Variance Swaps and Non-Constant Vega
David E. Kuenzi Head of Risk Management and Quantitative Research
Glenwood Capital Investments, LLC 123 N. Wacker Drive, Suite 2800 Chicago, IL 60606
Phone (312) 881-6520 Fax (312) 881-6501
Abstract
Variance swaps are often touted as pure volatility plays, and in many senses they are.
There is, however, one very important aspect in which they are not. While they provide relatively stable gamma to investors, they do not provide a stable vega, but rather a vega exposure that varies dramatically over the lifetime of the contract. Combining a variance swap with an intermittently adjusted exposure to a pure vega product, such as a VIX futures contract, produces a position that has vega and gamma exposures that are generally stable relative to changes in both the value of the underlying and in the time to expiry. Disaggregating this combination into its pure vega and pure gamma components gives rise to a gamma contract that may be of use to some investors. 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 made very clearly by Demeterfi, et. al. (1999a) is that variance vega of a variance swap (sensitivity to maturity-scaled variance, similar to standard vega) remains stable despite movements in the underlying. A standard straddle, on the other hand, experiences quick shifts in vega (or variance vega) as soon as the underlying moves in either direction. The position then requires continuous rehedging if one wishes to maintain a stable vega. The same is generally true for gamma. While the gamma of a variance swap explodes as the price of the underlying goes to zero, it remains rather stable for high-probability values of the underlying. By contrast, the gamma of a standard straddle is peaked around at-the- money forward levels of the underlying (as is vega). This stability of gamma and variance vega for moves in the underlying represents the core advantage of variance swaps—and a driving factor behind their increased use. The VIX futures contract offers a similar advantage in that it provides constant vega exposure for changes in the underlying.
An issue arises, however, when an investor wishes to maintain constant gamma and constant vega across time. The VIX futures contract has no gamma and stable variance vega across time. While variance swaps provide constant gamma exposure across time, their variance vega exposure is a linear function of time to expiration—exhibiting significant exposure to changes in implied volatility on a mark-to-market basis early in the life of the contract with little exposure to changes in implied volatility just prior to maturity. As such, an investor in a variance swap experiences very significant shifts in
2 mark-to-market risk over the life of a variance swap and very different volatility
exposures depending on the swap’s time 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 simple strategy for the maintenance of generally stable vega and gamma. Second,
we propose a new instrument that will maintain constant gamma and zero vega. A
combination of this new instrument with a VIX futures contract would provide fairly
stable volatility exposure (gamma and vega) for both changes in time 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 minus a
variance “strike” agreed upon at contract initiation. Its payoff at maturity is:
2 ⎛⎞M 1 ⎛⎞SSii− −1 Payoff =−N ⎜⎟⎜⎟Kvar ⎜⎟Tt− ∑ S ⎝⎠01i=1 ⎝⎠i− M ⎛⎞1 2 =NR⎜∑()i −Kvar ⎟ (1) ⎝⎠Tt− 0 i=1
=−NV()(,t0vT) Kar
where M is the total number of monitoring periods between swap inception at time t0 to
swap maturity at time T. If we assume daily monitoring, M would be the number of
3 trading days between swap inception and maturity. Si is the price of the underlying on
day i, Ri is return of the underlying on day i, V(t0,T) is the realized variance over the
period, Kvar is the initially agreed upon variance strike expressed in volatility points squared,1 and N is the notional amount.
