I have long been partial to linear strategies due to their simplicity and relative immunity to overfitting. They can be used quite easily to profit from mean-reversion. However, there is a serious problem: they are quite fragile, i.e. vulnerable to tail risks. As we move from mean-reverting strategies to momentum strategies, we immediately introduce a nonlinearity (stop losses), but simultaneously remove certain tail risks (except during times when markets are closed). But if we want to enjoy anti-fragility and are going to introduce nonlinearities anyway, we might as well go full-monty, and consider options strategies. (It is no surprise that Taleb was an options trader.)
It is easy to see that options strategies are nonlinear, since options payoff curves (value of an option as function of underlying stock price) are plainly nonlinear. I personally have resisted trading them because they all seem so complicated, and I abhor complexities. But recently a reader recommended a little book to me: Jeff Augen's "Day Trading Options" where the Black-Scholes equation (and indeed any equation) is mercifully absent from the entire treatise. At the same time, it is suffused with qualitative ideas. Among the juicy bits:
1) We can find distortions in the 2D implied volatility surface (implied volatility as z-axis, expiration months as x, and strike prices as y) which may mean revert to "smoothness", hence presenting arbitrage opportunities. These distortions are present for both stock and stock index options.
2) Options are underpriced intraday and overpriced overnight: hence it is often a good idea to buy them at the market open and sell them at market close (except on some special days! See 4 below.). In fact, there are certain days of the week where this distortion is the most drastic and thus favorable to this strategy.
3) Certain cash instruments have unusually high kurtosis, but their corresponding option prices consistently underprice such tail risks. Thus structures such as strangles or backspreads can often be profitable without incurring any left tail risks.
4) If there is a long weekend before expiration day (e.g. Easter weekend), the time decay of the options value over 3 days is compressed into an intraday decline on the last trading day before the weekend.
Now, as quantitative traders, we have no need to take his word on any of these assertions. So, onward to backtesting!
(For those who may be stymied by the lack of affordable historical intraday options data, I recommend Nanex.net.)
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There are still 2 slots available in my online Mean Reversion Strategies workshop in May. The workshop will be conducted live via Adobe Connect, and is limited to a total of 4 participants. Part of the workshop will focus on how to avoid getting hurt when a pair or a portfolio of instruments stop cointegrating.
Showing posts with label Book reviews. Show all posts
Showing posts with label Book reviews. Show all posts
Friday, May 03, 2013
Thursday, March 14, 2013
What Can Quant Traders Learn from Taleb's "Antifragile"?
It can seem a bit ironic that we should be discussing Nassim Taleb's best-seller "Antifragile" here, since most algorithmic trading strategies involve predictions and won't be met with approval from Taleb. Predictions, as Taleb would say, are "fragile" -- they are prone to various biases (e.g. data snooping bias) and the occasional Black Swan event will wipe out the small cumulative profits from many correct bets. Nevertheless, underneath the heap of diatribes against various luminaries ranging from Robert Merton to Paul Krugman, we can find a few gems. Let me start from the obvious to the subtle:
1) Momentum strategies are more antifragile than mean-reversion strategies.
Taleb didn't say that, but that's the first thought that came to my mind. As I argued in many places, mean reverting strategies have natural profit caps (exit when price has reverted to mean) but no natural stop losses (we should buy more of something if it gets cheaper), so it is very much subject to left tail risk, but cannot take advantage of the unexpected good fortune of the right tail. Very fragile indeed! On the contrary, momentum strategies have natural stop losses (exit when momentum reverses) and no natural profit caps (keep same position as long as momentum persists). Generally, very antifragile! Except: what if during a trading halt (due to the daily overnight gap, or circuit breakers), we can't exit a momentum position in time? Well, you can always buy an option to simulate a stop loss. Taleb would certainly approve of that.
2) High frequency strategies are more antifragile than low frequency strategies.
Taleb also didn't say that, and it has nothing to do with whether it is easier to predict short-term vs. long-term returns. Since HF strategies allow us to accumulate profits much faster than low frequency ones, we need not apply any leverage. So even when we are unlucky enough to be holding a position of the wrong sign when a Black Swan hits, the damage will be small compared to the cumulative profits. So while HF strategies do not exactly benefit from right tail risk, they are at least robust with respect to left tail risk.
