Reinforcement Learning (RL) aims at the automatic acquisition of skills or some other form of intelligence, to behave appropriately and wisely in comparable situations and potentially on situations that are slightly different from the ones seen in training. When it comes to real world situations, there are two challenges: having a data-efficient learning method and being able to handle complex and unknown dynamical systems that can be difficult to model and are too far away from the systems observed during the training phase. Because the dynamic nature of the environment may be challenging to learn, a first stream of RL methods has consisted in modeling the environment with a model. Hence it is called model-based RL. Model-based methods tend to excel in learning complex environments like financial markets. In mainstream agents literature, examples include, robotics applications, where it is highly desirable to learn using the lowest possible number of real-world trails Kaelbling_1996. It is also used in finance where there are a lot of regime changes Freitas_2009; Niaki2013; Heaton_2017. A first generation of model-based RL, relying on Gaussian processes and time-varying linear dynamical systems, provides excellent performance in low-data regimes deisenroth2011learning; deisenroth2011pilco; deisenroth2014gaussian; Levine_Koltun_2013; Kumar_2016. A second generation, leveraging deep networks Gal_2016; Depeweg_2016; Nagabandi_2018
, has emerged and is based on the fact that neural networks offer high-capacity function approximators even in domains with high-dimensional observationsOh_2015; Ebert_2018; Kaiser_2019 while retaining some sample efficiency of a model-based approach. Recently, it has been proposed to adapt model-based RL via meta policy optimization to achieve asymptotic performance of model-free models Clavera_2018. For a full survey of model-based RL model, please refer to moerland2020modelbased. In finance, it is common to scale portfolio’s allocations based on volatility and correlation as volatility is known to be a good proxy for the level of risk and correlation a standard measure of dependence. It is usually referred as volatility targeting. It enables the portfolio under consideration to achieve close to constant volatility through various market dynamics or regimes by simply sizing the portfolio’s constituents according to volatility and correlation forecasts.
In contrast, the model-free approach aims to learn the optimal actions blindly without a representation of the environment dynamics. Works like Mnih_2015; Lillicrap_2016; Haarnoja_2018 have come with the promise that such models learn from raw inputs (and raw pixels) regardless of the game and provide some exciting capacities to handle new situations and environments, though at the cost of data efficiency as they require millions of training runs.
Hence, it is not surprising that the research community has focused on a new generation of models combining model-free and model-based RL approaches. A first idea has been to combine model-based and model-free updates for Trajectory-Centric RL. Chebotar_2017. Another idea has been to use temporal difference models to have a model-free deep RL approach for model-based control Pong_2018. van_Hasselt_2018
answers the question of when to use parametric models in reinforcement learning. Likewise,Janner_2019 gives some hints when to trust model-based policy optimization versus model-free. Feinberg_2018
shows how to use model-based value estimation for efficient model-free RL.
All these studies, mostly applied to robotics and virtual environments, have not hitherto been widely used for financial time series. Our aim is to be able to distinguish various financial models that can be read or interpreted as model-based RL methods. These models aim at predicting volatility in financial markets in the context of portfolio allocation according to volatility target methods. These models are quite diverse and encompass statistical models based on historical data such as simple and naive moving average models, multivariate generalized auto-regressive conditional heteroskedasticity (GARCH) models, high-frequency based volatility models (HEAVY) Noureldin12multivariatehigh-frequency-based and forward-looking models such as implied volatility or PCA decomposition of implied volatility indices. To be able to decide on an allocation between these various models, we rely on deep model-free RL. However, using just the last data points does not work in our cases as the various volatility models have very similar behaviors. Following Benhamou2020bridging and Benhamou2021knowledge, we also add contextual information like macro signals and risk appetite indices to include additional information in our DRL agent hereby allowing us to choose the pre-trained models that are best suited for a given environment.
1.1. Related works
The literature on portfolio allocation in finance using either supervised or reinforcement learning has been attracting more attention recently. Initially, Freitas_2009; Niaki2013; Heaton_2017 use deep networks to forecast next period prices and to use this prediction to infer portfolio allocations. The challenge of this approach is the weakness of predictions: financial markets are well known to be non-stationary and to present regime changes (see Salhi_2016; Dias_2015; benhamou2018trend; Zheng_2019).
