Wednesday, July 02, 2014

Another "universal" capital allocation algorithm

Financial engineers are accustomed to borrowing techniques from scientists in other fields (e.g. genetic algorithms), but rarely does the borrowing go the other way. It is therefore surprising to hear about this paper on a possible mechanism for evolution due to natural selection which is inspired by universal capital allocation algorithms.

A capital allocation algorithm attempts to optimize the allocation of capital to stocks in a portfolio. An allocation algorithm is called universal if it results in a net worth that is "similar" to that generated by the best constant-rebalanced portfolio with fixed weightings over time (denoted CBAL* below), chosen in hindsight. "Similar" here means that the net worth does not diverge exponentially. (For a precise definition, see this very readable paper by Borodin, et al. H/t: Vladimir P.)

Previously, I know only of one such universal trading algorithm - the Universal Portfolio invented by Thomas Cover, which I have described before. But here is another one that has proven to be universal: the exceedingly simple EG algorithm.

The EG ("Exponentiated Gradient") algorithm is an example of a capital allocation rule using "multiplicative updates": the new capital allocated to a stock is proportional to its current capital multiplied by a factor. This factor is an exponential function of the return of the stock in the last period. This algorithm is both greedy and conservative: greedy because it always allocates more capital to the stock that did well most recently; conservative because there is a penalty for changing the allocation too drastically from one period to the next. This multiplicative update rule is the one proposed as a model for evolution by natural selection.

The computational advantage of EG over the Universal Portfolio is obvious: the latter requires a weighted average over all possible allocations at every step, while the former needs only know the allocation and returns for the most recent period. But does this EG algorithm actually generate good returns in practice? I tested it two ways:

1) Allocate between cash (with 2% per annum interest) and SPY.
2) Allocate among SP500 stocks.

In both cases, the only free parameter of the model is a number called the "learning rate" η, which determines how fast the allocation can change from one period to the next. It is generally found that η=0.01 is optimal, which we adopted. Also, we disallow short positions in this study.

The benchmarks for comparison for 1) are, using the notations of the Borodin paper,

a)  the buy-and-hold SPY portfolio BAH, and
b) the best constant-rebalanced portfolio with fixed allocations in hindsight CBAL*.

The benchmarks for comparison for 2)  are

a) a constant rebalanced portfolio of SP500 stocks with equal allocations U-CBAL,
b) a portfolio with 100% allocation to the best stock chosen in hindsight BEST1, and
c) CBAL*.

To find CBAL* for a SP500 portfolio, I used Matlab Optimization Toolbox's constrained optimization function fmincon.

There is also the issue of SP500 index reconstitution. It is complicated to handle the addition and deletion of stocks in the index within a constrained optimization function. So I opted for the shortcut of using a subset of stocks that were in SP500 from 2007 to 2013, tolerating the presence of surivorship bias. There are only 346 such stocks.

The result for 1) (cash vs SPY) is that the CAGR (compound annualized growth rate) of EG is slightly lower than BAH (4% vs 5%). It turns out that BAH and CBAL* are the same: it was best to allocate 100% to SPY during 2007-2013, an unsurprising recommendation in hindsight.

The result for 2) is that the CAGR of EG is higher than the equal-weight portfolio (0.5% vs 0.2%). But both these numbers are much lower than that of BEST1 (39.58%), which is almost the same as that of CBAL* (39.92%). (Can you guess which stock in the current SP500 generated the highest CAGR? The answer, to be revealed below*, will surprise you!)

We were promised that the EG algorithm will perform "similarly" to CBAL*, so why does it underperform so miserably? Remember that similarity here just means that the divergence is sub-exponential: but even a polynomial divergence can in practice be substantial! This seems to be a universal problem with universal algorithms of asset allocation: I have never found any that actually achieves significant returns in the short span of a few years. Maybe we will find more interesting results with higher frequency data.

So given the underwhelming performance of EG, why am I writing about this algorithm, aside from its interesting connection with biological evolution? That's because it serves as a setup for another, non-universal, portfolio allocation scheme, as well as a way to optimize parameters for trading strategies in general: both topics for another time

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Workshops Update:

My next online workshop will be on  Mean Reversion Strategies, August 26-28. This and the Quantitative Momentum workshops will also be conducted live at Nanyang Technological University in Singapore, September 18-21.