3) Parameter estimation errors and vulnerability to them should be explicitly incorporated in a backtest performance measurement.
Suppose your trading model has a few parameters which you estimated/optimized using some historical data set. Based on these optimized parameters, you compute the Sharpe ratio of your model on this same data. No doubt this Sharpe ratio will be very good, due to the in-sample optimization. If you apply this model with those optimized the parameters on out-of-sample data, you would probably get a worse Sharpe ratio which is more predictive. But why stop at just two data sets? We can find N different data sets of the same size, calculate the optimized parameters on each of them, but compute the Sharpe ratios over the N-1 out-of-sample data sets. Finally, you can average over all these Sharpe ratios. If your trading model is fragile, you will find that this Sharpe ratio is quite low. But more important than Sharpe ratios, you should compute the maximum drawdown based on each set of parameters, and also the maximum of all these max drawdowns. If your trading model is fragile, this maximum of maximum drawdowns is likely to be quite scary.
The scheme I described above is called cross-validation and is well-known before Taleb, though his book reminds me of its importance.
4) Notwithstanding 3) above, a true estimate of the max drawdown is impossible because it depends on the estimate of the probability of rare events. As Taleb mentioned, even in case of a normal distribution, if the "true" standard deviation is higher than your estimate by a mere 5%, the probability of a 6-sigma event will be increased by 5 times over your estimate! So really the only way to ensure that our maximum drawdown will not exceed a certain limit is through Constant Proportion Portfolio Insurance: trading risky assets with Kelly-leverage in a limited liability company, putting money that you never want to lose in a FDIC-insured bank, with regular withdrawals from the LLC to the bank (but not the other way around).
5) Correlations are impossible to estimate/predict. The only thing we can do is to short at +1 and buy at -1.
Taleb hates Markowitz portfolio optimization, and one of the reasons is that it relies on estimates of covariances of asset returns. As he said, a pair of assets that may have -0.2 correlation over a long period can have +0.8 correlation over another long period. This is especially true in times of financial stress. I quite agree on this point: I believe that manually assigning correlations with values of +/-0.75, +/-0.5, +/-0.25, 0 to entries of the correlation matrix based on "intuition" (fundamental knowledge) can generate as good out-of-sample performance as any meticulously estimated numbers.The more fascinating question is whether there is indeed mean-reversion of correlations. And if so, what instruments can we use to profit from it? Perhaps this article will help.
6) Backtest can only be used to reject a strategy, not to predict its success.
This echoes the point made by commenter Michael Harris in a previous article. Since historical data will never be long enough to capture all the possible Black Swan events that can occur in the future, we can never know if a strategy will fail miserably. However, if a strategy already failed in a backtest, we can be pretty sure that it will fail again in the future.
===
The online "Quantitative Momentum Strategies” workshop that I mentioned in the previous article is now fully booked. Based on popular demand, I will offer a "Mean Reversion Strategies" workshop in May. Once again, it will be conducted in real-time through Skype, and the number of attendees will be similarly limited to 4. See here for more information.
1) Momentum strategies are more antifragile than mean-reversion strategies.
Taleb didn't say that, but that's the first thought that came to my mind. As I argued in many places, mean reverting strategies have natural profit caps (exit when price has reverted to mean) but no natural stop losses (we should buy more of something if it gets cheaper), so it is very much subject to left tail risk, but cannot take advantage of the unexpected good fortune of the right tail. Very fragile indeed! On the contrary, momentum strategies have natural stop losses (exit when momentum reverses) and no natural profit caps (keep same position as long as momentum persists). Generally, very antifragile! Except: what if during a trading halt (due to the daily overnight gap, or circuit breakers), we can't exit a momentum position in time? Well, you can always buy an option to simulate a stop loss. Taleb would certainly approve of that.
2) High frequency strategies are more antifragile than low frequency strategies.
Taleb also didn't say that, and it has nothing to do with whether it is easier to predict short-term vs. long-term returns. Since HF strategies allow us to accumulate profits much faster than low frequency ones, we need not apply any leverage. So even when we are unlucky enough to be holding a position of the wrong sign when a Black Swan hits, the damage will be small compared to the cumulative profits. So while HF strategies do not exactly benefit from right tail risk, they are at least robust with respect to left tail risk.
