## Wednesday, January 05, 2011

### Shorting the VIX calendar spread

Lately there were a few interesting discussions in the blogosphere on the profitability of shorting the VXX-VXZ spread. (See Quantum Blog and The Speculator's Ball.) For background, VXX is an ETN that tracks the first and second month of the VIX future, which in turn tracks the VIX volatility index, which in turn tracks the volatility of SPX. VXZ is the ETN that tracks the 4th - 7th months of the VIX future. During the period 2009-2010, there were 2 different reasons why shorting this "calendar spread" was profitable:

1) The VIX futures were/are in contango: i.e. the back months' futures are more expensive than the front months'.
2) The volatility of SPX was decreasing with time.

However, some traders seem to think that either one of these conditions is enough to ensure the profitability of shorting a calendar spread.  It is not. (Otherwise, life as a futures trader would be too easy!)

To see this, let's resort to a simplistic linear approximation to a model of futures prices. From John Hull's book on derivatives, section 3.12, the price of  a future which matures at time T is

F(t, T)=E(ST)exp(c(T-t)),

where E(ST) is the expected value of the spot price at maturity, c is a constant, and t is the current time. If the futures are in contango, then c > 0.

If we assume that abs(c) is small, and T-t is also small (i.e. not too far from maturity), and that the expected value of the spot price changes slowly, we can linearize this formula as

F=(a+b(T-t))*(1+c(T-t))

If the market expects the future spot price to increase, then b > 0.

After a few simple algebraic steps, you can verify that the calendar spread's price is proportional to

F(t, T1)-F(t, T2) ~ bct

where T1 < T2 (i.e. F(t, T1) is the front month's price, and F(t, T2) the back month's).

This is a satisfyingly illustrative result. It says that shorting this calendar spread will be profitable if

A) futures are in contango and the expected spot price in the future is decreasing; or else
B) futures are in backwardation and the expected spot price in the future is increasing.

So what is the situation today? Will it still be profitable to short this spread? As our fellow bloggers have pointed out, VIX futures are still in contango, but the market is expecting volatility to increase in the future over the last month or so. So this may no longer be a profitable trade anymore.

Pollux Technicals said...

Great post Mr. Chan! With an understanding of the general concept mentioned above, I often hear about slippage in the ETF world. I am wondering what the cost of slippage, if any, is associated with a trading strategy like this vs. buying the strategy outright? Thanks

Ernie Chan said...

Hi Pollux,
I haven't found any slippage trading ETF that are different from the usual slippage in trading stocks, especially not for liquid ETF's like VXX and VXZ which have very tight spreads.
Ernie

KHG said...

Hy, are you sure you can just take Hull's ordinary formula and apply it to VIX Futures? According to the VIX primer of CBOE (http://cfe.cboe.com/education/vixprimer/Features.aspx): 'Since there is no carry between VIX and a position in VIX futures, the fair value of VIX futures cannot be derived by a similar relationship.'. They give 100*sqrt(365/30*(P_t-var_t[F_T])) as fair value for VIX-Futures (P_t: forward variance price, and -var_t as concavity adjustement).

Ernie Chan said...

Hi KHG,
1) I don't think Hull's formula depends on any assumptions of carry-cost. It is based on fairly general arbitrage arguments.
2) The CBOE page talked about "fair value" of the futures contract, which is the price of the future given that we know the empirical implied volatility of the SPX. However, this formula doesn't tell us the analytical formula for this implied volatility as a function of the expiration date. That is, it doesn't propose a term-structure model. So to understand whether the calendar spread trade is profitable or not, we still need such a term-dependent model, the simplest of which is a linear one given in my blog post.
Ernie

Paul Teetor said...

Thank you for your post, Ernie. This is a good subject that warrants some discussion.

For several months, I've had good luck with a nuanced version of this spread: trading the 1-month roll-down in VIX futures. The futures curve is quite steep in the front months, and it's quite flat in the back months. The natural trade, then, is selling a near-by month (but not the front month) and buying a farther-out month.

For instance, I am currently short March and long May. The expected P&L is \$1,000 per spread over the next month. The risk is that the futures curve will flatten and the trade will be a bust -- or worse. So far, so good.

In a month, I'll close the spread, then reevaluate the curve and choose a new roll-down pair, if any.

KHG said...

