### What is Stop-Loss Trading?

While it must be said at once that Retail Backtest is dedicated to the minimization of losses and of the attendant costs of trading, for the purpose of ultimately profiting, it's nonetheless obvious that you can't simply trade out of a deteriorating position and thereby come out of it without a loss. The order of events doesn't work that way: you buy the stock or ETF and then, after the price drops, you may sell it. That would certainly “stop the bleeding”, but you'd have a loss.

But, as the price drops further you may think that if it begins to rise that you should buy those shares back. Why not? If the price is now about at the price at which you sold, then you'll come out about the same as those who simply bought and held on— yet you will have avoided the downside risk. But... wait a minute! What if the price goes back up immediately after you sell your shares? In that case you'd have to buy back in at a price higher than the sale price. In that case you'd not be doing as well as the buy-and-hold investor. So if you were to keep attempting that sort of thing, then how exactly would it all work out? That's the question. And... surprise! Herein you'll get a rather complete and accurate answer. The question is not an idle thought of some doofus. There are innate costs associated with any sort of stop-loss trading, and every investor should know what is meant by that.

It's not the case that you get nothing back in return for the costs that you sustain when you trade so as to systematically avoid losses. You do get some downside risk reduction, potentially quite a lot of it. Furthermore, while there are pristine academic theories that say that your risk-adjusted returns can't thereby be improved there is every reason to believe that a well-constructed portfolio optimization program can do just that. Here the discussion is partly on just the nature of the challenge, especially with regard to the associated trading costs being greater than most would assume and difficult to avoid.

#### The Bid-Ask Spread

What else? Well, stock trading is really an auction in which some people who are not in a big hurry and want to sell some stock try to get more for it than the last sale price, while some others who are also not in a big hurry but want to buy the stock try to do so for less than the last sale price. And then there are other folks who are in a big hurry. If you're one of those you place a market order, which is an order for immediate execution at the best available price. It takes all four kinds of participants to make a market: in-a-hurry-buy, in-a-hurry-sell, not-in-a-hurry-buy, not-in-a-hurry-sell.

Now then, if we are placing a stop order, where are we? Ultimately we're in the crowd that's in a hurry and placing market orders because when the market price reaches or goes through our stop price our stop order is converted into a market order, for immediate execution. Thus if we are buying we'll get the lowest ask price, which is often going to be a tad above our stop price; if we're selling we'll get the highest bid price which will often be below our stop price.

It's clear that if we go round and round through many cycles of buying and selling in the course of using our program to avoid losses by trading the bid-ask spread will cost us some money. How much? It depends upon the liquidity of the security. If it's very heavily traded the bid-ask spread could be a mere penny per share and you won't even have to pay that every round trip.

#### Slippage

Finally, as far as other possibilities for our plans not working out exactly as we had hoped, there is such a thing as the quality of the “fills” that you'll get. Stocks and ETFs are sold on multiple exchanges which means that your broker has a choice of exchanges to which to route your order. There is even a question of which exchange to use as the source of data on the last sale price when the broker's computer tests to see if your stop price has been reached. Some of the exchanges have more liquidity than others and the best price may be revealed to be not quite the best price by the time that your order reaches the chosen exchange.

This slippage is a minor cost. It can be minimized to insignificance by not adopting, as a retail investor, a trading program that requires frequent trading of low-liquidity securities.

#### In All

And now for the kicker! All of these costs, summed up, are substantially less than the natural built-in cost of loss-avoidance trading. Yes. This article is really about an additional, more substantial cost. We started to see what the additional cost is all about above when we realized that you can't always buy at a price equal to or lower than the price that you sold. That's the ticket.

### Proven: Natural Costs of Stop-Order Trading Exist

You will find a breezy introduction to options in the first two paragraphs of a page on this very website. An option is a security that is derived from another underlying security such as a stock, an ETF or a futures contract— or sometimes from an index such as the S&P500. Call options in particular are generally settled at expiration by the holder being allowed to pay only 100 times the strike price to buy 100 shares of the underlying security if its market price is then equal to or greater than the strike price— the latter circumstance amounting to a payout equal to 100 times the difference between the final market price for the stock and the strike price. Or if the call finishes “out of the money”, if the price of the underlying security falls below the strike price at expiration, then it just expires worthless.

