher strike than the ‘fair’ volatility2
, a phenomenon which is amplified
when volatility skew is steep. Thus, the fair strike of a variance swap is often in line with the
implied volatility of the 90% put.
Exhibit 1.1.2 — Variance swaps are convex in volatility
-$3,000,000
-$2,000,000
-$1,000,000
$0
$1,000,000
$2,000,000
$3,000,000
$4,000,000
$5,000,000
0 10 20 30 40 5
Realized
Volatility
Payoff
0
Strike =24
Variance
Volatility
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2
Readers with a mathematical background will recall Jensen’s inequality: E( Variance) ≤ E(Variance) .
Rules of thumb
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D T O KN OW A B O U T VARIANCE SWAPS
Demeterfi—Derman—Kamal—Zou (1999) derived a rule of thumb for the fair strike of a
variance swap when the skew is linear in strike:
2 Kvar ≈ σ ATMF 1+ 3T × skew
where σ ATMF is the at-the-money-forward volatility, T is the maturity, and skew is the slope
of the skew curve. For example, with σ ATMF = 20%, T = 2 years, and a 90-100 skew of 2
vegas, we have Kvar ≈ 22.3%, which is in line with the 90% put implied volatility normally
observed in practice.
For log-linear skew, similar techniques give the rule of thumb:
( ) 2 4 2
2
2 3
var 12 5
4
K ATMF ATMFT σ ATMFT σ ATMFT β ≈ σ + βσ + +
where σ ATMF is the at-the-money-forward volatility, T is the maturity, and β is the slope of
the log skew curve3
. For example, with σ ATMF = 20%, T = 2 years, and a 90-100 skew of 2
vegas, we have 0.19
ln(0.9)
2% β = − ≈ and Kvar ≈ 22.8%.
Note that these two rules of thumb produce good results only for non-steep skew.
1.2. Applications
Volatility Trading
Variance swaps are natural instruments for investors taking directional bets on volatility:
Realized volatility: unlike the trading P&L of a delta-hedged option position, a long
variance position will always benefit when realized volatility is higher than implied at
inception, and conversely for a short position [see Section 2.1 on P&L path-dependency.]
Implied volatility: similar to options, variance swaps are fully sensitive at inception to
changes in implied volatility
Variance swaps are especially attractive to volatility sellers for the following two reasons:
Implied volatility tends to be higher than final realized volatility: ‘the derivative house has
the statistical edge.’
Convexity causes the strike to be around the 90% put implied volatility, which is slightly
higher than ‘fair’ volatility.
Forward volatility trading
Because variance is additive, one can obtain a perfect exposure to forward implied volatility
with a calendar spread. For example, a short 2-year vega exposure of €100,000 on the
EuroStoxx 50 starting in 1 year can be hedged as follows [levels as of 21 April, 2005]:
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3
The skew curve is thus assumed to be of the form: (K) ln(K / F ) where F is the forward price. ATMF σ = σ − β
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D T O KN OW A B O U T VARIANCE SWAPS
Long 2-year variance struck at 19.50 on a Vega Notional of €200,000 (i.e. a Variance
Notional of 5,128)
Short 1-year variance struck at 18.50 on a Variance Notional of 5,128 / 2 = 2,564 (i.e. a
Vega Notional of €94,868)
Implied forward volatility on this trade is approximately4
:
{ { 19.50 2 – 18.50 1 20.5
2 year vol tenor 1 year vol tenor
× × =
− −
123 123 .
Therefore, if the 1-year implied volatility is above 20.5 in one year’s time, say at 21, the
hedge will be approximately up ½ a vega, or €50,000, while the exposure will be down by the
same amount.
However, keep in mind that the fair value of a variance swap is also sensitive to skew.
Forward volatility trades are interesting because the forward volatility term structure tends to
flatten for longer forward-start dates, as illustrated in Exhibit 1.2.1 below. In this example,
we can see that the 1-year forward volatilities exhibit a downard sloping term structure.
Thus, an investor who believes that the term structure will revert to an upward sloping shape
might want to sell the 12×1 and buy the 12×12 implied volatilities, or equivalently sell 13m
and buy 24m, with appropriate notionals:
Buy 12×12 = Buy 24m and Sell 12m
Sell 12×1 = Sell 13m and Buy 12m
Buy spread = Buy 24m and Sell 13m
Exhibit 1.2.1 — Spot and forward volatility curves derived from fair variance swap strikes
13
14
15
16
17
18
19
20
21
22
23
24
1m 2m 3m 4m 5m 6m 7m 8m 9m 10m 11m 12m
Spot 3m fwd 6m fwd 12m fwd
Source: JPMorgan.
