The Gaussian model
1. The Gaussian model
People in finance use models. Models are abstractions of reality that can be used to extrapolate or forecast from past data. Now you will see a simple but powerful model for stock return: The Gaussian model. It is widely used in finance because it is a simple model to depict the behavior of financial returns.2. The Gaussian model
We also call the Gaussian model "the Normal model" because it relies on the Normal distribution used in statistics. This is the well-known bell-shaped plot.3. The Gaussian model
When displayed as a histogram, the Gaussian model provides the theoretical probabilities of observing returns belonging to a range of values. The bar corresponding to a bin indicates the probability of observing a return falling within a bin.4. The Gaussian model
It is also common to display the Gaussian model with a continuous curve, called the density. The density does not report a probability, but a density value. We will see that both the Gaussian histogram and the density curve are related.5. The Gaussian model
Under the Gaussian model, there is a 50% chance of observing returns below the average and a 50% chance of observing returns above the average.6. The Gaussian model
Also, the probability of observing values around the average is higher than observing extreme returns.7. Function NORMDIST()
The Gaussian density curve is computed with the function NORMDIST(). The function requires 4 arguments.8. Function NORMDIST(): density curve
The first argument is the value on the horizontal axis for which we want to compute the density of the Gaussian model. In this case, the density evaluated at -1 is 0.242. Be careful, this is not a probability!9. Function NORMDIST(): density curve
The second and third arguments are respectively the location and the dispersion of the Gaussian model. We will see how we can play with them later.10. Function NORMDIST(): density curve
The last argument is a logical TRUE/FALSE indicating whether we want to compute the cumulative probability (TRUE) or the density (FALSE). In this case, we set it to FALSE as we are looking for the density.11. Function NORMDIST(): probability
When we set the last argument to TRUE, NORMDIST() provides the surface below the density up to the desired point on the horizontal axis. Here the function provides the surface from -Infinity to 0. The value is 50%. Let's see now how we can use this to compute the theoretical probability for a given bin. We proceed in two steps.12. Function NORMDIST(): probability
We define the boundaries of a given bin as [LB, UB]. In our example the lower bound LB is set to -0.25 and the upper bound UB is set to 0. First, we compute the surface below the density up to the upper bound at 0. We use NORMDIST() evaluated at 0 with the fourth argument set to TRUE, as before.13. Function NORMDIST(): probability
Next, we compute the surface up to the lower bound at -0.25.14. Function NORMDIST(): probability
The theoretical probability of observing a value lying between -0.25 and 0 is then obtained as the difference between the two surfaces. In this case it is 9.87%.15. Function NORMDIST(): probability
We can then represent the Gaussian model and the associated probabilities with a histogram, as with the empirical distribution. With the Gaussian model we have probabilities. In the empirical distribution we have relative frequencies. Both can be compared!16. Gaussian parameters
You can make the Gaussian model more flexible by changing its location "m" and playing with its dispersion parameter "s".17. Modifying the location
By changing "m", you shift the location of the model.18. Modifying the dispersion
By modifying "s", you render the dispersion larger or smaller.19. Let's practice!
You will see in the next lesson how you can match the model with the data. But first, let's play a bit with the Gaussian model! Time to practice!Create Your Free Account
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