Index options work in an analogous way except that they are “cash settled”: e.g., If the index finishes 10 points above the strike price then a call option on it becomes worth $1,000 which is 100 times$10. That's the same payout as would be computed for a call option contract on some stock whose price finished $10 in the money. (Here I've used a “contract multiplier” of 100, which is ubiquitous but not universal. “Contract” refers to the agreement between the option buyer and seller. It is established and enforced by the exchange on which trading occurs.) Said payout is really the value of the call option to the holder at expiration, and while the value can decline to zero it can't go negative. In practice the impossibility of the option expiring with a negative value can be seen for what it is, which is a stop-loss aspect that is not unlike the circumstance of the investor who has put in a stop-loss order on a previously purchased stock. To see that, to put the two schemes on an equal footing for purposes of comparison, in place of the purchase of the stock and the subsequent stop-loss order at a particular stop price we might have call options purchased, and cash held in reserve for each of the call options in the amount of 100 times the strike price of the option— with the strike price playing the role of the stop price. With neither scheme will the final value of the positions held be less than that implied by the stop or strike price, except perhaps for slight slippage with the stop-loss order. In effect, the call option arrangement doesn't ever put the cash for the strike price up at risk; the stop-loss order approach involves getting out in time. #### Eggheads Economists sometimes give each other Nobel prizes for constructs and theories that are specious. But Fischer Black was a physics student before turning applied mathematician and the work that he and Myron Scholes did to calculate the proper price of a stock option is not in that category. Oh they did allow themselves some academic conveniences, such as assuming that the market behaves in a particular way— it roughly does what they assumed— and disregarding every one of the aforementioned ho-hum costs of trading. They even disregarded the fact that the future volatility of the marketplace is not predictable! But, among other things they conclusively demonstrated that call options, which represent a particularly ideal form of loss control, can't be free— and by implication neither can any stop-loss-like trading scheme that seeks to bring about the same performance be free of comparable attendant costs. In particular, whereas the payout of a call option at expiration is always zero if the market price is not then greater than the strike price and is otherwise 100 times the amount by which the market price exceeds the strike price, at any time prior to expiration the price of the option is always greater than that. That additional component of the cost of the option is what we're talking about. #### Poor Man's Call Option Definition and Limitations You may already rather fully understand that there is a similarity between the strike prices of options and the stop prices that we have discussed above as to the intended effect. That will be come clearer still if we try to create a sort of Poor Man's Call Option— one that we hope at the outset to pay nothing for except some of the piddling trading costs of the ho-hum variety that we discussed above. Let it be a trading scheme to avoid substantial losses that simply consists of the policy of selling the security if the price drops below a stated stop price and buying it back if the price goes above that price. Note especially that we have but one stop price that will serve as both a sell-stop and buy-stop price. Then let's add to that the intention to carry that scheme on for only a predetermined set period of time before readjusting the stop price. So with our scheme that set period of time until readjustment corresponds to the time until expiration of a call option, and the stop price looks awfully like a strike price. Indeed, doesn't it seem at first that our Poor Man's Call Option would cost us nothing but perform pretty much like a real call option? More exactly, if we were managing 100 shares this way with an investor-implemented stop price wouldn't that be like buying a call option (which pertains to 100 shares) with a strike price equal to the stop price and keeping 100 times the strike price as a cash reserve? With the latter scheme we would incur that additional cost component that is discussed in the second paragraph above about the Black and Scholes finding. Don't we avoid it here if we do the Poor Man's Call Option instead? Would that it could be that way, but it can't. And we already touched on the reason when we realized that of course there will be times when the market price rises above the price at which we sold the security before we have an opportunity to buy it back, which with our scheme will happen in the vicinity of but not usually exactly at the stop price. Thus, whereas we might hope to find ourselves whole on all occasions when we reach the preset future stop-price readjustment time with the price of the security above the stop price, very often we will find ourselves out a bit of cash— the result of having had to buy back at a higher price than we sold. That untoward happening is called “getting whipsawed”. Utter avoidance of that in the trading of securities is not possible. (With a buy-and-hold policy, yes. But that is not trading.) #### Stochasticity Note especially that although this getting whipsawed business will be attended by those minor ho-hum trading costs that we discussed above, at base the whipsaw losses have nothing to do with those costs. Even if we were to somehow always be able to trade at the last market price, at the very price that triggered a move on our part due to the stop price having been reached or gone past, with no fee or commission paid, with no bid-ask spread or slippage, that would not prevent us from fairly often having to buy the security back at a price in excess of that at which we previously sold. This is really the result of the innate stochasticity of the market— the principally random and unpredictable character of the price histories of securities. If the market price arrives at our stop price or goes just past it, the odds of the next quote being a move in the opposite direction are not much different from the odds of a further move in the same direction. That's why we will sometimes find ourselves unavoidably paying more to get back in the market than we were paid to get out, to our detriment. #### Poor Man vs. Nobel Laureate But still, it's not as though we would be looking like fools all of the time were we to implement the Poor Man's Call Option. Fairly often the price of the security would stay above our stop price all the way to the preset stop-price readjustment time, giving us returns exactly equal to those of the buy-and-hold investor; fairly often the price would dive below the stop price and not come back up above it prior to readjustment time, meaning that we would have suffered a small loss whereas that of the buy-and-hold investor might be substantial. Indeed, while the market is mainly stochastic in character that is not to say that momentum doesn't exist, especially during panics and it's really only the steep downdrafts of big panics that we need to avoid to stay quite happy in the long run. But what exactly is the “proof” of innate costs that was said to be in the offing? Well, whereas we defined a poor man's way of simulating the performance of a call option, a scheme that we've yet to see worked out but will below, Black and Scholes came up with a Nobel-prize-winning way. Whereas with our version we would occasionally suffer sizable whipsaws at the stop price by alternating between being 100% in the position and 100% out of it, their method of simulating the performance of an option involves beginning to continuously reduce the size of the position in the underlying security in small decrements, starting from well above the strike price as the price of the underlying security approaches the strike price, to where the position is about cut in half when the strike price is reached and with the position size diminishing further the deeper the excursion below the strike price. Now their scheme also sustains whipsaws with attendant costs, lots of small whipsaws for them versus a few big ones for the poor man's scheme. But crucially their scheme's expiration-time performance exactly matches that of the terms of a call option contract and with it they are able to sum up their whipsaw losses— which by the design of their way of calculating position sizes accumulate to a fixed, predicted trading cost that depends upon the volatility of security's price, the starting price/strike price relationship, the time until expiration (and in a minor way, the interest rate on risk-free loans). Whereas our Poor Man's Call Option would fairly often fail to exactly replicate the terms of the call option contract as it would have fluctuating trading costs due to the occasionally gross whipsaw effect we'll soon see that the Black and Scholes scheme is also attended by variations in overall performance due to the impossibility of accurately predicting the future volatility. (Continued...) ### Returns of Different Strategies— Real S&P500 Data ##### Data: Daily Closing Prices of the Last Two Decades #### The Number of Samples #### Trading Days per Sample #### Select Strategies #### Strike/Stop Price ##### Percent from Starting Market Price #### Redraw Chart Strategy Excess Return / Volatility Geometric Averages: 1-Year Poor Man's Call B-S Call + Cash RB's New Program Buy-and-Hold Strategy Excess Return / Volatility Geometric Averages 10-Year Poor Man's Call B-S Call + Cash RB's New Program Buy-and-Hold  Chart and table both pertain to returns net of that on cash. An Excess Return/Volatility ratio of, say, 1.25/0.45 means that the most-likely final outcome of let-it-ride trading would be an increase of 125%, a most-likely final value of 2.25 times what cash alone would have yielded, with about one out of six final values falling more than 45% below that. ### So Show Me the Costs of Stop-Loss Trading The fact that Black and Scholes demonstrated that the trading costs of their option simulation scheme accumulate to a fixed and substantial sum proves the existence of trading costs over and beyond the ho-hum costs that we discussed on the previous page (which, you may recall, they disregarded). But what of our pitiful old Poor Man's Call Option? Did it really seem so bad after all? Maybe we could stomach the fluctuating trading costs. It seems that with the right choice of a stop price we might have a small price to pay for avoiding just the most precipitous downturns. Well, having been told about a theoretical proof of the existence of innate costs of stop-loss trading, albeit one that has a firm place only in the real marketplace of listed options, how about some empirical evidence? How would the Poor Man's Call Option perform in the real marketplace? Would our trading costs really be something like the prices of real call options? Who will win: the eggheads with their Black-Scholes options pricing theory or the poor man? And should we really care? #### The Chart and Table The chart and table above show data from samples of real historical data at randomly-chosen starting dates— samples equal in number to the Number of Samples and of duration equal to the Trading Days per Sample as specified on the control panel on the right-hand side of the chart (or below it if your screen is narrow). So for example you might be viewing 100 samples involving 20 trading days each, which would be roughly a month each. On the chart each dot represents the margin by which the return over the chosen number of Trading Days per Sample exceeds that of cash, with the horizontal axis pertaining to the buy-and-hold strategy. So if over, say, 120 trading days$1 held as cash would have become $1.02 but the same dollar invested in the S&P500 using one of the listed strategies would have become$1.06, then the return in excess of cash would be 0.04 on the chart, or four percent. (Returns on cash were much larger back at the start of the two-decade period from which the samples are drawn than they are now.)

