Derivatives Week www.derivativesweek.com October 9, 2006

Copying prohibited without the permission of the publisher.10

Conditional variance swaps are similar tostandard variance swaps but variance exposure is

limited to a predefined range of underlying levels. As an illustration, the graph below shows how conditional

variance swaps only include variance observations on those dayswhere yesterday’s closing price satisfied the condition—in thiscase, above the 3800-trigger for the up-variance swapconsidered here.

The payoff at expiry of a conditional variance swap is:Payoff = Multiplier x ( FCRV – Strike ) x POThe Final Conditional Realized Variance (FCRV) iscalculated as:

where “Pt” is the closing price level on the currentobservation day, and Ct equals 1 if the condition LowerBarrier≤ Pt-1 ≤ UpperBarrier is met, and 0 otherwise. The percentageof occurrences (“PO”) is the ratio of number of days where theconditional criteria is realized, NR, over the expected numberof observation days over the life of the trade.

The multiplier calculation converts the desired vega exposureto variance units, but it is calculated using the normal variance

swap strike (not the conditional strike). Due to the scaling backof exposure by percent of occurrences the eventual exposuremay be significantly less than the normal variance swap anddepends on the path of the underlying.

Motivation For Conditional Variance SwapsTraditional variance swaps allow investors to benefit from thecompetitive advantage of broker dealers in accessing optionmarkets and the active management of a specific delta-hedgedoptions portfolio. Conditional variance swaps take this benefitone step further and provide a formulaic return for skewtrading that helps investors isolate a specific volatility view—and importantly skew—in an easily tradable form.

As a typical equity skew implies lower volatility for strikes athigher underlying levels, truncating the variance exposure fromdownside underlying levels should therefore cheapen the up-variance swap versus the normal variance swap level. As anillustration, the graph below shows this effect with an up-variance swap level that is below the standard swap level.

Conditional Var Swap Pricing: Initial, Final And Mid-LifeConditional variance swap models take from option prices boththe implied volatility skew of an underlying as well as theexpected number of trading days the underlying should meetthe conditional criteria.

The conditional variance swap expiry settlement calculationis essentially the same as for a standard variance swap, exceptthat it only includes observations that satisfy the conditionalcriteria. Thus, the realized variance calculation reflects a scalingdown of the standard variance swap exposure by the percent ofobservations that satisfy the conditional criteria.

The mid-life valuation calculation adds another level of

L E A R N I N G C U R V E ®

Introduction To Conditional Variance Swaps

Derivatives Week is now accepting submissions from industry professionals for the Learning Curve® section. For details and guidelines on writing a Learning Curve®,

please call Elinor Comlay in New York at 212-224-3208 or Matthew Tremblay in Hong Kong at 852-2912-8097.

3400

3500

3600

3700

3800

3900

4000

Jan-

06

Feb-

06

Mar

-06

Apr-

06

Date

Inde

x Le

vel

Trigger level

variance swap taken in green zoneVariance observations for 3800-strike up-

3700

3750

3800

3850

3900

1-M

ar

3-M

ar

5-M

ar

7-M

ar

9-M

ar

11-M

ar

13-M

ar

15-M

ar

17-M

ar

19-M

ar

21-M

ar

23-M

ar

25-M

ar

27-M

ar

29-M

ar

31-M

ar

2-Ap

r

4-Ap

r

6-Ap

r

8-Ap

r

10-A

pr

12-A

pr

14-A

pr

16-A

pr

18-A

pr

20-A

pr

Date

Inde

x Le

vel

Variance counted if yesterday's close was above trigger observations count toward variance. Other observations are ignored.

Source: Citigroup

==

n

1t

2

1-t

tlnNR

25210,000 tC

P

PFCRV

1002=

NormVar

VegaMultiplier

Comparison of implied volatility pricing vs underlying level

10%

15%

20%

25%

2400 2800 3200 3600 4000 4400 4800 5200

Underlying level

Impl

ied

vola

tility

Implied volatilityBy strike

Variance swap level(1/K^2 weighting)

3800-UP conditional variance swap level(1/K^2 weighting truncated)

Source: Citigroup

dw1009in 10/5/06 2:25 PM Page 10

Copying prohibited without the permission of the publisher. 11

October 9, 2006 www.derivativesweek.com Derivatives Week

complexity to normal variance swap valuations. This is becauseit is difficult to model the future distribution of realizedvariance when this is conditional upon the underlying beingwithin a certain range.

