Positional Option Trading

(1.3)

If this option was initially delta-hedged, the P/L over this price move would be

(1.4)

Next note that

(1.5)

so that the profit in hedging over each time interval is

(1.6)

(Although equation 1.6 is only asymptotically true, if we worked with an infinitesimal price change, this derivation would be exact.

This is the first term of the BSM differential equation. Literally, BSM says that these profits from rebalancing due to gamma are exactly equal to the theta of the option. Expected movement cancels time decay. The only way a directionally neutral option position will make money is if the option's implied volatility (which governs theta) is not the same as the underlying's realized volatility (which determines the rebalancing profits). This is true no matter which structure is chosen and the particulars of the hedging scheme.

If we can identify situations where this volatility mismatch occurs, the expected profit from the position will be given by

(1.7)

This is the fundamental equation of option trading. All the “theta decay” and “gamma scalping” profits and losses are tied up in this relationship.

Note also that this vega P/L will affect directional option trades. If we pay the wrong implied volatility level for an option, we might still make money but we would have been better off replicating the option in the underlying.

The BSM equation depends on a number of financial and mathematical assumptions.

• The underlying is a tradable asset.

• There is a single, risk-free interest rate.

• The underlying can be shorted.

• Proceeds from short sales can be invested at the risk-free rate.

• All cash flows are taxed at the same rate.

• The underlying's returns are continuous and normally distributed with a constant volatility.

Traders have devised various workarounds to address these limiting assumptions (see Appendix One). The most important of these is the concept of the implied volatility surface. If the BSM were an accurate descriptive theory, all options on a given underlying would have one volatility. This is not true. For the BSM equation to reproduce market option prices, options with different strikes have different implied volatilities (the smile) and options with different maturities have different implied volatilities (the term structure). These implied volatilities make up the IV surface. An example is in shown in Figure 1.1.

The IV surface exists partially because the BSM is mathematically misspecified. The underlying does not have returns that are continuous and normally distributed with a constant volatility. However, even a model that perfectly captured the underlying dynamics would need a fudge factor like the implied volatility surface. Some of the reasons for its existence have nothing to do with the underlying. Different options have different supply and demand, and these distort option prices. Because of this, there is often an edge in selling options with high volatilities relative to others on the same underlying (see the section on the implied skewness premium in Chapter Four).

Equation 1.7 gives the average PL of any hedged option position, but there is a wide dispersion of results for this mean, and the spread of this distribution decreases with the number of hedges. Figure 1.2 shows the PL distribution of a short straddle that is never re-hedged, and Figure 1.3 shows the distribution when the straddle is re-hedged every day. The underlying paths were generated from 10,000 realizations of a GBM. The implied and realized volatility were equal so we expect an average PL of zero.

FIGURE 1.1 The implied volatility surface for SPY on September 10, 2019.

FIGURE 1.2 The terminal PL distribution of a single short one-year ATM straddle that is never re-hedged. Stock price is $100, rates are zero, and both realized and implied volatilities are 30%.

FIGURE 1.3 The terminal PL distribution of a single one-year ATM straddle that is hedged daily. Stock price is $100, rates are zero, and both realized and implied volatilities are 30%.

The dependence of the standard deviation of the PL distribution on the number of hedges is shown in Figure 1.4.

FIGURE 1.4 The standard deviation of the terminal PL distribution of a single one-year ATM straddle as a function of the number of hedges. Stock price is $100, rates are zero, and both realized and implied volatilities are 30%.

The reason to hedge less frequently and accept a wider standard deviation of results is that hedging costs money. All hedges incur transaction costs (brokerage, exchange fees, and infrastructure costs). Costs like this are an easily forgotten drain on a portfolio. Individually they are small, but they accumulate. To emphasize this point, Table 1.1 compares the summary statistics of results for the daily hedged short straddle when there is a transaction cost of $.10 per share and when hedges are costless.

The difference between these two cases is roughly equivalent to misestimating volatility by two points.

In practice, aggressive re-hedging is done by market-making firms and some volatility specialists. The vast majority of retail and buy-side users seldom or never hedge. The relevant theory for those hoping to approximate continuous hedging is discussed in Sinclair (2013). In this book we will generally assume that no re-hedging takes place. These results are also applicable to those who hedge infrequently. They can just assume that the original position has been closed and a new one opened. So, a one-year position that is hedged after a month would thereafter have the expected distribution of an 11-month option.

TABLE 1.1 Statistics for the Short One-Year ATM Daily Hedged Straddle With and Without Hedging Costs (stock price is $100, rates are zero, and both realized and implied volatilities are 30%.)