Variance swaps “accrue” variance based on each day’s realized return innovation. Mark-
to-market pricing at time t (tt0 < variance” and the expectation of variance looking ahead. As such, the value of the contract can change dramatically based on the change in expected future variance. As Chriss and Morokoff (1999) show, the mark-to-market value of a variance swap can be written as: −−rT()t MTM =−Ne ((λV (t0v,t) K ar)+(1−λ)(Kt −Kvar)) (2) where V(t0,t) is the realized variance from time t0 to time t, Kt is the strike for a new variance swap running from time t to time T, and λ is the proportion of time elapsed between t0 and t: tt− τ λ()t ==0 1− (3) Tt− 00T− t where τ = T-t. 1 For calculation of the variance strike, Kvar, see Demeterfi, et. al. (1999a and 1999b), Carr and Wu (2004), Carr and Madan (1998), and Britten-Jones and Neuberger (2000). 4 We use the symbol ϒ to denote variance vega.2 We can then write: ∂()MTM −rT()−t ⎛⎞τ −r(T−t) ϒ= =(1 −λ(tN)) e =⎜⎟Ne (4) ∂−KTt ⎝⎠t0 So the exposure of the variance swap to changes in fair market variance is positively related to the proportion of time remaining to expiration of the swap. If we then assume zero interest rates and a notional of $1, we can consider the sensitivity of variance vega to changes in time to expiration: ∂ϒ 1 = (5) ∂τ Tt− 0 Equation (5) says that the change in variance vega for a change in time to maturity is related to the inverse of the total period covered by the contract, so that the closer we move to expiry, the lower the variance vega. If, for example, we have two year variance swap, its variance vega will fall by half after one year. Equation (5) also makes it clear that an investor rolling into a new variance swap on, say, a quarterly basis would experience varying levels of variance vega over time. This can best be seen by taking a rolling correlation of the returns of a three-month S&P 500 variance swap, rolled into a new variance swap every three months, and a VIX futures contract, as shown in figure 1.3 2 This is defined as in Demeterfi, et. al. 1999a. 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 mark-to-market value of the variance swap with respect to the level of fair variance for a new swap with a maturity equal to the remaining time to expiry of the existing swap. 3 The CBOE changed its methodology for calculation of the VIX index on September 22, 2003 such that the level of the VIX is now calculated so that (VIX)2 is equal to the variance strike of a one-month variance swap. See CBOE (2003) and Harmstone (2004). 5 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 moves toward maturity, however, its variance vega approaches zero, and thus its returns show very little correlation with the VIX. Given these observations, it is instructive to more carefully compare the volatility exposures of a standard straddle,4 a variance swap, and the VIX futures. One might perhaps think of volatility exposure as consisting of two elements: gamma (exposure to realized versus initially expected volatility) and vega (exposure to changes in expected and market-traded levels of volatility). In table A, we consider the gamma and variance vega exposures of these instruments and how they change in response both to changes in the value of the underlying and to changes in time to expiry. It is first useful to note the straightforward values for the variance swap and VIX futures, as compared to the standard straddle. The VIX futures has just one very simple volatility exposure—to variance vega—which is invariant for changes in the underlying or time to expiry. The shapes of the gamma and variance vega functions of the standard straddle are dominated by the normal probability density function and generally have quite a large (and not straightforward) response to changes in the underlying and time to expiry. Variance swaps contain both gamma and variance vega exposures. The gamma exposure remains constant with changes in time to expiry but could get quite large were the underlying to move down substantially. The variance vega of the variance swap remains constant 4 This is defined as a long at-the-money put combined with a long an at-the-money call; this might be considered a rather “impure” way of obtaining volatility exposure. 