Whereas the chart compares the excess returns of each strategy with that of buying and holding, the table evaluates each strategy on its own based on just the returns in excess of cash. In effect, the table pertains to the spread of values on the chart in the vertical direction only. Though it is a “geometric” average, the quotients in the column of 1-Year ratios of Excess Return/Volatility correspond very closely, quantitatively speaking, to the famous “Sharpe ratio”. William Sharpe won a Nobel prize for his several contributions to the Capital Assets Pricing Model and related matters. Here's what he said about his ratio not too long ago (2014) in an interview that was published in the American Association of Individual Investors (AAII) Journal:

I have two minds about the Sharpe ratio. I originally called it the reward-to-variability ratio, which I think at least captures its meaning better. What I set out to do a long time ago was to answer the question: If you’re going to look at the performance of an investment and you can have only one number to do it with, what number might be useful? The idea was that you really need to look at the expected return and you really need to look at the risk. If you can only use one number, then maybe what you ought to do is to compare the investment with some relevant alternative and divide the expected excess return by the relative risk. In the original version, which is the one generally called the Sharpe ratio, the alternative was a riskless investment like a Treasury bill. So you divided the expected excess return over the Treasury bill by the risk relative to the Treasury bill. I still think that if you’re going to look at only one number, this is a useful one.

And so but for the use of the geometrical average, which upon compounding is more representative of the most-likely final outcome, that's what was done to create the chart and table— the 13-week US Treasury bill bank discount rate, which is generally streamed under the symbol IRX, was used to compute the interest earned by cash. Calendar days, not trading days, were used to properly calculate the interest (and yes, the fact that the US Treasury year has 360 days, not 365 or 366, was taken into account).

#### First Impressions

You may prefer about now to open up another copy of this page in a separate tab or window so as to have ready access to the chart without scrolling. Let's just go to the control panel to the right of the chart (or below it if your screen is not very wide) and select 50 for the Number of Samples and 20 for the Trading Days per Sample. If the prior settings were something else you will see notice to the effect that the chart must be redrawn. That happened because you changed the sampling parameters and so we need another sample. So click on the Resample button under Redraw Chart.

Now let's clear up the chart by selecting just the Buy-and-Hold strategy under the Select Strategies heading of the control panel. Click on that button until it's darker in shade, and click on any of the other strategy buttons that are darker in shade, so as to deselect them. You should now see only red dots on the chart along the diagonal, forming a reference line that shows what Buy-and-Hold looks like. If you click the Resample button rapidly a number of times you will gain the distinct impression that, happily, the red dots most often fall on the right-hand side of the cross (the plus sign) in the middle that marks the point of zero excess return for every strategy.

That's why we like to buy and hold stocks, for the long term. If however we're worried about the short term that can indeed be problematical as we see that even over sample durations of as brief as 20 trading days the S&P500 suffered declines in excess of 20% on rare occasions while declines of more than 10% frequently occurred every 50 samples. The declines over 120 trading-day periods have occasionally been absolutely frightful and we'd surely like to avoid that.

The arms of the cross in the center extend 0.05 from the center, which will allow you to more easily gauge the separations from the Buy-and-Hold line as we continue.

