Note

The following implementations and documentation closely follow the work of Tim Leung: Tim Leung and Xin Li Optimal Mean reversion Trading: Mathematical Analysis and Practical Applications.

Trading Under the Cox-Ingersoll-Ross Model

Warning

Alongside with Leung’s research we are using \(\theta\) for mean and \(\mu\) for mean-reversion speed, while other sources e.g. Wikipedia, use \(\theta\) for mean reversion speed and \(\mu\) for mean.

Model fitting

Note

We are solving the optimal stopping problem for a mean-reverting portfolio that is constructed by holding \(\alpha = \frac{A}{S_0^{(1)}}\) of a risky asset \(S^{(1)}\) and shorting \(\beta = \frac{B}{S_0^{(2)}}\) of another risky asset \(S^{(2)}\), yielding a portfolio value:

\[Y_t^{\alpha,\beta} = \alpha S^{(1)} - \beta S^{(2)}, t \geq 0\]

Since in terms of mean-reversion we care only about the ratio between \(\alpha\) and \(\beta\), without the loss of generality we can set \(\alpha = const\) and A = $1, while varying \(\beta\) to find the optimal strategy \((\alpha,\beta^*)\)

We establish a Cox-Ingersoll-Ross process driven by the SDE:

\begin{gather*} dY_t = \mu(\theta - Y_t)dt + \sigma \sqrt{Y_t} dB_t,\\ \theta, \mu, \sigma > 0,\\ B\ -\text{a standard Brownian motion} \end{gather*}

  • \(\theta\) βˆ’ long term mean level, all future trajectories of Y will evolve around a mean level πœƒ in the long run.

  • \(\mu\) - speed of reversion, characterizes the velocity at which such trajectories will regroup around \(\theta\) in time.

  • \(\sigma\) - instantaneous volatility, measures instant by instant the amplitude of randomness entering the system. Higher values imply more randomness.

To fit the model to our data and find optimal parameters we define the average log-likelihood function as in Borodin and Salminen (2002):

\begin{gather*} \ell (\theta,\mu,\sigma|y_0^{\alpha\beta},y_1^{\alpha\beta},\cdots,y_n^{\alpha\beta}) := \frac{1}{n}\sum_{i=1}^{n} ln f^{CIR}(y_i|y_{i-1};\theta,\mu,\sigma)\\ \end{gather*}
\[\begin{gather*} = -ln(\tilde{\sigma}) - \frac{1}{n\tilde{\sigma}}\sum_{i=1}^{n} [y_i +y_{i-1}e^{-\mu \Delta t}] - \frac{1}{n} \sum_{i=1}^{n} [\frac{q}{2}ln(\frac{y_i}{y_{i-1}e^{-\mu\Delta t}}) - ln I_q(\frac{2}{\tilde{\sigma}^2}\sqrt{y_i y_{i-1}e^{-\mu\Delta t}}) \end{gather*}\]

Then, maximizing the log-likelihood function by applying maximum likelihood estimation(MLE) we are able to determine the parameters of the model and fit the observed portfolio prices to a CIR process. Let’s denote the maximized average log-likelihood by \(\hat{\ell}(\theta^*,\mu^*,\sigma^*)\). Then for every \(\alpha\) we choose \(\beta^*\), where:

\[\beta^* = \underset{\beta}{\arg\max}\ \hat{\ell}(\theta^*,\mu^*,\sigma^*|y_0^{\alpha\beta},y_1^{\alpha\beta},\cdots, y_n^{\alpha\beta})\]

Optimal Timing of Trades

To find the optimal timing of trades we have implemented two approaches - optimal stopping and optimal switching. The main difference between the two is the assumption about the number of trades in regards to which we are searching for the optimal timing of trades. In case of an optimal stopping problem, we assume that we want to search for such optimal levels that will maximize the output of a single entry-exit pair of trades. On the other hand, the optimal switching approach is aimed at maximizing the infinite amount of entry-exit trades, thus accounting for any number of consequent trades our investor might want to carry out.

