TRADING THE ODDS » Studies/Survey

Williams’ VIX Fix (WVF)

admin — Sun, 05 Sep 2010 21:59:17 +0000

Most recently Michael Stokes at MarketSci and MINDMONEYMARKET (I don’t know who is behind the blog) had – as always – some very inspiring postings about the Williams’ VIX Fix (WVF) (here and here). William’s VIX Fix is a synthetic VIX (CBOE Volatility Index) calculation which can be used in any market to mimic the performance (but not the quotes) of the well-known volatility index (the WVF is not based on option’s implied volatility but derived from historical and intraday prices only).

William’s original formula:

WVF = [Highest (Close,22) - Low) / (Highest(Close,22)] * 100

Michael and MINDMONEYMARKET already showed that at least with respect to the S&P 500, relatively high WVF readings were frequently (means with a probability significantly above the respective at-any-time probability) followed by a higher close on the then following session. Buying into high WVF readings (when the delta between the current intraday low and the highest close of the previous 22-trading days) is at positive extremes is therefore buying into strength, not betting on a short-term mean reversion tendency.

But with respect to the WFV‘s usefulness for any kind of trading strategy, Michael noted two shortcomings as well:

(1) The WMF‘s dependency on intraday data (the ‘Low‘), and (2) the WVF‘s potential lack of robustness due to the fact that less extreme WVF readings might significantly (negatively) impact a potential trading strategy. Both bullet points incited me to check if – and how to – slight adjustements to the formula cited above could solve these issues.

Table I below shows the SPY‘s historical performance assumed one went long on close of a session when the WVF closed

Strat. #1: among the top 4 of all readings over the previous 22 trading days,
Strat. #2: among the top 3 of all readings over the previous 22 trading days,
Strat. #3: among the top 2 of all readings over the previous 22 trading days,
Strat. #4: as the highest of all readings over the previous 22 trading days,

otherwise no position is taken (move to cash). The 22-trading day period is slightly deviating from what Michael and MINDMONEYMARKET used, but matches the one month period William’s chose for his original formula (see above).

Interesting to note that by increasing the WVF entry level (and therefore lowering the respective number of occurrences and time in market), compounded returns remain almost unchanged due to the fact that going long on close of a sessions where the WVF closed among the top 2 of all readings over the previous 22 trading days (Strat. #3 ) showed the highest median trade, profit factor and distribution of returns and the lowest drawdown among all entry levels. Lowering the entry level – at least with respect to the set of parameters utilized for this evaluation – wouldn’t turn a profitable strategy into a losing one, but would increase risk and expenditures (maximum drawdown, time in market, transaction costs …) with no additional reward.

The second question was if utilizing the intraday low as part of the original formula is a must, or if substituting the intraday low by the daily close (therefore relying on daily closing prices only) would negatively impact the formula’s usefulness and quality of forecast. Table II below shows the SPY‘s historical performance assumed one went long on close of a session when a ‘redefined‘ WVF closed among the top 2 of all readings over the previous 22-trading days (previous strat. #3), with

Strat. #1: WVF = [Highest (Close,22) - High) / (Highest(Close,22)] * 100,
Strat. #2: WVF = [Highest (Close,22) - Low) / (Highest(Close,22)] * 100,
Strat. #3: WVF = [Highest (Close,22) - Close) / (Highest(Close,22)] * 100,

otherwise no position is taken (move to cash). Strategy #2 represents William’s original formula.

And now going into one more extreme (shortened time frame): Table III below shows the SPY‘s historical performance assumed one went long on close of a session when a ‘redefined‘ WVF closed among the top 2 of all readings over the previous 10-trading days (instead of 22-trading days), with again

Strat. #1: WVF = [Highest (Close,22) - High) / (Highest(Close,22)] * 100,
Strat. #2: WVF = [Highest (Close,22) - Low) / (Highest(Close,22)] * 100,
Strat. #3: WVF = [Highest (Close,22) - Close) / (Highest(Close,22)] * 100,

otherwise no position is taken (move to cash). Strategy #2 represents William’s original formula.