An approximate mark-to-market valuation (“MTM_Value”),similar to the expiry settlement calculation, can be used thatinvolves so-called blended variance:

MTM_Value ≈ Multiplier x (BlendedVar – Strike) x POBlended Var and PO are modified formulae that combine

the realized variance at mid-life with the model derivedremaining implied variance, as per the formulae:

The modification in the formulae includes a newvariable/model output required for the calculation,“NImpliedInRange”, that is the estimated number of observationsuntil maturity expected to count toward the final realized variance.This figure will depend mostly on the so-called moneyness of thetrigger level, but also to some extent on the volatility skew. In abinomial tree context, this figure can be derived from the numberof nodes of the binomial tree that will satisfy the trigger condition.The graph below illustrates this concept, as well as the impliedquote’s contribution to “blended variance”. For simplicity, we haveassumed that: (i) valuation date is half way through the trade, (ii)50% of realized observations satisfy the trigger condition, (iii)realized volatility and implied volatility quotes are equal, (iv) spotlevel is equal to forward level and (v) no volatility skew.

By looking at the cone of potential future stock paths derivedfrom a binomial tree, it can intuitively be seen that about 50% ofremaining observations should satisfy the trigger conditions, andblended variance will be a mix of 50% implied and 50% realized.As for the variance multiplier, it will be scaled back by PO,estimated at 50% around the valuation date.

Trading StrategiesVarious trading strategies using conditional variance swaps canbe implemented.

Isolating skew: skew trading is very dynamic and requiresactive management. Conditional variance swaps transfer therisk of this position management to dealers with competitiveadvantage. Skew can, however, be a poor predictor of volatilityfor structural and fundamental reasons.

Carry trades: Conditional variance swaps can be used byinvestors that have views on volatility and market trends. Forinstance, up-variance swaps can cut out some of the lowerstrike exposure that produces a higher normal variance strike.Therefore, those with a bullish view on the market canconsider a carry trade that buys up-variance (13%) and sellsnormal variance (15%). While the index remains above theup-variance trigger, this accrues the strike spread of the twoswaps as P&L (2%). If the market moves below the trigger, theposition is short realized volatility however. A similar trade canbe implemented to capture more of the volatility skew inrange-bound markets. For example, given a normal skew, bybuying an up-variance swap with 95% trigger at strike 12%and selling a down-variance swap with 105% trigger at strike15%, a 3% volatility profit will be achieved if the indexremains between the triggers.

Portfolio protection: empirical evidence shows that volatilityusually increases as the market decreases. This is priced intovolatility skews somewhat but for structural (structured productsissuance) or fundamental reasons (market trend changes), theskew may not accurately reflect volatility risks. Long-biasedportfolio managers could utilize conditional variance swaps bybuying down-variance with a trigger below current marketlevels. If the market still rallies, no payment will ever need to bemade. If the market reverses, the manager will have longvolatility exposure in a declining market that could offset lossesin their equity portfolio if the skew under priced the risk.

Hedging Risks And ConstructionA conditional variance swap is hedged in much the same wayas a normal variance swap—through the daily delta hedging ofa portfolio of options weighted by the inverse of the optionstrike squared. Because variance exposure is only required atcertain underlying levels, however, the portfolio of options istruncated at the trigger level. Thus, whereas for a standardvariance swap, the 1/K2 weighting of the options in the hedgeproduces constant variance exposure, for a conditional varianceswap the variance exposure is not constant at the trigger level.At this point, the required gamma step-change is impossible tofully replicate with listed options. This imperfection is the riskof the dealer.

This week’s Learning Curve was written by Gerry Fowler and AlexisCollomb, equity derivatives strategists at Citigroup in London.

++

+=

angeImpliedInRRangeRealisedIn

angeImpliedInR2Implied

angeImpliedInRRangeRealisedIn

RangeRealisedIn2Realised

2

NN

N

NN

NBlended

Expected

angeImpliedInRRangeRealisedIn

N

NNPO

+=

Valuation equally implied and realized

2800

3200

3600

4000

4400

4800

24-M

ar

31-M

ar

7-Ap

r

14-A

pr

21-A

pr

28-A

pr

5-M

ay

12-M

ay

19-M

ay

Already realized (NInRange ~50%) Still implied (NInRange ~50%)

Half way through trade and PO = 50% = (50%+50%)/2

Source: Citigroup

dw1009in 10/5/06 2:25 PM Page 11