7%20equity%20variance%20swaps.pdf

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

[email protected]

+44(0) 20 7742 2537

April 2013

V A

R I A

N C

E

S W

A P

S

1

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

V A

R I A

N C

E

S W

A P

S

2

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

V A

R I A

N C

E

S W

A P

S

3

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)

V A

R I A

N C

E

S W

A P

S

4

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)

V A

R I A

N C

E

S W

A P

S

5

Realised Volatility: Definition and Characteristics (IV)

EuroStoxx 50 (SX5E) Index Volatility: Realised vs. (BS) Implied


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.

V A

R I A

N C

E

S W

A P

S

6

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

V A

R I A

N C

E

S W

A P

S

7

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?

V A

R I A

N C

E

S W

A P

S

8

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

V A

R I A

N C

E

S W

A P

S

9

Trading volatility via straddles and delta-hedged options

V A

R I A

N C

E

S W

A P

S

10

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


-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.

V A

R I A

N C

E

S W

A P

S

11

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)

V A

R I A

N C

E

S W

A P

S

12

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

V A

R I A

N C

E

S W

A P

S

13

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

V A

R I A

N C

E

S W

A P

S

14

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

V A

R I A

N C

E

S W

A P

S

15

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

V A

R I A

N C

E

S W

A P

S

16

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

)(

V A

R I A

N C

E

S W

A P

S

17

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

V A

R I A

N C

E

S W

A P

S

18

Dollar Gamma is not constant

Dollar Gamma & Stock Price Dollar Gamma & Time to Expiry

Example used


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

V A

R I A

N C

E

S W

A P

S

19

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

V A

R I A

N C

E

S W

A P

S

20

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)


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.

V A

R I A

N C

E

S W

A P

S

21

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)

V A

R I A

N C

E

S W

A P

S

22

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?


V A

R I A

N C

E

S W

A P

S

23

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%!


V A

R I A

N C

E

S W

A P

S

24

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

V A

R I A

N C

E

S W

A P

S

25

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)


2500-strike option 4000-strike option

V A

R I A

N C

E

S W

A P

S

26

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

V A

R I A

N C

E

S W

A P

S

27

Dollar Gamma: How to make it constant

V A

R I A

N C

E

S W

A P

S

28

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.

V A

R I A

N C

E

S W

A P

S

29


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?



V A

R I A

N C

E

S W

A P

S

30


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?


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


V A

R I A

N C

E

S W

A P

S

31


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.


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


V A

R I A

N C

E

S W

A P

S

32


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).

V A

R I A

N C

E

S W

A P

S

33

Dollar Gamma of each option (1/K2) Total dollar gamma

Dollar Gamma of each option (1) Dollar Gamma of each option (1/K)


V A

R I A

N C

E

S W

A P

S

34


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.

V A

R I A

N C

E

S W

A P

S

35

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.


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


V A

R I A

N C

E

S W

A P

S

36

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).


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

V A

R I A

N C

E

S W

A P

S

37


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!!

V A

R I A

N C

E

S W

A P

S

38


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.

V A

R I A

N C

E

S W

A P

S

39


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

V A

R I A

N C

E

S W

A P

S

40


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.


V A

R I A

N C

E

S W

A P

S

41

Variance Swaps

V A

R I A

N C

E

S W

A P

S

42

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

V A

R I A

N C

E

S W

A P

S

43

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.

V A

R I A

N C

E

S W

A P

S

44

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.

V A

R I A

N C

E

S W

A P

S

45

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

V A

R I A

N C

E

S W

A P

S

46

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

V A

R I A

N C

E

S W

A P

S

47

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


V A

R I A

N C

E

S W

A P

S

48

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

V A

R I A

N C

E

S W

A P

S

49

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.

V A

R I A

N C

E

S W

A P

S

50

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

V A

R I A

N C

E

S W

A P

S

51

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

&

V A

R I A

N C

E

S W

A P

S

52

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


-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

V A

R I A

N C

E

S W

A P

S

53

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

V A

R I A

N C

E

S W

A P

S

54

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

V A

R I A

N C

E

S W

A P

S

55

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 .


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 ])[(

V A

R I A

N C

E

S W

A P

S

56

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)

V A

R I A

N C

E

S W

A P

S

57

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

V A

R I A

N C

E

S W

A P

S

58

Mark-to-Market - Exercise II (Solution a)



Trade 2: Sell variance notional , at time t, strike , expiry .


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)

V A

R I A

N C

E

S W

A P

S

59

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

V A

R I A

N C

E

S W

A P

S

60

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 2: Sell variance notional , at time t, strike , expiry .


TK ,0VarN T

][ 2,02

,0 TTVar KN

TtK , T

][ 2,2

, TtTtVar KT

tTN

T

tTNVar

V A

R I A

N C

E

S W

A P

S

61

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

V A

R I A

N C

E

S W

A P

S

62

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 ,

V A

R I A

N C

E

S W

A P

S

63

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)

V A

R I A

N C

E

S W

A P

S

64

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.


