Visualizing Stock Pricing Simulations
One of the simplest methods of simulating fluctuations of equity pricing over time is by using a binomial return distribution.
At each interval, there can only be two outcomes: the price rises by a fixed percentage or falls by a fixed percentage. We can define 3 inputs to the model:
- initial price
- p
- delta
Where initial price refers to the assets initial market value, p is the probability of the price increasing at a single interval, and delta is the amount that the stock’s price will increase or decrease at each interval.
In other words, at each interval the prices will either be multiplied by 1+delta with probability p or decrease by 1-delta with probability 1-p.
Each interval is a bernoulli trail. Bernoulli trials date back to the 17th century.
Conducting many simulations can provide insight into the probability distributions associated with this simulations. This technique is often known as Monte Carlo simulation. A Monte Carlo simulation with p=0.5 is visualized below.
Of course, increasing p increases the price’s upward drift. A similar experiment with p=0.8 is visualized below.
The increase in p clearly increases the expected returns.
Sample Calculation
We can calculate the probability of beating a certain return using the cumulative distribution function of the binomial distribution.
When delta=0.1, as in the simulations above, at least 53 of the 100 intervals need to be positive to obtain a total return of at least 10%.
Using k=53, n=100, and p=0.8, we find that obtaining a total return of at least 10% is almost certain as the probability is more than 0.9999. However, if we use p=0.5, the probability drops to just 0.2421.
Theoretically, this means that out of 1000 simulations with p=0.5 & delta=0.1, approximately 242 should obtain returns of at least 10%. In fact, in the simulation shown below, 10% returns were obtained by exactly 244 of the 1000 simulations.