Spreads on indices
Variance swaps can also be used to capture the volatility spread between two correlated
indices, for instance by being long 3-month DAX variance and short 3-month EuroStoxx 50
variance. Exhibit 1.2.2 below shows that in the period 2000-2004 the historical spread was
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4
An accurate calculation would be: (2 )
(1 ) 1 2 2 1 2 2
PV y
PV y y vol × − y vol × × , where PV(t) is the present value of €1 paid at time
almost always in favor of the DAX and sometimes as high as 12 vegas, while the implied
spread5
ranged between -4 and +4 vegas.
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D T O KN OW A B O U T VARIANCE SWAPS
Exhibit 1.2.2 — Volatility spread between DAX and EuroStoxx 50: historical (a) and implied (b)
a)
b)
Source: JPMorgan—DataQuery.
Correlation trading: Dispersion trades
A popular trade in the variance swap universe is to sell correlation by taking a short position
on index variance and a long position on the variance of the components. Exhibit 1.2.3 below
shows the evolution of one-year implied and realized correlation.
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5
Measured as the difference between the 90% strike implied volatilities. Actual numbers may differ depending on
skew, transaction costs and other market conditions.
Exhibit 1.2.3 — Implied and realized correlation of EuroStoxx 50
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D T O KN OW A B O U T VARIANCE SWAPS
Source: JPMorgan—DataQuery.
More formally the payoff of a variance dispersion trade is:
w Notional Notional Residual Strike Index Index
n
i 1
∑ i i i − − =
2 2 σ σ
where w’s are the weights of the index components, σ’s are realized volatilities, and notionals
are expressed in variance terms. Typically, only the most liquid stocks are selected among the
index components, and each variance notional is adjusted to match the same vega notional as
the index in order to make the trade vega-neutral at inception.
1.3. Mark-to-market and Sensitivities
Mark-to-market
Because variance is additive in time dimension the mark-to-market of a variance swap can be
decomposed at any point in time between realized and implied variance:
( )
( )
− − +
= × × ×
2 2
2
Implied Vol( , )
Notional ( ) Realized Vol(0, )
t T Strike
T
T t
t
T
t VarSwapt PVt T
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where Notional is in variance terms, PVt(T) is the present value at time t of $1 received at
maturity T, Realized Vol(0, t) is the realized volatility between inception and time t, Implied
Vol(t, T) is the fair strike of a variance swap of maturity T issued at time t.
For example, consider a one-year variance swap issued 3 months ago on a vega notional of
$200,000, struck at 20. The 9-month zero-rate is 2%, realized volatility over the past 3 months
was 15, and a 9-month variance swap would strike today at 19. The mark-to-market of the
one-year variance swap would be:
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D T O KN OW A B O U T VARIANCE SWAPS
$359,619
19 20
4
3 15
4
1
(1 2%)
1
2 20
200,000 2 2 2
0.75
= −
× × + × − +
×
×
VarSwapt =
Note that this is not too far from the 2 vega loss which one obtains by computing the weighted
average of realized and implied volatility: 0.25 x 15 + 0.75 x 19 = 18, minus 20 strike.
Vega sensitivity
The sensitivity of a variance swap to implied volatility decreases linearly with time as a direct
consequence of mark-to-market additivity:
T
T t Notional VarSwap Vega implied
implied
t − = × ×
∂
∂ = (2σ )
σ
Note that Vega is equal to 1 at inception if the strike is fair and the notional is vega-adjusted:
Strike
Vega Notional Notional
× = 2
Skew sensitivity
As mentioned earlier the fair value of a variance swap is sensitive to skew: the steeper the
skew the higher the fair value. Unfortunately there is no straightforward formula to measure
skew sensitivity but we can have a rough idea using the rule of thumb for linear skew in
Section 1.1:
( ) 2 2 2 Kvar ≈ σ ATMF 1+ 3T × skew
skew
T
T t Skew Sensitivity Notional ATMF × − ≈ × × × 2 6 σ
For example, consider a one-year variance swap on a vega notional of $200,000, struck at 15.
At-the-money-forward volatility is 14, and the 90-100 skew is 2.5 vegas. According to the rule
of thumb, the fair strike is approximately 14 x (1 + 3 x (2.5/10)2
) = 16.62. If the 90-100 skew
steepens to 3 vegas the change in mark-to-market would be:
$100,000
10
3 2.5
10
2.5 14
2 15
200,000 6 2 ≈
− × × ×
×
∆ ≈ ×
∆
14 2 444 4 3 444 14243 Sensitivity Skew
MTM
Dividend sensitivity
Dividend payments affect the price of a stock, resulting in a higher variance. When dividends
are paid at regular intervals, it can be shown that ex-dividend annualized variance should be
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adjusted by approximately adding the square of the annualized dividend yield divided by the
number of dividend payments per year6
. The fair strike is thus:
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D T O KN OW A B O U T VARIANCE SWAPS
Nb Divs Per Year
Div Yield K K ex div
2
2
var var
( ) ≈ ( ) + −
From this adjustment we can derive a rule of thumb for dividend sensitivity:
T
T t
K
Div Yield Nb Divs Per Year Notional
Div Yield
VarSwapt − ≈ × ×
∂
∂ =
var
µ
For example, consider a one-year variance swap on a vega notional of $200,000 struck at 20.
