November 2020 - QUANTITATIVE RESEARCH AND TRADING

November 24, 2020

Can Machine Learning Techniques Be Used To Predict Market Direction? The 1,000,000 Model Test.

During the 1990’s the advent of Neural Networks unleashed a torrent of research on their applications in financial markets, accompanied by some rather extravagant claims about their predicative abilities. Sadly, much of the research proved to be sub-standard and the results illusionary, following which the topic was largely relegated to the bleachers, at least in the field of financial market research.

With the advent of new machine learning techniques such as Random Forests, Support Vector Machines and Nearest Neighbor Classification, there has been a resurgence of interest in non-linear modeling techniques and a flood of new research, a fair amount of it supportive of their potential for forecasting financial markets. Once again, however, doubts about the quality of some of the research bring the results into question.

Against this background I and my co-researcher Dan Rico set out to address the question of whether these new techniques really do have predicative power, more specifically the ability to forecast market direction. Using some excellent MatLab toolboxes and a new software package, an Excel Addin called 11Ants, that makes large scale testing of multiple models a snap, we examined over 1,000,000 models and model-ensembles, covering just about every available non-linear technique. The data set for our study comprised daily prices for a selection of US equity securities, together with a large selection of technical indicators for which some other researchers have claimed explanatory power.

In-Sample Equity Curve for Best Performing Nonlinear Model

The answer provided by our research was, without exception, in the negative: not one of the models tested showed any significant ability to predict the direction of any of the securities in our data set. Furthermore, our study found that the best-performing models favored raw price data over technical indicator variables, suggesting that the latter have little explanatory power.

As with Neural Networks, the principal difficulty with non-linear techniques appears to be curve-fitting and a failure to generalize: while it is very easy to find models that provide an excellent fit to in-sample data, the forecasting performance out-of-sample is often very poor.

Out-of-Sample Equity Curve for Best Performing Nonlinear Model

Some caveats about our own research apply. First and foremost, it is of course impossible to prove a hypothesis in the negative. Secondly, it is plausible that some markets are less efficient than others: some studies have claimed success in developing predictive models due to the (relative) inefficiency of the F/X and futures markets, for example. Thirdly, the choice of sample period may be criticized: it could be that the models were over-conditioned on a too- lengthy in-sample data set, which in one case ran from 1993 to 2008, with just two years (2009-2010) of out-of-sample data. The choice of sample was deliberate, however: had we omitted the 2008 period from the “learning” data set, it would be very easy to criticize the study for failing to allow the algorithms to learn about the exceptional behavior of the markets during that turbulent year.

Despite these limitations, our research casts doubt on the findings of some less-extensive studies, that may be the result of sample-selection bias. One characteristic of the most credible studies finding evidence in favor of market predictability, such as those by Pesaran and Timmermann, for instance (see paper for citations), is that the models they employ tend to incorporate independent explanatory variables, such as yield spreads, which do appear to have real explanatory power. The finding of our study suggest that, absent such explanatory factors, the ability to predict markets using sophisticated non-linear techniques applied to price data alone may prove to be as illusionary as it was in the 1990’s.

ONE MILLION MODELS

November 17, 2020

The F/X Momentum Strategy

Our approach is based upon the idea that currencies tend to be range bound, that momentum ultimately exhausts itself and that prices tend to fall faster than they rise. The strategy seeks to exploit these characteristics with short trades that may be closed within a few hours, or continue over several days when the exhaustion pattern emerges more slowly.

November 10, 2020April 23, 2024

Covered Writes, Covered Wrongs

What is a Covered Call?

A covered call (or covered write or buy-write) is a long position in a security and a short position in a call option on that security. The diagram below constructs the covered call payoff diagram, including the option premium, at expiration when the call option is written at a $100 strike with a $25 option premium.

Equity index covered calls are an attractive strategy to many investors because they have realized returns not much lower than those of the equity market but with much lower volatility. But investors often do the trade for the wrong reasons: there are a number of myths about covered writes that persist even amongst professional options traders. I have heard most, if not all of them professed by seasons floor traders on the American Stock Exchange and, I confess, I have even used one or two of them myself. Roni Israelov and Larn Nielsen of AQR Capital Management, LLC have done a fine job of elucidating and then dispelling these misunderstandings about the strategy, in their paper Covered Call Strategies: One Fact and Eight Myths, Financial Analysts Journal, Vol. 70, No. 6, 2014.