6 despite changes in the underlying, but is quite sensitive to changes in the time to expiry. It is this last characteristic of variance swaps that is of concern. To provide more clarity, figure 2 shows how the volatility exposures of a standard straddle and variance swap change based on changes in the value of the underlying and in time to expiry. While the exposures of the standard straddle prove highly sensitive to changes in both underlying and time to expiry, the variance swap’s exposures are constant with two exceptions. First, gamma explodes when the underlying falls dramatically (into low probability regions), as shown in panel 1 of figure 2. Second, as shown in panel 4 of figure 2, variance vega exhibits large and predictable changes as the contract moves to maturity. It should now be quite clear that variance swaps provide fairly stable gamma 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 maintain generally stable gamma and nearly constant vega could do so by entering into a variance swap and simultaneously entering into a VIX futures contract. The notional of the futures position would need to be adjusted then on a daily basis (assuming daily monitoring in the variance calculation) in order to match the initial variance vega exposure of the variance swap. Conceptually, as the variance swap loses variance vega due to its movement toward maturity, we add variance vega by purchasing a little more 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 remains in excess of 0.93. 7 In order to further explore the characteristics of variance swaps with ease of exposition, we make two key assumptions. First, we assume a flat term structure of volatility— clearly an unrealistic assumption that we will later relax. Related to this assumption is a change of notation: we parameterize Kvar to Kvar(t), where Kvar(t) is the fair market variance (for all expiries, as we are assuming a flat term structure of volatility) as known at time t. So our previous Kvar becomes K . Relatedly, we define an intermediate day number var(t0 ) m, where 0 < tt− m λ =0 = (6) Tt− 0 M So we can write Kvar(m) in place of K var(t ) , and we can write Kvar(T) = Kvar(M) to denote fair market variance at the time of the variance swap’s expiration. Second, in order to avoid convexity issues, we assume that there is an over-the-counter fair variance forward contract with payoff (K− K) at maturity rather than continuing to refer to VIX var(T) var(t0 ) futures. The fair variance forward is similar to a VIX futures contract except that it is a forward (no daily settle) and the payoff involves market quoted variances rather than volatilities. On any given day, the P&L of this fair variance forward will be ()KKvar(mm) − var( −1) . 8 Using equations (2) and (6) and the above assumptions, and again assuming zero interest rates and unit notional, we can write the P&L of the combined variance swap and fair variance forward with accreting notional: 1 m MTM =−λλ((V t ,t) K )+(1−)(K −K )+i(K −K ) (7) 0 var(tt00) var()var(t) ∑ var(i)var(i−1) M i=1 The last term 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 amount equal to (1/M) to assure that we always have λ exposure to the pure variance vega product. In this way, the sum of the second and third terms on the right assure that we always have unit exposure to variance vega. On day m, for instance, the second term shows that we have 1(− λ =−M m)/Mdollars of exposure to changes in market variance through the variance swap, while the third term shows that we have m / M dollars of exposure to changes in market variance through our fair variance forward position. Using such a trading strategy, an investor can assure