## What's This About Innate Costs? All Hail the “Poor Man's Call Option” Trader?
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## Returns of Different Strategies— Real S&P500 Data## Data: Daily Closing Prices of the Last Two Decades
## So Show Me the Costs of Stop-Loss TradingThe 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 ## The Chart and TableThe 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:
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 ImpressionsYou 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 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 ManWe are near the moment of truth. After all, the 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 The latter disappointment, the failure of the ## Listed Call Option Buyer Even Poorer?We now turn our attention to the Thus, as we elect the 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 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 ## “Moneyness” and Actually Owning OptionsBasically 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:
We have yet to fully explore the dependencies of our 1-Year Sharpe-like ratios for the 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 And so now it's time to examine the performance of (Continued...) Mike O'Connor is a physicist who now develops and tests computerized systems for optimizing portfolio performance. | |||||||||||||||||||||||||||||||||||

## A Better WayNote especially, from the chart on the Home page, that trading with Nonetheless, we will now go over how ## RB's New ProgramSo 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 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 The outcomes are generally better than any of the other strategies, including
## The 10-Year PerspectiveGeometric 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 CriterionIf 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 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 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 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 StrategiesOnce 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 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
## It's an Ex Ante WorldAnd 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 ## The Ex Post Optimal Commitments by StrategyWell, 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 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 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 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. 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 RB's New Program f* And the f* for However, apart from the risk of single-day plunges, it's notable that while ## In SumWhat 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 But — 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) |