# Trading a synthetic replication of the VIX index

One cannot directly buy and sell the VIX index. Theoretically, however, one could approximate the index by purchasing an at-the-money straddle on the SP500, then delta-hedging the straddle.

Does anyone have experience with such a "synthetic" replication of the index? It might be very useful for betting on volatility or for spreads against the VIX futures (a sort of basis trade), but I can see potential problems if the replication is too inaccurate.

(To anticipate your comments: I'm aware of the many VIX-related ETFs; but, no, I would not consider using them. I'm also aware that the VIX calculation uses other strikes beyond the ATM options; this proposed synthetic is admittedly an approximation.)

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Are you aware of CBOE's VIX futures and options? cboe.com/micro/VIX/vixintro.aspx ... If you don't like futures, you can always synthetic by buying a call and selling a put. –  barrycarter Feb 3 '11 at 8:41
Why not use the ETNs? You can go long or short and I'm pretty sure that their tracking error is less than we could do as a solo trader. ipathetn.com/VXX-overview.jsp?investorType=pro –  Richard Herron Feb 3 '11 at 12:50
@barrycarter He mentions the VIX futures in the second paragraph, so I would infer that he's familiar... –  Shane Feb 3 '11 at 14:42
@shane @pteetor My bad.. I read the "VIX-related ETFs" paragraph, but missed the "spreads against VIX futures". I'll pretend I meant to say VXX :) –  barrycarter Feb 3 '11 at 15:24
@richardh According to my calculations, VXX has an 80% correlation with the spot VIX index... and their returns are only 44% correlated. I suppose VXX would be better than nothing, but I am exploring the alternatives and looking for a smaller tracking error. –  pteetor Feb 3 '11 at 15:38

A synthetic model for the VIX would be quite useful. I just mention this since it has been covered elsewhere in the past, although I don't think that it's a real solution to your problem (for a number of reasons).

Several blogs posted on the "William's VIX Fix" (WVF) in the past: marketsci, trading the odds, mindmoneymarkets. The WVF is intended to be a synthetic VIX calculation, derived by Larry Williams (see the original article here), and is represented by the following formula:

$wvf = \frac{Highest(Close, 22) - Low}{Highest(Close, 22)} * 100$

In R, this can be represented as:

wvf <- function(x, n=22) {
hc <- as.xts(rollmax(as.zoo(Cl(x)), k=n, align="right"))
100*(hc-Lo(x))/hc
}


This has had a reasonable correlation to the VIX: from 1995-2010 it was +0.75:

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@shane Nice work! Thank you for such a detailed reply. I've seen the VIX Fix before, and I like it for markets where no one publishes the daily implied volatilties. The problem here is that I cannot trade the VIX Fix either, just like I cannot (directly) trade the VIX itself. I'm looking for a tradable synthetic, leading to opportunities for spreads and out-right positions. –  pteetor Feb 3 '11 at 15:43
@pteetor Great point. I have heard of short term strategies that trade the S&P that replicate this, but I'm not going to venture into how that can be done (assuming that it's even desirable). My other suggestion: you would need to be an institutional investor (with an ISDA), but you might be able to do a variance swap with a bank based on the VIX. –  Shane Feb 3 '11 at 15:49
@pteetor Just to add one final comment: I think that you've already listed the primary vehicles (futures, ETF's) for this. Short of either (a) doing a swap or (b) developing an algorithm, I think that you may be out of options. :) –  Shane Feb 3 '11 at 15:55
@shane Thanks, Shane. At this point, I think an experiment is in order. I will simply trade my proposed synthetic on a (very) small scale, monitor the results, and hope to learn from the outcome. In the words of Robert Stovall, "Selling a soybean contract short is worth two years at the Harvard Business School." It's time to get some hands-on experience. –  pteetor Feb 3 '11 at 16:45
@pteetor + $\infty$ –  Shane Feb 3 '11 at 16:49

One final thought: you want something that depends on volatility, but not on price. In other words, if SPX goes up, your VIX replicant wouldn't necessarily change at all.

Would an SPX option calendar spread behave something like this, since option prices change w/ volatility, and short-term options change more than long-term options?

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to match the constant 30-day VIX horizon, I think you would want to trade two straddles in the first and second expiration cycles and delta hedge, gradually rolling the weight towards the second month straddle and then finally to a new straddle at/near expiration each month. Here are some problems I can imagine for this approximation.

1. Hedging error - the details of the delta hedging regime matter as there seems to be a wide range of BSM etc. hedging errors in the literature. This paper looks promising.
2. Missing the boat on large moves - beyond a certain % move in the underlying, the hedged straddles just become a synthetic long or short position, i.e. they "run out" of gamma. This matters in cases of a major rally or crash, where the VIX is priced with the most weight at some strike at 80% or 120% of the moneyness of the strike of your existing straddles. This would also be a problem for a replication of VXO.
3. Skew changes - There are also those cases where the market is moving modestly, price-wise, and yet VIX changes not because traders are repricing volatility but because the IV skew is changing. Even though OTM options have less weight, they can cause the index to move if the bids change enough to steepen or flatten the curve. Here a replication of VXO would be easier.
4. There are some calendar and calculation quirks with VIX and its component indices VIN and VIF that can be ignored if your replication needs are relaxed enough, but otherwise would introduce some tracking error, esp. intraday.

The CFE is (re)listing its S&P 500 variance futures soon. Those also have fixed settlement dates, so you would need to roll them continuously to maintain a constant time exposure matching VIX, but if they attract liquidity I expect they'll be a better fit than the straddles.

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At my previous job, I have, actually, replicated VIX index for a structured note. Straddles are not going to work - mainly because of differential decay and gamma creep. You can try replicating the strip, but on such a short-dated horizon (30 days) you will be pretty hard pressed to keep the weighting perfect. –  Strange Oct 9 '12 at 4:05
@Jared Thank you very much for your thoughtful and complete reply. These are excellent observations. Thanks, too, for bringing the variance futures to my attention. The "if they attract liquidity" thing is a problem. Does any contract but the Big VIX really trade on the CFE? –  pteetor Oct 11 '12 at 0:44
@Strange Thanks for contributing your real experience. All in all, I have concluded that replicating the cash VIX is too difficult. I am sticking with VIX curve trades using only the futures. –  pteetor Oct 11 '12 at 0:46
@Paul - Nothing trades like the big VIX, but the new Brazil, EEM, oil, and gold products have been trading pretty regularly, sometimes several hundreds a day, which is promising. –  jaredwoodard Oct 11 '12 at 19:36
@Strange So any thoughts on alternative approaches? –  user915 Oct 15 '12 at 3:53

the true way to replicate the vix is to use an infinite strip of out of the money calls and puts and actually, this is the definition of vix. it is $\sqrt{\int^{T+\Delta}_T v_s ds}$ where $v_s$ is realized variance.

Peter Carr showed that we can value any exotic payoff, free of any option model by using the spanning formula.

Let $g(S_T)$ be the exotic payoff that you are trying to replicate, then: $\mathbb{E} [g(S_T)] = g(F) + \int^F_0 dK \tilde{P}_K g''(K) + \int^\infty_F dK \tilde{C}_K g''(K)$ where $C_K, P_K$ are the values of the call options and put options which we can get from the market. Now, the funky payoff that turns out the most interesting is $log\frac{S_T}{S_0}$ since this payoff replicates the total variance. If you are familiar with stochastic calculus, you can perform ito's lemma on this and then you can value exactly the vix spot index. I know for a fact that this is what many banks are doing :).

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How about trading the full string of options? http://www.cboe.com/data/variancestrips/intro.aspx

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