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Do follow me @chanep on Twitter, as I often post links to interesting articles there.

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*The SP500 stock that generated the highest return from 2007-2013 is AMZN.

Friday, May 09, 2014

Short Interest as a Factor

Readers of zerohedge.com will no doubt be impressed by this chart and the accompanying article:

Cumulative Returns of Most Shorted Stocks in 2013

Indeed, short interest (expressed as the number of shares shorted divided by the total number of shares outstanding) has long been thought to be a useful factor. To me, the counter-intuitive wisdom is that the more a stock is shorted, the better is its performance. You might explain that by saying this is a result of the "short squeeze", when there is jump in price perhaps due to news and stock lenders are eager to sell the stock they own. If you have borrowed this stock to short, your borrowed stock may be recalled and you will be forced to buy cover at this most inopportune time. But this is an unsatisfactory explanation, as this will result only in a short term (upward) momentum in price, not the sustained out-performance of the most shorted stocks. This long-term out-performance seems to suggest that short sellers are less informed than the average trader, which is odd.

Whatever the explanation, I am intrigued to find out if short interest really is a good factor to incorporate into a comprehensive factor model over the long term.

The result? Not particularly impressive. It turns out that 2013 was one of the best years for this factor (hence the impressive chart above). For that year, a daily-rebalanced long-short portfolio (long 50 most shorted stocks and short 50 least shorted stocks in the SPX) returned 6.9%, with a Sharpe ratio of 2 and a Calmar ratio of 2.9. However, if we extend our backtest to 2007, the APR is only 2.8%, with a Sharpe ratio of 0.5 and a Calmar ratio of 0.3. This backtest was done using survivorship-bias-free data from CRSP, with short interest data provided by Compustat.

Here is the cumulative returns chart from 2007-2013:

Cumulative Returns of LS Portfolio based on Short Interest: 2007-2013


Interesting, trying this on the SP600 small-cap universe yielded negative returns, possibly meaning that short-sellers of small caps do have superior information.

I promise, this will be the last time I talk about factors in a while!

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Tech Update:

I was shocked to learn that Matlab now offers licenses for just $149 - the so-called Matlab Home  (h/t: Ken H.) In addition, its Trading Toolbox now offers API connection to Interactive Brokers, in addition to a few other brokerages. I am familiar with both Matlab and R, and while I am impressed by the large number of free, sophisticated statistical packages in R, I still stand by Matlab as the most productive platform for developing our own strategies. The Matlab development (debugging) environment is just that much more polished and easy-to-use. The difference is bigger than Microsoft Word vs. Google Docs.
A reader Ravi B. told me that there is a website called www.seasonalgo.com if you want to try out different seasonal futures strategies.
Finally, a startup at inovancetech.com offers machine learning algorithms to help you find the best combination of technical indicators for trading FX.
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Workshops Update:

I am now offering the Millisecond Frequency Trading (MFT) Workshop as an online course on June 26- 27. Previously, I have only offered it live in London and to a few institutional investors. It has two main parts:

Part 1: introducing techniques for traders who want to avoid HFT predators.

Part 2: how to backtest a strategy that requires tick data with millisecond resolution using Matlab.

The example strategy used is based on order flow. For more details, please visit epchan.com/my-workshops.

Additionally, I will be teaching the Mean Reversion and Momentum (but not MFT) workshops in Hong Kong on June 17-20.

Thursday, March 27, 2014

Update on the fundamentals factors: their effect on small cap stocks

In my last post, I reported that the fundamental factors used by Lyle and Wang seem to generate no returns on SP500 large cap stocks. These fundamental factors are the growth factor return-on-equity (ROE), and the value factor book-to-market ratio (BM).

I have since studied the effect of these factors on SP600 small cap stocks since 2004, using a survivorship-bias-free database combining information from both Compustat and CRSP. This time, the factors do produce an annualized average return of 4.7% and a Sharpe ratio of 0.8. Though these numbers are nowhere near the 26% return that Lyle and Wang found, they are still statistically significant. I have plotted the equity curve below.