Interesting to note that at least with respect to utilizing the SPY for the time frame since 01/01/1990, relying on closing prices alone (no intraday high or low) would (partly significantly) improve the overall profitability (compound returns, probability of a higher close, median trade, distribution of returns), by having to accept a slightly larger drawdown (but significantly smaller number of maximum sessions in a drawdown). For shorter periods (e.g. the previous 10-trading days instead of the previous 22-trading days) using the intraday ‘High‘ instead of William’s intraday ‘Low’ would’ve worked best the majority of the time (see equity curve above, blue line), out-performing the ‘Low‘ in the original formula by a wide margin.

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And in order to ensure that utilizing closing prices only instead of using intraday low or high for the computation of the WVF may be the better choice in principle and not only applicable for the SPY, I performed the same stats for the QQQQ (Nasdaq 100) as well (see table IV below, data since 01/01/2000).

It is again strategy #3, utilizing closing prices only instead of intraday data, which provides the highest rate of return, profit factor, median trade and distribution of returns.

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Summary: Utilizing percentage rankings, and at the extreme (high) end of the then recent previous x readings, William’s VIX Fix seems to be a reliable (with respect to it’s quality of forecast) and very robust indicator forecasting the bullish side of the market (unfortunately WVF readings at the low end of the then recent previous x readings are not able to forecast weakness in the markets by any kind of statistically significance). But using daily closing prices instead of the intraday low may not only simplify the handling and computation but will probably improve the quality of forecast (with respect to probabilites and odds) as well.

Successful trading,
Frank

Disclaimer: No position in the securities mentioned in this post at time of writing.

Remarks: Due to their conceptual scope – and if not explicitely stated otherwise -, all models/setups/strategies do not account for slippage, fees and transaction costs, do not account for return on cash, do not use position sizing (e.g. Kelly, optimal f) – they’re always ‘all in‘ -, do not use leverage (e.g. leveraged ETFs) - but a marginable account is mandatory -, do not utilize any kind of abnormal market filter (e.g. during market phases with extremely elevated volatility) , do not use intraday buy/sell stops (end-of-day prices only), and models/setups/strategies are not ‘adaptive‘ (do not adjust to the ongoing changes in market conditions like bull and bear markets).

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Option Expiration and Down-Days

admin — Sat, 21 Aug 2010 07:38:52 +0000

In full compliance to historical probabilities and odds (see SPY’s Option Expiration Seasonalities), the SPY posted another down day on option expiration, on a back-to-back session to Thursday’s serious down-day (-1.74%). In addition, trading lower (than the previous session’s close) at the start of the final hour of option expiration’s session (see my Twitter Feed), (compliant to historical odds) the SPY was not able to recoup more than +0.50% of it’s intraday losses and couldn’t turn an intraday loss into a gain.

Table I below shows the SPY‘s performance (since 01/01/1990) on thoses sessions immediately following option expiration (in this event the session on Monday, August 23, 2010) assumed one went long on close …

Strat. #1: of a(ny) option expiration session,
Strat. #2: of a(ny) option expiration session when the SPY closed lower ,
Strat. #3: of a(ny) option expiration session when the SPY closed lower on a back-to-back session (two consecutive lower closes),
Strat. #4: of a(ny) option expiration session when the SPY closed lower on a back-to-back session (two consecutive lower closes), but reduced it’s losses on option expiration (a smaller loss than the one of the previous session).

All four setups were triggered on close of Friday, August 20, 2010.

There seems to be something special with those session at and/or around option expiration. Although at-any-time historical probabilities show a positive (short-term mean-reversion) tendency on a session immediately following a serious down day (e.g. SPY lost at least -1.50%) and/or following two consecutive down-days, not so on and immediately following option expiration. Long-term probabilities (winning percentage) and odds (expectancy) are (partly significantly) tilt in favor of another (a third) lower close, but at least most recent occurrences show a remarkable positive tendency.

Table II below shows the most recent twenty historical occurrences (the SPY‘s performance on a session immediately following option expiration) and their respective returns, assumed one went long on close of an option expiration session when the SPY closed lower (Strat. #2):

With respect to setup #2, the SPY closed higher on all of the most recent nine occurrences, and higher on 16 out of the last twenty occurrences, with thirteen out of those sixteen gains exceeding +0.50%.

