Variance Swaps and Non-Constant Vega

Transcription

1 Variance Swaps and Non-Constant Vega David E. Kuenzi Head of Risk anageent and Quantitative Research Glenwood Capital Investents, LLC 3 N. Wacker Drive, Suite 8 Chicago, IL 666 dkuenzi@glenwood.co Phone ( Fax ( Abstract Variance swaps are often touted as pure volatility plays, and in any senses they are. There is, however, one very iportant aspect in which they are not. While they provide relatively stable gaa to investors, they do not provide a stable vega, but rather a vega exposure that varies draatically over the lifetie of the contract. Cobining a variance swap with an interittently adjusted exposure to a pure vega product, such as a VIX futures contract, produces a position that has vega and gaa exposures that are generally stable relative to changes in both the value of the underlying and in the tie to expiry. Disaggregating this cobination into its pure vega and pure gaa coponents gives rise to a gaa contract that ay be of use to soe investors.

2 One of the touted advantages of variance and volatility products is that they provide investors with an efficient source of constant volatility exposure. A point ade very clearly by Deeterfi, et. al. (999a is that variance vega of a variance swap (sensitivity to aturity-scaled variance, siilar to standard vega reains stable despite oveents in the underlying. A standard straddle, on the other hand, experiences quick shifts in vega (or variance vega as soon as the underlying oves in either direction. The position then requires continuous rehedging if one wishes to aintain a stable vega. The sae is generally true for gaa. While the gaa of a variance swap explodes as the price of the underlying goes to zero, it reains rather stable for high-probability values of the underlying. By contrast, the gaa of a standard straddle is peaked around at-theoney forward levels of the underlying (as is vega. This stability of gaa and variance vega for oves in the underlying represents the core advantage of variance swaps and a driving factor behind their increased use. The VIX futures contract offers a siilar advantage in that it provides constant vega exposure for changes in the underlying. An issue arises, however, when an investor wishes to aintain constant gaa and constant vega across tie. The VIX futures contract has no gaa and stable variance vega across tie. While variance swaps provide constant gaa exposure across tie, their variance vega exposure is a linear function of tie to expiration exhibiting significant exposure to changes in iplied volatility on a ark-to-arket basis early in the life of the contract with little exposure to changes in iplied volatility just prior to aturity. As such, an investor in a variance swap experiences very significant shifts in

3 ark-to-arket risk over the life of a variance swap and very different volatility exposures depending on the swap s tie to expiration. The purpose of this article is twofold. First, we explore the relationships between standard straddles, variance swaps, and VIX futures contracts (or a close proxy and point out a siple strategy for the aintenance of generally stable vega and gaa. Second, we propose a new instruent that will aintain constant gaa and zero vega. A cobination of this new instruent with a VIX futures contract would provide fairly stable volatility exposure (gaa and vega for both changes in tie and the value of the underlying without the need on the part of investors to continually rehedge. Variance Swaps Non-Constant Vega A variance swap is a contract whose payoff is the realized variance of an asset inus a variance strike agreed upon at contract initiation. Its payoff at aturity is: Si S i Payoff = N Kvar T t i= Si = N R K T t = N V t T ( i var ( i= ( (, K var where is the total nuber of onitoring periods between swap inception at tie t to swap aturity at tie T. If we assue daily onitoring, would be the nuber of 3

4 trading days between swap inception and aturity. S i is the price of the underlying on day i, R i is return of the underlying on day i, V(t,T is the realized variance over the period, K var is the initially agreed upon variance strike expressed in volatility points squared, and N is the notional aount. Variance swaps accrue variance based on each day s realized return innovation. arkto-arket pricing at tie t is therefore a weighted average of the accrued variance and the expectation of variance looking ahead. As such, the value of the contract can change draatically based on the change in expected future variance. As Chriss and orokoff (999 show, the ark-to-arket value of a variance swap can be written as: (t < t < T rt ( t T = Ne ( λ( V ( t, t K + ( λ( K K ( var t var where V(t,t is the realized variance fro tie t to tie t, K t is the strike for a new variance swap running fro tie t to tie T, and λ is the proportion of tie elapsed between t and t: λ t t τ = = T t T t ( t (3 where τ = T-t. For calculation of the variance strike, K var, see Deeterfi, et. al. (999a and 999b, Carr and Wu (4, Carr and adan (998, and Britten-Jones and Neuberger (. 4