2004-2013
Equity curve of long-short small-cap portfolio based on regression on ROE and BM factors (2004-2013)
One may wonder whether ROE or BM is the more important factor. So I run a simpler model which uses one factor at a time to rank stocks every day. We buy stocks in top decile of ROE, and short the ones in the bottom decile. Ditto for BM. I found an annualized average return of 5% with a Sharpe ratio of 0.8 using ROE only, and only 0.8% with a Sharpe ratio of 0.09 using BM only. The value factor BM is almost completely useless! Indeed, if we were to first sort on ROE, pick the top and bottom deciles, and then sort on BM, and pick the top and bottom halves, the resulting average return is almost the same as sorting on ROE alone. I plotted the equity curve for sorting on ROE below.

Equity curve of long-short small-cap portfolio based on top and bottom deciles of ROE (2004-2013)

Notice the sharp drawdown from 2008-05-30 to 2008-11-04, and the almost perfect recovery since then. This mirrors the behavior of the equity market itself, which raises the question of why we bother to construct a long-short portfolio at all as it provides no hedge against the downturn. It is also interesting to note that this factor does not exhibit "momentum crash" as explained in a previous article: it does not suffer at all during the market recovery. This means we should not automatically think of a fundamental growth factor as similar to price momentum.

My conclusion was partly corroborated by I. Kaplan who has written a preprint on a similar topic. He found that a long-short portfolio created using the ratio EBITA/Enterprise Value on large caps generates a Sharpe ratio of about 0.6 but with very little drawdown unlike the ROE factor that I studied above as applied to small caps.

As Mr. Kaplan noted, these results are in some contradiction not only with Lyle and Wang's paper, but also with the widely circulated paper by Cliff Asness et al. These authors found the the BM factor works in practically every asset class. Of course, the timeframe of their research is much longer than my focus above. Furthermore, they have excluded financial and penny stocks, though I did not find such restrictions to have great impact in my study of large cap portfolios. In place of a fundamental growth factor, these authors simply used price momentum over an 11-month period (skipping the most recent month), and found that this is also predictive of future quarterly returns.

Finally, we should note that the ROE and BM factors here are quite similar to the Return-on-Capital and Earnings Yield factors used by Joel Greenblatt in his famous "Little Book That Still Beats The Market". One wonders if those factors suffer a similar drawdown during the financial crisis.

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My online Momentum Workshop will be offered on May 5-7. Please visit epchan.com/my-workshops for registration details. Furthermore, I will be teaching my Mean Reversion, Momentum, and Millisecond Frequency Trading workshops in Hong Kong on June 17-20.


Saturday, February 08, 2014

Fundamental factors revisited, with a technology update

Contrary to my tradition of alerting readers to new and fancypants factors for predicting stock returns (while not necessarily endorsing any of them),  I report that Lyle and Wang have recently published new research demonstrating the power of two very familiar factors: book-to-market ratio (BM) and return-on-equity (ROE).

The model is simple: at the end of each calendar quarter, compute the log of BM and ROE for every stock based on the most recent earnings announcement, and regress the next-quarter return against these two factors. One subtlety of this regression is that the factor loadings (log BM and ROE) and the future returns for stocks within an industry group are pooled together.  This makes for a cross-sectional factor model, since the factor loadings (log BM and ROE) vary by stock but the factor returns (the regression coefficients) are the same for all stocks within an industry group. (A clear elucidation of cross-sectional vs time-series factor models can be found in Section 17.5 of Ruppert.) If we long stocks within the top decile of expected returns and short the bottom decile and hold for a quarter, the expected annualized average returns of this model is an eye-popping 26% or so.

I have tried to replicate these results, but unfortunately I couldn't. (My program generated a measly, though positive, APR.) The data requirement and the program are both quite demanding. I am unable to obtain the 60 quarters of fundamental data that the authors recommended - I merely have 40. I used the 65 industry groups defined by the GIC industry classifications, while the authors used the 48 Fama-French industry groups. Finally, I am unsure how to deal with stocks which have negative book values or earnings, so I omit those quarterly data. If any of our readers are able to replicate these results, please do let us know.

