Note

The following documentation closely follows the article by Alexandr Lipton and Marcos Lopez de Prado: “A closed-form solution for optimal mean-reverting trading strategies”

As well as the book by Marcos Lopez de Prado: “Advances in Financial Machine Learning”

A closed-form solution for optimal mean-reverting trading strategies

As stated in the paper by Marcos Lopez de Prado: “When prices reflect all available information, they oscillate around an equilibrium level. This oscillation is the result of the temporary market impact caused by waves of buyers and sellers.” Which means that if we can consider the asset prices to be mean-reverted, then there is a clear toolkit for its approximation, namely, the Ornstein-Uhlenbeck process.

Attempting to monetize this oscillations market makers provide liquidity by entering the long position if the asset is underpriced - hence, the price is lower than the equilibrium, and short position if the situation is reversed. The position is then held until one of the three outcomes occurs:

they achieve a targeted profit
they experience a maximum tolerated loss
the position is held longer then the maximum tolerated horizon

The main problem that arises now is how to define the optimal profit-taking and stop-loss levels. In this module to obtain the solution we utilize the approach introduced by Alexandr Lipton and Marcos Lopez de Prado that applies the method of heat potentials, which is widely applied in physics, to Sharpe ratio maximization problem. The maximization is performed with respect to the border values of our exit corridor.

Problem definition

We suppose an investment strategy S invests in \(i = 1,...I\) opportunities or bets. At each opportunity \(i\), S takes a position of \(m_i\) units of security X, where \(m_i \in (-\infty; +\infty)\). The transaction that entered such opportunity was priced at a value \(m_i P_{i,0}\), where \(P_{i,0}\) is the average price per unit at which the \(m_i\) securities were transacted.

Note

In this approach, we use the volume clock metric instead of the time-based metric. More in that in the paper “The Volume Clock: Insights into the High Frequency Paradigm” by David Easley, Marcos Lopez de Prado, and Maureen O’Hara.

As other market participants continue to transact security X, we can mark-to-market (MtM) the value of that opportunity \(i\) after \(t\) observed transactions as \(m_i P_{1,t}\). Where \(m_i P_{1,t}\) represents the value of opportunity \(i\) if it were liquidated at the price observed in the market after \(t\) transactions. Accordingly, we can compute the MtM p/l of opportunity \(i\) after \(t\) transactions as \(\pi_{i,T_i}=m(P_{i,t}-P_{i,0})\) The exiting opportunity arises in the following scenarios:

\(\pi_{i,T_i}\geq \bar{\pi}\), where \(\bar{\pi}>0\) is a profit-taking threshold
\(\pi_{i,T_i}\leq \underline{\pi}\), where \(\underline{\pi}<0\) is a stop-loss level

Consider the OU process representing the \({P_i}\) series of prices

\[P_{i,t} - \mathbb{E}_0[P_{i,t}] = \mu(\mathbb{E}_0 - P_{i,t-1}) + \sigma\epsilon_{i,t}\]

In order to calculate the Sharpe ratio we need to reformulate our problem in terms of heat potentials.

Suppose S - long investment strategy with p/l driven by the OU process:

\[dx' = \mu'(\theta'-x')dt'+\sigma'dW_{t'}, x'(0) = 0\]

and a trading rule \(R = \{ \bar{\pi}',\underline{\pi}',T' \}\). Now we transform it to use its steady-state by performing scaling to remove superfluous parameters.

\[ \begin{align}\begin{aligned}t = \mu't',\ T = \mu'T',\ x = \frac{\sqrt{\mu'}}{\sigma'} x',\\\theta = \frac{\sqrt{\mu'}}{\sigma'} \theta',\ \bar{\pi} = \frac{\sqrt{\mu'}}{\sigma'} \bar{\pi}', \ \underline{\pi} = \frac{\sqrt{\mu'}}{\sigma'} \underline{\pi}'\end{aligned}\end{align} \]

and get

\[dx = (\theta-x)dt + dW_t, \ \bar{\pi}' \leq x \leq \underline{\pi},\ 0 \leq t \leq T\]

where \(\theta\) is an expected value and its standard deviation is given by \(\Omega=\frac{1}{\sqrt{2}}\) .