5 We use the sybol ϒ to denote variance vega. We can then write: ( T τ ϒ= = ( λ( t Ne = Ne Kt T t rt ( t rt ( t (4 So the exposure of the variance swap to changes in fair arket variance is positively related to the proportion of tie reaining to expiration of the swap. If we then assue zero interest rates and a notional of $, we can consider the sensitivity of variance vega to changes in tie to expiration: ϒ = τ T t (5 Equation (5 says that the change in variance vega for a change in tie to aturity is related to the inverse of the total period covered by the contract, so that the closer we ove to expiry, the lower the variance vega. If, for exaple, we have two year variance swap, its variance vega will fall by half after one year. Equation (5 also akes it clear that an investor rolling into a new variance swap on, say, a quarterly basis would experience varying levels of variance vega over tie. This can best be seen by taking a rolling correlation of the returns of a three-onth S&P 5 variance swap, rolled into a new variance swap every three onths, and a VIX futures contract, as shown in figure. 3 This is defined as in Deeterfi, et. al. 999a. In their article, it is a Black Scholes vega (except using variance based on the value of the portfolio of options that replicates the variance swap. It as also be derived as above as the derivative of the ark-to-arket value of the variance swap with respect to the level of fair variance for a new swap with a aturity equal to the reaining tie to expiry of the existing swap. 3 The CBOE changed its ethodology for calculation of the VIX index on Septeber, 3 such that the level of the VIX is now calculated so that (VIX is equal to the variance strike of a one-onth variance swap. See CBOE (3 and Harstone (4. 5

6 When a new contract is first initiated, the returns of the variance swap are highly correlated with the VIX futures due to the significant variance vega of the variance swap. As the variance swap oves toward aturity, however, its variance vega approaches zero, and thus its returns show very little correlation with the VIX. Given these observations, it is instructive to ore carefully copare the volatility exposures of a standard straddle, 4 a variance swap, and the VIX futures. One ight perhaps think of volatility exposure as consisting of two eleents: gaa (exposure to realized versus initially expected volatility and vega (exposure to changes in expected and arket-traded levels of volatility. In table A, we consider the gaa and variance vega exposures of these instruents and how they change in response both to changes in the value of the underlying and to changes in tie to expiry. It is first useful to note the straightforward values for the variance swap and VIX futures, as copared to the standard straddle. The VIX futures has just one very siple volatility exposure to variance vega which is invariant for changes in the underlying or tie to expiry. The shapes of the gaa and variance vega functions of the standard straddle are doinated by the noral probability density function and generally have quite a large (and not straightforward response to changes in the underlying and tie to expiry. Variance swaps contain both gaa and variance vega exposures. The gaa exposure reains constant with changes in tie to expiry but could get quite large were the underlying to ove down substantially. The variance vega of the variance swap reains constant 4 This is defined as a long at-the-oney put cobined with a long an at-the-oney call; this ight be considered a rather ipure way of obtaining volatility exposure. 6

7 despite changes in the underlying, but is quite sensitive to changes in the tie to expiry. It is this last characteristic of variance swaps that is of concern. To provide ore clarity, figure shows how the volatility exposures of a standard straddle and variance swap change based on changes in the value of the underlying and in tie to expiry. While the exposures of the standard straddle prove highly sensitive to changes in both underlying and tie to expiry, the variance swap s exposures are constant with two exceptions. First, gaa explodes when the underlying falls draatically (into low probability regions, as shown in panel of figure. Second, as shown in panel 4 of figure, variance vega exhibits large and predictable changes as the contract oves to aturity. It should now be quite clear that variance swaps provide fairly stable gaa exposure for changes in the underlying, but that variance vega is highly variable throughout the life of the contract. Given these observations, it is clear that an investor wishing to aintain generally stable gaa and nearly constant vega could do so by entering into a variance swap and siultaneously entering into a VIX futures contract. The notional of the futures position would need to be adjusted then on a daily basis (assuing daily onitoring in the variance calculation in order to atch the initial variance vega exposure of the variance swap. Conceptually, as the variance swap loses variance vega due to its oveent toward aturity, we add variance vega by purchasing a little ore of the VIX futures contract, and this occurs on a daily basis. Figure 3 shows that the rolling correlation of such a position with the VIX futures now reains in excess of.93. 7