The fair strike ex-dividend is 20 and the annual dividend yield is 5%, paid semi-annually. The
adjusted strike is thus (202
+ 52
/ 2)0.5 = 20.31. Were the dividend yield to increase to 5.5% the
change in mark-to-market would be:
( ) 5.5 5 $12,310
20.31
5/ 2 ∆ ≈ 200,000× × − ≈
∆
14243 14 24 4 34 Div Yield
skew sensitivity
MTM
However, in the presence of skew, changes in dividend expectations will also impact the
forward price of the underlying which in turns affects the fair value of varianc. This
phenomenon will normally augment the overall dividend sensitivity of a variance swap.
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6
More specifically the adjustment is M
T
D
M
Τ d T
M
j
j = ×
∑ × =
2
2
1
1 where d1, d2, …, dM are gross dividend yields and D is
the annualized ‘average’ dividend yield. See Section 3.3 for more details.
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2. Valuation and Hedging in Practice
2.1. Vanilla Options: Delta-Hedging and P&L Path-Dependency
Delta-Hedging
Option markets are essentially driven by expectations of future volatility. This results from
the way an option payoff can be dynamically replicated by only trading the underlying stock
and cash, as described in 1973 by Black—Scholes and Merton.
More specifically, the sensitivity of an option price to changes in the stock price, or delta, can
be entirely offset by continuously holding a reverse position in the underlying in quantity equal
to the delta. For example, a long call position on the S&P 500 index with an initial delta of
$5,000 per index point (worth $6,000,000 for an index level of 1,200) is delta-neutralized by
selling 5,000 units of the S&P 500 (in practice 20 futures contracts: 6,000,000/(250 x 1,200))
Were the delta to increase to $5,250 per index point, the hedge should be adjusted by selling
an additional 250 units (1 contract), and so forth. The iteration of this strategy until maturity
is known as delta-hedging.
Once the delta is hedged, an option trader is mostly left with three sensitivities:
Gamma: sensitivity of the option delta to changes in the underlying stock price ;
Theta or time decay: sensitivity of the option price to the passage of time ;
Vega: sensitivity of the option price to changes in the market’s expectation of future
volatility (i.e. implied volatility.)7
The daily P&L on a delta-neutral option position can be decomposed along these three factors:
Daily P&L = Gamma P&L + Theta P&L + Vega P&L + Other (Eq. 1)
Here ‘Other’ includes the P&L from financing the reverse delta position on the underlying, as
well as the P&L due to changes in interest rates, dividend expectations, and high-order
sensitivities (e.g. sensitivity of Vega to changes in stock price, etc.)
Equation 1 can be rewritten:
Daily P&L = ) ( ) ( ) … 2
1 2 Γ× (∆S + Θ× ∆t + V × ∆σ +
where ∆S is the change in the underlying stock price, ∆t is the fraction of time elapsed
(typically 1/365), and ∆σ is the change in implied volatility.
We now consider a world where implied volatility is constant, the riskless interest rate is zero,
and other P&L factors are negligible. In this world resembling Black-Scholes, we have the
reduced P&L equation:
Daily P&L = ) ( ) 2
1 2 Γ× (∆S + Θ× ∆t (Eq. 2)
We proceed to interpret Equation 2 in terms of volatility, and we will see that in this world
the daily P&L of a delta-hedged option position is essentially driven by realized and implied
volatility.
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7
Note that in Black-Scholes volatility is assumed to remain constant through time. The concept of Vega is thus
inconsistent with the theory, yet critical in practice.
We start with the well-known relationship between theta and gamma:
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D T O KN OW A B O U T VARIANCE SWAPS
2 2
2
1 Θ ≈ − ΓS σ (Eq. 3)
where S is the current spot price of the underlying stock and σ the current implied volatility of
the option.
In our world with zero interest rate, this relationship is actually exact, not approximate.
Appendix B presents two derivations of Equation 3, one based on intuition and one which is
more rigorous.
Equation 3 is the core of Black-Scholes: it dictates how option prices diffuse in time in relation
to convexity. Plugging Equation 3 into Equation 2 and factoring S2
, we obtain a
characterization of the daily P&L in terms of squared return and squared implied volatility:
Daily P&L =
∆
∆
t
S
S S 2
2
2
1 Γ × −σ 2 (Eq. 4)
The first term in the bracket, S
∆S , is the percent change in the stock price — in other words,
the one-day stock return. Squared, it can be interpreted as the realized one-day variance.