According to the trading rule, we exit the trade in one of the three scenarios:

Price hits a targeted profit \(\bar{\pi}\)
Price hits \(\underline{\pi}\) stop-loss level
The trade expires at \(t=T\)

Tip

Short strategy reverses the roles of \({\bar{\pi}',\underline{\pi}'}\):

\(-\underline{\pi}\) equals the profit taken when the price hits \(\underline{\pi}\) and

\(-\bar{\pi}\) losses are incurred while price hits \(-\bar{\pi}\)

Hence, we can restrict ourself to the case with \(\theta \geq 0\) .

Sharpe ratio calculation

To compute the approximate SR we need to perform the four-step numerical evaluation.

Step 1: Define a calculation grid

First of all, we define the grid \(\upsilon\) based on which we will perform our numerical calculation:

\[0=\upsilon_0<\upsilon_1<...<\upsilon_n=\Upsilon,\ \upsilon(t) = \frac{1 - e^{-2(T-t)}}{2}\]

Step 2: Numerically calculate helper functions \(\bar{\epsilon}, \underline{\epsilon}, \bar{\phi}, \underline{\phi}\)

We are going to use the classical method of heat potentials to calculate the SR. As a preparation, in this step, we solve the two sets of Volterra equations by using the trapezoidal rule of integral calculation.

Step 3: Calculate the values of \(\hat{E}(\Upsilon,\bar{\omega})\) and \(\hat{F}(\Upsilon,\bar{\omega})\)

We need to compute these functions at one point \(\Upsilon,\bar{\omega}\), which can be done by approximation of the integrals using the trapezoidal rule:

\[ \begin{align}\begin{aligned}\hat{E}(\Upsilon,\bar{\omega}) = \frac{1}{2} \sum_{i=1}^k(\underline{w}_{n,i}\underline{\epsilon}_i + \underline{w}_{n,i-1}\underline{\epsilon}_{i-1} + \bar{w}_{n,i}\bar{\epsilon}_i + \bar{w}_{n,i-1}\bar{\epsilon}_{i-1})(\upsilon_i - \upsilon_{i-1})\\\hat{F}(\Upsilon,\bar{\omega}) = \frac{1}{2} \sum_{i=1}^k(\underline{w}_{n,i}\underline{\phi}_i + \underline{w}_{n,i-1}\underline{\phi}_{i-1} + \bar{w}_{n,i}\bar{\phi}_i + \bar{w}_{n,i-1}\bar{\phi}_{i-1})(\upsilon_i - \upsilon_{i-1})\end{aligned}\end{align} \]

Where \(w\) are the weights.

Step 4: Calculate the SR using the obtained values

The previously computed functions \(\hat{E}(\Upsilon,\bar{\omega})\) and \(\hat{F}(\Upsilon,\bar{\omega})\) are substituted into the following formula to calculate the Sharpe ratio.

\[SR = \frac{\hat{E}(\Upsilon,\bar{\omega}) - \frac{2 (\bar{\omega}-\theta)}{ln(1-2\Upsilon)}}{\sqrt{\hat{F}(\Upsilon,\bar{\omega}) - (\hat{E}(\Upsilon,\bar{\omega}))^2 + \frac{4(\Upsilon + ln(1-2\Upsilon)(\bar{\omega}+\theta)\hat{E}(\Upsilon,\bar{\omega})}{(ln(1-2\Upsilon))^2}}}\]

To find the optimal thresholds for the data provided by the user we maximize the calculated SR with respect to \(\bar{\pi}\geq0,\underline{\pi}\leq0\) :

\[{\bar{\pi}*,\underline{\pi}*}=\underset{\bar{\pi}\geq0,\underline{\pi}\leq0}{\arg\max}\ SR\]

The HeatPotentials module of the ArbitrageLab package allows the user to calculate the threshold levels that establish the trading rule for their data, the provided parameters are transformed to a steady-state solutions internally and the reverse transformation is performed for the optimal threshold values.