The Cover Call Strategy and its Benefits for Investors

The covered call strategy has generated attention due to its attractive historical risk-adjusted returns. For example, the CBOE S&P 500 BuyWrite Index, the industry-standard covered call benchmark, is commonly described as providing average returns comparable to the S&P 500 Index with approximately two-thirds the volatility, supported by statistics such as those shown below.

The key advantages of the strategy (compared to an outright, delta-one position) include lower volatility, beta and tail risk. As a consequence, the strategy produces higher risk-adjusted rates if return (Sharpe Ratio). Note, too, the beta convexity of the strategy, a topic I cover in this post:

http://jonathankinlay.com/2017/05/beta-convexity/

Although the BuyWrite Index has historically demonstrated similar total returns to the S&P 500, it does so with a reduced beta to the S&P 500 Index. However, it is important to also understand that the BuyWrite Index is more exposed to negative S&P 500 returns than positive returns. This asymmetric relationship to the S&P 500 is consistent with its payoff characteristics and results from the fact that a covered call strategy sells optionality. What this means in simple terms is that while drawdowns are somewhat mitigated by the revenue associated with call writing, the upside is capped by those same call options.

Understandably, a strategy that produces equity-like return with lower beta and lower risk attracts considerable attention from investors. According to Moringstar, growth in assets under management in covered call strategies has been over 25% per year over the 10 years through June 2014 with over $45 billion currently invested.

Myths about the Covered Call Strategy

Many option strategies are the subject of investor myths, partly, I suppose, because option strategies are relatively complicated and entail risks in several dimensions. So it is quite easy for investors to become confused. Simple anecdotes are attractive because they appear to cut through the complexity with an easily understood metaphor, but they can often be misleading. An example is the widely-held view – even amongst professional option traders – is that selling volatility via strangles is a less risk approach than selling at-the-money straddles. Intuitively, this makes sense: why wouldn’t selling straddles that have strike prices (far) away from the current spot price be less risky than selling straddles that have strike prices close to the spot price? But, in fact, it turns out that selling straddles is the less risky the two strategies – see this post for details:

http://jonathankinlay.com/2016/11/selling-volatility/

Likewise, the covered call strategy is subject to a number of “urban myths”, that turn out to be unfounded:

Myth 1: Risk exposure is described by the payoff diagram

That is only true at expiration. Along the way, the positions will be marked-to-market and may produce a substantially different payoff if the trade is terminated early. The same holds true for a zero-coupon bond – we know the terminal value for certain, but there can be considerable variation in the value of the asset from day to day.

Myth 2: Covered calls provide downside protection

This is partially true, but only in a very limited sense. Unlike a long option hedge, the “protection” in a buy-write strategy is limited to only the premium collected on the option sale, a relatively modest amount in most cases. Consider a covered call position on a $100 stock with a $10 at-the-money call premium. The covered call can potentially lose $90 and the long call option can lose $10. Each position has the same 50% exposure to the stock, but the covered call’s downside risk is disproportionate to its stock exposure. This is consistent with the covered call’s realized upside and downside betas as discussed earlier.

Myth 3: Covered calls generate income.

Remember that income is revenue minus costs.

It is true that option selling generates positive cash flow, but this incorrectly leads investors to the conclusion that covered calls generate investment income. Just as is the case with bond issuance, the revenue generated from selling the call option is not income (though, like income, the cash flows received from selling options are considered taxable for many investors). In order for there to be investment income or earnings, the option must be sold at a favorable price – the option’s implied volatility needs to be higher than the stock’s expected volatility.

Myth 4: Covered calls on high-volatility stocks and/or shorter-dated options provide higher yield.

Though true that high volatility stocks and short-dated options command higher annualized premiums, insurance on riskier assets should rationally command a higher premium and selling insurance more often per year should provide higher annual premiums. However, these do not equate to higher net income or yield. For instance, if options are properly priced (e.g., according to the Black-Scholes pricing model), then selling 12 at-the-money options will generate approximately 3.5 times the cash flow of selling a single annual option, but this does not unequivocally translate into higher net profits as discussed earlier. Assuming fairly priced options, higher revenue is not necessarily a mechanism for increasing investment income.

