Limits of Replicating VIX with Delta-Hedged SPX Straddles

# Trading a synthetic replication of the VIX index


# 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.)

## Answer by Shane (score 19, accepted)

https://quant.stackexchange.com/a/143

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:

## Answer by jaredwoodard (score 4)

https://quant.stackexchange.com/a/4295

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.

- 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.

- 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.

- 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.

- 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.

## Answer by Andrew (score 3)

https://quant.stackexchange.com/a/4831

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 :).

## Answer by user59 (score 2)

https://quant.stackexchange.com/a/387

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?

## Answer by anonymous (score 1)

https://quant.stackexchange.com/a/10580

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