generally stable gamma and vega exposures across both time and asset price movements. The Constant Gamma, Zero Vega Product As noted above, the market has access to a constant vega product in the form of the VIX futures contract. The market also has access to fairly stable gamma exposure—albeit mixed with varying amounts of vega—in the form of a variance swap. The market does not currently have a widely traded pure gamma product. Such a product can be derived 9 from equation (7). As equation (7) represents a combination of products such that the portfolio achieves both unit gamma and unit variance vega, it should be possible to disaggregate this combination such that two “pure” products result: 1 m MTM =−λλ((V t ,t) K )+(1−)(K −K )+i(K −K ) 0 var(tt00) var( ) var(t) ∑ var(i) var(i−1) M i=1 1 m =+λλVt(,t) (1−)K −K +i(K −K ) 0 var(tt) var( 0 ) ∑ var(i) var(i−1) M i=1 1 m−1 =+λλVt(,t) (1−)K −K +λK −λK +i(K −K ) (8) 0 var(tt) var( 0 ) var(t) var(m−1) ∑ var(i) var(i−1) M i=1 ⎡⎤1 m−1 =−λλVt(,t) K +i(K −K )+⎡⎤K −K ⎢⎥0 var(mi−−1) ∑ var( ) var(i1) ⎣⎦var(t) var(t0 ) ⎣⎦M i=1 =+Gamma Derivative Vega Derivative In equation (8) we isolate the gamma derivative and the vega derivative. The vega derivative is simply unit exposure to the fair variance forward contract, as we would have expected. Now we manipulate the equation for the gamma derivative: 10 1 m−1 Gamma Derivative =−λλV (,t0 t) Kvar(mi−−1) +∑i(Kvar( ) −Kvar(i1) ) M i=1 ⎡⎤⎛⎞1 m−1 =−λ ⎢⎥Vt(,0 t) ⎜⎟Kvar(mi−−1) +∑i(Kvar( 1) −Kvar(i) ) ⎣⎦⎝⎠m i=1 mm−1 ⎡⎤112 ⎛⎞m − 1 =−λ ⎢⎥∑∑RKim⎜⎟var( −−1) −Kvar(m1) +Kvar(i−1) ) ⎣⎦tt− 0 ii==11⎝⎠m m (9) mm ⎡⎤M 112 = λ ⎢⎥∑∑RKii− var( −1) ⎣⎦Tt− 0 mii==11m m ⎡⎤1 M ⎛⎞2 Tt− 0 =−λ ⎢⎥∑⎜⎟RKiivar( −1) ⎣⎦Tt− 0 mi=1 ⎝⎠M m 1 ⎛⎞2 Tt− 0 =−∑⎜⎟RKiivar( −1) Tt− 0 i=1 ⎝⎠M Equation (9) provides a very intuitive result; the pure gamma product is the sum of the squared daily asset returns minus 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 monitoring period. The intrusion of vega into the variance swap contract results from taking the fair variance over a period different from the length of the monitoring period. The value of the gamma derivative has no dependence on expected future volatility because the fair market variance just prior to each monitoring period is the level that makes the expected payoff of that “leg” of the contract equal to zero. If a client wants to go long gamma through a gamma 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 mark-to-market value of the gamma contract as: 11 ⎛⎞1 m Gamma Derivative =+Variance Swap i()K −K −(K −K ) ⎜⎟M ∑ var(ii) var( −1) var(t) var(t0 ) ⎝⎠i=1 (10) 1 m =−Variance Swap ∑()M −i (Kvar(ii) −Kvar( −1) ) M i=1 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 (1-λ) = (M-i)/M, so shorting this amount in the fair variance forward contract will give us a payoff identical to that of the gamma contract. Before we explore the implications of fully relaxing our prior assumptions, we might question the extent to which the gamma contract and a vega instrument are correlated, thus potentially obviating the need for the gamma contract. Over the period from February 1, 2000 to January 31, 2005, daily correlation of a three-month gamma contract P&L and changes in the VIX were 0.09, indicating that in a very broad sense, the two contracts have little correlation.5 Figure 4, which shows the rolling correlation of the gamma swap and the VIX, indicates that the correlation between a vega product and a gamma swap can move quite rapidly between extremes, with a high 21-day correlation of 0.86 and low of -0.62. The periods of extreme high correlations are generally those in which both actual and implied volatility are rising together—e.g., days on which daily changes in implied volatility tend to follow the daily changes in realized volatility. The periods of low correlation tend to be in range bound markets—in terms of both 5 As we do not have data for one-day vol, we use the VIX minus the first-month / second-month term spread for SPX at-the-money options (squared) as available from Bloomberg. This assumes that the slope between the first two options contracts is the same as the slope between one-day and one-month. 