8 In order to further explore the characteristics of variance swaps with ease of exposition, we ake two key assuptions. First, we assue a flat ter structure of volatility clearly an unrealistic assuption that we will later relax. Related to this assuption is a change of notation: we paraeterize K var to K var(t, where K var(t is the fair arket variance (for all expiries, as we are assuing a flat ter structure of volatility as known at tie t. So our previous K var becoes K var( t. Relatedly, we define an interediate day nuber, where < <. So after one day, λ = / and after days we have: λ t t T t = = (6 So we can write in place of, and we can write Kvar( K var( t Kvar( T Kvar( = to denote fair arket variance at the tie of the variance swap s expiration. Second, in order to avoid convexity issues, we assue that there is an over-the-counter fair variance forward contract with payoff K K ( var( T var( t at aturity rather than continuing to refer to VIX futures. The fair variance forward is siilar to a VIX futures contract except that it is a forward (no daily settle and the payoff involves arket quoted variances rather than volatilities. On any given day, the P&L of this fair variance forward will be ( Kvar( Kvar(. 8

9 Using equations ( and (6 and the above assuptions, and again assuing zero interest rates and unit notional, we can write the P&L of the cobined variance swap and fair variance forward with accreting notional: T = λ( V ( t, t K + ( λ( K K + i( K K i (7 var( t var( t var( t var( i var( i= The last ter on the right hand side of equation (7 is the accrued P&L of the position in the fair variance forward. Each trading day we need to increase the size of this position by an aount equal to (/ to assure that we always have λ exposure to the pure variance vega product. In this way, the su of the second and third ters on the right assure that we always have unit exposure to variance vega. On day, for instance, the second ter shows that we have λ = ( / dollars of exposure to changes in arket variance through the variance swap, while the third ter shows that we have / dollars of exposure to changes in arket variance through our fair variance forward position. Using such a trading strategy, an investor can assure generally stable gaa and vega exposures across both tie and asset price oveents. The Constant Gaa, Zero Vega Product As noted above, the arket has access to a constant vega product in the for of the VIX futures contract. The arket also has access to fairly stable gaa exposure albeit ixed with varying aounts of vega in the for of a variance swap. The arket does not currently have a widely traded pure gaa product. Such a product can be derived 9

10 fro equation (7. As equation (7 represents a cobination of products such that the portfolio achieves both unit gaa and unit variance vega, it should be possible to disaggregate this cobination such that two pure products result: T = λ( V ( t, t K + ( λ( K K + i( K K var( t var( t var( t var( i var( i i= = λvt (, t + ( λ K K + ik ( K var( t var( t var( i var( i i= = λvt (, t + ( λ K K + λk λk + ik ( K var( t var( t var( t var( var( i var( i i= = λvt (, t λkvar( + ik ( var( i Kvar( i Kvar( t Kvar( t + i= = Gaa Derivative + Vega Derivative (8 In equation (8 we isolate the gaa derivative and the vega derivative. The vega derivative is siply unit exposure to the fair variance forward contract, as we would have expected. Now we anipulate the equation for the gaa derivative:

