Category / Local volatility

Vexed Vix February 24, 2011 at 10:09 am

FT alphaville asks Can you trust the Vix?

The answer rather depends on what you mean by ‘trust’. ‘Can you trust the price of VIX futures to provide the markets’ best estimate of what realised S&P volatility will be?’ is a different and more difficult question to ‘Can you trust the the price of VIX futures to reflect the price at which people are willing to buy and sell VIX futures?’. The connection between the first and second of these is problematic, as we will see.

Why might the price of a Futures contract on a commodity reflect the expectation of the spot price in the future? Because if not, the naive argument goes, you can buy or sell the spot vs. the Future and profit from that difference. That is all very well if you can indeed do both transactions, and if your arbitrage bound includes the costs of doing that (i.e. storage of the commodity or its borrow cost, interest on borrowed or invested money, and so on). But how do you take a position in volatility?

The simple answer is to trade an option. If you delta hedge (at least in the Black-Scholes world), your P/L is a function of the difference between the vol you paid, implied, and the vol that is realised. If the VIX suggests vol is going to be higher than you think it is, and if you believe that the VIX will reflect implieds in the future, then sell an option and delta hedge to the VIX expiry.

Unfortunately that doesn’t work that well as the sensitivity of the option to volatility is a function of spot. It’s like buying a commodity then finding there is a different amount in the warehouse every day. This isn’t helpful. Moreover the VIX prices off the whole volatility smile not just the at-the-moneys, so even if you knew what the at-the-money level was going to be at expiry, you would still be exposed to smile. You could just trade the vol (or var) swap, but that’s OTC so there is a significant amount of infrastructure needed to do the trade.

The other alternative is to take some risk. Let’s look at the VIX curve. It is, as Alphaville notes, in steep contango. If we were to view this as wrong, then we would want to sell the VIX futures and hope that volatility will be lower at expiry than the future predicts. Who would be the other side in this trade? Well, dealers who are short volatility through selling options can hedge by buying the VIX, and they might be keeping the curve in contango. It is much more common to worry about being short vol than long as the OTC equity derivatives business tends to be more about selling options than buying them, so often the industry position is structurally short. What all this means is that there is a good reason for the VIX curve to be in contango, and a good reason to just keep rolling a futures position front or second month vs. one year. If it were me, then, I’d forget the no arb arguments – the arb is too hard to get on, and there is too much money on the other side for it to come in – and just play the curve.

Reverse converts do what they say on the tin January 9, 2011 at 6:06 am

Yep, they reverse profit into loss for the investor, or so Bloomberg reports.

Now admittedly with reverse converts the key issue is selecting the strike, given the smile for the stock: get that wrong, and the product will suck. Sadly it seems that that happened all too often.

Warren’s puts November 25, 2008 at 10:53 pm

There is a very nice post on Financial Crookery about Warren Buffett’s written puts.

Let us assume that BRK sold $40bn notional 20 year puts (over 4 indices) in 2006-2007 at an average equivalent S&P 500 level of 1400. At the prevailing swap rate and dividend yields, and implied volatility of around 24%, this would have realised premia of approximately $4.5bn, close enough to the premia actually received not to worry too much about the exact details of the transactions.

The undiscounted future value of this liability, ie the fair value expectation of payment in 2027, is presently around $19bn. (At the money long dated volatility has expanded to 38%; this option now is well in the money and the skewed volatility for 1400 strike is more like 33%). The present value of this liability, before the impact of credit spreads, is around $10bn using the current swap curve.

So far, so simple. But this valuation does not take account of the credit spread of the writer of the put…

[The writer then goes on to estimate the credit effect and to speculate on whether BRK uses such credit-effected prices for its own mark to market. My reading of FAS 157/159 is not only that it can but that it must.]

The only issue I might take issue with it is that the article uses Black Scholes with vols that seem rather low (33%) to value Warren’s 18 year puts. These are far out of the money forward, and I am always a bit nervous about using Black Scholes for long-dated OTM puts – my guess would be that different process-theoretic assumptions would increase the value of the position (i.e. increase Warren’s loss). Kudos to Goldman though for buying these options: all that downside vol in size must make hedging their index books fun at the moment.