Conclusion

The BSM model gives the replication strategy for the option. The expected return of the underlying is irrelevant to this strategy. The only distributional property of the underlying that is used in the BSM model is the volatility. A hedged position will, on average, make a profit proportional to the difference between the volatility implied by the option market price (by inverting the BSM model) and the subsequent realized volatility. The choice of the option structure and hedging scheme can change the shape of the PL distribution, but not the average value. These choices are far from immaterial, but successful option trading depends foremost on finding situations in which the implied volatility is mispriced.

Summary

• Arbitrage-free option pricing models do not include the underlying return. BSM includes only volatility.

• Inverting the pricing model using the option's market price as an input gives the implied volatility.

• The average profit of a hedged option position is proportional to the difference between implied volatility and the subsequent realized volatility.

• Practical option hedging is designed to give an acceptable level of variance for a given amount of transaction costs.

CHAPTER 2

The Efficient Market Hypothesis and Its Limitations

A lot of trading books propagate the myth that successful trading is based on discipline and persistence. This might be the worst advice possible. A trader without a real edge who persists in trading, executing a bad plan in a disciplined manner, will lose money faster and more consistently than someone who is lazy and inconsistent. A tough but unskilled fighter will just manage to stay in a losing fight longer. All she will achieve is being beaten up more than a weak fighter would.

Another terrible weakness is optimism. Optimism will keep losing traders chasing success that will never happen. Sadly, hope is a psychological mechanism unaffected by external reality.

Emotional control won't make up for lack of edge. But, before we can find an edge, we need to understand why this is hard and where we should look.

The Efficient Market Hypothesis

The traders' concept of the efficient market hypothesis (EMH) is “making money is hard.” This isn't wrong, but it is worth looking at the theory in more detail. Traders are trying to make money from the exceptions to the EMH, and the different types of inefficiencies should be understood, and hence traded, differently.

The EMH was contemporaneously developed from two distinct directions. Paul Samuelson (1965) introduced the idea to the economics community under the umbrella of “rational expectations theory.” At the same time, Eugene Fama's studies (1965a, 1965b) of the statistics of security returns led him to the theory of “the random walk.”

The idea can be stated in many ways, but a simple, general expression is as follows:

A market is efficient with respect to some information if it is impossible to profitably trade based on that information.

And the “profitable trades” are risk-adjusted, after all costs.

So, depending on the information we are considering, there are many different EMHs, but three in particular have been extensively studied:

• The strong EMH in which the information is anything that is known by anyone

• The semi-strong EMH in which the information is any publicly available information, such as past prices, earnings, or analysts' studies

• The weak EMH in which the information is past prices

The EMH is important as an organizing principle and is a very good approximation to reality. But, it is important to note that no one has ever believed that any form of the EMH is strictly true. Traders are right. Making money is hard, but it isn't impossible. The general idea of the theory and also the fact it isn't perfect is agreed on by most successful investors and economists.

“I think it is roughly right that the market is efficient, which makes it very hard to beat merely by being an intelligent investor. But I don't think it's totally efficient at all. And the difference between being totally efficient and somewhat efficient leaves an enormous opportunity for people like us to get these unusual records. It's efficient enough, so it's hard to have a great investment record. But it's by no means impossible.”

—Charlie Munger

Even one of the inventors of the theory, Eugene Fama, qualified the idea of efficiency by using the word good instead of perfect.

“In an efficient market, at any point in time, the actual price of a security will be a good estimate of its intrinsic value.”

—Eugene Fama

There is something of a paradox in the concept of market efficiency. The more efficient a market is, the more random and unpredictable the returns will be. A perfectly efficient market will be completely unpredictable. But the way this comes about is through the trading of all market participants. Investors all try to profit from any informational advantage they have, and by doing this their information is incorporated into the prices. Grossman and Stiglitz (1980) use this idea to argue that perfectly efficient markets are impossible. If markets were efficient, traders wouldn't make the effort to gather information, and so there would be nothing driving markets toward efficiency. So, an equilibrium will form where markets are mostly efficient, but it is still worth collecting and processing information.

(This is a reason fundamental analysis consisting of reading the Wall Street Journal and technical analysis using well-known indicators is likely to be useless. Fischer Black [1986] called these people “noise traders.” They are the people who pay the good traders.)

There are other arguments against the EMH. The most persuasive of these are from the field of behavioral finance. It's been shown that people are irrational in many ways. People who do irrational things should provide opportunities to those who don't. As Kipling (1910) wrote, “If you can keep your head when all about you are losing theirs, … you will be a man, my son.”