More recently, Jiang_2016; Zhengyao_2017; Liang_2018; Yu_2019; Wang_2019; Liu_2020; Ye_2020; Li_2019; Xiong_2018; Benhamou2020detecting; Benhamou2020time; Benhamou2021knowledge; Benhamou2020bridging
have started using deep reinforcement learning to do portfolio allocation. Transaction costs can be easily included in the rules. However, these studies rely on very distinct time series, which is a very different setup from our specific problem. They do not combine a model-based with a model-free approach. In addition, they do not investigate how to rank features, which is a great advantage of methods in ML like decision trees. Last but not least, they never test the statistical difference between the benchmark and the resulting model.
Our contributions are precisely motivated by the shortcomings presented in the aforementioned remarks. They are four-fold:
The use of model-free RL to select various models that can be interpreted as model-based RL. In a noisy and regime-changing environment like financial time series, the practitioners’ approach is to use a model to represent the dynamics of financial markets. We use a model-free approach to learn from states to actions and hence distinguish between these initial models and choose which model-based RL to favor. In order to augment states, we use additional contextual information.
The walk-forward procedure. Because of the non stationary nature of time-dependent data, and especially financial data, it is crucial to test DRL model stability. We present a traditional methodology in finance but never used to our knowledge in DRL model evaluation, referred to as walk-forward analysis that iteratively trains and tests models on extending data sets. This can be seen as the analogy of cross-validation for time series. This allows us to validate that the selected hyper-parameters work well over time and that the resulting models are stable over time.
Features sensitivity procedure.
Inspired by the concept of feature importance in gradient boosting methods, we have created a feature importance of our deep RL model based on its sensitivity to features inputs. This allows us to rank each feature at each date to provide some explanations why our DRL agent chooses a particular action.
A statistical approach to test model stability. Most RL papers do not address the statistical difference between the obtained actions and predefined baselines or benchmarks. We introduce the concept of statistical difference as we want to validate that the resulting model is statistically different from the baseline results.
2. Problem formulation
Asset allocation is a major question for the asset management industry. It aims at finding the best investment strategy to balance risk versus reward by adjusting the percentage invested in each portfolio’s asset according to risk tolerance, investment goals and horizons.
Among these strategies, volatility targeting is very common. Volatility targeting forecasts the amount to invest in various assets based on their level of risk to target a constant and specific level of volatility over time. Volatility acts as a proxy for risk. Volatility targeting relies on the empirical evidence that a constant level of volatility delivers some added value in terms of higher returns and lower risk materialized by higher Sharpe ratios and lower drawdowns, compared to a buy and hold strategy Hocquard_2013; Perchet_2016; Dreyer_2017. Indeed it can be shown that Sharpe ratio makes a lot of sense for manager to measure their performance. The distribution of Sharpe ratio can be computed explicitly benhamou2019connecting. Sharpe ratio is not an accident and is a good indicator of manager performance benhamou2019testing. It can also be related to other performance measures like Omega ratio benhamou2019omega and other performance ratios benhamou2018incremental. It also relies on the fact that past volatility largely predicts future near-term volatility, while past returns do not predict future returns. Hence, volatility is persistent, meaning that high and low volatility regimes tend to be followed by similar high and low volatility regimes. This evidence can be found not only in stocks, but also in bonds, commodities and currencies. Hence, a common model-based RL approach for solving the asset allocation question is to model the dynamics of the future volatility.
To articulate the problem, volatility is defined as the standard deviation of the returns of an asset. Predicting volatility can be done in multiple ways:
Moving average: this model predicts volatility based on moving averages.
Level shift: this model is based on a two-step approach that allows the creation of abrupt jumps, another stylized fact of volatility.
GARCH: a generalized auto-regressive conditional heteroske-dasticity model assumes that the return can be modeled by a time series where is the expected return and
is a zero-mean white noise, and, where . The parameters are estimated simultaneously by maximizing the log-likelihood.
GJR-GARCH: the Glosten-Jagannathan-Runkle GARCH (GJR-GARCH) model is a variation of the GARCH model (see Glosten_1993) with the difference that
, the variance of the white noise, is modelled as: where if and 0 otherwise. The parameters are estimated simultaneously by maximizing the log-likelihood.
HEAVY: the HEAVY model utilizes high-frequency data for the objective of multi-step volatility forecasting Noureldin12multivariatehigh-frequency-based.