Whither volatility? October 28, 2008 at 6:19 am

The VIX opened on Friday 24th at 89.03. What does that mean? Using the root-t rule (which is extremely questionable at the moment, admittedly) that scales to an average daily move of 5.6%. Now admittedly things have been volatile recently. But even taking the period from 6th-24th October, the daily move in the S&P was only 4.4%. If you didn’t have any other reason to trade (such as hedging a short vega position that you are panicing about), you would only buy the VIX at 89 if you thought the average daily move in the next 30 days was going to be even higher than it has been over the last two weeks extreme though they were. I’m not saying that can’t happen, but it does seem unlikely to me.

Lessons from 2001 October 10, 2008 at 10:27 am


The last time there was this big a market rout, I learned a few lessons.

  • Vol is really expensive. Sell it around the money. But the wings are dangerous. I like being short vega in this kind of environment, but to get longer on big market falls or rallies.

  • Whipsaws happen. Live with it and plan your rehedge frequency accordingly: failing to capture the whipsaw if you are long gamma loses you a lot of potential upside. Look at your 1%, 2.5%, 5% and 10% deltas, not just the instantaneous one.

  • Gamma holes can kill you at any strike within 30% of the money. Fill them as cheaply as you can.

  • Correlation structures break down completely in environments like this one. Be especially careful of assumptions about cross gamma or strategies like dispersion trading that rely on stable correlations.

And if that doesn’t work, pull up the drawbridge.

Volatility smolatility January 19, 2008 at 7:43 pm

The FT points out what is perhaps obvious, that volatility of equity indices is rising. I hadn’t appreciated the size of the issue, though, until I saw this:

Wednesday’s intra-day whipsaw of 632 points on the Dow Jones Industrial Average is the fifth largest on record (the top four all occurred during the tech bubble collapse).
The Vix is rising off a long period of relative calm – it averaged 13 in the three years ending July 2007.

13? 13 is far, far too low. I remember FTSE five year ATM vols going to 40 at the peak of the LTCM debacle. Even in the glory days before that 13 was a low number for short dated vol (the VIX is ATM one month S&P vol). Still, that little episode must be causing some amusing calibration issues for mean reverting stochastic vol models…

Does a steeper skew presage a crash? October 19, 2007 at 6:38 pm

The S&P skew has steepened significantly of late. Does that mean anything, other than more buyers of downside options? In particular is there any evidence that a steeper smile is associated with large moves down? In the other direction it works – if there is a big fall then vols rise and the skew steepens. But I know of no studies that suggest skew steepening is a useful predictor of falls. It would be interesting to know though… after all, we are close to the anniversary. Certainly it seems likely that with all the press coverage, the big 2 0 is spooking investors. But of course that doesn’t mean we won’t have a crash.

Smoothly runs the Don June 21, 2007 at 3:35 pm

Earlier, I wrote something about changing distributions. More recently a couple of examples of this phenomena have come up, so let’s make things concrete.

Suppose x(t) is randomly distributed according to N(0,s(t)) [normal distribution with mean zero and standard deviation s(t)] where s(t) is a continuous, bounded and slow function of t. Suppose we sample x(t) discretely. (Take s(t) = 2 + sin(t) with t in years and daily sampling, for instance.)

The variables x(t) are not iid, but (under a bunch of smoothness conditions) they are locally iid: we can think of the distribution being fibred over time and a small change in time inducing a small change in the distribution. One might hope that one could import a lot of non parametric statistics into this setting, where we were trying to gain information about the variation of s(t) by sampling the xs.

Note that this is not a stochastic volatility model or a GARCH model: volatilities are not random, but rather determined by an initially unknown function.

Is this sort of thing well known I wonder? It’s similar to local volatility models, but there we have a situation where we can deduce s(t) with certainty as we know the prices of all vanilla options, whereas I’m more interested in a situation where we can only observe s(t) by sampling x(t).

One of the applic- ations, as in this picture of a large piece of metal hanging from a crane a hundred feet above a busy junction, is operational risk, but there are others which may be even more slippery.