3) Parameter estimation errors and vulnerability to them should be explicitly incorporated in a backtest performance measurement.
Suppose your trading model has a few parameters which you estimated/optimized using some historical data set. Based on these optimized parameters, you compute the Sharpe ratio of your model on this same data. No doubt this Sharpe ratio will be very good, due to the in-sample optimization. If you apply this model with those optimized the parameters on out-of-sample data, you would probably get a worse Sharpe ratio which is more predictive. But why stop at just two data sets? We can find N different data sets of the same size, calculate the optimized parameters on each of them, but compute the Sharpe ratios over the N-1 out-of-sample data sets. Finally, you can average over all these Sharpe ratios. If your trading model is fragile, you will find that this Sharpe ratio is quite low. But more important than Sharpe ratios, you should compute the maximum drawdown based on each set of parameters, and also the maximum of all these max drawdowns. If your trading model is fragile, this maximum of maximum drawdowns is likely to be quite scary.
The scheme I described above is called cross-validation and is well-known before Taleb, though his book reminds me of its importance.
4) Notwithstanding 3) above, a true estimate of the max drawdown is impossible because it depends on the estimate of the probability of rare events. As Taleb mentioned, even in case of a normal distribution, if the "true" standard deviation is higher than your estimate by a mere 5%, the probability of a 6-sigma event will be increased by 5 times over your estimate! So really the only way to ensure that our maximum drawdown will not exceed a certain limit is through Constant Proportion Portfolio Insurance: trading risky assets with Kelly-leverage in a limited liability company, putting money that you never want to lose in a FDIC-insured bank, with regular withdrawals from the LLC to the bank (but not the other way around).
5) Correlations are impossible to estimate/predict. The only thing we can do is to short at +1 and buy at -1.
Taleb hates Markowitz portfolio optimization, and one of the reasons is that it relies on estimates of covariances of asset returns. As he said, a pair of assets that may have -0.2 correlation over a long period can have +0.8 correlation over another long period. This is especially true in times of financial stress. I quite agree on this point: I believe that manually assigning correlations with values of +/-0.75, +/-0.5, +/-0.25, 0 to entries of the correlation matrix based on "intuition" (fundamental knowledge) can generate as good out-of-sample performance as any meticulously estimated numbers.The more fascinating question is whether there is indeed mean-reversion of correlations. And if so, what instruments can we use to profit from it? Perhaps this article will help.
6) Backtest can only be used to reject a strategy, not to predict its success.
This echoes the point made by commenter Michael Harris in a previous article. Since historical data will never be long enough to capture all the possible Black Swan events that can occur in the future, we can never know if a strategy will fail miserably. However, if a strategy already failed in a backtest, we can be pretty sure that it will fail again in the future.
===
The online "Quantitative Momentum Strategies” workshop that I mentioned in the previous article is now fully booked. Based on popular demand, I will offer a "Mean Reversion Strategies" workshop in May. Once again, it will be conducted in real-time through Skype, and the number of attendees will be similarly limited to 4. See here for more information.
Tuesday, June 19, 2012
Momentum strategies: a book review
As a devout mean-reversion trader, I find Mike Dever's new book "Jackass Investing" unexpectedly well-argued and readable.
You see, momentum and mean-reversion traders live in two separate universes, and they are often mutually incomprehensible to each other. Dever, as a CTA, inhabits the momentum universe. Example: my favorite performance measure, the Sharpe ratio, has been brusquely dispatched as a bad measurement of risk, and drawdown becomes king. But all for good reasons: Dever argues that Sharpe ratio measures only the daily volatility of returns, but disregarded the "black swan" events, which are much better captured by the maximum drawdown. I agree with the author on this point, but there are other uses of Sharpe ratio: a high Sharpe ratio strategy does indicate high statistical significance of the trading strategy, a claim that momentum strategies can seldom make. I often think of momentum strategies as being long options: you have to keep paying premium until one day, you make them all back with a home run. But when you are backtesting a strategy, how would you know that the rare, statistically insignificant, home run was not due to data snooping bias? Unless of course, like the author, you have fundamental insights into the traded instruments.
