Chapter 3
BSP and BOP Strategy
The most difficult situation to deal with is when the probabilities are both unknown and changing in time. This is the problem you are faced with in the financial markets. There is no doubt that the market experiences trends. There are periods of time when the probability of an upward move is high and periods when it is low. The trends can also change quickly and unexpectedly. In a situation like this, using a large amount of historical data to estimate probabilities is probably not very useful and may even be counterproductive.
The simplest way to deal with this situation is to assume that today's probabilities are the same as yesterday's. In other words, you bet that the market will do the same today as it did yesterday. If it went up yesterday then bet that it will go up today. If it went down then bet that it will go down. If a probability bias or trend lasts at least a few days then this simple strategy can be effective but errors will occur when the bias suddenly reverses.
Let's now analyze this strategy for the case of the stock market. The \(A\) outcome is that the market closes above the previous day's close and the \(B\) outcome is that it closes below. The analysis will be simplified by assuming that the size of the returns for up and down days are the same but just different in sign. This is of course unrealistic but not an unreasonable first approximation. It means that we are treating the stock market as being similar to the coin flip game discussed above with the only difference being a scale factor for the expectation that is equal to the assumed size of the returns.
The analysis will be further simplified if we define the \(A\) and \(B\) probabilities as follows
\begin{eqnarray}\label{eq50} P(A) & = & p = \frac{1}{2} + b\\ P(B) & = & 1-p = \frac{1}{2} - b\nonumber \end{eqnarray}The \(b\) parameter explicitly shows the degree of bias. It can range from -1/2 to +1/2. For \(b=-1/2\) a down move is guaranteed and for \(b=+1/2\) an up move is guaranteed. When \(b=0\) there is no bias and up or down moves are equally likely (see the Single Coin Model, section 15.2, for more discussion on bias).
Now if you bet that things will go the same as the previous day then you will win if the outcome for the two days is \(AA\) or \(BB\). The probability of this happening is
\begin{equation}\label{eq60} P(AA \mathrm{or} BB) = (\frac{1}{2} + b)^2 + (\frac{1}{2} - b)^2 \end{equation}You will lose if the outcome for the two days is \(AB\) or \(BA\). The probability of this happening is
\begin{equation}\label{eq70} P(AB \mathrm{or} BA) = 2(\frac{1}{2} + b)(\frac{1}{2} - b) \end{equation}Assuming the returns are \(\pm r\), the expected return is
\begin{eqnarray}\label{eq80} E & = & r((\frac{1}{2} + b)^2 + (\frac{1}{2} - b)^2) - r(2(\frac{1}{2} + b)(\frac{1}{2} - b))\nonumber\\ & = & r4b^2 \end{eqnarray}The extraordinary thing about this result is that since the bias parameter is squared, you will get a positive expectation if it is positive or negative. It makes no difference, and you don't have to know or guess, what the direction of the bias is. Just bet the same as the previous day and you will have a positive expectation as long as there is some bias, however small.
Of course this is not the whole story. The assumption in the above analysis is that the bias today is the same as yesterday. This will not be true when the bias switches. If the bias switches very often then the errors will probably erase the positive expectation. Another thing to consider is the variance of this strategy. To get the variance just add the win and lose probabilities and subtract the square of the expectation. This gives:
\begin{equation}\label{eq90} \mathrm{Var} = r^2(1 - 16b^4) \end{equation}If the bias is small the variance will be very large. This means that periods of large losses are possible even though the expectation is positive. With a small bias you can expect a lot of volatility.
There is a complement to the "bet the same as previous" (BSP) strategy and that is the "bet the opposite of previous" (BOP) strategy. If the market went up yesterday then bet that it will go down today and vice versa. The assumption here is that the bias switches from day to day. This means that the sign of the \(b\) parameter in the \(P(A)\) and \(P(B)\) formulas switches from one day to the next. Using this strategy you will now lose on \(AA\) or \(BB\) and win on \(AB\) or \(BA\). Due to the switching of the sign on \(b\), the probabilities for these two outcomes are:
\begin{eqnarray}\label{eq100} P(AA \mathrm{or} BB) & = & 2(\frac{1}{2} + b)(\frac{1}{2} - b)\\ P(AB \mathrm{or} BA) & = & (\frac{1}{2} + b)^2 + (\frac{1}{2} - b)^2\nonumber \end{eqnarray}The win and lose probabilities are the same as for BSP strategy and so the expectation and variance must be exactly the same.
The BSP and BOP strategies complement each other and every day one of the strategies will be successful. The market today will either move in the same direction as yesterday or the opposite. There are no other possibilities (remember we count no movement as a down day). The market will go through periods when the BSP strategy is dominant and periods when the BOP strategy dominates. The holy grail of trading systems is to come up with a way of knowing when to switch between them. One way of trying to deal with this trend switching process is by using Markov models. We will discuss some simple trading systems based on Markov models further on in the book but first we look at some examples of using a pure BSP or BOP strategy.