In his original work on the EMH, Fama mentioned three conditions that were sufficient (although not necessary) for efficiency:

• Absence of transaction costs

• Perfect information flow

• Agreement about the price implications of information

Helpfully for us, these conditions do not usually apply in the options market. Options, particularly when dynamically hedged, have large transaction costs. Information is not universally available and volatility markets often react slowly to new information. Further, the variance premium cannot be directly traded. Volatility markets are a good place to look for violations of the EMH.

Let's accept that the EMH is imperfect enough that it is possible to make money. The economists who study these deviations from perfection classify them into two classes: risk premia and inefficiencies. A risk premium is earned as compensation for taking a risk, and if the premium is mispriced, it will be profitable even after accepting the risk. An inefficiency is a trading opportunity caused by the market not noticing something. An example is when people don't account for the embedded options in a product.

There is a joke (not a funny one) about an economist seeing a $100 bill on the ground. She walks past it. A friend asks: “Didn't you see the money there?” The economist replies: “I thought I saw something, but I must've imagined it. If there had been $100 on the ground, someone would've picked it up.” We know that the EMH is not strictly true, but the money could be there for two different reasons. Maybe it is on a busy road and no one wants to run into traffic. This is a risk premium. But maybe it is outside a bar where drunks tend to drop money as they leave. This is an inefficiency. There is also the possibility that the note was there purely by luck.

It is often impossible to know whether a given opportunity is a risk premium or an inefficiency, and a given opportunity will probably be partially both. But it is important to try to differentiate. A risk premium can be expected to persist: the counterparty is paying for insurance against a risk. They may improve their pricing of the insurance, but they will probably continue to pay something.

By contrast, an inefficiency will last only until other people notice it. And failing to differentiate between a real opportunity and a chance event will only lead to losses.

Some traders will profit from inefficiencies. Not all traders will. A lot of traders will use meaningless or widely known information. Many forecasts are easy. I can predict the days the non-farm payroll will be released. I can predict what days fall on weekends. I can predict the stock market closes at 4 p.m. eastern time. In many cases, making a good prediction is the easy part. The hard part is that the forecast has to be better than the market's, which the consensus of everyone else's prediction is. For developed stock indices the correlation between the daily range on one day and the next is roughly between 65% and 70%. So a very good volatility estimator is that it will be what it was the day before (a few more insights like this will lead you to GARCH). It is both hard and profitable to make an even slightly better one-day forecast. And whether it is because the techniques that are used are published, employees leave and take information with them, or just that several people have a similar idea at the same time, these forecast edges don't last forever.

Aside: Alpha Decay

The extinction of floor traders is an example of a structural shift in markets destroying a job. Similar to most people, traders tend to think that their skills are special, and their jobs will always be around. This isn't true. The floors have gone. Fixed commissions have gone. Investment advisors are being replaced by robo-advisors. There are fewer option market-makers, each trading many more stocks than in the past. Offshoring will definitely come to trading, and it is quite possible that a market structure such as a once-a-day auction could replace continuous trading.

But as well as these structural changes, the alpha derived from market inefficiencies (as opposed to the beta of exposure to a mispriced risk factor) doesn't last forever. Depending on how easy it is to trade the effect, the half-life of an inefficiency-based strategy seems to be between 6 months and 5 years. Mclean and Pontiff (2016) showed that the publication of a new anomaly lessens its returns by up to 58%. And publication isn't the only thing that erodes alpha. Chordia et al. (2014) showed that increasing liquidity also reduces excess returns by about 50%. Sometimes the anomaly exists only because it isn't worth the time of large traders to get involved. A similar effect is that the easy access to data will kill strategies. Sometimes the alpha isn't due to a wrinkle in the financial market. It is due to the costs of processing information.

Just as some traders will profit by using a stupid idea like candlestick charting, some traders will succeed for a while with an overfit model. I'm in no way using this to condone data-mining, but we can learn a valid lesson from this. As Guns and Roses pointed out, “nothing lasts forever.” Lucky strategies will never last but even the best, completely valid strategy will have a lifetime. So, when you are making money don't think that being “prudent” is a good idea. The right thing to do is to be as aggressive as possible. Amateurs go broke for a lot of reasons, but professionals often suffer in bad times because they didn't fully capitalize on good times, instead thinking that making steady but small profits was the best thing to do.