Fundamental insights are in fact one of the delicious highlights of this book. Dever describes his orange juice futures strategy using the "marginal cost of production" as a fundamental valuation tool. He argues that orange juice cannot be sold below this cost, since farmers would have no incentive for production otherwise. And he was right: orange juice futures started to rebound from the 27-year low of 55 cents/pound in May 2004, to almost 90 cents/pound in September (thanks partly to hurricanes hitting Florida). Dever went long at 70 cents. Oh, how we quantitative traders would love to have the confidence that such insights inspire!
Of course, I don't agree with everything written in the book. For example, though the author rightly pointed out that the distribution of returns often have a positive kurtosis, he uses that as evidence of trending behavior. While I agree that price trends can indeed produce positive kurtosis, we can certainly construct mean-reverting price series with occasional catastrophes that have the same kurtosis. To us mean-reversion traders, positive kurtosis is not an invitation to "follow-the-trend", but as a warning sign to find risk management measures that protect us from catastrophes.
Even though momentum strategies in general are in a state of trauma right now (more on that later), Dever nevertheless makes a good case why we should include them as part of our portfolio of strategies. Comparing the S&P500 index (SPX) with the S&P Diversifed Trends Indicator (DTI, a simple trend-following strategies on 24 futures), he finds that the Sharpe ratio (though of course he refuses to use that hated term) of the DTI is more than double that of the SPX, with only about 1/3 of the maximum drawdown. But before you, the reader, decides to join the momentum bandwagon, I invite you to take a look at a plot of DTI's values since inception:
Since its high watermark in 2008/12/5, this representative momentum strategy has been in a relentless drawdown. Why? This is due to another well-studied and troubling property of momentum strategies: they always performed poorly for several years after a financial crisis.
You see, momentum and mean-reversion traders live in two separate universes, and they are often mutually incomprehensible to each other. Dever, as a CTA, inhabits the momentum universe. Example: my favorite performance measure, the Sharpe ratio, has been brusquely dispatched as a bad measurement of risk, and drawdown becomes king. But all for good reasons: Dever argues that Sharpe ratio measures only the daily volatility of returns, but disregarded the "black swan" events, which are much better captured by the maximum drawdown. I agree with the author on this point, but there are other uses of Sharpe ratio: a high Sharpe ratio strategy does indicate high statistical significance of the trading strategy, a claim that momentum strategies can seldom make. I often think of momentum strategies as being long options: you have to keep paying premium until one day, you make them all back with a home run. But when you are backtesting a strategy, how would you know that the rare, statistically insignificant, home run was not due to data snooping bias? Unless of course, like the author, you have fundamental insights into the traded instruments.
Fundamental insights are in fact one of the delicious highlights of this book. Dever describes his orange juice futures strategy using the "marginal cost of production" as a fundamental valuation tool. He argues that orange juice cannot be sold below this cost, since farmers would have no incentive for production otherwise. And he was right: orange juice futures started to rebound from the 27-year low of 55 cents/pound in May 2004, to almost 90 cents/pound in September (thanks partly to hurricanes hitting Florida). Dever went long at 70 cents. Oh, how we quantitative traders would love to have the confidence that such insights inspire!
Of course, I don't agree with everything written in the book. For example, though the author rightly pointed out that the distribution of returns often have a positive kurtosis, he uses that as evidence of trending behavior. While I agree that price trends can indeed produce positive kurtosis, we can certainly construct mean-reverting price series with occasional catastrophes that have the same kurtosis. To us mean-reversion traders, positive kurtosis is not an invitation to "follow-the-trend", but as a warning sign to find risk management measures that protect us from catastrophes.
Even though momentum strategies in general are in a state of trauma right now (more on that later), Dever nevertheless makes a good case why we should include them as part of our portfolio of strategies. Comparing the S&P500 index (SPX) with the S&P Diversifed Trends Indicator (DTI, a simple trend-following strategies on 24 futures), he finds that the Sharpe ratio (though of course he refuses to use that hated term) of the DTI is more than double that of the SPX, with only about 1/3 of the maximum drawdown. But before you, the reader, decides to join the momentum bandwagon, I invite you to take a look at a plot of DTI's values since inception:
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| S&P DTI index |
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