Although long-term probabilities and odds are tilt in favor of a potential third lower close on Monday’s session, and not taking into account volume and/or market breadth data but closing prices alone, time-weigthed (most recent) probabilities and odds favour a positive outcome (a higher close) on Monday, August 23,2010.

Successful trading,
Frank

Disclaimer: No position in the securities mentioned in this post at time of writing.

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SPY’s Option Expiration Seasonalities

admin — Fri, 20 Aug 2010 06:51:17 +0000

Ahead of Friday’s option expiration the SPY (S&P 500 SPDR.) posted a serious down-day, losing -1.74% on the close.

Table I below shows the SPY‘s performance (since 01/01/1990) on option expiration assumed one went long on close …

Strat. #1: of a(ny) session immediately preceding option expiration,
Strat. #2: of a(ny) session when the SPY had lost at least -1.50%,
Strat. #3: of a session when the SPY had lost at least -1.50% immediately preceding option expiration (like on Thursday, August 19, 2010).

Table II below shows all historical occurrences (the SPY‘s performance on option expiration) and their respetive returns, assumed one went long on close of a session when the SPY had lost at least -1.50% immediately preceding option expiration (Strat. #3):

From my perspective a favorable mean-reversion tendency on option expiration looks quite different (in contrast to the SPY‘s overall mean-reversion tendency after posting a down-day of that magnitude, see Strat. #2). On almost two out of every three (11 out of 17) occurrences (Strat. #3), the SPY posted another down-day on option expiration, and it is a single occurrence on 11/21/2008 (+5.39%) only which turns the compound return and profit factor into a positive number.

Option expiration – from a statistical and historical perspective - seems to be a session where generally a long trade is a trade against the odds: although the percentage of winning trades is slightly positive (51.85%), the distribution of returns (a ranking of a setup’s returns in comparison to at-any-time returns) undercuts the at-any-time median return of a buy-and-hold approach, and only 6.58% of the SPY‘s returns on option expiration fall into the top tenth of the SPY‘s at-any-time top tenth (best) returns (disproportionally small), while 10.70% fall into the bottom tenth of the SPY‘s at-any-time bottom tenth (worst) returns, slightly above-average.

Successful trading,
Frank

Disclaimer: No position in the securities mentioned in this post at time of writing.

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Pairs Trading Part II – SPY vs. RTH

admin — Mon, 16 Aug 2010 14:32:45 +0000

One of the most interesting findings dealt with in a previous posting (Pairs Trading (ETFs) was the RTHs (Retail HOLDR.) salient feature of being a favorable candidate for a potential mean-reversion strategy in conjunction with a major market or sector ETF. With respect to the primarily method used for cointegration (the augmented Dickey-Fuller test), the RTH showed a probability better than 90% of being cointegrated with the IWM (Russel 2000) and the SMH (Semiconductor HOLDR.), and missed being cointegrated with SPY and QQQQ by a hairbreadth only (two price series are called cointegrated if the pair has a consistent mean and standard deviation, both prices series never indefinitely wandering off in opposite directions and never drifting farther and farther away from its mean without eventually returning to the initial ratio or mean).

But the RTH doesn’t seem to be a favorable candidate for a longer-term (the half-life – the expected time to revert half of its deviation from the mean – is regualary measured in weeks or month) pairs trading strategy only, but may provide favorable short-term mean-reversion opportunities as well (market timing).

Table I below shows the performance metrics (since 06/01/2001 due to the RTH‘s inception in May 2001) for different pairs in conjunction with the RTH and – for demonstration puposes – different pairs of major market ETFs (SPY, QQQQ and IWM) and sector ETFs (XLY – Consumer Discretionary – and XLP – Consumer Staples -) based on an exemplary mean-reversion strategy, assumed one would’ve bought the pair (is equivalent to buying the first and selling short the second ETF in equal money amounts (number of shares in each ETF = 100% net asset value / share price)) on close of a session when the 4-day EMA (Exponential Moving Average) of the pair (the ratio of the closing prices) is less than the 4-day EMA of the ratio of closing prices for yesterday, and vice versa (selling short the first and buying the second ETF in equal money amounts in the event of a rising 4-day EMA of the ratio of closing prices).