Thanks for your reply! My understanding of the fair-value formula of the CBOE was that it contains at least (some) parts of a term-structure model as (a) prices of options with suitable expiries are multiplied with an time-dependent interest rate term in the determination of the forward price P_t and (b) the concavity adjustment also depends on the time to expiry.
Please correct, but if I understood you correctly your formula basically is meant to reflect a model for the term-structure of implied vol as a function of time to expiration? So I think, if this model is correct, one could plug it into the calculation of the forward price of de-annualized variance P_t .

Ernie Chan said...

Hi KHG,
Yes, the CBOE fair value does contain time value adjustment, but you would still need to plug the empirical values of various variances in order to come up with the fair value. The question remains: what is the time-dependence of these empirical variances?

But yes, if you use my linear model, you can certainly back out the forward price of the variances, provided you know the parameters a, b, and c!

Ernie

Anonymous said...

Dear Ernie,

Do most of your strategies revolve around linear modelization or do you venture in non-linear with neural nets?

Thanks,

Al

Ernie Chan said...

Hi Al,
I dislike most nonlinear models. You can search for my blog posts on artificial intelligence and you will see why.
Best,
Ernie

J.F. said...

Ernie,
Thank you for the informative post.
In your book, and on this blog, you concentrate on spreads that are mean reverting. VXX-VZX appears to be a spread that is in a trend (but, as you point out, for how long?).
Are you presenting this mainly as an academic example/ intellectual exercise, or do you think it is a good idea to attempt to profit off trending spreads?
Regards,
J.F.

Ernie Chan said...

Hi J.F.,
The analysis here is not purely academic, as it points out what I think is a common misconception among traders. For more debate on this topic, see http://sdanlovell.wordpress.com/2011/01/11/the-vxxvxz-trade/.

My personal view is that this spread will eventually mean-revert, but this is just a hunch with no theory behind it.

Ernie

Anonymous said...

When we are at it with the nonlienear stuff. I Have a question about SVM post. The model doesn't seem to be able to generate good sharp but beside that. You wrote once that the model that seem to work for you do not involve non linear fitting. Do you consider SVM as one of the artificial intelligence crap? Any Kernel based Method?

I'm starting to learn about kernel based pattern analysis and don't really know for now were I'm heading.

Thanks

Ernie Chan said...

Anon,

From my personal backtests, SVM has shown no ability to generate consistent profits.

If you find that a nonlinear method generates good out-of-sample backtest results in a consistent way, my kudos to you!

Ernie

drtom said...

Hi, in your book you present Thorp's formula for continuously compounded growth. I looked at Thorp's paper and in his derivation on pg23 he skips the steps where develops the previous formula into a power series. Despite efforts of me and my friends we could not figure out how he gets there. We were close but not quite there. This is the only thing in his paper I didn't manage to derive. I would be very grateful if you could point me to any source where this derivation is done properly. Thanks in advance for your help.

Ernie Chan said...

drtom,
You can direct your question to Dr. Thorp himself at edwardothorp.com. I have corresponded with him before and he has been very generous with his time.
Ernie

Anonymous said...

Hello Ernie!

I'm new to trading, could you give me a clue what trading platform which enables me to customize it, write my own scripts and manipulate data with less possible restrictions should I use? I just don't know what to begin with. Thank you!

Ernie Chan said...

Hi Anon,
When you said "write my own scripts", does that mean you are familiar with programming? If so, you can easily write Visual Basic programs and run them with Interactive Brokers.

Tradestation is of course another popular choice.

For forex and futures traders, I heard Metatrader is a popular platform that can be used with multiple brokerages. Maybe readers who have experience with Metatrader can comment on it?

Ernie

Anonymous said...

Yes, I'm familiar with programming. I have my own plan for developing a trading machine and all I need from the trading platform is that it be able to execute orders, and give me access to data.

I'll check these out, thank you!

Anonymous said...

Hi,

Great post! But, shouldn't

F(t, T)=E(ST)exp(c(T-t))

simple be

F(t, T) = St exp(c(T-t)) where St is the underlying's price at t?

Ernie Chan said...

Hi Anon,
No, the expected spot price E(ST) at time T is in general not equal to the current spot price St at time t.
Ernie

George said...

