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https://github.com/0xnu/fms

FMS implements a financial model for stock market simulation.
https://github.com/0xnu/fms

brownian-motion dynamic-portfolio-value monte-carlo monte-carlo-simulation ocaml probability-of-profit sharpe-ratio sr-lstm stock-market

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FMS implements a financial model for stock market simulation.

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README 

        
## FMS — Financial Modelling System



FMS implements a financial model for stock market simulation using [OCaml](https://en.wikipedia.org/wiki/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](https://arxiv.org/abs/2312.08948) (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](https://ocaml.org/) (version 4.08 or higher recommended)

- [OPAM](https://opam.ocaml.org/) (OCaml Package Manager)

- [Make](https://en.wikipedia.org/wiki/Make_(software))



### Building and Running



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



2. Build and run the simulation:



```sh

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](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Rosenbrock_function).



### License



This project is licensed under the [BSD 3-Clause](LICENSE) License.



### Citation



```tex

@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](https://finbarrs.eu). All Rights Reserved.