11 Gaa Derivative = λv ( t, t λk + i( K K var( var( i var( i i= = λ Vt (, t Kvar( + ik ( var( i Kvar( i i= = λ Ri Kvar( Kvar( Kvar( i t t + i= i= = λ Ri T t i = i= var( i T t = λ T t T t = T t Ri Kvar( i i= Ri Kvar( i i= K (9 Equation (9 provides a very intuitive result; the pure gaa product is the su of the squared daily asset returns inus the fair value of variance as of the previous night s close. It is the difference between realized variance and expected variance where the expected variance is observed afresh for each onitoring period. The intrusion of vega into the variance swap contract results fro taking the fair variance over a period different fro the length of the onitoring period. The value of the gaa derivative has no dependence on expected future volatility because the fair arket variance just prior to each onitoring period is the level that akes the expected payoff of that leg of the contract equal to zero. If a client wants to go long gaa through a gaa contract, a dealer can hedge this position by paying fixed in a variance swap and hedging the vega risk. Using equation (8, we can write the ark-to-arket value of the gaa contract as:

12 Gaa Derivative = Variance Swap + i( K K ( K K i= = Variance Swap ( i( Kvar( i Kvar( i i= var( i var( i var( t var( t ( We want a variance swap with all of the variance vega risk hedged out. On any given day the variance vega exposure of the swap is (-λ = (-i/, so shorting this aount in the fair variance forward contract will give us a payoff identical to that of the gaa contract. Before we explore the iplications of fully relaxing our prior assuptions, we ight question the extent to which the gaa contract and a vega instruent are correlated, thus potentially obviating the need for the gaa contract. Over the period fro February, to January 3, 5, daily correlation of a three-onth gaa contract P&L and changes in the VIX were.9, indicating that in a very broad sense, the two contracts have little correlation. 5 Figure 4, which shows the rolling correlation of the gaa swap and the VIX, indicates that the correlation between a vega product and a gaa swap can ove quite rapidly between extrees, with a high -day correlation of.86 and low of -.6. The periods of extree high correlations are generally those in which both actual and iplied volatility are rising together e.g., days on which daily changes in iplied volatility tend to follow the daily changes in realized volatility. The periods of low correlation tend to be in range bound arkets in ters of both 5 As we do not have data for one-day vol, we use the VIX inus the first-onth / second-onth ter spread for SPX at-the-oney options (squared as available fro Blooberg. This assues that the slope between the first two options contracts is the sae as the slope between one-day and one-onth.

13 underlying level and iplied. Overall, it is clear that the gaa and vega products provide very different types of exposures. When we relax the assuptions of a flat ter structure of volatility and the existence of a fair variance forward contract, we are faced with two issues. The first but less critical issue is that equations (8, (9, and ( will no longer be atheatically precise. 6 Regardless, the intuitions continue to hold and these equations could still be used (in conjunction with VIX futures in order to anage exposures. The second, ore critical issue is that there is not an easily agreed upon one-day variance swap rate. Such a readily available and quoted rate would give prospective clients interested in such a product cofort that the value of the gaa derivative would accrue in a copletely fair and objective anner. This could be a substantial obstacle to the creation of such a product. One possible solution would be to quote a gaa derivative based on the level of a related and widely quoted volatility index, such as the VIX, plus or inus soe spread in order to take account of the expected ter structure of volatility between one day and one onth volatility over the life of the contract. So a gaa contract on the S&P 5 ight be quoted at VIX s, where s is a spread. If a dealer believes that the ter structure between one day and one onth will average % over the life of the gaa derivative, then the contract would be quoted at VIX inus %. One drawback of this approach is 6 Without a flat ter structure of volatility, we ust rely on equations ( and ( in order to define the fair variance levels K t and K T. These equations would iply that Kt = [ E R i ], where E is ( T t i= + the conditional expectation with respect to the inforation available on day. The definition of K T is then probleatic, as there are no ters under the suation. 3

14 that there are few generally accepted volatility indices. Another drawback (fro the perspective of aintaining purity of exposure is the iplicit ter structure bet that both parties to the contract would be taking. The value of the contract would now be sensitive to the arket ter spread in relation to the initially agreed upon ter spread. If the initial ter spread agreed between the counterparties is s and the arket ter spread soe tie into the life of the contract is s t, then the value of the contract, again assuing unit notional and zero interest rates, will be: T t + ] ( T = Ri ( VIX i s + E[ G, s t T t i= where the final ter is the expectation with respect to the inforation available at tie of the gaa accrual (G +, fro day + to day given the currently quoted arket spread is s t. This final ter can be defined as: i= + [ +, t] = E G s E T t Ri ( VIXi s T t T t Ri ( VIXi st T t i= + = E ( VIX i st ( VIX i s i= + i= + = ( s t s + E ( s s t VIX i i= + ( = ( st s + ( s st E VIX, ( 4