#### Some Technicalities That You Can Skip But Probably Shouldn't

• The Excess Return/Volatility ratios of the table, all of them, are representative of 1-Year or 10-Year outcomes as indicated if the investor commits 100% of available capital to the strategy at all times. Note that this does not mean that the investor would never hold cash because all of the strategies except Buy-and-Hold involve cash positions of varying amounts, one of them at all times and two others on occasions.
• For the chart and table above price data for the Exchange Traded Fund whose ticker symbol is SPY were used for the Poor Man's Call Option, the RB's New Program (more on that later) and the Buy-and-Hold strategy; for the Black-Scholes Call + Cash the S&P500 index was used, along with implied volatility data for options on it which are generally streamed under the symbol VIX.
• Actual options prices were not used for the Black-Scholes Call + Cash strategy; prices were estimated using the Black-Scholes options pricing formula along with the IRX and VIX data for the chosen sample starting date. That limits the precision of the results so that no hair-splitting comparisons will be attempted as we examine the chart and table. However the VIX data are inferred from actual option prices, not by directly measuring the trailing historical volatility of the underlying security, so we are coming close to recreating the real option prices by computing them using the VIX. The S&P500 index was used for pricing the options (rather than SPY which has the complications of dividends) because you can indeed buy index options— the contracts are quite liquid— and because the VIX, for which data are readily available, is the true companion index of volatility for the S&P500 index (and not for SPY, though the values are necessarily almost the same). The absence of said complications encourage us to presume that the resultant formula value for the call option price is rather accurate, especially if the Strike Price is chosen to be near the starting market price because that circumstance confers liquidity.
• Although “listed” options for sale on exchanges have certain set days of expiration, often the Saturday after the third Friday of each month, that reality was disregarded and for each sample starting date it was assumed that options were available with the given number of Trading Days per Sample until expiration. (The goal here is not to test the validity of the options formula of Black and Scholes against market prices for options; it's to show how options limit downside risk like any other stop-loss scheme but also burden us with costs. For that the formula wielded in this way does suffice. Furthermore, there are firms that are in the business of writing private options contracts with arbitrary expiration dates.)
• Regarding the cash in the Black-Scholes Call Option + Cash strategy, cash was added for the purpose of preventing the calculated returns from being incomparable to those of the other strategies and to avoid the possibility that anyone might ever go absolutely broke using the strategy. For example, the return on a bare option without the cash buffer could vary from minus a hundred percent to plus several hundred percent in one trial; not so for, say, the Buy-and-Hold strategy. The assumed amount of cash was equal to the remaining equity in the account after purchasing call option contracts equal in number to the equity in the account divided by 100 times the starting value of the index, with which allocation if the calls were deep in the money then the equity in the account would vary just as though all of it were instead 100% invested in the S&P500 index. Of course there would have to be a substantial amount of money in the investor's account for the roundoff error to be insignificant, but investors and financial entities with such resources exist and no net bias is to be expected from the rounding off. Note especially that if holding call options is profitable, if the arithmetic average of the returns is positive, then holding this cash-buffered call option position is profitable (13-week US Treasuries have threatened to ultimately pay negative interest but they have yet to do so).
• Only closing prices were used for the chart and table. For the Poor Man's Call Option and RB's New Program strategies it was assumed to be possible to trade at the closing price— each trading day for the Poor Man's Call Option but only at the close of the first trading day of each week for RB's New Program. It would not really be exactly possible to get the reported closing price, however miraculous the execution, because the reported price is generally some sort of average over the various exchanges. The calculations of the performances of the various schemes assume that the reported price was actually realized. Of the discrepancies, about half would work to our disadvantage and about half would be advantageous, and so it should all “come out in the wash”.

• Our resort to day-end trading for the Poor Man's Call Option strategy couldn't be improved upon all that much by intraday trading, at least not in the stock market due to the frequent occurrences of overnight gaps. Furthermore, the analysis by Black and Scholes, which confirmed the existence of significant innate costs of stop-loss trading, was based on the assumption of a continuum— with trading being done with every minute change in the real-number value of the stock price, to an infinite number of decimal places. As with their scheme, increasing the frequency of trading would reduce the size of each whipsaw loss of the Poor Man's Call Option strategy but increase the number of whipsaw losses in rough proportion, amounting to hardly any difference in the cumulative cost of trading.
• Just as in the academic studies that are cited here, the ho-hum trading costs that are described on the previous page were not included in the calculations. That simply means that if with those costs neglected we find a portfolio optimization program that looks promising then we are obliged to introduce those costs in a subsequent rough computational step prior to making any use of that program— to confirm that the ho-hum-cost-affected program still performs better than the ho-hum-cost-free Buy-and-Hold strategy.

#### Poor Poor Man

We are near the moment of truth. After all, the Poor Man's Call Option was the obvious prototype of elemental stop-loss trading. Recall our early hopes (which we soon dashed through logic). Ideally we wanted to prevent any and all losses below the stop price, while getting 100% of the gains from outcomes above the stop price.

Let's first set the Stop Price on the control panel to 0% from the starting market price if that is not already the setting, so that the assumed Stop Price is equal to the starting market price. No redrawing will be prompted. Now with any settings for the Number of Samples and for the Trading Days per Sample that you prefer, what would we hope to see? We'd like to see the blue dots lined up along the 0.00 line to the left of the center cross— no losses on those occasions when the market declined.

But unfortunately that's not what we get. We instead sometimes do see the excess returns as high as 0.00 to the left of the center cross, when buy-and-hold has sustained a loss, but more often they are considerably below that and are occasionally even below the Buy-and-Hold line! And to the right of the center cross we had hoped to see the blue dots aligned along the Buy-and-Hold line. Some are but most are below it.

The latter disappointment, the failure of the Poor Man's Call Option to do as well to the right of the center cross as Buy-and-Hold, as not trading, could be said to be what we mean by the innate cost of trading; it's to the left of the center cross that we find benefits. The question then, is do the benefits outweigh the costs? And for that we turn to the table. To reduce the variations among the tallies let's select 500 for the Number of Samples. And let's look at the Sharpe ratio for the Poor Man's Call Option in the first (1-Year) data column of the table. Sadly, as we click away multiple times at the Resample button we see that it is fairly often not even positive. In contrast, the ratio for Buy-and-Hold is almost never negative and ranges above 0.50 fairly often.

#### Listed Call Option Buyer Even Poorer?

We now turn our attention to the Black-Scholes Call Option + Cash strategy. The investor buys call options equal in number to one-hundredth of the available account equity divided by the S&P500 index value, which calculated allocation means that when the calls are deep in the money the percentage change in account value will vary one-for-one with the percentage change in the value of the index. The cash balance is kept in a money-market account or the like that pays interest at a rate similar to that of the 13-week US Treasury bill. With this arrangement, should the market finish above the strike price the outcome will never be as good as for the Buy-and-Hold investor because the interest on the cash will not cover the expended price of the option.