Implementation

First of all the user has to use fit for the parameters to be scaled to remove superfluous parameters, and set up the delta for the grid calculation and maximum duration of the trade.

class HeatPotentials

This class implements the algorithm for finding profit-taking and stop-loss levels for p/l that follow an Ornstein-Uhlenbeck process using the heat potential method. The following implementation is based on the work of Alexandr Lipton and Marcos Lopez de Prado “A closed-form solution for optimal mean-reverting trading strategies”<https://ssrn.com/abstract=3534445>`_

__init__(): Initializes parameters

HeatPotentials.fit(ou_params: list, delta_grid: float, max_trade_duration: float | None = None)

Fits the steady-state distribution to the given OU model, assigns the grid density with respect to t, maximum duration of the trade. Calculates and assigns values for optimal stop-loss and profit-taking levels.

Parameters:

ou_params – (list) Parameters of the OU model. [theta, mu, sigma]
delta_grid – (float) Grid density with respect to t.
max_trade_duration – (float) Maximum duration of the trade.

To separately perform the optimization process we use the optimal_levels function.

HeatPotentials.optimal_levels() → list

Calculates optimal profit-taking and stop-loss levels by maximizing the Sharpe ratio.

Returns:: (list) The list that consists of profit-taking level, stop-loss level, max Sharpe values.

There is also a possibility to calculate the Sharpe ratio for chosen optimal levels and the maximum duration of the trade of choice.

HeatPotentials.sharpe_calculation(max_trade_duration: float, optimal_profit: float, optimal_stop_loss: float) → float

Calculates the Sharpe ratio. (p.6 )

Parameters:

max_trade_duration – (float) Maximum duration of the trade.
optimal_profit – (float) Optimal profit-taking threshold.
optimal_stop_loss – (float) Optimal stop-loss level.

Returns:

(float) Sharpe ratio.

To view the optimal levels scaled back to initial parameters the description function is used.

HeatPotentials.description() → Series

Returns the statistics of the model. The optimal thresholds are converted back to the terms of the original data.

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

Example

# Importing modules
from arbitragelab.optimal_mean_reversion.ou_model import OrnsteinUhlenbeck
from arbitragelab.optimal_mean_reversion import HeatPotentials
import numpy as np

# Generating the sample OU data
ou_data = OrnsteinUhlenbeck()

data = ou_data.ou_model_simulation(n=1000, theta_given=0.03711, mu_given=65.3333,
                                   sigma_given=0.3, delta_t_given=1/255)

# To get the model parameters we need to fit the OU model to the data.

# Assign the delta value
ou_data.delta_t = 1/252

# Model fitting
ou_data.fit_to_portfolio(data)

# Now we obtained the parameters to use for our optimization procedure
theta,mu,sigma = ou_data.theta, ou_data.mu, np.sqrt(ou_data.sigma_square)

# Establish the instance of the class
example = HeatPotentials()

# Fit the model and establish the maximum duration of the trade
example.fit(ou_params=(theta, mu, sigma), delta_grid=0.1, max_trade_duration=0.03)

# Calculate the initial optimal levels
levels = example.optimal_levels()
print(levels)

# We can also calculate the Sharpe ratio for given scaled parameters
sr = example.sharpe_calculation(max_trade_duration=1.9599, optimal_profit=5.07525, optimal_stop_loss=-3.41002)
print(sr)

# To get the results scaled back to our initial model we call the description function
example.description()

Research Notebook

The following research notebook can be used to better understand the concepts of the Heat Potentials module.

Heat Potentials