12 underlying level and implied. Overall, it is clear that the gamma and vega products provide very different types of exposures. When we relax the assumptions of a flat term 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 (10) will no longer be mathematically precise.6 Regardless, the intuitions continue to hold and these equations could still be used (in conjunction with VIX futures) in order to manage exposures. The second, more 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 comfort that the value of the gamma derivative would accrue in a completely fair and objective manner. This could be a substantial obstacle to the creation of such a product. One possible solution would be to quote a gamma derivative based on the level of a related and widely quoted volatility index, such as the VIX, plus or minus some spread in order to take account of the expected term structure of volatility between one day and one month volatility over the life of the contract. So a gamma contract on the S&P 500 might be quoted at VIX − s , where s is a spread. If a dealer believes that the term structure between one day and one month will average 2% over the life of the gamma derivative, then the contract would be quoted at VIX minus 2%. One drawback of this approach is 6 Without a flat term structure of volatility, we must rely on equations (1) and (2) in order to define the fair M 1 2 variance levels Kt and KT. These equations would imply that K= E[R], where Em is tmm ∑ i ()Tt− m im=+1 the conditional expectation with respect to the information available on day m. The definition of KT is then problematic, as there are no terms under the summation. 13 that there are few generally accepted volatility indices. Another drawback (from the perspective of maintaining purity of exposure) is the implicit term structure bet that both parties to the contract would be taking. The value of the contract would now be sensitive to the market term spread in relation to the initially agreed upon term spread. If the initial term spread agreed between the counterparties is s0 and the market term spread some time into the life of the contract is st, then the value of the contract, again assuming unit notional and zero interest rates, will be: m 1 ⎛22Tt− 0 ⎞ MTM =−∑⎜Rii()VIX −10−s ⎟+Em[Gm+1,M|st] (11) Tt− 0 i=1 ⎝⎠M where the final term is the expectation with respect to the information available at time m of the gamma accrual (Gm+1,M) from day m+1 to day M given the currently quoted market spread is st. This final term can be defined as: M ⎡⎤1 ⎛⎞22Tt− 0 ⎢⎥∑ ⎜⎟RVii−−()IX−10s − Tt− im=+1⎝⎠M EG[|s]= E⎢⎥0 mm+1,Mt m⎢⎥M 1 ⎛⎞22Tt− 0 ⎢⎥∑ ⎜⎟RVii−−()IX−1 st ⎣⎦Tt− 0 im=+1⎝⎠M MM 1 ⎡22⎤ (12) =−Emi⎢∑∑()VIX −1st−(VIX i−1−s0)⎥ M ⎣⎦im=+11i=+m M Mm− 22 2 ⎡ 2⎤ =−()sstm00+E⎢ ∑ (s−st)VIXi−1⎥ MM⎣im=+1 ⎦ Mm−−222(Mm) =−()ss+(s−s)E⎡VIXmM,1− ⎤ M tt00M m⎣ ⎦ 14 where EV⎡IXmM,−1⎤ is the conditional expectation of the average level of the VIX index m ⎣⎦ between day m+1 and day M. So that, combining equations (11) and (12), we have the mark-to-market value: 1 m ⎛⎞Tt− Mm− MTM = R 22−−()VIX s 0 +(s2−s2)+ Tt− ∑⎜⎟ii−10 M M t0 0 i=1 ⎝⎠ (13) 2(Mm− ) ()ss− E⎡⎤VIXmM,1− M 0 tm⎣⎦ The long gamma contract makes (loses) money when term spreads come in (widen), as the investor has locked in a lower (higher) daily volatility hurtle than that currently available in the market. This approach provides gamma exposure mixed with significant volatility term structure exposure. We have removed the impurity of vega from the variance swap and replaced it with sensitivity to the term spread of volatility. The sensitivity of this contract to a change in the term spread will be: ∂−()MTM 2(M m) = −Em[VIX mM,1− −st] (14) ∂sMt This is essentially the derivative of the value of the contract with respect to the average one-day implied volatility, weighted by the percentage of time remaining until contract expiration. 