HAR: this model is an heterogeneous auto-regressive (HAR) model that aims at replicating how information actually flows in financial markets from long-term to short-term investors.
Adjusted TYVIX: this model uses the TYVIX index to forecast volatility in the bond future market,
RM2006: RM2006 uses a volatility forecast derived from an exponentially weighted moving average (EWMA) metric.
2.1. Mathematical formulation
We have models. Each model predicts a volatility for the rolled U.S. 10-year
note future contract that we shall call "bond future" in the remainder of this paper.
The bond future’s daily returns are denoted by and typically range from -2
to 2 percents with a daily average value of a few basis points and a daily standard
deviation 10 to 50 times higher and ranging from 20 to 70 basis points. By these standards,
the bond future’s market is hard to predict and has a lot of noise making its forecast a
difficult exercise. Hence, using some volatility forecast to scale position makes a lot of sense.
These forecasts are then used to compute the allocation to the bond future’s models. Mathematically,
if the target volatility of the strategy is denoted by and if the
model predicts a bond future’s volatility , based on information
up to , the allocation in the bond future’s model at time is given
by the ratio between the target volatility and the predicted volatility: .
Hence, we can compute the daily amounts invested in each of the bond future volatility models and create a corresponding time series of returns , consisting of investing in the bond future according to the allocation computed by the volatility targeting model . This provides time series of compounded returns whose values are given by . Our RL problem then boils down to selecting the optimal portfolio allocation (with respect to the cumulative reward) in each model-based RL strategies such that the portfolio weights sum up to one and are non-negative and for any . These allocations are precisely the continuous actions of the DRL model. This is not an easy problem as the different volatility forecasts are quite similar. Hence, the time series of compounded returns look almost the same, making this RL problem non-trivial. Our aim is, in a sense, to distinguish between the indistinguishable strategies that are presented in figure 1. More precisely, figure 1 provides the evolution of the net value of an investment strategy that follows the different volatility targeting models.
Compared to standard portfolio allocation problems, these strategies’ returns are highly correlated and similar as presented by the correlation matrix 2, with a lowest correlation of 97%. The correlation is computed as the Pearson correlation over the full data set from 2004 to 2020.
, we formulate this RL problem as a Markov Decision Process (MDP) problem. We define our MDP with a 6-tuplewhere is the (possibly infinite) decision horizon, the discount factor, the state space, the action space,
the transition probability from the stateto given that the agent has chosen the action , and the reward for a state and an action .
The agent’s objective is to maximize its expected cumulative returns, given the start of the distribution. If we denote by the policy mapping specifying the action to choose in a particular state, , the agent wants to maximize the expected cumulative returns. This is written as: .
MDP assumes that we know all the states of the environment and have all the information to make the optimal decision in every state.
From a practical standpoint, there are a few limitations to accommodate. First of all, the Markov property implies that knowing the current state is sufficient. Hence, we modify the RL setting by taking a pseudo state formed with a set of past observations . The trade-off is to take enough past observations to be close to a Markovian status without taking too many observations which would result in noisy states.
In our settings, the actions are continuous and consist in finding at time the portfolio allocations in each volatility targeting model. We denote by
the portfolio weights vector.
Likewise, we denote by the closing price vector, and by the price relative difference vector, where denotes the element-wise division,
and by the returns vector which is also the percentage change of each closing prices . Due to price change in the market, at the end of the same period, the weights evolve according to where is the element-wise multiplication, and the scalar product.
The goal of the agent at time is hence to reallocate the portfolio vector from to by buying and selling the relevant assets, taking into account the transaction costs that are given by where is the percentage cost for a transaction (which is quite low for future markets and given by 1 basis point) and is the norm operator. Hence at the end of time , the agent receives a portfolio return given by . The cumulative reward corresponds to the sum of the logarithmic returns of the portfolio strategy given by
, which is easier to process in a tensor flow graph as a log sum expression and is naturally given by.