The authors and I used Compustat database for the fundamental data. If you do not have subscription to this database, you can consider a new, free, website called Thinknum.com. This website makes available all data extracted from companies' SEC filings starting in 2009 (2011 for small caps). There is also a neat integration with R described here.

*** Update ***

I forgot to point out one essential difference between the method in the cited paper and my own effort: the paper used the entire stock universe except for stocks cheaper than $1, while I did my research only on SP500 stocks (Hat tip to Prof. Lyle who clarified this). This turns out to be of major importance:  a to-be-published paper by our reader I. Kaplan reached the conclusion that "Linear models based on value factors do not predict future returns for the S&P 500 universe for the past fifteen years (from 1998 to 2013)."


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Speaking of new trading technology platforms that provide historical data for backtesting (other than Thinknum.com and the previously mentioned Quantopian.com), here is another interesting one: QuantGo.com. It provides institutional intraday historical data through its data partners from 1 minute bars to full depth of book in your own private cloud running on Amazon EC2 account for a low monthly rate. They give unlimited access to years of historical data for a monthly data access fee, for examples US equities Trades and Quotes (TAQ) for an unlimited number of years are $250 per month of account rental, OPRA TAQ $250 permonth and tagged news is $200. Subscribers control and manage their own computer instances, so can install and use whatever software they want on them to backtest or trade using the data. The only hitch is that you are not allowed to download the vendor data to your own computer, it has to stay in the private cloud.

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Follow @chanep to receive my occasional tweets on interesting quant trading industry news and articles.

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My online Mean Reversion Strategies Workshop will be offered on April 1-3. Please visit epchan.com/my-workshops for registration details. Furthermore, I will be teaching my Mean Reversion, Momentum, and Millisecond Frequency Trading workshops in London on March 17-21, and in Hong Kong on June 17-20.

Wednesday, January 08, 2014

Variance Risk Premium for Return Forecasting

Folklore has it that VIX is a reasonable leading indicator of risk. Presumably that means if VIX is high, then there is a good chance that the future return of the SP500 will be negative. While I have found some evidence that this is true when VIX is particularly elevated, say above 30, I don't know if anyone has established a negative correlation between VIX and future returns. (Contemporaneous VIX and SP500 levels do have a very nice linear relationship with negative slope.)

Interestingly, the situation is much clearer if we examine the Variance Risk Premium (VRP), which is defined as the difference between a model-free implied volatility (of which VIX is the most famous example) and the historical volatility over a recent period. The relationship between VRP and future returns is examined in a paper by Chevallier and Sevi in the context of OVX, which is the CBOE Crude Oil Volatility Index. They have found that there is a statistically significant negative linear relationship between VRP and future 1-month crude oil futures (CL) returns. The historical volatility is computed over 5-minute returns of the most recent trading day. (Why 5 minutes? Apparently this is long enough to avoid the artifactual volatility induced by bid-ask bounce, and short enough to truly sample intraday volatility.)  If you believe in the prescience of options traders, it should not surprise you that the regression coefficient is negative (i.e. a high VRP predicts a lower future return).

I have tested a simple trading strategy based on this linear relationship. Instead of using monthly returns, I use VRP to predict daily returns of CL. It is very similar to a mean-reverting Bollinger band strategy, except that here the "Bollinger bands" are constructed out of moving first and third quartiles of VRP with a 90-day lookback. Given that VRP is far from normally distributed, I thought it is more sensible to use quartiles rather than standard deviations to define the Bollinger bands. So we buy a front contract of CL and hold for just 1 day if VRP is below its moving first quartile, and short if VRP is above its moving third quartile. It gives a decent average annual return of 17%, but performance was poor in 2013.

Naturally, one can try this simple trading strategy on the E-mini SP500 future ES also. This time, VRP is VIX minus the historical volatility of ES.  Contrary to folklore, I find that if we regress the future 1 day ES return against VRP, the regression coefficient is positive. This means that an increase of VIX relative to historical volatility actually predicts an increase in ES! (Does this mean investors are overpaying for put options on SPX for portfolio protection?) Indeed, the opposite trading rules from the above give positive returns: we should buy ES if VRP is above its moving third quartile, and short ES if VRP is below its moving first quartile. The annualized return is 6%, but performance in 2013 was also poor.