The key point here is that what matters is value, not price. In other words, expected investment profits are generated by the option’s richness, not the option’s price. For example, if you want to short a stock with what you consider to be a high valuation, then the goal is not to find a stock with a high price, but rather one that is overpriced relative to its fundamental value. The same principle applies to options. It is not appropriate to seek an option with a high price or other characteristics associated with high prices. Investors must instead look for options that are expensive relative to their fundamental value. Put another way, the investor should seek out options trading at a higher implied volatility than the likely futures realized volatility over the life of the option.

Myth 5: Time decay of options written works in your favor.

While it is true that the value of an option declines over time as it approaches expiration, that is not the whole story. In fact an option’s expected intrinsic value increases as the underlying security realizes volatility. What matters is whether the realized volatility turns out to be lower than the volatility baked into the option price – the implied volatility. In truth, an option’s time decay only works in the seller’s favor if the option is initially priced expensive relative to its fundamental value. If the option is priced cheaply, then time decay works very much against the seller.

Myth 6: Covered calls are appropriate if you have a neutral to moderately bullish view.

This myth is an over-simplification. In selling a call option you are expressing a view, not only on the future prospects for the stock, but also on its likely future volatility. It is entirely possible that the stock could stall (or even decline) and yet the value of the option you have sold rises due, say, to takeover rumors. A neutral view on the stock may imply a belief that the security price will not move far from its current price rather than its expected return is zero. If so, then a short straddle position is a way to express that view — not a covered call — because, in this case, no active position should be taken in the security.

Myth 7: Overwriting pays you for doing what you were going to do anyway

This myth is typically posed as the following question: if you have a price target for selling a stock you own, why not get paid to write a call option struck at that price target?

In fact this myth exposes the critical difference between a plan and a contractual obligation. If the former case, suppose that the stock hits your target price very much more quickly than you had anticipated, perhaps as a result of a new product announcement that you had not anticipated at the time you set your target. In those circumstances you might very well choose to maintain your long position and revise your price target upwards. This is an example of a plan – a successful one – that can be adjusted to suit circumstances as they change.

A covered call strategy is an obligation, rather than a plan. You have pre-sold the stock at the target price and, in the above scenario, you cannot change your mind in order to benefit from additional potential upside in the stock.

In other words, with a covered call strategy you have monetized the optionality that is inherent in any plan and turned it into a contractual obligation in exchange for a fee.

Myth 8: Overwriting allows you to buy a stock at a discounted price.

Here is how this myth is typically framed: if a stock that you would like to own is currently priced at $100 and that you think is currently expensive, you can act on that opinion by selling a naked put option at a $95 strike price and collect a premium of say $1. Then, if the price subsequently declines below the strike price, the option will likely be exercised thus requiring you to buy the stock for $95. Including the $1 premium, you effectively buy the stock at a 6% discount. If the option is not exercised you keep the premium as income. So, this type of outcome for selling naked put options may also lead you to conclude that the equivalent covered call strategy makes sense and is valuable.

But this argument is really a sleight of hand. In our example above, if the option is exercised, then when you buy the stock for $95 you won’t care what the stock price was when you sold the option. What matters is the stock price on the date the option was exercised. If the stock price dropped all the way down to $80, the $95 purchase price no longer seems like a discount. Your P&L will show a mark-to-market loss of $14 ($95 – $80 – $1). The initial stock price is irrelevant and the $1 premium hardly helps.

Conclusion: How to Think About the Covered Call Strategy

Investors should ignore the misleading storytelling about obtaining downside buffers and generating income. A covered call strategy only generates income to the extent that any other strategy generates income, by buying or selling mispriced securities or securities with an embedded risk premium. Avoid the temptation to overly focus on payoff diagrams. If you believe the index will rise and implied volatilities are rich, a covered call is a step in the right direction towards expressing that view.

If you have no view on implied volatility, there is no reason to sell options, or covered calls

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November 3, 2020November 3, 2020

Range-Based EGARCH Option Pricing Models (REGARCH)

The research in this post and the related paper on Range Based EGARCH Option pricing Models is focused on the innovative range-based volatility models introduced in Alizadeh, Brandt, and Diebold (2002) (hereafter ABD). We develop new option pricing models using multi-factor diffusion approximations couched within this theoretical framework and examine their properties in comparison with the traditional Black-Scholes model.