Actions are modeled by a multi-input, multi-layer convolution network whose details are given by Figure 5. It has been shown that convolution networks are better for selecting features in portfolio allocation problem Benhamou_DRPLECML, Benhamou2020detecting and Benhamou2020time. The goal of the model-free RL method is to find the network parameters. This is done by an adversarial policy gradient method summarized by the algorithm 1 using traditional Adam optimization so that we have the benefit of adaptive gradient descent with root mean square propagation kingma2014method with a learning rate of 1% and a number of iteration steps of 100,000 with an early stop criterion if the cumulative reward does not improve after 15 full episodes. Because each episode is run on the same financial data, we use on purpose a vanilla policy gradient algorithm to take advantage of the stability of the environment rather than to use more advanced DRL agents like TRPO, DDPG or TD3 that would add on top of our model free RL layer some extra complexity and noise.
In order to benchmark our DRL approach, we need to compare to traditional financial methods. Markowitz allocation as presented in Markowitz_1952 is a widely-used benchmark in portfolio allocation as it is a straightforward and intuitive mix between performance and risk. In this approach, risk is represented by the variance of the portfolio. Hence, the Markowitz portfolio minimizes variance for a given expected return, which is solved by standard quadratic programming optimization. If we denote by the expected returns for our model strategies and by the covariance matrix of these strategies’ returns, and by the targeted minimum return, the Markowitz optimization problem reads
Another classical benchmark model for indistinguishable strategies, is the arithmetic average of all the volatility targeting models. This seemingly naive benchmark is indeed performing quite well as it mixes diversification and mean reversion effects.
2.2.3. Follow the winner
Another common strategy is to select the best performer of the past year, and use it the subsequent year. It replicates the standard investor’s behavior that selects strategies that have performed well in the past.
2.3. Procedure and walk forward analysis
The whole procedure is summarized by Figure 3. We have models that represent the dynamics of the market volatility. We then add the volatility and the contextual information to the states, thereby yielding augmented states. The latter procedure is presented as the second step of the process. We then use a model-free RL approach to find the portfolio allocation among the various volatility targeting models, corresponding to steps 3 and 4. In order to test the robustness of our resulting DRL model, we introduce a new methodology called walk forward analysis.
2.3.1. Walk forward analysis
In machine learning, the standard approach is to do-fold cross-validation. This approach breaks the chronology of data and potentially uses past data in the test set. Rather, we can take sliding test set and take past data as training data. To ensure some stability, we favor to add incrementally new data in the training set, at each new step.
This method is sometimes referred to as "anchored walk forward" as we have anchored training data. Finally, as the test set is always after the training set, walk forward analysis gives less steps compared with cross-validation. In practice, and for our given data set, we train our models from 2000 to the end of 2013 (giving us at least 14 years of data) and use a repetitive test period of one year from 2014 onward. Once a model has been selected, we also test its statistical significance, defined as the difference between the returns of two time series. We therefore do a T-test to validate how different these time series are. The whole process is summarized by Figure 4.
2.3.2. Model architecture
The states consist in two different types of data: the asset inputs and the contextual inputs.
Asset inputs are a truncated portion of the time series of financial returns of the volatility targeting models and of the volatility of these returns computed over a period of 20 observations. So if we denote by the returns of model at time , and by the standard deviation of returns over the last periods, asset inputs are given by a 3-D tensor denoted by , with
This setting with two layers (past returns and past volatilities) is very different from the one presented in Jiang_2016; Zhengyao_2017; Liang_2018 that uses layers representing open, high, low and close prices, which are not necessarily available for volatility target models. Adding volatility is crucial to detect regime change and is surprisingly absent from these works.
Contextual inputs are a truncated portion of the time series of additional data that represent contextual information. Contextual information enables our DRL agent to learn the context, and are, in our problem, short-term and long-term risk appetite indices and short-term and long-term macro signals. Additionally, we include the maximum and minimum portfolio strategies return and the maximum portfolio strategies volatility. Similarly to asset inputs, standard deviations is useful to detect regime changes. Contextual observations are stored in a 2D matrix denoted by with stacked past individual contextual observations. The contextual state reads
2.3.3. Features sensitivity analysis
One of the challenges of neural networks relies in the difficulty to provide explainability about their behaviors. Inspired by computer vision, we present a methodology here that enables us to relate features to action. This concept is based on features sensitivity analysis. Simply speaking, our neural network is a multi-variate function. Its inputs include all our features, strategies, historical performances, standard deviations, contextual information, short-term and long-term macro signals and risk appetite indices. We denote these inputs by, which lives in where is the number of features. Its outputs are the action vector , which is an -d array with elements between 0 and 1. This action vector lives in an image set denoted by , which is a subset of . Hence, the neural network is a function with . In order to project the various partial derivatives, we take the L1 norm (denoted by ) of the different partial derivatives as follows: . The choice of the L1 norm is arbitrary but is intuitively motivated by the fact that we want to scale the distance of the gradient linearly.