As the authors of the paper noted, whether or not VRP is a strong enough stand-alone predictor of returns, it is probably useful as an additional factor in a multi-factor model for CL and ES. If any reader know of other volatility index like VIX and OVX, please do share with us in the comments section!

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My online Backtesting Workshop will be offered on February 18-19. Please visit epchan.com/my-workshops for registration details. Furthermore, I will be teaching my Mean Reversion, Momentum, and Millisecond Frequency Trading workshops in London on March 17-21, and in Hong Kong on June 17-20.

Friday, November 15, 2013

Cointegration Trading with Log Prices vs. Prices

In my recent book, I highlighted a difference between cointegration (pair) trading of price spreads and log price spreads. Suppose the price spread hA*yA-hB*yB of two stocks A and B is stationary. We should just keep the number of shares of stocks A and B fixed, in the ratio hA:hB, and short this spread when it is much higher than average, and long this spread when it is much lower. On the other hand, for a stationary log price spread hA*log(yA)-hB*log(yB), we need to keep the market values of stocks A and B fixed, in the ratio hA:hB, which means that at the end of every bar, we need to rebalance the shares of A and B due to price changes.

For most cointegrating pairs that I have studied, both the price spreads and the log price spreads are stationary, so it doesn't matter which one we use for our trading strategy. However, for an unusual pair where its log price spread cointegrates but price spread does not (Hat tip: Adam G. for drawing my attention to one such example), the implication is quite significant. A stationary price spread means that prices differences are mean-reverting, a stationary log price spread means that returns differences are mean-reverting. For example, if stock A typically grows 2 times as fast as B, but has been growing 2.5 times as fast recently, we can expect the growth rate differential to decrease going forward. We would still short A and long B, but we would exit this position when the growth rates of A vs B return to a 2:1 ratio, and not when the price spread of A vs B returns to a historical mean. In fact, the price spread of A vs B should continue to increase over the long term.

This much is easy to understand. But thanks to a reader Ferenc F. who referred me to a paper by Fernholz and Maguire, I realize there is a simple mathematical relationship between stock A and B in order for their log prices to cointegrate.

Let us start with a formula derived by these authors for the change in log market value P of a portfolio of 2 stocks: d(logP) = hA*d(log(yA))+hB*d(log(yB))+gamma*dt.

The gamma in this equation is

gamma=1/2*(hA*varA + hB*varB), where varA is the variance of stock A minus the variance of the portfolio market value, and ditto for varB.

Note that this formula holds for a portfolio of any two stocks, not just when they are cointegrating. But if they are in fact cointegrating, and if hA and hB are the weights which create the stationary portfolio P, we know that d(logP) cannot have a non-zero long term drift term represented by gamma*dt. So gamma must be zero. Now in order for gamma to be zero, the covariance of the two stocks must be positive (no surprise here) and equal to the average of the variances of the two stocks. I invite the reader to verify this conclusion by expressing the variance of the portfolio market value in terms of the variances of the individual stocks and their covariance, and also to extend it to a portfolio with N stocks. This cointegration test for log prices is certainly simpler than the usual CADF or Johansen tests! (The price to pay for this simplicity? We must assume normal distributions of returns.)

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My online Quantitative Momentum Strategies workshop will be offered on December 2-4. Please visit epchan.com/my-workshops for registration details.

Thursday, October 24, 2013

How Useful is Order Flow and VPIN?

Can short-term price movement be predicted? (I am speaking of  seconds or minutes here.) This is a question not only relevant to high frequency traders, but to every long-term investor as well. Even if  one plans to buy and hold a stock for years,  nobody likes to suffer short-term negative P&L immediately after entry into position.

