Chapter 8 - Python for Finance¶

In this notebook I am working through chapter 8 of the book Python for Finance, see https://home.tpq.io/books/py4fi/.

I provide additionnal comments with further interesting thoughts I had about the exercises and possible relations to other stuff I've learned in the past.

Fundamentals¶

Load data from five S%P 500 companies, the S&P 500 index itself, as well as gold price, volatility and exchange rate indicators from 01st January 2010 to 29th of June 2018.

Get general statistical metrics about the dataset.

We now plot the absolute returns, that is to say $P_i-P_{i-1}$, units in Dollars. Then we plot the dimensionless relative returns, so $\frac{P_i-P_{i-1}}{P_{i-1}}$.

Interestingly to note, absolute returns are like numeric differentiation (Backwards Euler method) with a constant timestep (that we ignore). Because we are working with noisy data this leads to a very noisy result in the absolute returns.

More commonly used in Finance than absolute or relative returns are logarithimic (short: "log") returns. Their main advantage is being additive over time, so: $R_i = \sum_k^i R_k $ where $R_i = ln(\frac{P_i}{P_0})$ are the returns accumulated at time $t_i$. This is a lot more convenient vis-à-vis relative returns, those have to be multiplied together to get the final return.

Another advantage is being dimensionless and revealing the uniform distributions of the securities in multiple orders of magnitude.

Nice analogy: the difference between log returns and relative returns is the same as between strain and true strain in elasto-mechanics! Another way to say this is that relative returns are the first order Taylor approximation of log returns, so they are equivalent over small timescales.

Below we check whether the formula $R_{log} = ln(1+R_{rel})$ holds true by comparing it to the definition of the logarithmic returns.

Now we use the addivity over time property to easily visualize the performances of the securities by performing a cumulative sum.

Here you can see the great advantage of log returns, all the graphs start out at the same place and you can easily see how much gains you would have had if you had put in 1 $ in 2010, specifically for Apple you would have made over 10 $, so a > 1000% return.

Because the current data is too finely distributed in time we perform moving averages, in pandas so-called rolling averages. The syntax is really simply here, you just call the method .rolling() and it automatically adjusts all methods appended after itself.

Now we compute the exponentially moving average EWMA. It's very useful because it takes into account the entire history, just with lower and lower weights (exponentially decreasing). One can decide by the parameter alpha whether to favor current or past data.

It is computed like this: $EWMA_i = \alpha x_i + (1-\alpha) EWMA_{i-1}$

We see that it is basically a linear interpolation but recursively applied which means that the past values do never "fall out the window" like in the rolling average.

If you are wondering: the rolling average applies its computed value to the last index inside of its window. It is only looking backward, this eliminates any "foresight-bias" that we want to avoid when performing financial calculations.

Below we look at the standard rolling averages. Look at the high local maxima and notice that they are "forgotten" after a certain amount of time (the window size). Also interesting: the min and max hug the graph from the left to the right.

Here we do the same thing but just for the last 200 days of the dataset. We also include the rolling mean and median. The median might be more appropiate here because it is less sensitive to outliers.

Testing trading strategies¶

Now we actually want to apply what we learned before in order to develop trading strategies that can be used to turn a profit. At least that's we aim for.

The first and simplest one is the Two-Rolling-Window strategy. It consists of one short and one long rolling mean of the stock price. The very simple theory behind this is that if the shorter one exceeds the longer one then this means that we are in an uncharacteriscally high upswing and should "go long" on the stock. The reverse means that we should "go short".

Going long and short refers to actions that benefit us if the stock price rises or falls respectively. This could be simply buying the stock, buying an inverse fund or using derivatives.

Now we want to visualize the strategy we expressed in words above. We do this by creating a "trading signal" that is defined by a boolean statement.

Below you can see the visualization of the trading signal as a square wave above the stock price and the rolling means.

We see that this strategy is not particulary efficient because we only perform four trades in the span of more than seven years.

Volatility¶

In this section we will further explore the measure of volatility which is defined as the standard deviation of logarithmic returns.

We want to investigate the hypothesis whether or not the S&P 500 is negatively correlated with its volatility. The reasoning behind it is that when the price falls, trading volume increases due to panic which decreases the price further and thus volatility increases also. It's important to note that this is not a causal effect.

We first compute the correlation between the rolling S&P 500 and the volatility for each one-year time period, the rolling window is also automatically applied to the volatility.

We see that the values of the correlation vary wildly throughout time, sometimes even becoming positive. In general though the observation holds true, with a mean correlation of $-58\%$.

Below you can see the chart for the one-year period with the strongest correlation right at the beginning. The relationship is striking.

We now investigate the relationship when we use log returns instead of the stock price.

Log returns give same correlation as relative returns.

We now have a look at the covariance between the two.

Log returns are used here because they have the big advantage of transforming both indicators to the same dimensionless scale, making comparison very easy.

We perform a linear regression to quantify their covariance.

We want to prove that using the data directly without performing the log transformation is not useful. We see in the plot below that there is a bias for certain values in the S&P 500 price, it is not uniformly distributed. This means that the correlation values would be misleading.

This reminds me of Benford's law, should be further investigated.

High Frequency Data¶

Next we are going to take a look at "tick" data. This means raw data from the financial trading day with no resampling applied. It essentially contains every transaction done on the market. Each transaction comes with its timestamp and the bid and ask prices at that moment.

The bid price is the highest price that a buyer is willing to buy the security for at this moment. The ask price is the lowest price that a seller is willing to sell for.

We add a third column called "mid" which contains the mean between the bid and ask prices.

We resample to 5 minutes, this makes the interval i.e. the sampling time constant, which is quite useful for further development or analysis.

Conclusion¶

We explored the fundamentals of technical analysis, focusing on how to apply the Python pandas package for this task.

We learned about the different types of returns, rolling averages, the connection between volatility and returns and finally tick data.

Next time we will dive deeper into how one could evaluate the performance of trading strategies using backtests and permutation testing of the price history to detect overfitting.

Thanks for reading! Best, Marius