In order to measure the sensitivity of the outputs, simply speaking, we change the initial feature by its mean value over the last periods. This is inspired by a "what if" analysis where we would like to measure the impact of changing the feature from its mean value to its current value. In computer vision, the practice is not to use the mean value but rather to switch off the pixel and set it to the black pixel. In our case, using a zero value would not be relevant as this would favor large features. We are really interested here in measuring the sensitivity of our actions when a feature deviates from its mean value.
The resulting value is computed numerically and provides us for each feature a feature importance. We rank these features importance and assign arbitrarily the value to the largest and to the lowest. This provides us with the following features importance plot given below 6. We can notice that the HAR returns and volatility are the most important features, followed by various returns and volatility for the TYVIX model. Although returns and volatility are dominating among the most important features, macro signals 0d observations comes as the 12th most important feature over 70 features with a very high score of 84.2. The features sensitivity analysis confirms two things: i) it is useful to include volatility features as they are good predictors of regime changes, ii) contextual information plays a role as illustrated by the macro signal.
3. Out of sample results
In this section, we compare the various models: the deep RL model (DRL1) using states with contextual inputs and standard deviation, the deep RL model without contextual inputs and standard deviation (DRL2), the average strategy, the Markowitz portfolio and the "the winner" strategy. The results are the combination of the 7 distinct test periods: each year from 2014 to 2020. The resulting performance is plotted in Figure 7. We notice that the deepRL model with contextual information and standard deviation is substantially higher than the other models in terms of performance as it ends at 157, whereas other models (the deepRL with no context, the average, the Markowitz and "the winner" model) end at 147.6, 147.8, 145.5, 143.4 respectively.
3.1. Results description
3.1.1. Risk metrics
We provide various statistics in Table 1 for different time horizons: 1, 3 and 5 years. For each horizon, we put the best model, according to the column’s criterion, in bold. The Sharpe and Sortino ratios are computed on daily returns. Maximum drawdown (written as mdd in the table), which is the maximum observed loss from a peak to a trough for a portfolio, is also computed on daily returns. DRL1 is the DRL model with standard deviations and contextual information, while DRL2 is a model with no contextual information and no standard deviation. Overall, DLR1, the DRL model with contextual information and standard deviation, performs better for 1, 3 and 5 years except for three-year maximum drawdown. Globally, it provides a 1% increase in annual net return for a 5-year horizon. It also increases the Sharpe ratio by 0.1 and is able to reduce most of the maximum drawdowns except for the 3-year period. Markowitz portfolio selection and "the winner" strategy, which are both traditional financial methods heavily used by practitioners, do not work that well compared with a naive arithmetic average and furthermore when compared to the DRL model with context and standard deviation inputs. A potential explanation may come from the fact that these volatility targeting strategies are very similar making the diversification effects non effective.
|DRL1||22.659||2.169||2.419||- 6.416||- 0.614|
|DRL2||20.712||2.014||2.167||- 6.584||- 0.640|
|Average||20.639||2.012||2.166||- 6.560||- 0.639|
|Markowitz||19.370||1.941||2.077||- 6.819||- 0.683|
|Winner||17.838||1.910||2.062||- 6.334||- 0.678|
|DRL1||8.056||0.835||0.899||- 17.247||- 1.787|
|DRL2||7.308||0.783||0.834||- 16.912||- 1.812|
|Average||7.667||0.822||0.876||- 16.882||- 1.810|
|Markowitz||7.228||0.828||0.891||- 16.961||- 1.869|
|Winner||6.776||0.712||0.754||- 17.770||- 1.867|
|DRL1||6.302||0.651||0.684||- 19.794||- 2.044|
|DRL2||5.220||0.565||0.584||- 20.211||- 2.187|
|Average||5.339||0.579||0.599||- 20.168||- 2.187|
|Markowitz||4.947||0.569||0.587||- 19.837||- 2.074|
|Winner||4.633||0.508||0.526||- 19.818||- 2.095|
3.1.2. Statistical significance
Following our methodology described in 4
, once we have computed the results for the various walk-forward test periods, we do a T-statistic test to validate the significance of the result. Given two models, we test the null hypothesis that the difference of the returns running average (computed asfor various times ) between the two models is equal to 0. We provide the T-statistic and, in parenthesis, the p-value. We take a p-value threshold of 5%, and put the cases where we can reject the null hypothesis in bold in table 2. Hence, we conclude that the DRL model with context (DRL1) model is statistically different from all other models. These results on the running average are quite intuitive as we are able to distinguish the DRL1 model curve from all other curves in Figure 7. Interestingly, we can see that the DRL model without context (DRL2) is not statically different from a pure averaging of the average model that consists in averaging allocation computed by model-based RL approaches.
