# Trading realized variance using listed vanillas

#### Published in Automated Trader Magazine Issue 40 Q3 2016

## Listed futures on VIX and its cousins give exposure to implied variance. But getting exposure to realised variance is very different and usually has been the realm of OTC variance swaps. Here we examine strategies to trade the realised variance using only listed instruments, with simple time-independent formulas not requiring models such as Black-Scholes.

Buy-side agents have been trading realised variance with banks since at least 1996, typically selling it to them using OTC variance swaps. In recent years variations such as OTC variance options have also been very popular. It seems that many participants find it easier to take a trading view on realised variance rather than implied variance, especially over fairly long time scales and slower frequencies.

Several exchanges have tried to jump on this bandwagon, creating listed contracts replicating OTC realised variance swaps, but all of these have failed to gain traction. This might be because although OTC contracts are highly standardized, the operational details, surprisingly, turn out to be too complex. As an example, Eurex tried to list variance futures several years ago with a very complex system on the back-end for mirroring the OTC flow of payments, but these never took off.

While OTC contracts provide exposure in a simple, trouble-free way, they also suffer from various limitations: Some agents cannot use them at all, exact tailoring to a particular exposure may not be available, and their liquidity has reduced in recent years (i.e. wider bid/ask spreads and lower volumes). Here we shall discuss how to avoid these issues and trade various realised variance strategies directly in the listed vanilla option markets.

In this treatment, the level of implied volatility in the market is only involved at the inception of the trade: we implicitly assume that Mark-to-Market (MtM) considerations are not relevant, so that the agent does not care about daily fluctuations in the MtM P/L caused by changes in the market implied volatility, and that the only relevant P/L is the one at maturity.

The way banks typically model and hedge OTC variance contracts is somewhat different: In addition to attempting to match the precise contract specifications, they must also continue to MtM the contract after inception and, consequently, must take into account market option prices for the lifetime of the trade.

In practice of course, every agent has stop-loss limits or margin calls, so the relevant criterion is whether these limits (or equivalently, the pain threshold for margin calls) are wide enough that the probability they are triggered by a worst-case move in implied volatility is acceptably low. In particular, the possible worst-case scenario will increase with longer maturity, and hence restrict the maturity T tradable for a given agent.

### Generalizing realized variance

We focus on the variance of a tradable futures contract, Ft on underlying St with expiry Tfut, in the period t = 0 (i.e 'now') to T, where T≤Tfut.

Consider the series of times 0 = t0 < t1 < ... < tn = T. Let Fi be the futures prices at these times, δFi = Fi+1 - Fi the change in the futures price from ti to ti+1, and δti= ti+1 - ti. The choice of ti is arbitrary, and hence the frequency of the trading strategies discussed later.

It is important to note that when considering a trading strategy the Fi will be the executed prices, net of trading costs (exchange fees, broker commissions, etc) at the trading times ti.

If targeting a particular reference (e.g. the closing daily prices), they will also include 'slippage', specifically the possible difference between targeted execution price and actual execution price.

We define a family of variance-like measures:

(1)

where g is an arbitrary function. The following examples are of particular interest.

#### Standard Variance

Choosing g(Fi) = gives the standard realized variance formula:

(2)

The realised volatility is its square root, sometimes called the 'lognormal volatility'.

#### Corridor Standard Variance

This is an important variation, particularly relevant for this article, but also frequently used in OTC contracts. Given a range between a low strike KL and a high strike KH, define the corridor variance as

(3)

where means 1 in the range and 0 outside.

We will see below that in practice it is (3) rather than (2) that is actually tradable via the strategy discussed in the section 'Model-Independent Strategy'.

#### Normal Variance

Choose g in (1) to be a constant, F~0. Then:

(4)

Taking F~0 = 1 gives the square of what fixed-income traders would call 'normal' volatility; not a pure number anymore, but with the same dimension as F. In the following examples we will choose F~0 to be of the same order as F, keeping (4) dimensionless.