The second term in the bracket, , is the squared daily implied volatility, which one
could name the daily implied variance.
∆t 2 σ
Thus, Equation 4 tells us that the daily P&L of a delta-hedged option position is driven by the
spread between realized and implied variance, and breaks even when the stock price
movement exactly matches the market’s expectation of volatility.
In the following paragraph we extend this analysis to the entire lifetime of the option.
P&L path-dependency
One can already see the connection between Equation 4 and variance swaps: if we sum all
daily P&L’s until the option’s maturity, we obtain an expression for the final P&L:
Final P&L = ∑ [ ] =
− ∆
n
t
t tr t
0
2 2
2
1
γ σ (Eq. 5)
where the subscript t denotes time dependence, rt the stock daily return at time t, and gt the
option’s gamma multiplied by the square of the stock price at time t, also known as dollar
gamma.
Equation 5 is very close to the payoff of a variance swap: it is a weighted sum of squared
realized returns minus a constant that has the role of the strike. The main difference is that
in a variance swap weights are constant, whereas here the weights depend on the option
gamma through time, a phenomenon which is known to option traders as the path-dependency
of an option’s trading P&L, illustrated in Exhibit 2.1.1.
It is interesting to note that even when the stock returns are assumed to follow a random walk
with a volatility equal to σ, Equation 5 does not become nil. This is because each squared
return remains distributed around ∆t rather than equal to . However this particular 2 σ ∆t 2 σ
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path-dependency effect is mostly due to discrete hedging rather than a discrepancy between
implied and realized volatility and will vanish in the case of continuous hedging8
.
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D T O KN OW A B O U T VARIANCE SWAPS
Exhibit 2.1.1 — Path-dependency of an option’s trading P&L
In this example an option trader sold 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-heged his position daily. The realized volatility was 27.50%, yet his final
trading P&L is down $150k. Furthermore, we can see (Figure a) that the P&L was up $250k until a month before
expiry: how did the profits change into losses? One indication is that the stock price oscillated around the strike in
the final months (Figure a), triggering the dollar gamma to soar (Figure b.) This would be good news if the volatility
of the underlying remained below 30% but unfortunately this period coincided with a change in the volatility regime
from 20% to 40% (Figure b.) Because the daily P&L of an option position is weighted by the gamma and the volatility
spread between implied and realized was negative, the final P&L drowned, even though the realized volatility over
the year was below 30%!
a)
0%
20%
40%
60%
80%
100%
120%
140%
0
15
30
45
60
75
90
105
120
135
150
165
180
195
210
225
240
-250,000
–
250,000
500,000
750,000
Stock Trading P&L ($) Price (Initial = 100)
trading days
Strike = 110
Stock Price
‘Hammered at the strike’ !
Trading P&L
b)
43%
31%
21%
0%
20%
40%
60%
80%
100%
120%
140%
0
15
30
45
60
75
90
105
120
135
150
165
180
195
210
225
240
0%
10%
20%
30%
40%
50%
60%
70%
Stock Price (Initial = 100) Volatility
trading days
Strike = 110
Stock Price
50-day Realized
Volatility
Dollar Gamma
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8
See Wilmott (1998) for a theoretical approach of discrete hedging and Allen—Harris (2001) for a statistical analysis of
this phenomenon. Wilmott notes that the daily Gamma P&L has a chi-square distribution, while Allen—Harris
include a bell-shaped chart of the distribution of 1000 final P&Ls of a discretely delta-hedged option position.
Neglecting the gamma dependence, the central-limit theorem indeed shows that the sum of n independent chisquare variables converges to a normal distribution.
2.2. Static Replication of Variance Swaps
AT YO U NEE
D T O KN OW A B O U T VARIANCE SWAPS
In the previous paragraph we saw that a vanilla option trader following a delta-hedging
strategy is essentially replicating the payoff of a weighted variance swap where the daily
squared returns are weighted by the option’s dollar gamma9
. We now proceed to derive a
static hedge for standard (‘non-gamma-weighted’) variance swaps. The core idea here is to
combine several options together in order to obtain a constant aggregate gamma.
Exhibit 2.2.1 shows the dollar gamma of options with various strikes in function of the
underlying level. We can see that the contribution of low-strike options to the aggregate
gamma is small compared to high-strike options. Therefore, a natural idea is to increase the
weights of low-strike options and decrease the weights of high-strike options.
Exhibit 2.2.1 — Dollar gamma of options with strikes 25 to 200 spaced 25 apart
K = 25
K = 50
K = 75
K = 100
K = 125
K = 150
K = 175
K = 20