|DRL1||72.1 (0%)||14 (0%)||44.1 (0%)||79.8 (0%)|
|DRL2||1.2 (22.3%)||24.6 (0%)||10 (0%)|
3.1.3. Results discussion
It is interesting to understand how the DRL model achieves such a performance as it provides an amazing additional 1% annual return over 5 years, and an increase in Sharpe ratio of 0.10. This is done simply by selecting the right strategies at the right time. This helps us to confirm that the adaptive learning thanks to the model free RL is somehow able to pick up regime changes. We notice that the DRL model selects the GARCH model quite often and, more recently, the HAR and HEAVY model (Figure 8). When targeting a given volatility level, capital weights are inversely proportional to the volatility estimates. Hence, lower volatility estimates give higher weights and in a bullish market give higher returns. Conversely, higher volatility estimates drive capital weights lower and have better performance in a bearish market. The allocation of these models evolve quite a bit as shown by Figure 10, which plots the rank of the first 5 models.
We can therefore test if the DRL model has a tendency to select volatility targeting models that favor lower volatility estimates. If we plot the occurrence of rank by dominant model for the DRL model, we observe that the DRL model selects the lowest volatility estimate model quite often (38.2% of the time) but also tends to select the highest volatility models giving a U shape to the occurrence of rank as shown in figure 11. This U shape confirms two things: i) the model has a tendency to select either the lowest or highest volatility estimates models, which are known to perform best in bullish markets or bearish markets (however, it does not select these models blindly as it is able to time when to select the lowest or highest volatility estimates); ii) the DRL model is able to reduce maximum drawdowns while increasing net annual returns as seen in Table 1. This capacity to simultaneously increase net annual returns and decrease maximum drawdowns indicates a capacity to detect regime changes. Indeed, a random guess would only increase the leverage when selecting lowest volatility estimates, thus resulting in higher maximum drawdowns.
3.2. Benefits of DRL
The advantages of context based DRL are numerous: (i) by design, DRL directly maps market conditions to actions and can thus adapt to regime changes, (ii) DRL can incorporate additional data and be a multi-input method, as opposed to more traditional optimization methods.
3.3. Future work
As nice as this may look, there is room for improvement as more contextual data and architectural networks choices could be tested as well as other DRL agents like DDPG, TRPO or TD3. It is also worth mentioning that the analysis has been conducted on a single financial instrument and a relatively short out-of-sample period. Expanding this analysis further in the past would cover more various regimes (recessions, inflationary, growth, etc.) and potentially improve the statistical relevance of this study at the cost of losing relevance for more recent data. Another lead consists of applying the same methodology to a much wider ensemble of securities and identify specific statistical features based on distinct geographic and asset sectors.
In this work, we propose to create an adaptive learning method that combines model-based and model-free RL approaches to address volatility regime changes in financial markets. The model-based approach enables to capture efficiently the volatility dynamics while the model-free RL approach to time when to switch from one to another model. This combination enables us to have an adaptive agent that switches between different dynamics. We strengthen the model-free RL step with additional inputs like volatility and macro and risk appetite signals that act as contextual information. The ability of this method to reduce risk and profitability are verified when compared to the various financial benchmarks. The use of successive training and testing sets enables us to stress test the robustness of the resulting agent. Features sensitivity analysis confirms the importance of volatility and contextual variables and explains in part the DRL agent’s better performance. Last but not least, statistical tests validate that results are statistically significant from a pure averaging method of all model-based RL allocations.