Optimal stopping problem

Suppose the investor already has a position with a value process \((Y_t)_{t>0}\) that follows the CIR process. When the investor closes his position at the time \(\tau\) he receives the value \((Y_{\tau})\) and pays a constant transaction cost \(c_s \in \mathbb{R}\) To maximize the expected discounted value we need to solve the optimal stopping problem:

\[V^{\chi}(y) = \underset{\tau \in T}{\sup} \mathbb{E}({e^{-r \tau} (Y_{\tau} - c_s)| Y_0 = y})\]

where \(T\) denotes the set of all possible stopping times and \(r > 0\) is our subjective constant discount rate. \(V^{\chi}(y)\) represents the expected liquidation value accounted with y.

Current price plus transaction cost constitute the cost of entering the trade and in combination with \(V^{\chi}(y)\) we can formalize the optimal entry problem:

\[J^{\chi}(y) = \underset{\nu \in T}{\sup} \mathbb{E}({e^{-\hat{r} \tau} (V^{\chi}(Y_{\nu}) - Y_{\nu} - c_b)| Y_0 = y})\]

with

\[\hat{r}>0,\ c_b \in \mathbb{R}\]

To sum up this problem, we, as an investor, want to maximize the expected difference between the current price of the position - \(Y_{\nu}\) and its’ expected liquidation value \(V^{\chi}(Y_{\nu})\) minus transaction cost \(c_b\)

Note

Following part of the chapter presents the analytical solution for the optimal stopping problem, both the default version and the version with the inclusion of the stop-loss level.

To solve this problem we denote the CIR process infinitesimal generator:

\[L = \frac{\sigma^2y}{2} \frac{d^2}{dy^2} + \mu(\theta - y) \frac{d}{dy}\]

and recall the classical solution of the differential equation

\[L u(y) = ru(y), \text{for}\ y \in R_{+}\]
\[F^{\chi}(y) = M(\frac{r}{\mu},\frac{2\mu\theta}{\sigma^2},\frac{2\mu y}{\sigma^2})\]
\[G^{\chi}(y) = U(\frac{r}{\mu},\frac{2\mu\theta}{\sigma^2},\frac{2\mu y}{\sigma^2})\]
\[where\ M(a,b,z) = \sum_{n=0}^{\infty}\frac{a_n z^n}{b_n n!},\ \ a_0=1, a_n=a(a+1)(a+2)...(a+n-1)\]
\[U(a,b,z) = \frac{\Gamma(1-b)}{\Gamma(a-b+1)}M(a,b,z) + \frac{\Gamma(b-1)}{\Gamma(a)}z^{1-b}M(a-b+1,2-b,z)\]

Then we are able to formulate the following theorems (proven in Optimal Mean reversion Trading: Mathematical Analysis and Practical Applications by Tim Leung and Xin Li) to provide the solutions to the following problems:

Theorem 4.2 (p.85):

The optimal liquidation problem admits the solution:

\begin{gather*} V^{\chi}(x) = \begin{cases} (b^{\chi*} - c_s) \frac{F^{\chi}(x)}{F^{\chi}(b^{\chi*})} , & \mbox{if } x \in [0,b^{\chi*})\\ \\ x - c_s, & \mbox{ otherwise} \end{cases}\\ \end{gather*}

The optimal liquidation level \(b^{\chi*}\) is found from the equation:

\[\begin{split}F^{\chi} (b^{\chi}) - (b^{\chi} - c_s)F'^{\chi}(b^{\chi}) = 0\\\end{split}\]

Corresponding optimal liquidation time is given by

\[\tau^{\chi*} = inf [t\geq0:Y_t \geq b^{\chi*}]\]

Theorem 4.4 (p.86):

The optimal entry timing problem admits the solution:

\begin{gather*} J(x) = \begin{cases} V^{\chi}(x) - x - c_b, & \mbox{if } x \in [0,d^{\chi*})\\ \\ \frac{V^{\chi}(d^{\chi*}) - d^{\chi*} - c_b}{\hat{G^{\chi}}(d^{\chi*})}, & \mbox{if } x \in (d^{\chi*}, \infty) \end{cases} \end{gather*}

The optimal entry level \(d^{\chi*}\) is found from the equation:

\[\hat{G}^{\chi}(d^{\chi})(V'^{\chi}(d^{\chi}) - 1) - \hat{G}'^{\chi}(d^{\chi})(V^{\chi}(d^{\chi}) - d^{\chi} - c_b) = 0\]

Where β€œ\(\hat{\ }\)” represents the use of transaction cost and discount rate of entering.