They also spend too much in good times, forgetting that they won't last. I've had a floor trader tell me about his new Ferrari about an hour before laughing about the stupid spending habits of NFL and NBA players (the last I heard he was selling houses). Many times, traders have short careers because a valid strategy dies. Amateurs blow up, but professionals don't allow for alpha-decay. For example, many floor traders didn't survive the death of the open-outcry pits. Their edge disappeared, and their previous spending habits left them with little. (In this case “trickle-down” economics was correct, as profits from market-making trickled down to prostitutes, strippers, and cocaine dealers. At least it wasn't wasted.)

Behavioral Finance

Think about how stupid the average person is, then realize half of them are stupider than that.

—George Carlin

The history of markets is nowhere near as big as we often assume. For example, equity options have only been traded in liquid, transparent markets since the CBOE opened in 1973. S&P 500 futures and options have only been traded since 1982. The VIX didn't exist until 1990 and wasn't tradable until 2004. And the average lifetime of an S&P 500 company is only about 20 years. In the long term, values are related to macro variables such as inflation, monetary policy, commodity prices, interest rates, and earnings. And these change on the order of months and years. Even worse, they are all co-dependent.

So, what might seem like a decent length of history that we can study and look for patterns, quite possibly isn't (this does not apply to HFT or market-making where a huge number of data points can be collected in what is essentially a stationary environment). When it comes to volatility markets, I think that although there appear to be many thousands of data points, there might only be dozens. A better way to think of market data might be that we are seeing a small number of data points, and that they occur a lot of times.

I think this makes quantitative analysis of historical data much less useful than is commonly thought.

But there is something that has been constant: human nature.

Humans have been essentially psychologically unchanged for 300,000 years when Homo sapiens (us) first appeared. This means that any effect that can conclusively be attributed to psychology will effectively have 300,000 years of evidence behind it. This seems to be potentially a much better source for gaining clarity.

The problem with psychological explanations (for anything) is that they are incredibly easy to postulate. As the baseball writer Bill James was reported to say, “Twentieth-century man uses psychology exactly like his ancestors used witchcraft; anything you don't understand, it's psychology.” The finance media is always using this kind of pop psychology to justify what happened that day. “Traders are exuberant” when the market goes up a lot; “Traders are cautiously optimistic” when it goes up a little, and so on. I try not to do this, but I'm as guilty as anyone else. I think psychology could be incredibly helpful, but we have to be very careful in applying it. Ideally, we want several psychological biases pointing to one tradeable anomaly, and we want them to have been tested on a very similar situation to the one we intend to trade.

Further, traders aren't psychologists and reading behavioral finance at any level from pop psychology to real scientific journals is probably just going to lead to hunches and guesses. To be fair, traders currently make the same mistakes from reading articles about geopolitics or economics. One week, traders will be experts on the effects of tariffs on soybeans and the next week they will be talking about Turkish interest rates. It is far easier to sound knowledgeable than to actually be so. It isn't obvious that badly applied behavioral psychology is any more useful than badly applied macroeconomics. And it is obvious that traders can't do better than misapply, either.

After I explained this nihilistic view to an ex-employer he said, “Well, I have to do something.” And what we do is exactly what I've said isn't very good: we apply statistics and behavioral finance. These are far from perfect tools, but they are the best we have. The edges they give will be small, but some edges can be found. We will always know only a small part of what can be known. Making money is hard.

Proponents of behavioral finance contend that various psychological biases cause investors to systematically make mistakes that lead to market inefficiencies. Behavioral psychology was first applied to finance in the 1980s, but for decades before that psychologists were studying the ways people actually made decisions under uncertainty.

The German philosopher Georg Hegel is famous (as much as any philosopher can be famous) for his triad of thesis, antithesis, and synthesis. A thesis is proposed. An antithesis is the negation of that idea. Eventually, synthesis occurs, and the best part of thesis and antithesis are combined to form a new paradigm. Ignoring the fact that Hegel never spoke about this idea, the concept is quite useful for describing the progress of theories. A theory is proposed. Evidence is found that supports the theory. Eventually it becomes established orthodoxy. But after a period, either for theoretical reasons or because new evidence emerges, a new theory is proposed that is strongly opposed to the first one. Arguments ensue. Many people become more dogmatic and hold on tightly to their side of the divide, but eventually aspects of both thesis and antithesis are used to construct a new orthodoxy.

From the early 1960s until the late 1980s the EMH was the dominant paradigm among finance theorists. These economists modeled behavior in terms of rational individual decision-makers who made optimal use of all available information. This was the thesis.