15 where E VIX, is the conditional expectation of the average level of the VIX index between day + and day. So that, cobining equations ( and (, we have the ark-to-arket value: T t T = R VIX s + s s T t i ( i ( t + i= ( ( s s t E VIX, (3 The long gaa contract akes (loses oney when ter spreads coe in (widen, as the investor has locked in a lower (higher daily volatility hurtle than that currently available in the arket. This approach provides gaa exposure ixed with significant volatility ter structure exposure. We have reoved the ipurity of vega fro the variance swap and replaced it with sensitivity to the ter spread of volatility. The sensitivity of this contract to a change in the ter spread will be: ( T ( = s t E [ VIX, s t ] (4 This is essentially the derivative of the value of the contract with respect to the average one-day iplied volatility, weighted by the percentage of tie reaining until contract expiration. 5

16 It is also interesting to note that the correlation between the changes in the VIX and changes in the ter spread (we use available Blooberg delta point data June 3, to January 3, 5 to deterine the ter spread is Given that this particular for of a gaa contract benefits fro ter spread tightening, it therefore continues to have positive volatility exposure although indirectly and significantly uted. Finally, while one goal in this paper has been to propose a pure gaa derivative, it should be noted that the ter structure exposure of this contract could prove beneficial to those who wish to hedge this type of exposure. One solution in such a situation would be to benchark against a longer-dated VIX futures contract inus a spread. In this case, the spread would be reflective of the ter structure of volatility for perhaps six onths rather than one. Conclusion While variance swaps allow investors to trade an at-the-oney delta hedged position without delta hedging and without adjusting for the oneyness of the options, they do not as often purported represent pure volatility exposure in that variance vega varies draatically over the life of a variance swap. An investor wishing to aintain generally stable gaa and vega can do so by entering a variance swap and an interittently adjusted position in a fair variance forward contract (or roughly equivalently, a VIX futures contract. Given that a fair variance forward or VIX futures contract can be considered pure vega derivatives, these observations beg the question as to whether there is a pure gaa derivative. Cobining a variance swap and an 6

17 interittently adjusted fair variance forward, and then disaggregating into the gaa and vega portions, shows that there is. The two resulting derivatives are a static position in a fair variance forward and a variance swap in which the variance strike is reset to fair variance just prior to each variance onitoring period. If such a gaa contract is sold by a dealer, it can in turn be hedged through a variance swap and an interittently changing position in a fair variance forward or VIX futures position. The advent of a gaa contract would enable investors to be ore specific in ters of the volatility exposure they are taking on and would thus contribute to overall arket copletion. The ost significant obstacle for such a contract is the lack of a readily quoted and transparent level for one-day volatility. A potential solution would be to quote a onthly volatility index plus or inus a spread. This solution, however, introduces ter spread risk into the contract. Further research ight focus on other potential solutions to the lack of a one-day iplied volatility tie series. 7

18 References Britten-Jones,. and A. Neuberger,, Option Prices, Iplied Price Processes, and Stochastic Volatility, Journal of Finance, 55(, April, pp Carr, P. and D. adan,998, Towards a Theory of Volatility Trading, Volatility: New Estiation Techniques for Pricing Derivatives, Ed. R. Jarrow, Risk Books, NY, pp Carr, P. and L. Wu, 4, Variance Risk Preia, Working Paper, CBOE, 3, VIX CBOE Volatility Index, vixwhite.pdf. Chriss, N., and W. orokoff, 999, arket Risk for Variance Swaps, Risk October, pp Deeterfi, K., E. Deran,. Kaal, and J. Zou, 999a, A Guide to Volatility and Variance Swaps, The Journal of Derivatives, Suer, pp Deeterfi, K., E. Deran,. Kaal, and J. Zou, 999b, A Guide to Variance Swaps, Risk June, pp Harstone, A., 4 Investing in Iplied Volatility, Lehan Brothers, 9 February. 8