This variance gives a softer exposure than the standard one in the case where realized volatility increases on the downside, i.e. when δF and F are negatively correlated. This is often the case in equity markets and hence we will refer to it as the 'equity' case in the following.

A corridor version of (4) can trivially be defined as for the standard variance, and again will be the practical one used when trading.

### Model-independent strategy

Choose a T which is expiry date for both listed options and futures, i.e. T = Tfut.

Our aim is to capture the variance of Ft in the period t = 0 to T. Our strategy will involve trading at the series of times

0 = t0 < t1 < ... < tn = T. As mentioned before, the corresponding levels Fi should be regarded as execution prices, net of costs.

Let f(F) be some smooth function of F (which we shall specify later) and let fP (ST) be the portfolio of European-style options which we design to approximately replicate the European exotic payoff defined by f(FT) (see the appendix at the end of the article for details of its construction).

Note first the purely mathematical result, that

(5)

where the error term e1 = ∑O(δF3) represents higher order corrections.

To buy realised variance, choose the following trading strategy:

1. Buy the portfolio of options fP (ST) for premium X

2. At each time ti, be short a number of futures contracts

(6)

This is achieved by trading the quantity Δi-1 - Δi.

3. Close out all positions at time T i.e. let them expire.

Then, ignoring funding costs, the total P/L of the trade is

(7)

where X~ = X - f(F0). By the previous result this becomes

(8)

where e2 = fP (ST ) - f(FT ) and

(9)

In other words, assuming the error terms can be neglected (we shall return to them later), we have synthesized a long swap on the generalized measure of variance Vf , where g = f′′/2. The short variance strategy is obtained trivially by selling the portfolio of options rather than buying it and reversing the other signs.

Subject to the funding caveat below:

The strategy is model-independent, i.e. we have not needed to make any assumptions on how F moves (as in e.g. Black-Scholes). To determine Δi, only the form of f(F) is required.

The Δi has no explicit dependence on time, which allows us to trade 'instantaneous' variance (see the equally named section later), and for sensible choices of g will be a simple and computationally light formula.

In theory any generalized variance Vg could be captured, as given a reasonable g, f follows by double integration.

Caveat: If we take funding into account the same analysis goes through but the option premium should be multiplied by FV(0,T), and the delta at time t by 1 = FV(t,T), where FV(t,T) is the forward value factor from t to T. This introduces very mild model-dependence.

### Replicating the standard variance

For a standard variance contract we would take

(10)

which generates the standard variance (2). The corresponding delta is

This choice of f can be well-replicated (see the Appendix) with a 1/K2 portfolio of F~0-OTM calls and puts. This portfolio should extend over an arbitrarily wide strike range, but since in practice one can only trade a finite range, effectively one usually deals with the corridor variance case mentioned on the previous page.

We also remark in passing that this choice of f(F) is special: In the limit of a perfect replicating portfolio, hedging the options in the portfolio using the Black-Scholes delta calculated at a constant volatility across different strikes would have produced precisely the same result. This property is not true for general f(S).

### Replicating the corridor variance

To capture (3) we take

(11)

where F = min(max(F,KL),KH) with corresponding delta

which means that the futures position does not change for moves taking place outside the range and so variance is only fully captured inside the range. Essentially, no more trading happens while F is outside the corridor. Some more details on this can be found in the appendices.

f is replicated by restricting the strike range of the 1/K2 portfolio to the range (KL,KH), and the exposure generated is essentially that of a corridor variance contract.

In the 'equity' case it can be softer or heavier than the standard case, depending on the choice of range.

Note also that in this case, unlike the standard one, hedging the replicating portfolio of options with the Black-Scholes delta would not have achieved the same result since there would still be option gamma outside the strike range and consequently an exposure to the realised volatility.

### Replicating the normal variance

Taking

(12)

generates the normal variance (4), and makes the calculations for the strategy strikingly simple:

The error term for large moves e1 is exactly 0;

In the common case where the strikes are equally spaced, the corresponding portfolio fP has the same number of options at each strike (hence we call it the 'flat' or 'constant weight' portfolio);

Δi is just a linear function of F, hence the number of futures to trade at each time is proportional to the change of future price since last trading time δF: .

