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Applying Statistics to Speculation

By Lawrence G. McMillan

This article was originally published in The Option Strategist Newsletter Volume 14, No. 10 on May 26, 2005. 

Statistically-based trading is normally applied to hedged positions. It could be pairs trades for stock traders, or option spreads for option traders, or intramarket spreads for futures traders. But generally, the position is one that is based on a relationship between the entities involved – whether that relationship be a price-based relationship or a volatilitybased relationship. The position can be evaluated using assumptions about price relationships or about volatility, and those assumptions are based in historic fact, upon which mathematical calculations can be made (expected return, for example, and then the Kelly Criterion).

However, speculators face a different task. It may be very difficult to generate expectations for different speculative positions. In this article, we’ll look at some of those difficulties to see if speculators might be able to benefit from some of the insights that expected return calculations afford a trader: position size and an exit point, to name two.

In order to apply statistics to a trade, it is necessary to be able to calculate its expected return or at least to estimate the probability of success of the trade. If one has a specific speculative trading system, then he has statistics from that system with which to make the desired estimates (average gain of winning trade, average loss of losing trade, and percentage of trades that win, are enough to generate a statistical estimate).

But for many speculative trades, the trade itself is unique enough or there are so many variables that it is difficult to estimate the probability of success – much less estimate the magnitude of that success. However, it is not impossible to make estimates, and if that can be done, then a statistical approach to speculation can be developed.

Expected Return

As you know, we often apply expected return to our volatility-based strategies – in particular covered call writes and calendar spreads. In those strategies, we are often establishing positions in which volatility is out of line. It is normally “too high” in our covered call writing situations, and it is normally skewed (and perhaps low as well) in calendar spreads.

It is necessary to make a volatility estimate for the future when calculating expected return. We can do this for covered writes and calendar spreads by using a volatility estimate that is near the long-term historic volatility for the underlying stock. Thus, if the current implied volatility is not out of line, then the expected return will be minimal or negative and we would not establish the trade.

For speculation, however, things are often different. Normally a speculation is based on factors other than an aberrant implied volatility reading. It may have to do with put-call ratios, chart patterns, and so forth. If one buys an option that is “expensive” as a speculation, it is still quite possible that he will make money. The expensiveness of the option can easily be overcome by favorable movement in the underlying instrument.

Of course, one can plug a volatility into an expected return calculator to see what returns a prospective call purchase will generate. Let’s use the call purchase on last week’s Hotline as an example: we bought the Coca Cola (KO) Aug 45 calls for 1.30 with the stock at 45.25. That’s an implied volatility of 13.4% for the call option. Over the past two years, the average actual volatility of Coca Cola stock has been about 16%. Using that volatility, we can arrive at a positive expected return for the purchase of this call since the projected volatility is greater than the volatility we are paying. In this particular case, the expected profit is 22 cents (not including commissions). In other words, if we were able to consistently buy options with a 13.4% volatility and the underlying traded randomly with a 16% volatility, we could expect to average 22 cents profit per option purchased – mainly because of the increased volatility of the stock’s movement during the life of the option. The expected return is thus .21/1.30 = 16%.

The Kelly Criterion

If you will recall, the Kelly Criterion is a method of determining the “optimal” amount of your capital to risk on a trade, given the probability of profiting on the trade. If we know the expected return, we can estimate the probability of the trade being a winning trade as:

  Pwin = (1 + e) / (2 + e)
              Where e is the expected return

So, in this case e = 16%, thus pwin = 1.16 / 2.16 = 53.7%.
By inference, plose = 1 - pwin = 46.3%. Finally, the Kelly Criterion, simplified, states that the amount of your capital to invest is the difference in winning minus losing:
Kelly = pwin – plose = 7.4%
So, if we had a $25,000 trading account, then we’d put 7.4% of it, or $1,850 into this trade.
As our account grew or shrunk and various trades had better or worse expected returns, the Kelly Criterion would aid us in determining how much capital to put into each trade.

