FMS — Financial Modelling System

FMS implements a financial model for stock market simulation using OCaml. It models the behaviour of a £5,000 investment over 3 years, providing an understanding of potential outcomes and risk metrics for both individual stocks and the overall portfolio.

Warning

Disclaimer: FMS is for educational purposes only! Please do not use it for actual investment decisions. Always consult a qualified financial advisor for investment choices.

Features

• Monte Carlo simulation of stock price movements using both traditional and SR-LSTM models
• Portfolio modelling with multiple stocks (TSLA, GOOGL, NVDA, IBM, MSFT, AMZN, and many more)
• Risk metrics calculation (Value at Risk, Expected Return, Probability of Profit, Sharpe Ratio)
• Configurable parameters (investment amount, time horizon, stock characteristics)
• Dynamic Portfolio Value calculation to estimate future portfolio worth
• SR-LSTM (Self-Regulating Long Short-Term Memory) implementation for advanced stock price prediction
• Comparative analysis between traditional Monte Carlo and SR-LSTM models
• Individual stock performance metrics for both traditional and SR-LSTM models
• Identification of best-performing stocks based on Sharpe Ratio
• Calculation of the average Sharpe Ratio across the portfolio for both models
• Detailed output of overall portfolio metrics and individual stock metrics
• Extensive portfolio with a mix of established tech giants and emerging technology companies

Mathematical Model

Stock Price Simulation

Stock prices follow geometric Brownian motion:

$$dS = \mu S dt + \sigma S dW$$

where $S$ is the stock price, $\mu$ is the drift, $\sigma$ is the volatility, and $dW$ is a Wiener process.

The discrete simulation uses:

$$S(t + \Delta t) = S(t) \exp\left(\left(\mu - \frac{\sigma^2}{2}\right)\Delta t + \sigma \sqrt{\Delta t} Z\right)$$

where $Z$ is a standard normal random variable.

Portfolio Value

The portfolio value $V$ at time $t$ is:

$$V(t) = \sum_{i=1}^{n} q_i S_i(t) + C$$

where $q_i$ is the quantity of stock $i$, $S_i(t)$ is the price of stock $i$ at time $t$, and $C$ is the cash holding.

Risk Metrics

Value at Risk (VaR)

VaR is calculated using the percentile method:

$$VaR_\alpha = -\text{percentile}(1-\alpha, {V_1, V_2, ..., V_n})$$

where ${V_1, V_2, ..., V_n}$ are the simulated portfolio values.

Expected Return

The expected return is:

$$E[R] = \frac{1}{n} \sum_{i=1}^{n} \frac{V_i - V_0}{V_0}$$

where $V_0$ is the initial portfolio value.

Probability of Profit

Calculated for both individual stocks and the overall portfolio.

Sharpe Ratio

The Sharpe Ratio is calculated as:

$$\text{Sharpe Ratio} = \frac{E[R] - R_f}{\sigma}$$

where $E[R]$ is the expected return, $R_f$ is the risk-free rate, and $\sigma$ is the standard deviation of returns.

Dynamic Portfolio Value

The portfolio value $V$ at time $t$ remains:

$$V(t) = \sum_{i=1}^{n} w_i(t) S_i(t) + C(t)$$

where $w_i(t)$ is the dynamic weight of stock $i$, $S_i(t)$ is the price of stock $i$ at time $t$, and $C(t)$ is the cash holding at time $t$.

Machine Learning Integration with SR-LSTM

We incorporate a Self-Regulating Long Short-Term Memory (SR-LSTM) neural network to predict future stock prices:

• Regulatory Factor Calculation:

  $$R_t = \rho(W_r \cdot h_{t-1} + b_r)$$

  Here, $R_t$ is the regulatory factor for time step $t$, $W_r$ is the weight matrix, $b_r$ is the bias term, and $\rho$ is a non-linear activation function.

• Modified Forget and Input Gates:

  $$f'_t = f_t \odot R_t$$ $$i'_t = i_t \odot R_t$$

  We modify the standard forget gate $f_t$ and input gate $i_t$ by the regulatory factor $R_t$ to adjust the information flow dynamically.

• Cell State and Output Gate:

  The cell state update and output gate equations remain the same as in standard LSTM, but they operate on the modified gates $f'_t$ and $i'_t$.