As mentioned previously, this strategy gives a slightly softer exposure than the standard variance in the 'equity' case, as there is an under-weighting of variance on the downside and an over-weighting on the upside. This would normally be reflected in a lower variance strike X, as usually this case is associated with a negative implied volatility skew, and the 'normal' option portfolio has fewer puts at low strikes than the 'standard' one, so that when selling (buying) variance, less premium is received (paid) at inception of the trade.

Finally, for the standard variance, in practice one would usually trade the corridor normal variance, i.e. restrict the range of strikes in fP. The essential difference is that trading of F stops once it stays outside the range.

### The error for large moves e1

The first error term e1 can generally be ignored even for large moves. It is easy to see this when trading the normal variance (4), as e1 then is exactly 0.

In the case of the standard variance:

(13)

For a long variance strategy its sign is opposite to the direction of the δF move (so it is positive for a downward move).

As a fraction of the corresponding contribution from the variance term this is which, even for a 15% gap down, is only 10% of the variance term.

Consequently, though in a variance trade a single large move can dominate the P/L (as in the following example), it will mainly be from the variance, not e1.

### Example

Suppose every move is roughly the same size:

The standard variance is roughly σ2 and so the P/L ≈ σ2 - X~.

Now suppose there is one exceptional percentage return J. Then the P/L≈ (1 - 1/n) σ2 + J2/T - X~ + e1, and as a fraction of the extra P/L, e1 is 2J/3.

Now take a three month trade on daily variance, i.e. T ≈ 0.25, n ≈ 60 and σ ≈ 20% with a large jump J ≈ 20%. Then the extra P/L is very large (i.e. to achieve the same without a jump, assuming X~ ≈ 20%, would require the realised σ to roughly double), though only ≈ 13% of it is due to e1.

Portfolio Replication Error e2

The second error term e2 = fP(ST) - f(FT) arises from the fact that options cannot be traded at every strike so that fP might not match f (or f~) very well. This is only significant if the strikes available are very sporadic or narrow in range and the latter becomes less of an issue in the case where the target payoff f(FT) only requires a replicating portfolio with a finite range of strikes. For more details refer to the Appendix.

### The frequency

As mentioned, buy-side agents often sell realised variance using OTC variance swaps. This normally limits the choice of frequency to daily variance on the close. In the approach here the choice of re-hedging times, i.e. frequency, is more or less arbitrary, and different choices will lead to exposure to a different variance.

Hence, choosing the frequency will be a key decision point in the pre-trade analysis, together with choosing which type of generalised variance and which maturity.

Note that, apart from the important exception discussed next, any choice of frequency will require sending a buy or sell order on F at the time ti. As the quantity will nearly always be a function of the current Fi in the market, it is clear why in a fast market the computational lightness of Δ could be an advantage to avoid slippage.

### Instantaneous variance

Let us define the 'instantaneous' generalized variance as the limit of (1) when δt→0 with ti+1=ti+δt. Naively, we could think this choice implies sending an order to the market at every instant, but this is not necessary because the possible prices of F are discrete by the nature of the tick size, and Δ is essentially not a function of time.

If the strategy is long variance, then every day before market open the agent sends limit orders at each possible level of F, as the quantity can be pre-calculated, being a function of F and the last traded level. As soon as one sell (buy) order is filled, the opposite buy (sell) order will be sent one tick lower (higher). Essentially, one is following the market as it moves up and down, with pre-calculated quantities.

An appealing feature of the long instantaneous variance is that slippage risk is minimised. Clearly, one can choose to send orders at an interval wider than the tick: We leave the details to our esteemed readers.

The short variance case is in theory equivalent, by changing

- limit order to stop order
- buy to sell (and vice versa)
- higher to lower (and vice versa)

though its practicality is dubious since the use of stop orders could generate considerable slippage.