V A

R I A

N C

E

S W

A P

S

65

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


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


-20

0

20

40

60

80

100

0% 5% 10% 15% 20% 25% 30% 35% 40%


Capped Var. Swap 2.5x P&L


20 strike, 1 vega notional on both swaps

V A

R I A

N C

E

S W

A P

S

66

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.

V A

R I A

N C

E

S W

A P

S

67

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

V A

R I A

N C

E

S W

A P

S

68

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)

V A

R I A

N C

E

S W

A P

S

69

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.

V A

R I A

N C

E

S W

A P

S

70

Two offsetting capped var swaps Net payoff position

Two offsetting var swaps Net payoff position


-20

0

20

40

60

80

100

0% 5% 10% 15% 20% 25% 30% 35% 40%




-75

-55

-35

-15

5

25

45

65

0% 5% 10% 15% 20% 25% 30% 35% 40%

Net payoff at expiry


-75

-55

-35

-15

5

25

45

65

0% 10% 20% 30% 40% 50% 60% 70% 80%



-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)



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.

V A

R I A

N C

E

S W

A P

S

71

Two offsetting capped var swaps Net payoff position

Two offsetting var swaps Net payoff position


-30

-20

-10

0

10

20

30

40

0% 5% 10% 15% 20% 25% 30% 35% 40%



Final realised volatility-75

-55

-35

-15

5

25

45

65

0% 5% 10% 15% 20% 25% 30% 35% 40%



-40

-20

0

20

40

60

80

100

120

140

0% 20% 40% 60% 80% 100%




-75

-55

-35

-15

5

25

45

65

0% 20% 40% 60% 80% 100%



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.

V A

R I A

N C

E

S W

A P

S

72

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

V A

R I A

N C

E

S W

A P

S

73

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)

V A

R I A

N C

E

S W

A P

S

74

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).


V A

R I A

N C

E

S W

A P

S

75

Flat and linear skews SX5E 1y Imp. Vol Skew


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

V A

R I A

N C

E

S W

A P

S

76

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 ,,

V A

R I A

N C

E

S W

A P

S

77

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)

V A

R I A

N C

E

S W

A P

S

78

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

V A

R I A

N C

E

S W

A P

S

79

Variance Swap Pricing: Approximations (III)


No Exam

V A

R I A

N C

E

S W

A P

S

80

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


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


V A

R I A

N C

E

S W

A P

S

81

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)

V A

R I A

N C

E

S W

A P

S

82

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



1 vega notional on all swaps

V A

R I A

N C

E

S W

A P

S

83

SX5E 1y Var Swap vs. Implied ATM Vol SX5E 1y Var Swap vs. Implied ATM Vol

SX5E Variance Swaps SX5E Variance Swaps Term Structure


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%


1y minus 6m Var Swaps

0%

10%

20%

30%

40%

50%

60%


Implied 1y ATM Vol. 1y Variance Swap

-5%

0%

5%

10%

15%

20%

25%


1y Var Swap minus Implied ATM Vol.

V A

R I A

N C

E

S W

A P

S

84 Source: J.P. Morgan.

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%


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%


1y 90/100 Skew

V A

R I A

N C

E

S W

A P

S

85 Source: J.P. Morgan.

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.

V A

R I A

N C

E

S W

A P

S

86

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.

V A

R I A

N C

E

S W

A P

S

87

Variance Swap Indices (II)

VIX vs. Implied Vol VIX minus Implied Vol


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

V A

R I A

N C

E

S W

A P

S

88

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.

V A

R I A

N C

E

S W

A P

S

89

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.

V A

R I A

N C

E

S W

A P

S

90

Variance Swaps: Hedging and 2008 Crisis

V A

R I A

N C

E

S W

A P

S

91

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)

V A

R I A

N C

E

S W

A P

S

92

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)

V A

R I A

N C

E

S W

A P

S

93

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


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


One can imagine what happened in 2008/2009 market crash For a detailed

explanation, see J.P. Morgan, Volatility Swaps, 2010, Section 4.

V A

R I A

N C

E

S W

A P

S

94

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)

V A

R I A

N C

E

S W

A P

S

95

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)

V A

R I A

N C

E

S W

A P

S

96

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)

V A

R I A

N C

E

S W

A P

S

97

Vol Swaps

V A

R I A

N C

E

S W

A P

S

98

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.

V A

R I A

N C

E

S W

A P

S

99

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

V A

R I A

N C

E

S W

A P

S

100

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)

V A

R I A

N C

E

S W

A P

S

101

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)

V A

R I A

N C

E

S W

A P

S

102

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

V A

R I A

N C

E

S W

A P

S

103

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

7%20equity%20variance%20swaps.pdf

Documents