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 [email protected] 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 <<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 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.