The two-factor version of the model, which I have applied successfully in various option arbitrage strategies, encapsulates the intuively appealing idea of a trending long term mean volatility process, around which oscillates a mean-reverting, transient volatility process. The option pricing model also incorporates asymmetry/leverage effects and well as correlation effects between the asset return and volatility processes, which results in a volatility skew.

The core concept behind Range-Based Exponential GARCH model is Log-Range estimator discussed in an earlier post on volatility metrics, which contains a lengthy exposition of various volatility estimators and their properties. (Incidentally, for those of you who requested a copy of my paper on Estimating Historical Volatility, I have updated the post to include a link to the pdf).

We assume that the log stock price s follows a drift-less Brownian motion ds = sdW. The volatility of daily log returns, denoted h= s/sqrt(252), is assumed constant within each day, at ht from the beginning to the end of day t, but is allowed to change from one day to the next, from ht at the end of day t to ht+1 at the beginning of day t+1. Under these assumptions, ABD show that the log range, defined as:

is to a very good approximation distributed as

where N[m; v] denotes a Gaussian distribution with mean m and variance v. The above equation demonstrates that the log range is a noisy linear proxy of log volatility ln ht. By contrast, according to the results of Alizadeh, Brandt,and Diebold (2002), the log absolute return has a mean of 0.64 + ln ht and a variance of 1.11. However, the distribution of the log absolute return is far from Gaussian. The fact that both the log range and the log absolute return are linear log volatility proxies (with the same loading of one), but that the standard deviation of the log range is about one-quarter of the standard deviation of the log absolute return, makes clear that the range is a much more informative volatility proxy. It also makes sense of the finding of Andersen and Bollerslev (1998) that the daily range has approximately the same informational content as sampling intra-daily returns every four hours.

Except for the model of Chou (2001), GARCH-type volatility models rely on squared or absolute returns (which have the same information content) to capture variation in the conditional volatility h_t_. Since the range is a more informative volatility proxy, it makes sense to consider range-based GARCH models, in which the range is used in place of squared or absolute returns to capture variation in the conditional volatility. This is particularly true for the EGARCH framework of Nelson (1990), which describes the dynamics of log volatility (of which the log range is a linear proxy).

ABD consider variants of the EGARCH framework introduced by Nelson (1990). In general, an EGARCH(1,1) model performs comparably to the GARCH(1,1) model of Bollerslev (1987). However, for stock indices the in-sample evidence reported by Hentschel (1995) and the forecasting performance presented by Pagan and Schwert (1990) show a slight superiority of the EGARCH specification. One reason for this superiority is that EGARCH models can accommodate asymmetric volatility (often called the “leverage effect,” which refers to one of the explanations of asymmetric volatility), where increases in volatility are associated more often with large negative returns than with equally large positive returns.

The one-factor range-based model (REGARCH 1) takes the form:

where the returns process R_t is conditionally Gaussian: R_t ~ N[0, h_t²]

and the process innovation is defined as the standardized deviation of the log range from its expected value:

Following Engle and Lee (1999), ABD also consider multi-factor volatility models. In particular, for a two-factor range-based EGARCH model (REGARCH2), the conditional volatility dynamics) are as follows:

and

where ln q_t can be interpreted as a slowly-moving stochastic mean around which log volatility ln h_t makes large but transient deviations (with a process determined by the parameters k_h, f_h and d_h).

The parameters q, k_q, f_q and d_q determine the long-run mean, sensitivity of the long run mean to lagged absolute returns, and the asymmetry of absolute return sensitivity respectively.

The intuition is that when the lagged absolute return is large (small) relative to the lagged level of volatility, volatility is likely to have experienced a positive (negative) innovation. Unfortunately, as we explained above, the absolute return is a rather noisy proxy of volatility, suggesting that a substantial part of the volatility variation in GARCH-type models is driven by proxy noise as opposed to true information about volatility. In other words, the noise in the volatility proxy introduces noise in the implied volatility process. In a volatility forecasting context, this noise in the implied volatility process deteriorates the quality of the forecasts through less precise parameter estimates and, more importantly, through less precise estimates of the current level of volatility to which the forecasts are anchored.