Prerequisite

• OCaml (version 4.08 or higher recommended)
• OPAM (OCaml Package Manager)
• Make

Building and Running

1. Clone the repository and navigate to the project directory.

2. Build and run the simulation:

eval $(opam env) && make clean && make all VERBOSE=1 && ./fms

## OR ##

eval $(opam env) && make clean && make all VERBOSE=1 && make write-output

Output

The simulation displays:

• Initial investment amount and investment horizon
• Overall portfolio metrics for both Traditional and SR-LSTM models:
  • Expected Return
  • Value at Risk (VaR) at 95% and 99% confidence levels
  • Probability of Profit
  • Sharpe Ratio
• Individual stock metrics for both Traditional and SR-LSTM models:
  • Current Price
  • Expected Return
  • Probability of Profit
  • Sharpe Ratio
• Dynamic Portfolio Value estimation after the investment horizon
• Comparison of Total Expected Returns between Traditional and SR-LSTM models
• Difference in expected returns between the two models
• Average Sharpe Ratios for both Traditional and SR-LSTM models across the portfolio
• Best Performing Stocks based on Sharpe Ratio for both models

Customisation

Modify fms.ml to adjust:

• Initial investment and investment horizon
• Stock portfolio composition and characteristics
• Number of Monte Carlo simulations

Future Enhancements (Maybe)

a. Singular Value Decomposition (SVD)

SVD decomposes a matrix $A$ into three matrices:

$$A = U\Sigma V^T$$

Where:

• $A$ is an $m \times n$ matrix
• $U$ is an $m \times m$ orthogonal matrix
• $\Sigma$ is an $m \times n$ diagonal matrix with non-negative real numbers on the diagonal
• $V^T$ is the transpose of an $n \times n$ orthogonal matrix

The diagonal entries $\sigma_i$ of $\Sigma$ are known as the singular values of $A$. For dimensionality reduction, we can approximate $A$ by keeping only the $k$ largest singular values:

$$A_k = \sum_{i=1}^k \sigma_i u_i v_i^T$$

Where $u_i$ and $v_i$ are the $i$-th columns of $U$ and $V$ respectively.

Benefit: We could use SVD to analyse and potentially reduce the dimensionality of our stock data. It would be beneficial if we expanded to a larger number of stocks.

b. QR Decomposition

QR decomposition factors a matrix $A$ into a product of an orthogonal matrix $Q$ and an upper triangular matrix $R$:

$$A = QR$$

Where:

• $A$ is an $m \times n$ matrix
• $Q$ is an $m \times m$ orthogonal matrix
• $R$ is an $m \times n$ upper triangular matrix

For solving least squares problems $Ax = b$, we can use QR decomposition:

$$QRx = b$$ $$Rx = Q^Tb$$

Enables the system to solve efficiently due to $R$ being upper triangular.

Benefit: We can use it to solve least squares problems more efficiently, which might be useful for fitting models to historical stock price data.

c. Trust Region Methods

Trust region methods solve optimisation problems of the form:

$$\min_{x \in \mathbb{R}^n} f(x)$$

At each iteration, we solve the subproblem:

$$\min_{p \in \mathbb{R}^n} m_k(p) \quad \text{subject to} \quad |p| \leq \Delta_k$$

Where:

• $m_k(p)$ is a quadratic approximation of $f(x_k + p)$:

$$m_k(p) = f(x_k) + g_k^T p + \frac{1}{2} p^T B_k p$$

• $g_k$ is the gradient of $f$ at $x_k$
• $B_k$ is an approximation of the Hessian of $f$ at $x_k$
• $\Delta_k$ is the trust region radius

The next iterate is then:

$$x_{k+1} = x_k + p_k$$

Where $p_k$ is the solution to the subproblem. The trust region radius $\Delta_k$ is adjusted based on the agreement between the model $m_k$ and the actual function $f$.

Benefit: It could potentially improve the optimization of our portfolio allocation, especially if we're dealing with non-convex objective functions.

License

This project is licensed under the BSD 3-Clause License.

Citation

@misc{fmsafo2024,
  author       = {Oketunji, A.F.},
  title        = {FMS — Financial Modelling System},
  year         = 2024,
  version      = {0.0.4},
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.13910428},
  url          = {https://doi.org/10.5281/zenodo.13910428}
}

Copyright

(c) 2024 Finbarrs Oketunji. All Rights Reserved.