Even if substituted with a more efficient limit order calculated as the market moves, slippage would remain a serious risk anyway as it might not be filled if the market moves away before order arrival.

#### Figure 01: OESX Implied Volatility for 17 June 2016 Expiration

### The 'frequency swap'

It is worthwhile to draw attention to the 'frequency swap', a generalized variance trade which can be executed using no options at all: The agent buys (sells) variance at a given T and frequency f1 and sells (buys) variance on the same underlying at the same T but a different frequency f2.

The option portfolios cancel each other out and only step 2 is left. If the slower frequency sampling points are a subset of the faster frequency set, then the net Δ will be zero on all sampling points of the former.

We note that:

- Other frequencies could be traded in the same way, including the instantaneous one, particularly so for the 'normal' variance case.
- Since no options are involved, this strategy can be applied to any asset, extend to any maturity, with no need to limit it to corridors.
- The long fast/short slow frequency swap can be seen as 'mean-reversion' strategy, the opposite position as a 'momentum' strategy.

For example, buy-side agents have traded such weekly/daily frequency swaps against banks in OTC format. In that case, f1 is daily on the close and f2 weekly on the close.

To replicate without need for OTC, Monday to Thursday on the close the agent trades the change in Δ of the daily strategy, and on the Friday's close the position is closed. The trade realises daily minus weekly variance of the type chosen. This is shown in the second page of the spreadsheet in next section.

### Example: Daily Variance Trade

This is contained in a spreadsheet that can be downloaded from www.eqfltd.com. Registration is free, and the filename is 'Examples for article on Model-independent strategies etc' (in the folder available to standard members).

A variance trade is started on 21 March 2016, on the futures of the Eurostoxx 50 index, expiring 17 June 2016, using options with same expiry. The market prices used are daily settlement prices from Eurex, though we disregard the ones where settlement is equal to the tick (0.1 in this case), as they are unreliable and probably also very difficult to trade. In Figure 01 we show the implied Black volatilities from the settlement prices and the hypothetical bid/ask, using the implied discount factors and forwards. In Figure 02 we can see the evolution of the underlying FESX futures over the three months.

#### Figure 02: FESX Daily Settlement Prices

The sheet allows one to configure the details of the trade:

- Whether to trade the initial portfolio at the settlement prices or add a trading cost (suggested hypothetical bid/ask are there).
- Whether to trade the futures exactly at the settlement price, or to add some slippage/cost (e.g. 1 EUR, though more complex choices could be easily coded in as a function of size, for example).
- The size of the trade can be specified in vega notional or just simply notional. Note that the combination of trade size and multiplier can have some effect because of the needed rounding.
- Either the standard or normal variance can be chosen. The initial portfolio of options is constructed taking into account the fact that the strikes are not evenly spaced. The edges of the corridor can be specified. Other generalised variances can be coded by the reader relatively easily.

Given this particular price history, a wide corridor will lead to profit (loss) if short (long) variance, while a very tight range around the ATM could have shown the opposite, as shown in Table 01. There we choose six different configurations of trades, all short variance: Either standard or normal; on different ranges (wide with 81 strikes, tight with 7); with or without trading costs. We show P/L as a % of notional, the realised volatility of executed futures prices, and the volstrike i.e. the square root of the variance strike X. Note how the latter changes considerably with corridor width.

Wide Range: 1200-3950 - No Trading Costs | |||

Variance Type | P/L (in %) | Realised Vol | Vol Strike |

Standard | 2.25% | 19.3% | 24.4% |

Normal | 1.55% | 19.1% | 22.8% |

Wide Range: 1200-3950 - With Trading Costs | |||

Variance Type | P/L (in %) | Realised Vol | Vol Strike |

Standard | 1.45% | 19.7% | 23.0% |

Normal | 0.75% | 19.5% | 21.5% |

Tight Range: 2850-3000 - With Trading Costs | |||

Variance Type | P/L (in %) | Realised Vol | Vol Strike |

Standard | -0.60% | 15.2% | 12.8% |

Normal | -0.65% | 14.9% | 12.6% |