Thus, as we elect the Black-Scholes Call Option + Cash strategy and click many times on the Resample button with the Strike Price set at 0%, meaning at the starting market price of the underlying security, we see that the picture looks rather like the circumstances of the Poor Man's Call Option except that there is less variance in the vertical direction on the chart. And also, all of the orange dots are below the Buy-and-Hold line to the right of the center cross and below the 0.00 line to the left of the center cross.

You may recall the prior mention of the fact that Black and Scholes found that there was a fixed cost to the trading that their scheme did to simulate an option— the sum of all of those tiny whipsaws. But on the chart we see scattered values for the orange dots; they are not a set distance from the 0.00  line or from the Buy-and-Hold line. How so? Well, it's because over the years there have been tremendous variations in the volatility of the market and the Black and Scholes option price formula has a strong dependency upon volatility (except for deep-in-the-money options)— the higher the volatility the higher the price. Hence the vertically scattered look to the orange dots.

So again we understand that there are costs to this particular stop-loss scheme— they are simply the up-front costs of the options— and the question is whether or not the costs are outweighed by the benefits. Sadly the verdict is even a bit worse than it is for the Poor Man's Call Option. The Sharpe ratio in the first (1-Year) data column is often negative. Is is supposed to be like that? Who would ever buy a call option?

#### “Moneyness” and Actually Owning Options

Basically some of the academicians who study finance got tired of writing “in-the-money” and “out-of-the-money” all of the time when referring to options. It's particularly taxing when you have to write something like “the dependency upon the degree to which the option is in-the-money or out-of-the-money”. So they invented “moneyness”. It refers to that characteristic of the option. There is a high level of moneyness if the option is deep-in-the-money, meaning that the market price of the security far exceeds the strike price, and vice versa of course.

Here's the brief abstract of an article by Ryan McKeon that was recently published (2013) in the Journal of Investing:

While much work has been done on pricing of options and expected returns theory, far less has been written on the actual returns that investors receive from trading options. In the case of call options on equity, standard asset pricing theory suggests large, positive average returns, while articles and comments in the popular press suggest that many investors lose money trading options. Empirical results in the academic literature are few, and results are mixed. In this article, I study call option returns, analyzing how returns vary by levels of moneyness and different holding periods for both equity index and individual stock call options. I find a general and consistent result that call option returns are low on average and decreasing in the strike price. Only in-the-money options held for a month exhibit positive returns. Deep out-the-money options deliver large negative returns on average, consistent with risk-seeking investing on the part of buyers. I discuss the implications for some common options trading strategies.

We have yet to fully explore the dependencies of our 1-Year Sharpe-like ratios for the Poor Man's Call Option and Black-Scholes Call Option + Cash strategies. If instead of selecting 0% for the difference between the Strike Price and the starting market price we elect -10%, that's a deep-in-the-money option that is being simulated. Note that with repeated hits on the Redraw button the Sharpe ratios for both strategies are much improved. And they are worse if the difference is set to +10% which is the far-out-of-the-money case.

Thus our findings generally agree with those of Ryan McKeon (who wrote “deep out-of-the-money” where we would have written “far-out-of-the-money”). By “empirical” he means that he used only actual pricing histories for the options.

There are a number of ways that the failure of call ownership to show profits in a market history of rising prices could come about and for our present purposes we don't need to know the actual causes. But the prices of deep-in-the-money options have only a small dependence upon volatility whereas those of far-out-of-the-money options are extremely dependent upon volatility. And higher volatility means higher option prices. This suggests the following hypothesis: it could simply be that options sellers— they are not generally the same crowd as buyers— systematically overestimate the future volatility of the marketplace and that buyers, whose ranks include speculating retail investors, are not diligent enough to catch onto that.

It almost looks as though we could refer to said options sellers as the “smart money”. This seems to be in the general direction of what Ryan McKeon is hinting at in the second-to-last sentence of his abstract. Such systematic mispricing due to the assumption of a too-high forward volatility— it's called the “implied volatility”— might be expected to happen during very steep downturns. And indeed we have had two such downturns in the period that is covered by the chart and table. That implied volatility is often not well aligned with the actual future volatility has been well understood by traders and academicians alike. See for example this article by Federal Reserve Bank of Boston economist Peter Fortune— that's his real name— at about page 27. His article is generally informative about the Black-Scholes formula and its limitations.

And what could we say about the Poor Man's Call Option, given that it behaves rather similarly to the Black-Scholes Call Option + Cash strategy in that regard, even with respect to moneyness. Well, when our Poor Man's Call Option is operating with the market price of the security well above the stop price, deep-in-the-money, there isn't any trading— no whipsaws, hence no losses being generated by trading. At-the-money the whipsaws happen, hence losses. The far-out-of-the-money situation does usually tend to produce zero gains in excess of interest on cash unless the security's market price ultimately spends some time near or just above the stop price, which occurrence would lead to whipsaws, hence the often adverse results even for the Poor Man's Call Option that starts out well out-of-the-money.

And so now it's time to examine the performance of RB's New Program. An example of it in action can be found on the Home page of this site and other results appear on the RB's New Scheme Overview item of the Performance menu. On the next page we'll use the chart and table above to take a closer look.

(Continued...)

Mike O'Connor is a physicist who now develops and tests computerized systems for optimizing portfolio performance.

### A Better Way

Note especially, from the chart on the Home page, that trading with RB's New Program would seem to be a very good thing to do. However, the program is still undergoing development and testing and so the current version is for show as an example of work in progress, not for use. Position sizes for it are not shown anywhere on this site. The further testing of the new program will proceed along the same lines as in the article Chance or Discovery?.

Nonetheless, we will now go over how RB's New Program compares with the other strategies and further on we'll inquire about the implications of compounding using the Kelly criterion.

#### RB's New Program

So again we simply click repeatedly on the Resample button, with the Select Strategies buttons set appropriately and look at the chart. It's fairly uninspiring, but note that to the left of the center cross the green dots are seldom below the Buy-and-Hold line and are sometimes even rather far above the 0.00 line. To the right of the center cross there are unfortunately a few dots below the 0.00 line but note that there are even a few above the Buy-and-Hold line.