Hi Ernie,

Could you direct me to some literature deriving the linear approximation of the exponential? I imagine it is a very useful tool! However I am not sure I have an intuitive understanding where the coefficients a and b come from. Thanks for posting your insight!

Ernie Chan said...

Hi George,
See Taylor series expansion of the exponential function. For e.g.
http://en.wikipedia.org/wiki/Taylor_series

Ernie

Anonymous said...

Hi Ernie,
Thanks for the interesting post. I got a question. The condition for contango should be

ac + b > 0

because that's the coef of linear term for futures price. Let me know if I miss something.

Best,
VV

Ernie Chan said...

Hi VV,
Yes, you are right. Contango is determined by ac+b > 0 instead of just c > 0.
However, the main conclusion of my post is still correct. If b>0 and c>0, then we have contango (since a > 0 always, as futures price cannot be negative), and also the calendar spread will be profitable. If b<0 and b<0, then we have backwardation, but the calendar spread is still profitable!
Ernie

Anonymous said...

Ernie,
Thanks for the explanation. I think it is still true that your conclusion is much weaker now. One cannot argue that because term structure is contango(ac+b>0) and expectation of underlying is increasing(b>0) shorting the spread would not be profitable(bc>0).

VV

Anonymous said...

p.s. Therefore you cannot forecast the profitability of the trade by just observing the term structure and market expectation of the underlying trend

Ernie Chan said...

Anonymous,
Actually, I think we can still forecast the profitability of shorting the calendar spread based on the term structure and the sign of b.

Here is my argument. If we have contango as well as b < 0, then ac+b > 0 will mean ac/b < -1,
which means ac/b < 0, which means
bc < 0 since a > 0 as always, which implies shorting the calendar spread will be profitable.

(In my previous answer to you, I incorrectly stated that if b>0 and c>0, then shorting the spread would be profitable. I should have said "unprofitable".)

Ernie

Anonymous said...

Ernie,
I totally agree with you on the case where term structures of futures and expectation on underlying are opposite. However when they are in the same direction like the case you consider at the end of the post one cannot say one way or the other.

VV

Ernie Chan said...

Anon,
Yes, when we have contango and b>0, the profit is indeterminate, just as when we have backwardation and b<0.
But when we have contango and b<0, or backwardation and b>0, then shorting the calendar spread will be profitable.

Thanks for the excellent point and the discussion!
Ernie

gray13685 said...

Excellent topic. I've been doing research into the structures of these ETNs for similar trades, namely short the VXX by itself as a way to profit from the contango on the front end of the curve, essentially shorting the M2-M1 calendar spread.

Many make the assumption that contango is the market's way of saying E(ST) > spot VIX. However, I feel there are structural reasons keeping a persistent contango in the VIX futures market.

One of my observations, and something I'd appreciate your insight on, is that the VXX has become very large in proportion to the first and second month futures contracts it replicates. Back of envelope math shows that if iPath were to actually hedge its exposure by rolling its Month1 contracts into Month2 contracts bit by bit, they'd account for 30-40% of daily trading volume on the VIX futures.

This volume of buying would certainly push down the price of the contract being sold and push up the price of contract being bought, tipping the scales towards contango, which has been borne out by the past 6 or 7 years' data.

You're certainly right that contango by itself isn't enough to "ensure the profitability of shorting a calendar spread" but is enough to tip the odds in your favor. Proper risk controls are essential however, since the VIX can take the escalator down, but the elevator back up!

Ernie Chan said...

Hi gray,
Excellent point!

Yes, quite a few strategies involving ETF and ETN's are due to this pressure to rebalance or rollover. A strategy for trading the ultras (ETF's that deliver 2x or 3x the daily returns of SPX) comes to mind -- there is pressure to buy SP futures if SPX goes up, and to sell if SPX goes down. I have also heard of rollover pressures affecting commodities ETF's that hold commodities futures, but could never find the effect through backtest. Perhaps you have seen articles on this rollover effect? If so, please share with us here!

Ernie

gray13685 said...

I haven't seen any other sources commenting on the structural tendency towards contango in VIX, let alone the ETN's own impact on it.

I just posted an article here: http://seekingalpha.com/article/267950-ipath-s-p-500-vix-short-term-futures-etn-swimming-against-the-currents on the subject where I include data on the Month 1 to Month 2 roll yield going back to 2004 (over 75% of the time is contango)

I'm interested in any feedback of the theory