15 It is also interesting to note that the correlation between the changes in the VIX and changes in the term spread (we use available Bloomberg delta point data—June 3, 2002 to January 31, 2005—to determine the term spread) is -0.46. Given that this particular form of a gamma contract benefits from term spread tightening, it therefore continues to have positive volatility exposure—although indirectly and significantly muted. Finally, while one goal in this paper has been to propose a “pure” gamma derivative, it should be noted that the term 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 benchmark against a longer-dated VIX futures contract minus a spread. In this case, the spread would be reflective of the term structure of volatility for perhaps six months rather than one. Conclusion While variance swaps allow investors to trade an at-the-money delta hedged position without delta hedging and without adjusting for the moneyness of the options, they do not—as often purported—represent “pure” volatility exposure in that variance vega varies dramatically over the life of a variance swap. An investor wishing to maintain generally stable gamma and vega can do so by entering a variance swap and an intermittently 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 gamma derivative. Combining a variance swap and an 16 intermittently adjusted fair variance forward, and then disaggregating into the gamma and vega portions, shows that there is. The two resulting derivatives are 1) a static position in a fair variance forward and 2) a variance swap in which the variance strike is reset to fair variance just prior to each variance monitoring period. If such a gamma contract is sold by a dealer, it can in turn be hedged through a variance swap and an intermittently changing position in a fair variance forward or VIX futures position. The advent of a gamma contract would enable investors to be more specific in terms of the volatility exposure they are taking on and would thus contribute to overall market completion. The most 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 monthly volatility index plus or minus a spread. This solution, however, introduces term spread risk into the contract. Further research might focus on other potential solutions to the lack of a one-day implied volatility time series. 17 References Britten-Jones, M. and A. Neuberger, 2000, “Option Prices, Implied Price Processes, and Stochastic Volatility”, Journal of Finance, 55(2), April, pp. 839-866. Carr, P. and D. Madan,1998, “Towards a Theory of Volatility Trading”, Volatility: New Estimation Techniques for Pricing Derivatives, Ed. R. Jarrow, Risk Books, NY, pp. 417- 427. Carr, P. and L. Wu, 2004, “Variance Risk Premia”, Working Paper, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=577222 . CBOE, 2003, “VIX CBOE Volatility Index”, http://www.cboe.com/micro/vix/ vixwhite.pdf . Chriss, N., and W. Morokoff, 1999, “Market Risk for Variance Swaps”, Risk October, pp. 55-59. Demeterfi, K., E. Derman, M. Kamal, and J. Zou, 1999a, “A Guide to Volatility and Variance Swaps”, The Journal of Derivatives, Summer, pp. 9-32. Demeterfi, K., E. Derman, M. Kamal, and J. Zou, 1999b, “A Guide to Variance Swaps”, Risk June, pp. 54-59. Harmstone, A., 2004 “Investing in Implied Volatility”, Lehman Brothers, 19 February. 18 Table A. Comparison of Standard Straddle, Variance Swap, and VIX Futures— Gamma, Variance Vega, and the Derivatives of These with Respect to the Value of the Underlying and to Time to Maturity Standard Straddle Variance VIX Swap Futures ∂2 P ⎡ Nd'( ) ⎤ 2 0 2 1 Γ= 2 ⎢ ⎥ 2 ∂S ⎣ Sσ τ ⎦ ST()− t0 ∂P ⎡⎤ τ 1 ϒ= Nd'( 1)Sτ ∂σ 2 2 ⎢⎥ Tt− 2σ ⎣⎦2σ 0 ∂Γ ⎡⎤Nd'( ) dN'(d) 2 0 2 11− 1 − ⎢⎥2 22 3 ∂S ⎣⎦S στ S σ τ ST()− t0 ∂ϒ ⎡Nd'( ) τ dN'(d) ⎤0 0 2 11− 1 ∂S ⎢⎥2 ⎣⎦22σσ ∂Γ ⎡⎤−Nd'( ) dN'(d) ⎛⎞σ 2 log(SK/ ) 0 0 2 11−−1 ∂τ ⎢⎥3/2 ⎜⎟3/2 ⎣⎦22στS Sστ⎝⎠4 τ στ ∂ϒ ⎡⎤SN '(d ) d N '(d )S τ ⎛⎞σ 2 log(SK/ ) 1 0 2 11−−1 ∂τ ⎢⎥⎜⎟3/2 Tt− ⎣⎦44στ 22σσ⎝⎠τ τ 0 The above assumes zero interest rates and notional of $1. P denotes the value of the position, S is the underlying, K is the option strike, τ is equal to T-t or the time remaining to expiry, t0 is the time of inception log(SK/ ) + (1/ 2)σ 2 of the variance swap, d = , N '(⋅) is the normal probability density function, and 1 στ Γ and ϒ are as defined in the first column of the table. 