But again we have to go to the table to see the real story, and it's once again preferable to set the Number of Samples to 500  and to click the Resample button a number of times.

It doesn't matter what the Strike/Stop Price setting is because the program does not use any such price. (If you do nonetheless change the Stop/Strike Price setting the chart may be redrawn with the range of the vertical scale changed— a rescaling, not a resampling. That would be to permit you to at any time and without recalculation display well-scaled data for the Poor Man's Call Option and Black-Scholes Call + Cash strategies that do depend upon that setting.)

The outcomes are generally better than any of the other strategies, including Buy-and-Hold. The excess return is most often considerably better than Buy-and-Hold and the volatility is reliably smaller, better. Hence the Sharpe ratio of the first (1-Year) data column is substantially better than that of any other strategy.

RB's New Program does not include short selling. Consequently the green dots on the chart at the top of the previous page never form a “smile” shape. The choice to not have the program engage in short selling was more of a policy choice, an orientation toward programs of a character that would make their adoption feasible for many retail investors and for the professionals who serve them. The choice did not come about because trials of short-selling alternatives generally looked bad because hardly any such attempts were made anyway.

#### The 10-Year Perspective

Geometric averages take into consideration the effect of compounding. That is made clear in this site's article on the Kelly criterion. If we only go out to a year of compounding, as with the 1-Year Excess Return/Volatility column, the first data column of the table on the previous page, then the arithmetic average— it was the proper choice of by Sharpe for the context in which he was then working— yields very nearly the same numerical result as the geometric average. Hence we have referred to the ratios of the 1-Year column as though they were literally the Sharpe ratios.

Now if we go on to the 10-Year, second data column of the table on page 2 the computed ratios are substantially larger in magnitude due to compounding. This characterizes the result of an investor staying fully invested in the strategy (though it may sometimes involve remaining partly or wholly in cash). The 10-Year geometric Excess Return/Volatility ratio is a slightly unorthodox choice, but it has the meaning that is declared on the information panel on the right-hand side of that table (which is is also correct for interpreting the 1-Year ratio, though the magnitudes in the example would not occur in the span of just a year).

Not unexpectedly, the 10-Year values for the ratio fail to show qualitative distinctions among the strategies that are substantially different from those that are revealed by the 1-Year ratios. But they tell the story for that time horizon.

### Applying the Kelly Criterion

If the reader has read all of the Kelly criterion web article on this site then so much the better, but full understanding of the criterion is not needed to understand the Kelly ideas and findings that are presented below. This section of this web article is particularly academic in character. It is a thrust in a direction that could take us off a cliff if we were not to heed the cautions that are expressed. And some impractical things are given consideration, at least temporarily, as though they were not that.

We here introduce the Kelly criterion as a sort of overlay, on top of the pursuit of the strategies that we have already defined. Some of those strategies involve keeping some of the capital that is alloted to the strategy in the form of cash at least some of the time, yet the Kelly mathematics of fractional bet sizes also involves an allocation to cash. We had better clarify that.

Let us suppose that we open a brokerage account with a certain amount of cash, starting capital, with the purpose of implementing, the Kelly way, a single one of the strategies that we have defined— involving a single risky security, either the ETF SPY or call options on the S&P500 per the chosen strategy. We could think of a two-step procedure that is to be done each day to properly adjust the allocations so as to implement the chosen strategy with a Kelly criterion overlay. First the liquidation value of the account is computed. It's the available capital, amounting to the value of the held securities and cash where the cash is net of loans. It's our equity in the account.

Let f be the fixed fraction of all of the available equity in the account that is to be committed to trading a strategy the Kelly way, with the rest to be allocated to an accompanying pot of cash. So each day we allocate f times the available equity to the strategy and (1-f) times the available equity to cash. That's the first step. The second step would be to ensure that the amount allocated to the strategy that is being given the Kelly treatment is further divided between the risky security and cash as specified by the strategy (but of course this step is skipped with the Buy-and-Hold strategy of this article which has a 100% allocation to the ETF SPY and nothing to cash).

With the Kelly mathematics it is found that if the arithmetic average of the forward returns of the strategy that is to be given the Kelly treatment is positive— we can count on that being the circumstance of both the Buy-and-Hold and RB's New Program strategies in the long run— then there is an optimum fraction that is a positive number, customarily denoted by f*, with which the most likely outcome of compounding the returns over the long run is maximal.

This layering of one strategy, in this case the use of the Kelly criterion, on top of another may seem to be a complication that should be avoided by mixing the mathematical procedures of the underlying strategy and the overlaid strategy together. That could work. However, it's difficult to conclude that it would be a particularly golden way to proceed. Most of the Retail Backtest coding for the development and testing of programs to optimize portfolio performance involves layering the code using functional programming methods, with which any number of layers can be handled rather easily.

#### The Use of the Kelly Optimum Commitment with Strategies

Once again, in the present discussion we are talking about dividing the capital between the chosen strategy (which may by design itself incorporate cash) and a separate pot of cash— each day. But, the chart and table of the previous page assume f = 1, meaning 100% commitment to the strategy, with no Kelly criterion overlay. Hence the typed-in entries of the “assumed f” column of the table below are there as reminders of the meaning of f in application to what we have already done.

We now consider f = f* where f* is an optimal value of f and so we may have the fraction f* allotted to the strategy and 1 - f* allotted to the pot of cash. And it's even possible that f* > 1, in which case we mean that we should borrow cash and allocate those funds to the strategy— if we insist that the most likely final value of our wealth be maximal.

Now if you're thinking at first that f*=infinity could be maximal, no it couldn't. With any investment that can sustain so much as a one-day loss, there is of course a level of leveraging with which total ruin is possible. See the article on the Kelly criterion for further understanding, and we encounter and deal with this single-day loss risk issue below.