Here is the link to the stats in a more ‘readable’, original size: Statistics 1

While – with respect to the specific setup defined – the SPY vs. QQQQ, the SPY vs. IWM, the XLY vs. XLP and the SPY itself (buy and hold) as a benchmark virtually went nowhere (or even closed in the red) over the course of the last 9 years – especially after accounting for fees and transaction costs -, the RTH as a pairs trading component in conjunction with the SPY, the QQQQ, the IWM and the SMH not only easily out-performed a (S&P 500) buy-and-hold approach by a wide margin (and almost year by year as well, see ‘Periodic Returns‘ in the stats above), but comes up with a smoother equity curve as well, meaning there are much less dramatic departures from a gradually/geometrically increasing trendline (R-squared, maximum drawdown, maximum sessions in drawdown) in comparison to a SPY‘s buy and hold approach.

Interestingly the SPY vs. RTH and SMH vs. RTH pairs trading strategies and a S&P 500 buy-and-hold approach do NOT differ with respect to the probability of a winning trade (the probability is almost always slightly above 50% only). The reason for the deviation in total returns is the fact that – in contrast to the SPY‘s buy and hold approach – the median winning trade (+0.51%) now equals or slightly exceeds the median losing trade (-0.50% ), significantly improving the respective expectancy (probability of winning * average gain – probability of losing * average loss).

But a SPY vs. RTH‘s pairs trading strategy has another advantage as well: Chosing a slightly different setup in order to especially exploit those reversal opportunities where the pair is (from a historical and statistical perspective) exceptionally stretched to one or the other side would be sufficient to not only surpass previous compounded returns, but to cut in half the time in market and the maximium drawdown as well. Table II below shows the performance metrics (since 06/01/2001 due to the RTH‘s inception in May 2001) for the same pairs, assumed one would’ve bought the pair (buying the first and selling short the second ETF in equal money amounts) on close of a session when the pair (the ratio of the closing prices) closed at least -0.50% below its 4-day EMA, and vice versa (selling short the first and buying the second ETF in equal money amounts in the event of a close at least +0.50% above the 4-day EMA of the ratio of closing prices).

Here is the link to the stats in a more ‘readable’, original size: Statistics 2

With respect to SPY vs. RTH (Strat. #1), time in market and maximum drawdown have been exactly cut in half (giving you the chance to earn an additional return on cash) while the geometric growth rate per trade doubled. Although the probability of a winning trade (again) only slightly improved (from 54.63% to 57.44%), it is (again) the effectiviness (doing things right instead of doing the right thing only, meaning increasing your gains when you’re right and cutting your losses when you’re wrong) of the strategy which makes for the improvement in key performance metrics.

But what about the robustness of a SPY vs. RTH pairs trading strategy ? It it works with a -0.50%/+0.50% level below/above a 4-day EMA, it should work with a -/+0.30% up to a -/+0.70% level and a 3-day and 5-day EMA as well showing some gradual – no radical - changes with respect to the key performance metrics only.

Table III below shows the performance metrics for the SPY vs. RTH pairs trading strategy, assumed one would’ve bought the pair (buying the first and selling short the second ETF in equal money amounts) on close of a session when the pair (the ratio of the closing prices) closed at least

Strat. #1: -0.30% below (long) and +0.30% above (short) its 4-day EMA,
Strat. #2:-0.40% below (long) and +0.40% above (short) its 4-day EMA,
Strat. #3:-0.50% below (long) and +0.50% above (short) its 4-day EMA,
Strat. #4:-0.60% below (long) and +0.60% above (short) its 4-day EMA,
Strat. #5:-0.70% below (long) and +0.70% above (short) its 4-day EMA.

SPY vs. RTH (Strat. #6) represents a buy-and-hold approach (assumed one would always be long the SPY and short the RTH).

Here is the link to the stats in a more ‘readable’, original size: Statistics 3

And last but not least, table IV below shows the performance metrics for the SPY vs. RTH pairs trading strategy, assumed one would’ve bought the pair (buying the first and selling short the second ETF in equal money amounts) on close of a session when the pair (the ratio of the closing prices) closed at least

Strat. #1:-0.50% below (long) and +0.50% above (short) its 3-day EMA,
Strat. #2:-0.50% below (long) and +0.50% above (short) its 4-day EMA,
Strat. #3:-0.50% below (long) and +0.50% above (short) its 5-day EMA.