Alternate Method

If our purchases are part of a trading system, we can often use the statistics of the system itself to generate a Kelly Criterion figure for determining how much to invest. This is common with futures-based trading systems. For example, suppose we know these statistics from a back-testing or longterm usage of a particular trading system: average win (W): $2500, average loss (L): $1500, percent of winning trades (p): 52%.
In the case of a trading system, the Kelly formula is:
(W + L) * p – L
and one expresses L in terms of W.

In the above statistics L = 0.6W (i.e, 1500 = 0.6 x 2500). So Kelly reads: (1.6W) * .52 – .6W = .23W, which means that we should “invest” 23% of our bankroll in each trade. That is a very large amount, but with a system that hits more winners than losers, and where the wins are substantially greater than the losers, Kelly is necessarily going to want you to trade such a profitable system heavily.

Other than the above methods, one has no way to estimate the profitability of a trade. That is, if he pays above the expected volatility for a call option, then one must fall back on other methods – such as a fixed percent of the profits – until statistics for the trading system can be developed.

The Probability of Ruin

One might think that he could use a probability calculator to estimate the chances of any option purchase being a winning trade: merely calculate the probability of the stock ever being above the strike by an amount equal to the cost of the call. However, merely knowing the probability of winning is not the same as expected return, since call option buying does not have fixed results, like a sports bet might. That is, we don’t win a fixed ratio of what we lose. Rather, a call purchase – while having a fixed maximum loss – does not have a fixed win.

Using a method such as this is likely to show a large difference between the probability of winning versus the probability of losing. Therefore, the Kelly Criterion may ask you to invest an overly large amount in a single trade.

In fact, one should realize that the Kelly Criterion is designed to maximize profits, not to necessarily protect your capital.

Hence, one should also be aware of what is called the Probability of Ruin. This is the chance that you could go broke (or at least lose so much money as to be rendered powerless) – no matter how good your system is.

For example, suppose that we define “ruin” as losing 80% of your capital. Furthermore suppose we decide to invest very heavily in a system, placing 40% of our available capital in each trade. Also, assume that these trades are sequential, so that we are only in one position at a time.

If we lose everything on the first trade, we will have 60 % of our capital left. If we invest 40% of that amount and lose it all again, we will have 36% of our original capital left. Finally, if we invest 40% of that on the third trade and lose it all again, we would be left with 22% of our original capital. In effect, we would be “ruined.” This is why many conservative investors just place 2% or 3% of their capital in any one trade and don’t bother trying to optimize things.

In fact, if we define “ruin” as being left with only 20% of our original bankroll, then the following formula defines the situation, if the entire “bet” is lost when a loss occurs (as in sports betting, or as in a call purchase expiring worthless):

              0.2= (1 – r) n
   where r = % of assets risked on each trade
   and n = number of consecutive losses that can be
           sustained before you are ruined.
 Here are some examples:
   r   n           r   n
   2%  79 losses   10% 15 losses
   3   52          15   9
   4   39          20   7
   5   31          25   5
   6   26          30   4
   7   22          40   3
   8   19          50   2

So, if you only risk 2% of your account on any one trade, you could lose 79 trades in a row before you are ruined. The chances of that happening are infinitesimal. However, if you risk 25% of your capital on each trade, it only takes 5 consecutive losses to ruin you.

Summary

Speculators may be able to maximize the use of their capital in successful systems, by employing the Kelly Criterion – using either system statistics or expected return as the building blocks. Traders should be careful to not trade any one position too heavily, though, for the Probability of Ruin is a significant consideration.

However, in many cases, the expected return may be negative (if the implied volatility of the initial purchase is higher than the expected volatility of the underlying) and there may not be any available statistics on the “system” – if there even is a system. In these cases, one is best served by risking a fixed percentage of one’s capital on any given trade – keeping the risk small, especially if there is a significant possibility of losing all the invested capital on any particular trade.

 

This article was originally published in The Option Strategist Newsletter Volume 14, No. 10 on May 26, 2005.  

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