Optimal switching problem

To find the optimal levels, first, two critical constants have to be denoted:

\[\begin{split}y_s:=\frac{\mu\theta+rc_s}{\mu+r} \\ y_b:=\frac{\mu\theta-rc_b}{\mu+r}\end{split}\]

Theorem 4.7 (p.56):

Under the optimal switching approach it is optimal to re-enter the market if and only if all of the following conditions hold true:

  1. If \(y_b>0\)

  2. The following inequality must hold true:

\[c_b < \frac{b^{\chi*}-c_s}{F^{\chi}(b^{\chi*})}\]

In case any of the conditions are not met - re-entering the market is deemed not optimal. It would be advised to exit at the optimal liquidation price without re-entering, or not enter the position at all. The difference between the options depends on whether the investor had already entered the market beforehand, or did he or she start with a zero position.

How to use this submodule

This module provides you tools for obtaining the solutions to optimal stopping and optimal switching problems for the CIR model.

Step 1: Model fitting

During the module fitting stage, we need to use fit function to fit the CIR model to our training data. The data provided should consist of a log-price of an already created mean-reverting portfolio.

Implementation

class CoxIngersollRoss

This class implements the algorithms for optimal stopping and optimal switching problems for assets with mean-reverting tendencies based on the Cox-Ingersoll-Ross model mentioned in the following publication:’Tim Leung and Xin Li Optimal Mean reversion Trading: Mathematical Analysis and Practical Applications(November 26, 2015)’ <https://www.amazon.com/Optimal-Mean-Reversion-Trading-Mathematical/dp/9814725919>`_

Constructing a portfolio with mean-reverting properties is usually attempted by simultaneously taking a position in two highly correlated or co-moving assets and is labeled as β€œpairs trading”. One of the most important problems faced by investors is to determine when to open and close a position.

Optimal stopping problem is established on the premise of maximizing the output of one optimal enter-exit pair of trades, optimal switching, on the other hand, aims to maximize the cumulative reward of an infinite number of trades made at respective optimal levels.

To find the liquidation and entry price levels for both optimal switching and optimal stopping for each approach we formulate an optimal double-stopping problem that gives the optimal entry and exit level rules.

__init__()

Initializes the module

CoxIngersollRoss.fit(data: DataFrame, data_frequency: str, discount_rate: tuple, transaction_cost: tuple, start: str | None = None, end: str | None = None, stop_loss: float | None = None)

Fits the Cox-Ingersoll-Ross model to given data and assigns the discount rates, transaction costs and stop-loss level for further exit or entry-level calculation.

Parameters:

  • data – (np.array/pd.DataFrame) An array with time series of portfolio prices / An array with time series of of two assets prices. The dimensions should be either nx1 or nx2.

  • data_frequency – (str) Data frequency [β€œD” - daily, β€œM” - monthly, β€œY” - yearly].

  • discount_rate – (float/tuple) A discount rate either for both entry and exit time or a list/tuple of discount rates with exit rate and entry rate in respective order.

  • transaction_cost – (float/tuple) A transaction cost either for both entry and exit time or a list/tuple of transaction costs with exit cost and entry cost in respective order.

  • start – (Datetime) A date from which you want your training data to start.

  • end – (Datetime) A date at which you want your training data to end.

  • stop_loss – (float/int) A stop-loss level - the position is assumed to be closed immediately upon reaching this pre-defined price level.

Tip

To retrain the model just use one of the functions fit_to_portfolio or fit_to_assets. You have a choice either to use the new dataset or to change the training time interval of your currently used dataset.

CoxIngersollRoss.fit_to_portfolio(data: array | None = None, start: str | None = None, end: str | None = None)

Fits the Cox-Ingersoll-Ross model to time series for portfolio prices.

Parameters:

  • data – (np.array) All given prices of two assets to construct a portfolio from.

  • start – (Datetime) A date from which you want your training data to start.

  • end – (Datetime) A date at which you want your training data to end.