SPY vs. RTH (Strat. #4) represents a buy-and-hold approach (assumed one would always be long the SPY and short the RTH).

Neither a slight variation in the percentage level below/above the 4-day EMA nor a variation in the duration of the EMA itself affects any of the strategy’s key performance indicators in a significant way, except – but expectedly – the so called opportunity factor (total number of sessions and time in market).

Summary: A SPY vs. RTH‘s pairs trading strategy, assumed one would’ve bought the pair (buying the SPY and selling short the RTH in equal money amounts) on close of a session when the pair (the ratio of the closing prices) closed at least -0.50% below its 4-day EMA, and vice versa (selling short the SPY and buying the RTH in equal money amounts in the event of a close at least +0.50% above the 4-day EMA of the ratio of closing prices), historically provided a (consistently) profitable market timing strategy (a median annual return of +15.65%), (consistently, in 8 out of the last 9 years) out-performing a S&P 500 buy-and-hold approach, with a smooth equity curve (R-squared, maximum drawdowns on a week/month/year end basis, maximum time in a drawdown), meeting at least basic requirements for robustness and reliability. Unfortunately a shortcoming is the deviation in yearly returns (one standard deviation = 32.72%).

A favorable basis for some further investigations and refinements (accounting for return on cash, position sizing, and making the strategy adaptiv to changing market conditions – if necessary).

to be continued …

Successful trading,
Frank

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Disclaimer: Long SMH and short XRT at time of writing.

Pairs Trading (ETFs)

admin — Mon, 09 Aug 2010 21:30:36 +0000

First of all thanks for your patience, and from now on I’ll be posting again on a more frequent basis.

And furthermore I’d like to advise those interested in quantitative research of a new blog I just came across: Engineering Returns by Frank Hassler.

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Due to the fact that I’m a big fan of statistical arbitrage (and trading it for a living), I thought it would be interesting to check if – and to what extend – there are pairs of ETFs (Exchange Traded Funds) which – as always based on historical data, statistical anomalies, regularities and irregularities, … – would provide a favorable and tradable edge maintaining a market neutral position.

I personally prefer ETFs to individual stocks due to the fact that the latter are much more sensitive to unforeseeable events and/or outcomes like earnings, fundamentals, crew changes (CEO, CFO, …), rate disputes, strikes, take-overs, force majeure (casualties, disasters, …). And I speak from my own experience …

In conjunction with pairs trading, you’ll probably hear about two (quite different) concepts: correlation and cointegration.

Correlation states the degree to which the (daily, weekly, monthly …) returns of two series of prices (e.g. the S&P 500 and the Nasdaq 100) will move in the same direction most days/weeks/month over a period of time (but probably drifting farther and farther away from each other due to deviations in the magnitude of daily returns), while a pair (being long one and short the other series of prices in the right proportion) is called cointegrated if it has a consistent mean and standard deviation, both prices series never indefinitely wandering off in opposite directions and never drifting farther and farther away from its mean without eventually returning to the initial ratio or mean (mean-reversion).

But – unfortunately – the so-called half-life (the expected time to revert half of its deviation from the mean) concerning a cointegrated pair of price series will regularly be measured in weeks or month (see stats below), and I’m more a high-frequency trader looking for opportunities on a day-by-day basis.

The following are the ETF’s I’ve been utilizing for my investigations, meeting the necessary requirements like adequate liquidity (daily trading volume), as low as possible transaction costs (narrow bid/ask spreads), adequate volatility (in order to justify the arising high transaction costs), among others:

SPY: S&P 500
QQQQ: Nasdaq 100
IWM: Russel 2000
SMH: Semiconductor
RTH: Retail

Other ETFs may be subject to a follow-up posting.