One short-term prediction method that has long found favor with academic researchers and traders alike is order flow. Order flow is just signed transaction volume: if a transaction of 100 shares is classified as a "buy", the order flow is +100; if it is classified as a "sell", the order flow is -100. This might strike some as rather strange: every transaction has a buyer and seller, so what does it mean by a "buy" or a "sell"? Well, the "buyer" is defined as the one who is the "aggressor", i.e. one that is using a market order to buy at the ask price. (And vice versa for the seller, whom I will henceforth omit in this discussion.) The intuitive reason why a series of large "buy" market orders are predictive of short-term price increase is that if someone is so eager to go long, s/he is likely to know something about the market that others don't (either due to superior fundamental knowledge or technical model), so we better join her/him! Such superior traders are often called "informed traders", and their order flow is often called "toxic flow". Toxic, that is, to the uninformed market maker.

In theory, if one has a tick data feed, one can tell whether an execution is a "buy" or "sell" by comparing the trade price with the bid and ask price: if the trade price is equal to the ask, it is a "buy". This is called the "Quote Rule". But in practice, there is a hitch. If the bid and ask prices change quickly, a buy market order may end up buying at the bid price if the market has fortuitously moved lower since the order was sent. Besides, perhaps 1/3 of trading in the US equities markets take place in dark pools or via hidden orders, so the quotes are simply invisible and order flow non-computable. So this classification scheme is not foolproof. Therefore, a number of researchers (see "Flow Toxicity and Volatility in a High Frequency World" by Easley, et. al.) proposed an alternative, "easier", method to compute order flow. Instead of checking the trade price of each tick, they just need the "open" and "close" trade prices of a bar, preferably a volume bar, and assign a fraction of the volume in that bar to "buy" or "sell" depending on whether the close price is higher or lower than the open price. (The assignment formula is based on the cumulative probability density of a Gaussian distribution, which incidentally models price changes of volume bars, but not time bars, pretty well.) The absolute difference between buy and sell volume expressed as a fraction of the total volume is called "VPIN" by the authors, or Volume-Synchronized Probability of Informed Trading. The higher VPIN is, the more likely we will experience short-term momentum due to informed trading.

Theory and intuition aside, how well does order flow work in practice as a short-term predictor in various markets? And how predictive is VPIN as compared to the old Quote Rule?  In my experience, while this indicator is predictive of price change, the change is often too small to overcome transaction costs including the bid-ask spread. And more disturbingly, in those markets where both Quote Rule and VPIN should work (e.g. futures markets), VPIN has so far underperformed Quote Rule, despite (?) it being patented and highly touted. I have informally polled other investment professionals on their experience, and the answer usually come back indifferent as well.

Do you have live experience with VPIN? Or more generally, do you find strategies built using volume bars superior to those using time bars? If so, please leave us your comments!

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My online Quantitative Momentum Strategies workshop will be offered in December. Please visit epchan.com/my-workshops for registration details.


Tuesday, August 20, 2013

Guest Post: A qualitative review of VIX F&O pricing and hedging models

By Azouz Gmach

VIX Futures & Options are one of the most actively traded index derivatives series on the Chicago Board Options Exchange (CBOE). These derivatives are written on S&P 500 volatility index and their popularity has made volatility a widely accepted asset class for trading, diversifying and hedging instrument since their launch. VIX Futures started trading on March 26th, 2004 on CFE (CBOE Future Exchange) and VIX Options were introduced on Feb 24th, 2006.


VIX Futures & Options

VIX (Volatility Index) or the ‘Fear Index’ is based on the S&P 500 options volatility. Spot VIX can be defined as square root of 30 day variance swap of S&P 500 index (SPX) or in simple terms it is the 30-day average implied volatility of S&P 500 index options. The VIX F&O are based on this spot VIX and is similar to the equity indexes in general modus operandi. But structurally they have far more differences than similarities. While, in case of equity indices (for example SPX), the index is a weighted average of the components, in case of the VIX it is sum of squares of the components. This non-linear relationship makes the spot VIX non-tradable but at the same time the derivatives of spot VIX are tradable. This can be better understood with the analogy of Interest Rate Derivatives. The derivatives based on the interest rates are traded worldwide but the underlying asset: interest rate itself cannot be traded.

The different relation between the VIX derivatives and the underlying VIX makes it unique in the sense that the overall behavior of the instruments and their pricing is quite different from the equity index derivatives. This also makes the pricing of VIX F&O a complicated process. A proper statistical approach incorporating the various aspects like the strength of trend, mean reversion and volatility etc. is needed for modeling the pricing and behavior of VIX derivatives.


