## Monte Carlo Simulation

On a nice day 2 years ago, when I was on financial field. My boss sent our team an email. In this email, he would like to us propose some machine learning techniques to predict stock price.

So, after accepting the assignment from my manager, our team begin to research and apply some approaches for prediction. When we talk about Machine Learning, we often think of supervised and unsupervised learning. But one of the algorithms we applied is one that we usually forgotten  however equally highly effective algorithm: Monte Carlo Simulation.

## What is Monte Carlo simulation

Monte Carlo method is a technique that uses random numbers and probability to solve complex problems. The Monte Carlo simulation, or probability simulation, is a technique used to understand the impact of risk and uncertainty in financial sectors, project management, costs, and other forecasting machine learning models.[1]

Now let’s jump into python implementation to see how it applies,

## Python Implementation

In this task we used data of DXG stock dataset from 2017/01/01 to 2018/08/24 and we would like to know what is stock price after 10 days, 1 months and 3 months, respectively

We will simulate the return of stock and next price will be calculated by

P(t) = P(0) * (1+return_simulate(t))

Calculate mean and standard deviation of stock returns

miu = np.mean(stock_returns, axis=0)
dev = np.std(stock_returns)

Simulation process

simulation_df = pd.DataFrame()
last_price = init_price
for x in range(mc_rep):
count = 0
daily_vol = dev
price_series = []
price = last_price * (1 + np.random.normal(miu, daily_vol))
price_series.append(price)
for y in range(train_days):
if count == train_days-1:
break
price = price_series[count] * (1 + np.random.normal(miu, daily_vol))
price_series.append(price)
count += 1
simulation_df[x] = price_series

Visualization Monte Carlo Simulation

fig = plt.figure()
fig.suptitle('Monte Carlo Simulation')
plt.plot(simulation_df)
plt.axhline(y = last_price, color = 'r', linestyle = '-')
plt.xlabel('Day')
plt.ylabel('Price')
plt.show()

Now, let’s check with actual stock price after 10 days, 1 month and 3 months

plt.hist(simulation_df.iloc[9,:],bins=15,label ='histogram')
plt.axvline(x = test_simulate.iloc[10], color = 'r', linestyle = '-',label ='Price at 10th')
plt.legend()
plt.title('Histogram simulation and last price of 10th day')
plt.show()

We can see the most frequent occurrence price is pretty close with the actual price after 10th

If the forecast period is longer, the results is not good gradually

Simulation for next 1 month

After 3 months

## Conclusion

Monte Carlo simulation is used a lot in finance, although it has some weaknesses, but hopefully through this article, you will have a new look on the simulation application for forecasting.

## Reference

[1] Pratik Shukla, Roberto Iriondo, “Monte Carlo Simulation An In-depth Tutorial with Python”, medium, https://medium.com/towards-artificial-intelligence/monte-carlo-simulation-an-in-depth-tutorial-with-python-bcf6eb7856c8

Please also check Gaussian Samples and N-gram language models,
Bayesian Statistics  for statistics knowledges.

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## Hiring- Data Scientist (Algorithm Theory)

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Currently, we are looking for candidates with experiences in Algorithms, Natural Language Processing (NLP), but any other fields of AI will be considered too.

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### More on MTI – what is it like to work in MTI?

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## Bayesian estimator of the Bernoulli parameter

In this post, I will explain how to calculate a Bayesian estimator. The taken example is very simple: estimate the parameter θ of a Bernoulli distribution.

A random variable X which has the Bernoulli distribution is defined as

with

In this case, we can write

.

In reality, the simplest way to eatimate θ is to sample X, count how many time the event occurs, then estimate the probability of occuring of event. This is exactly what the frequestists do.

In this post, I will show how do the Bayesian statisticians estimate θ. Although this doesn’t have a meaningful application, but it helps to understand how do the Bayesian statistics work. Let’s start.

## The posterior distribution of θ

Denote Y as the observation of the event. Given the parameter θ, if we sample the event n time, then the probability that the event occurs k time is (this is called the probability density function of Bernoulli )

In Bayesian statistics, we would like to calculate

$p(\theta|Y=y)$

By using the Bayesian formula, we have

With the prior distribution of theta as an Uniform distribution, p(θ) = 1, and it is easy to prove that

where Γ is the Gamma distribution. Hence, the posterior distribution is

Fortunately, this is the density function of the Beta distribution:

We use the following properties for evaluating the posterior mean and variance of theta.

If , then

## Simulation

In summary, the Bayesian estimator of theta is the Beta distribution with the  mean and variance as above. Here is the Python codes for simulating data and estimating theta

def bayes_estimator_bernoulli(data, a_prior=1, b_prior=1, alpha=0.05):
'''Input:
data is a numpy array with binary value, which has the distribution B(1,theta)    a_prior, b_prior: parameters of prior distribution Beta(a_prior, b_prior)    alpha: significant level of the posterior confidence interval for parameter    Model:
for estimating the parameter theta of a Bernoulli distribution    the prior distribution for theta is Beta(1,1)=Uniform[0,1]    Output:
a,b: two parameters of the posterior distribution Beta(a,b)
pos_mean: posterior estimation for the mean of theta
pos_var: posterior estimation for the var of theta'''
n = len(data)
k = sum(data)
a = k+1
b = n-k+1
pos_mean = 1.*a/(a+b)
pos_var = 1.*(a*b)/((a+b+1)*(a+b)**2)
## Posterior Confidence Interval
theta_inf, theta_sup = beta.interval(1-alpha,a,b)
print('Prior distribution: Beta(%3d, %3d)' %(a_prior,b_prior))
print('Number of trials: %d, number of successes: %d' %(n,k))
print('Posterior distribution: Beta(%3d,%3d)' %(a,b))
print('Posterior mean: %5.4f' %pos_mean)
print('Posterior variance: %5.8f' %pos_var)
print('Posterior std: %5.8f' %(np.sqrt(pos_var)))
print('Posterior Confidence Interval (%2.2f): [%5.4f, %5.4f]' %(1-alpha, theta_inf, theta_sup))
return a, b, pos_mean, pos_var

# Example
n = 129 # sample size
data = np.random.binomial(size=n, n=1, p=0.6)
a, b, pos_mean, pos_var = bayes_estimator_bernoulli(data)

And the result is

Prior distribution: Beta(  1,   1)
Number of trials: 129, number of successes: 76
Posterior distribution: Beta( 77, 54)
Posterior mean: 0.5878
Posterior variance: 0.00183556
Posterior std: 0.04284341
Posterior Confidence Interval (0.95): [0.5027, 0.6703]

In the simulation, we simulated 129 data from the Bernoulli distribution with θ=0.6. And the Bayesian estimation of θ is the posterior mean which is 0.5878.
This is a very simple example of Bayesian estimation. In reality, it is usually tricky to determinate a closed form solution of the posterior distribution from the given prior distribution. In that case, Monte Carlo technique is one of solutions to approximate the posterior distribution.
Please also check Gaussian Samples and N-gram language models for statistics knowledges.

## Hiring Data Scientist / Engineer

We are looking for Data Scientist and Engineer.