19 Figure 1. Rolling 21-Day VIX Futures / Variance Swap P&L Correlations 1.2 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 0 0 0 0 0 000 000 001 001 002 002 003 003 004 004 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 / / / / / / / / / / /1/20 /1/20 /1/20 /1/20 /1/20 2 6 2 6 2 6 2 6 2 6 10 10 10 10 10 Rolling correlation of the P&L of a 3-month variance swap and a VIX futures contract. The variance swap is held to maturity, at which time another is purchased. For the purposes here, we assume a flat term structure of volatility from one to three months and thus use VIX2 as the variance swap rate and VIX as a proxy for the value of the VIX futures contract. 20 Figure 2. Derivatives of Gamma and Variance Vega with Respect to the Value of the Underlying and Time to Expiry for Both Variance Swaps and Standard Straddles Pa nel 1: Derivative of Gamma wrt Underlying Panel 2: Derivative of Var Vega wrt Underlying 0.025 5 0.020 4 a Straddle-- Straddle-- 0.015 ga 3 m d(Gamma)/dS e d(VarVega)/dS m V r a 0.010 a 2 G VarSw ap-- VarSw ap-- of d(Gamma)/dS d(VarVega)/dS 0.005 of V 1 ge nge n a 0.000 a 0 h 72 7476787 h 7 27476787 -0.005 -1 of C of C e t te a -0.010 a -2 R R -0.015 -3 -0.020 -4 Value of Underlying Value of Underlying Panel 3: Derivative of Gamma wrt Time to Expiry Panel 4: Derivative of Var Vega wrt Time to Expiry 0.0 0 0.1 0.2 0.3 0.4 0.5 200 -0.5 Straddle-- a -1.0 ga d(VarVega)/d(Tau) e m V m r 150 a -1.5 a VarSw ap-- V G f f -2.0 d(VarVega)/d(Tau) o o e ge g -2.5 100 n n a a h -3.0 C Ch f Straddle-- of o -3.5 d(Gamma)/d(Tau) 50 e te t a VarSw ap-- R Ra -4.0 d(Gamma)/d(Tau) -4.5 0 -5.0 0 0.1 0.2 0.3 0.4 0.5 Time to Expiry Time to Expiry The above are based on a standard straddle with a hypothetical underlying value of $50, strike price of $50, 3-month (0.25-year) expiry, 25% volatility, zero interest rates and zero dividend yields, and a variance swap with a 3-month expiration on the same underlying. The notional of the variance swap is set so that the variance vega of the standard straddle (when at-the-money) and the variance swap are equal. 21 Figure 3. Rolling 21-Day VIX Futures / Variance Swap + Accreting NotionalVIX P&L Correlations 1.2 1 0.8 0.6 0.4 0.2 0 0 0 1 1 2 2 3 3 4 4 0 1 2 3 4 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 / / / / / / / / / / / / / / / 2/1 6/1 2/1 6/1 2/1 6/1 2/1 6/1 2/1 6/1 10/1 10/1 10/1 10/1 10/1 The above graph is identical to that in figure 1, except that we add an intermittently adjusted amount of VIX futures to the variance swap position according to the scheme described in the last term of equation (7). (Equation (7) uses a fair variance forward, the graph shows a VIX futures position.) 22 Figure 4. Rolling 21-Day VIX Futures / Gamma Swap P&L Correlations 1 0.8 0.6 0.4 0.2 0 002 003 003 003 003 003 004 004 004 004 004 005 -0.2 002 003 004 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 / / / 4/ 4/ 4/ 4/ 4/ 4/ 4/ 4/ 4/ 4/ 4/ 4/ 4 4 4 / / / 9/ 1/ 3/ 5/ 7/ 9/ 1/ 3/ 5/ 7/ 9/ 1/ -0.4 1 1 1 1 1 1 -0.6 -0.8 -1 Rolling correlation of the P&L of a 3-month gamma swap and a VIX futures contract. The gamma swap is held to maturity, at which time another is purchased. For our purposes here, we assume that the one-day to one-month term structure is the same as the second-month vol minus first-month vol as provided by Bloomberg. 2 We thus use (VIXi-1-s0) on day i-1 (where s0 is the term spread) as the variance strike for the leg of the gamma swap being evaluated on day i. 23