And so f* has been computed for RB's New Program and Buy-and-Hold strategies, based on all of the data of the sampling period to which the chart and table of the previous page pertain— with no limitation on the number of samples but rather with a sample for every successive day. The first two strategies having already been shown to be rather unsuccessful, no f* values were computed for them. The computed f* values are in the last column of the table that follows, the column labeled “ex post optimal f*”.

Strategy Commitment Level
 Fixed strategy “bet” size:
 assumed f ex post optimal f*
Poor Man's Call
 1 n/a
B-S Call + Cash
 1 n/a
RB's New Program
 1 n/a
Buy-and-Hold
 1 n/a

#### It's an Ex Ante World

And so what is this “ex post” business and is it important? Yes it is. It refers to the fact that since all of the data of the two-decade sampling period were used to compute f* you could not have had the benefit of the calculated f* until the very end of the sampling period. Some would therefore call the thus-calculated f* “hypothetical”, and that's basically right.

Generally speaking, for Retail Backtest work on the development or testing of portfolio optimization programs we regard the ex post approach as something to be avoided, to the extent that it is possible to do so— we would ideally like any portfolio optimization program parameter value that is assumed, put in use, at a given point in time to be derived only from data that existed prior to that time. That would mean that we couldn't consider any applicability of our ex post f* to past data; we'd have to turn it into an ex ante f* for application to the future. But this is difficult subject matter. It is reasonable to set parameter values ex post if there is little sensitivity of outcomes to the parameter values and ultimately it's necessary to resort to that. This is explained at length starting on this page of another article on this website, under the subtitle “Statistics and Finalizing Considerations”.

While no results of augmenting RB's New Program or other strategies by applying the f* estimates of the table on this page are presented here, due to the ex post aspect, sorting out the implications of the allocations that the Kelly criterion would bring about should be interesting nonetheless. After all, RB's New Program incorporates elements that confer downside protection by design so that the observed reduction of downside risk is hardly likely to have been the result of chance occurrences and is likely to be persistent. That should permit the use of a bigger f. An ex post f* much bigger than 1 would suggest that we perhaps should investigate further to see if an ex ante version of the Kelly criterion should be implemented, to enhance returns. That was the motivation for producing the table.

#### The Ex Post Optimal Commitments by Strategy

Well, in a word, they are phenomenal... and ultimately untenable. We now refer to the last column of the table on this page, the last two lines of it. For the Buy-and-Hold and RB's New Program strategies it seems that the most-likely amount of wealth that we would have in the long term would be maximized if we effectively get leveraged more than 200% long the strategies, two major stock market plunges in the last two decades notwithstanding. That seems ridiculous, much too dangerous, and indeed nothing as risky as that will ever actually be offered here.

The Way of Leverage

Borrowing money to go about 200% long an asset like the ETF SPY is generally possible in the United States at brokerages, with a so-called “margin account”. And there are also leveraged ETFs which do the borrowing for the investor that are leveraged by as much as a factor of three that track the major indexes such as the S&P500. Leverage of about 20 times collateral can be had in a futures account and applied to futures contracts on the S&P500 and other indexes.

Investment advice is not dispensed here, but certain exceptions can be made: If you're a retail investor and not a financial professional with an established career doing that sort of thing, don't do any of that sort of thing... don't use leverage. Perhaps you can make an exception for yourself if you have a considerable amount of wealth outside of your trading account— well in excess of the risk you expose yourself to in that account (for in that case you wouldn't really be getting leveraged overall anyway.) We next see why a lot of leverage is a no-no.

The Cruel Fate of the Over-Leveraged

There was a 55% plunge from October of 2007 that was concluded in March of 2009. If we were to have been more than 200% long at the start of that and had simply kept that position then we would indeed have lost all of the capital that we had had at the start (after having to meet margin calls, demands by the broker for additional collateral).

But that's not how we actually put into effect fixed-fraction commitments such as for the implementation of the Kelly criterion. We instead readjust the allocations, perhaps as often as daily. So on the way down from the October 2007 high as prices declined we would have less and less equity from which to allocate more than 200% of to the risky asset, meaning that less and less equity would have been at risk and we would not have actually gone broke.

That's not obvious, so let's see the math. To keep it simple we'll suppose that we're just trading a risky security the Kelly way. Let $$\text{E}_i$$ be our account equity on day $$\scriptstyle i$$ and let $$\text{P}_i$$ be the price of the security on that day, with $$\scriptstyle i-1$$ referring to the previous trading day.

The first equation below shows the already-rebalanced allocation of the equity at the conclusion of the previous day: $$\text{f}\cdot\text{E}_{i-1}$$ to the risky security and $$(1-\text{f})\cdot\text{E}_{i-1}$$ to cash. The second equation shows what has become of those allocations by the end of the current day but prior to again rebalancing the allocation between the security and cash: the equity that has been used to purchase the risky security is shown multiplied by $$\frac{\text{P}_i}{\text{P}_{i-1}}$$ in order to show the effect of the price change that occurred between days $$\scriptstyle i-1$$ and $$\scriptstyle i$$. (Interest on the cash has been neglected here in order to simplify the equations; it was not neglected in the course of producing any of the numerical results that appear in this article.)

\begin{aligned} \text{E}_{i-1} &= \text{f}\cdot\text{E}_{i-1} + (1-\text{f})\cdot\text{E}_{i-1}\\ \text{E}_i &= \text{f}\cdot\text{E}_{i-1}\cdot\frac{\text{P}_i}{\text{P}_{i-1}}\\ &\quad\quad\quad + (1-\text{f})\cdot\text{E}_{i-1} \end{aligned}

Continuing, to get the next equation below we simply divided both sides of the last equation above by $$\text{E}_{i-1}$$, concluding with the ratio of day $$\scriptstyle i$$ equity to day $$\scriptstyle i-1$$ equity.

\begin{aligned} \frac{\text{E}_i}{\text{E}_{i-1}}&= 1-\text{f}+\text{f}\cdot\frac{\text{P}_i}{\text{P}_{i-1}} \end{aligned}

But the equity in our account on any given day, here day $$\scriptstyle n$$, is a product of such ratios, going back to starting day $$\scriptstyle 0$$.

\begin{aligned} \text{E}_n&= \text{E}_0\cdot\frac{\text{E}_1}{\text{E}_{0}}\cdot\frac{\text{E}_2}{\text{E}_{1}}...\cdot\frac{\text{E}_n}{\text{E}_{n-1}} \end{aligned}

So $$\text{E}_n$$ can't become zero or negative unless one of the $$\frac{\text{E}_i}{\text{E}_{i-1}}$$ ratios does, unless $$1-\text{f}+\text{f}\cdot\frac{\text{P}_i}{\text{P}_{i-1}}<=0$$. We can add f - 1 to both sides of this inequality and then if f is positive we can divide both sides by it without changing the “less than” character to “greater than”, which yields $$\frac{\text{P}_i}{\text{P}_{i-1}}<=1-\frac{1}{\text{f}}$$.