Research on Pricing Models

There has been a lot of research in deriving models for the VIX F&O pricing based on different approaches. These models have their own merits and demerits and it becomes a tough decision to decide on the most optimum model. In this regards, I find the work of Mr. Qunfang Bao titled ‘Mean-Reverting Logarithmic Modeling of VIX’ quite interesting. In his research, Bao not only revisits the existing models and work by other prominent researchers but also comes out with suggestive models after a careful observation of the limitations of the already proposed models. The basic thesis of Bao’s work involves mean-reverting logarithmic dynamics as an essential aspect of Spot VIX.

VIX F&O contracts don’t necessarily track the underlying in the same way in which equity futures track their indices. VIX Futures have a dynamic relationship with the VIX index and do not exactly follow its index. This correlation is weaker and evolves over time. Close to expiration, the correlation improves and the futures might move in sync with the index. On the other hand VIX Options are more related to the futures and can be priced off the VIX futures in a much better way than the VIX index itself.


Pricing Models

As a volatility index, VIX shares the properties of mean reversion, large upward jumps & stochastic volatility (aka stochastic vol-of-vol). A good model is expected to take into consideration, most of these factors.

There are roughly two categories of approaches for VIX modeling. One is the Consistent approach and the other being Standalone approach.

        I.            Consistent Approach: - This is the pure diffusion model wherein the inherent relationship between S&P 500 & VIX is used in deriving the expression for spot VIX which by definition is square root of forward realized variance of SPX.

      II.            Standalone Approach: - In this approach, the VIX dynamics are directly specified and thus the VIX derivatives can be priced in a much simpler way. This approach only focuses on pricing derivatives written on VIX index without considering SPX option.
Bao in his paper mentions that the standalone approach is comparatively better and simpler than the consistent approach.


MRLR model

The most widely proposed model under the standalone approach is MRLR (Mean Reverting Logarithmic Model) model which assumes that the spot VIX follows a Geometric Brownian motion process. The MRLR model fits well for VIX Future pricing but appears to be unsuited for the VIX Options pricing because of the fact that this model generates no skew for VIX option. In contrast, this model is a good model for VIX futures.


MRLRJ model

Since the MRLR model is unable to produce implied volatility skew for VIX options, Bao further tries to modify the MRLR model by adding jump into the mean reverting logarithmic dynamics obtaining the Mean Reverting Logarithmic Jump Model (MRLRJ). By adding upward jump into spot VIX, this model is able to capture the positive skew observed in VIX options market.


MRLRSV model

Another way in which the implied volatility skew can be produced for VIX Options is by including stochastic volatility into the spot VIX dynamics. This model of Mean Reverting Logarithmic model with stochastic volatility (MRLRSV) is based on the aforesaid process of skew appropriation.
Both, MRLRJ and MRLRSV models perform equally well in appropriating positive skew observed in case of VIX options.


MRLRSVJ model

Bao further combines the MRLRJ and MRLRSV models together to form MRLRSVJ model. He mentions that this combined model becomes somewhat complicated and in return adds little value to the MRLRJ or MRLRSV models. Also extra parameters are needed to be estimated in case of MRLRSVJ model.

MRLRJ & MRLRSV models serve better than the other models that have been proposed for pricing the VIX F&O. Bao in his paper, additionally derives and calibrates the mathematical expressions for the models he proposes and derives the hedging strategies based on these models as well. Quantifying the Volatility skew has been an active area of interest for researchers and this research paper addresses the same in a very scientific way, keeping in view the convexity adjustments, future correlation and numerical analysis of the models etc. While further validation and back testing of the models may be required, but Bao’s work definitely answers a lot of anomalous features of the VIX and its derivatives.

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Azouz Gmach works for QuantShare, a technical/fundamental analysis software.

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My online Mean Reversion Strategies workshop will be offered in September. Please visit epchan.com/my-workshops for registration details.

Also, I will be teaching a new course Millisecond Frequency Trading (MFT) in London this October.

-Ernie