CoxIngersollRoss.fit_to_assets(data: array | None = None, start: str | None = None, end: str | None = None)

Creates the optimal portfolio in terms of the Cox-Ingersoll-Ross model from two given time series for asset prices and fits the values of the model’s parameters. (p.13)

Parameters:

  • data – (np.array) All given prices of two assets to construct a portfolio from.

  • start – (Datetime) A date from which you want your training data to start.

  • end – (Datetime) A date at which you want your training data to end.

Step 2: Determining the optimal entry and exit values

To get the optimal liquidation or entry level for your data we need to call one of the functions mentioned below. They present the solutions to the equations established during the theoretical part. To choose whether to account for the stop-loss level or not choose the respective set of functions.

Implementation

\(b^{\chi*}\): - optimal level of liquidation:

CoxIngersollRoss.optimal_liquidation_level() → float

Calculates the optimal liquidation portfolio level. (p.85)

Returns:

(float) Optimal liquidation portfolio level.

\(d^{\chi*}\) - optimal level of entry, accounting for preferred stop-loss level:

CoxIngersollRoss.optimal_entry_level() → float

Calculates the optimal entry portfolio level. (p.86)

Returns:

(float) Optimal entry portfolio level.

Tip

General rule for the use of the optimal levels:

  • If not bought, buy the portfolio as soon as portfolio price reaches the optimal entry level (enters the interval).

  • If bought, liquidate the position as soon as portfolio price reaches the optimal liquidation level.

Step 3: Determining the optimal switching entry and exit values

To get the optimal switching values we need to use the optimal_switching_levels function that either returns the interval of optimal switching prices or an optimal liquidation level if it is deemed not optimal to re-enter the market.

Implementation

CoxIngersollRoss.optimal_switching_levels() → array

Calculates the optimal switching levels.

Returns:

(np.array) Optimal switching levels.

Step 4: (Optional) Plot the optimal levels on your data

Additionally, you have the ability to plot your optimal levels for default problem and optimal switching onto your out-of-sample data. You have a choice whether you want the optimal switching levels displayed or not.

../_images/cir_optimal_switching.png

An exemplary output of the cir_plot_levels function.

Implementation

CoxIngersollRoss.cir_plot_levels(data: DataFrame, switching: bool = False) → Figure

Plots the found optimal exit and entry levels on the graph alongside with the given data.

Parameters:

  • data – (np.array/pd.DataFrame) Time series of portfolio prices.

  • switching – (bool) A flag whether to take stop-loss level into account when showcasing the results.

Returns:

(plt.Figure) Figure with optimal exit and entry levels.

Tip

To view all the model stats, including the optimal levels call the cir_description function.

CoxIngersollRoss.cir_description(switching: bool = False) → Series

Returns all the general parameters of the model, training interval timestamps if provided, the goodness of fit, allocated trading costs and discount rates, which stands for the optimal ratio between two assets in the created portfolio, default optimal levels calculated. If re-entering the market is optimal shows optimal switching levels.

Parameters:

switching – (bool) Flag that signals whether to output switching data.

Returns:

(pd.Series) Summary data for all model parameters and optimal levels.

../_images/cir_description.png

Example

import numpy as np
from arbitragelab.optimal_mean_reversion import CoxIngersollRoss

example = CoxIngersollRoss()

# We establish our training sample
delta_t = 1/252
np.random.seed(30)
cir_example =  example.cir_model_simulation(n=1000, theta_given=0.2, mu_given=0.2,
                                            sigma_given=0.3, delta_t_given=delta_t)
# Model fitting
example.fit(cir_example, data_frequency="D", discount_rate=0.05,
            transaction_cost=[0.001, 0.001])

# You can separately solve optimal stopping
# and optimal switching problems

# Solving the optimal stopping problem
b = example.optimal_liquidation_level()

d = example.optimal_entry_level()

# Solving the optimal switching problem
d_switch, b_switch = example.optimal_switching_levels()

# You can display the results using the plot
fig = example.cir_plot_levels(cir_example, switching=True)

# Or you can view the model statistics
example.cir_description(switching=True)

Research Notebook

The following research notebook can be used to better understand the concepts of trading under the Cox-Ingersoll-Ross Model.

References

  1. Leung, T.S.T. and Li, X., 2015. Optimal mean reversion trading: Mathematical analysis and practical applications (Vol. 1). World Scientific.