So for example if f = 2.0 then any $$\frac{\text{P}_i}{\text{P}_{i-1}}<=0.5$$ will be fatal. A 50% one-day decline in the price of the risky security would lead to utter ruin. But nothing less than that would have that result, not even a huge number of daily percentage declines just short of that in magnitude.

Now is that some sort of saving grace? Hardly. Could a 50% decline ever happen in one day? Well in 1987 there was about a 23% plunge in one day and that was without a nuclear weapon going off anywhere. Analysts actually struggled to name the cause. Of course such a plunge could happen in a day, or over a few days in which trading opportunities would be extremely limited by exchange conditions or edicts.

It's interesting to see how the reallocations go. We can write $$\text{N}_i\cdot\text{P}_i=\text{f}\cdot\text{E}_i$$ where $$\text{N}_i$$ is the number of shares of the risky security that we must hold after reallocation on day  $$\scriptstyle i$$. Dividing that equation by the one for the prior trading day gives us the first equation below.

\begin{aligned} \frac{\text{N}_i}{\text{N}_{i-1}}\frac{\text{P}_i}{\text{P}_{i-1}}&=\frac{\text{E}_i}{\text{E}_{i-1}}\\ &= (1-\text{f})+\text{f}\cdot\frac{\text{P}_i}{\text{P}_{i-1}} \end{aligned}

And if we simply multiply both sides by $$\frac{\text{P}_{i-1}}{\text{P}_i}$$ we get the equation below. We see that if the price does not change then the number of shares isn't changed. However, if f >1 then the first term on the right-hand side is negative, in which circumstance if $$\text{P}_{i-1}>\text{P}_i$$ then the magnitude of the negative first term is increased by more than the value that it would have for $$\text{P}_{i-1}=\text{P}_i$$, with the consequence that $$\text{N}_i<\text{N}_{i-1}$$. So for  f > 1 a price decline causes us to sell some of the stock during the reallocation; for f < 1 we must buy some additional shares with every price decline.

\begin{aligned} \frac{\text{N}_i}{\text{N}_{i-1}}&=(1-\text{f})\frac{\text{P}_{i-1}}{\text{P}_i}+\text{f} \end{aligned}

The Buy-and-Hold f*

The 200+% factor for this strategy from the table on this page looks particularly absurd when we consult this chart— just the Buy-and-Hold line in red for now please, as the chart is otherwise about another portfolio management program. (If you're concerned with details, the red line on the chart supposes reinvestment of dividends and so it is slightly different from charts that have not been adjusted in that way.) On that chart you can confirm the 52% plunge from October of 2007 that was concluded in March of 2009 that is discussed above with regard to the dire consequences that would have come about if any such leverage were to have been used.

RB's New Program f*

And the f* for RB's New Program is even more absurd, being nearly three times the value for the Buy-and-Hold strategy. We can see the sense of it on the chart on the Home page of this site— the black line, not the red. Whether the application of the new program is to a portfolio as on that chart or to a single ETF such as SPY, there is a markedly smaller drop during the 2007-2008 debacle. But with such leverage a one-day drop in the general vicinity of about 20% would cause us to lose all of the equity in the account. Of course there have been a such drops in the history of the stock market. So the ultra-high f* for the RB's New Program is untenable for our purposes.

However, apart from the risk of single-day plunges, it's notable that while RB's New Program as it's currently configured does not necessarily respond at all to minor declines in prices it nonetheless never fails to bail out, to sell, during a pronounced downturn. With or without a Kelly criterion overlay it automatically deleverages itself, going to an f that is substantially less than 1 and sometimes all of the way to 0 during such crises.

### In Sum

What have we discovered? Well we have certainly seen that programs to avoid losses by simply trading out of a position when its price declines only to eventually take up the same position once again when the price rises do not automatically work as well as we might hope. And we include a program that involves purchasing listed call options in that category because their performance can be simulated by instead scaling out of and back into shares of the underlying security to similar effect.

However RB's New Program does seem to work. What's different about it? Well, with the random sampling from a two-decade price history that we have used on the previous page we are respectively setting stop and strike prices and expiration dates for the Poor Man's Call Option and the Black-Scholes Call + Cash strategies at random times in the market's price history and at arbitrarily chosen offsets of the stop and strike prices from the starting market price. We are not first studying the price history prior to the randomly chosen time of each sample in order to see if we should then really hold such a position and set the stop and strike prices and expiration dates to those particular values or to some other values. No. For those strategies and of course for the Buy-and-Hold strategy which does not permit trading we are instead proceeding as though the prior history were irrelevant.

But RB's New Program is starkly different. It does study the prior price history and draw deductions from it so as to determine the most statistically advantageous position size to assume on any given day, and the fact that we randomly sampled its history too doesn't change that. That's the difference. You need an edge to implement a successful program to reduce downside risks while still profiting during the good times. Necessarily that edge will involve making use of whatever information is available, and we must tolerate and work around the fact that it's difficult to distinguish useful information from noise.

— Mike O'Connor

Comments or Questions: write to Mike. Your comment will not be made public unless you give permission. Corrections are appreciated.

Update Frequency: Daily (chart data only)