To test for cointegration, the primarily method used is called the augmented Dickey-Fuller test. If two price series are cointegrated (with a probability of better than 90%), the Dickey-Fuller test would’ve to come up with a t-statistic exceeding the 90% critical value of -3.038 (in absolute terms), otherwise the hypothesis that those two price series are conintegrated would be rejected. The following table provides the respective t-statistics based on the augmented Dickey-Fuller test for the time frame between 01/01/2002 and 08/06/2010 (price series are adjusted for dividend and cash payments).

t-statistic	SPY	QQQQ	IWM	SMH	RTH
SPY	-	-0.8210	-2.7995	-1.8635	-2.8513
QQQQ	-	-	-2.6115	-1.0084	-2.0935
IWM	-	-	-	-1.9626	-3.2206
SMH	-	-	-	-	-3.3625
RTH	-	-	-	-	-

Interestingly there are only two pairs – IWM vs. RTH (gt. 90%) and SMH vs. RTH (gt. than 95%) – which are cointegrated with a probability of better than 90%, while the SPY (as a proxy for the S&P 500) and the QQQQ (as a proxy for the Nasdaq 100) show the least probability for being cointegrated. IWM vs. RTH shows a half-life of 194 sessions, and SMH vs. RTH a half-life of 92 sessions. Both pairs seem to be good candidates for a (longer-term) mean-reversion strategy.

A second interesting observation is that even in conjunction with SPY and IWM, the RTH (Retail HOLDRS) seems to be a favorable candidate for a potential mean-reversion strategy.

But fortunately cointegration is not mandatory in order to find a profitable mean-reversion strategy, and on a day-by-day basis even non-cointegrated pairs (like the SPY vs. QQQQ) may provide favorable short-term mean-reversion opportunities (better fitting my style of trading). So my next step was to check for the pair’s performance based on the easiest mean-reversion strategy:

Buy (on close) the pair in the event the ratio of ETF X and ETF Y closed lower (means ETF A under-performed ETF B on the respective session), and sell short (on close) in the event the ratio of ETF X and ETF Y closed up (means ETF A out-performed ETF B on the respective session). Due to the RTH‘s inception date in 2001 start date for the following stats is always Jan. 1, 2002.

“Buy” means buy ETF A and sell short ETF B (and vice versa), the number of respective shares specified by the ratio of closing prices (e.g. if the ratio of ETF A’s and ETF B’s closing prices is 3, one would sell short 3 shares of ETF B for every share bought of ETF A). A marginable account would be mandatory, especially due to the fact that it is assumed that one would invest 100% of the then current net liquidation value on both sides of the market (means 100% on the buy and 100% on the short side).

(FAQs and a glossary concerning the stats can be found at the FAQ/GLOSSARY page)

Here is the link to the stats in a ‘readable’ size: Statistics 1

Interestingly it is again the RTH in conjunction with every other ETF which delivers the best results, always exceeding the 200% mark for compounded returns (gross profits before applying commissions, slippage and fees). Unfortunately commissions, slippage and fees would regularly eat up a major part of the compounded return, due to the fact that one would always have a position in the market, with an exposure of 200% (100% on the buy and 100% on the short side), and reversing one’s position (switching from the long to the short side of the pair and vice versa) would quadruple the respective transaction costs in comparison to somone who simply closes a long or short position with an 100% exposure.

In a second step I utilized a little bit more sophisticated concept (Bollinger Bands %B with 4-days EMA and 1 standard deviation):

Buy (on close) the pair in the event the Bollinger Bands %B closed below 0.35, and sell short (on close) in the event the Bollinger Bands %B closed above 0.65.

For a detailed explanation of the Bollinger Bands %B concept see Stockcharts.com. In other words: Buy the pair in the event the ratio closed almost (< 0.35) one standard deviation below its 4-day exponential moving average (ETF A is short-term ‘oversold’ in comparison to ETF B), and sell short the pair in the event the ratio closed almost (> 0.65) one standard deviation above its 4-day exponential moving average (ETF A is short-term ‘overbought’ in comparison to ETF B). A classical mean reversion concept.

Here is the link to the stats in a ‘readable’ size: Statistics 2

Things are (partly) significantly improving: Compounded returns, t-score (vs. chance and benchmark) are increasing while transaction costs, maximum drawdowns are decreasing (now Time in Market is less than 100%, with a smaller frequency of closing or reverting one’s position), and especially the SPY vs. RTH and IWM vs. RTH pairs show promising results to be worth some further investigations.

More to come in a follow-up post (at time of writing it’s almost midnight in Germany) …

to be continued …

Successful trading,
Frank

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Disclaimer: No position in the securities mentioned in this post at time of writing.