19 Table A. Coparison of Standard Straddle, Variance Swap, and VIX Futures Gaa, Variance Vega, and the Derivatives of These with Respect to the Value of the Underlying and to Tie to aturity Standard Straddle Variance VIX Swap Futures P '( Γ= N d S Sσ τ S ( T t P ϒ= N'( d S τ τ σ σ T t σ Γ '( '( N d d N d S S σ τ S σ τ 3 S ( T t ϒ N'( d τ dn'( d S σ σ Γ τ N'( d d N'( d σ log( S/ K 3/ 3/ σsτ Sσ τ 4 τ στ ϒ SN '( d dn '( d S τ σ log( S/ K τ 3/ 4σ τ σ 4 τ στ T t The above assues zero interest rates and notional of $. P denotes the value of the position, S is the underlying, K is the option strike, τ is equal to T-t or the tie reaining to expiry, t is the tie of inception log( S/ K + (/ σ of the variance swap, d =, N '( is the noral probability density function, and Γ and ϒ are as defined in the first colun of the table. σ τ 9

20 Figure. Rolling -Day VIX Futures / Variance Swap P&L Correlations // 6// // // 6// // // 6// // //3 6//3 //3 //4 6//4 //4 Rolling correlation of the P&L of a 3-onth variance swap and a VIX futures contract. The variance swap is held to aturity, at which tie another is purchased. For the purposes here, we assue a flat ter structure of volatility fro one to three onths and thus use VIX as the variance swap rate and VIX as a proxy for the value of the VIX futures contract.

21 Figure. Derivatives of Gaa and Variance Vega with Respect to the Value of the Underlying and Tie to Expiry for Both Variance Swaps and Standard Straddles.5. Panel : Derivative of Gaa wrt Underlying 5 4 Panel : Derivative of Var Vega wrt Underlying Rate of Change of Gaa Straddle-- d(gaa/ds VarSw ap-- d(gaa/ds Rate of Change of Var Vega Straddle-- d(varvega/ds VarSw ap-- d(varvega/ds Value of Underlying -4 Value of Underlying Rate of Change of Gaa Panel 3: Derivative of Gaa wrt Tie to Expiry Tie to Expiry Straddle-- d(gaa/d(tau VarSw ap-- d(gaa/d(tau Rate of Change of Var Vega 5 5 Panel 4: Derivative of Var Vega wrt Tie to Expiry Tie to Expiry Straddle-- d(varvega/d(tau VarSw ap-- d(varvega/d(tau The above are based on a standard straddle with a hypothetical underlying value of $5, strike price of $5, 3-onth (.5-year expiry, 5% volatility, zero interest rates and zero dividend yields, and a variance swap with a 3-onth expiration on the sae underlying. The notional of the variance swap is set so that the variance vega of the standard straddle (when at-the-oney and the variance swap are equal.

22 Figure 3. Rolling -Day VIX Futures / Variance Swap + Accreting NotionalVIX P&L Correlations // 6// // // 6// // // 6// // //3 6//3 //3 //4 6//4 //4 The above graph is identical to that in figure, except that we add an interittently adjusted aount of VIX futures to the variance swap position according to the schee described in the last ter of equation (7. (Equation (7 uses a fair variance forward, the graph shows a VIX futures position.

23 Figure 4. Rolling -Day VIX Futures / Gaa Swap P&L Correlations /4/ /4/ /4/3 3/4/3 5/4/3 7/4/3 9/4/3 /4/3 /4/4 3/4/4 5/4/4 7/4/4 9/4/4 /4/4 /4/ Rolling correlation of the P&L of a 3-onth gaa swap and a VIX futures contract. The gaa swap is held to aturity, at which tie another is purchased. For our purposes here, we assue that the one-day to one-onth ter structure is the sae as the second-onth vol inus first-onth vol as provided by Blooberg. We thus use (VIX i- -s on day i- (where s is the ter spread as the variance strike for the leg of the gaa swap being evaluated on day i. 3