In accordance with various embodiments of the present disclosure, the methods described herein may be implemented using a hardware computer system that executes software programs. Further, in an exemplary, non-limited embodiment, implementations can include distributed processing, component/object distributed processing, and parallel processing. Virtual computer system processing can be constructed to implement one or more of the methods or functionalities as described herein, and a processor described herein may be used to support a virtual processing environment.

As described herein, various embodiments provide optimized methods and systems for dynamic request for quotation (RFQ) pricing using reinforcement learning and symbolic regression in order to obtain a pseudo-optimal nonlinear state-feedback controller for tracking a metric with increased interpretability.

Referring to FIG. 2, a schematic of an exemplary network environment 200 for implementing a method for dynamic RFQ pricing using reinforcement learning and symbolic regression in order to obtain a pseudo-optimal nonlinear state-feedback controller for tracking a metric with increased interpretability is illustrated. In an exemplary embodiment, the method is executable on any networked computer platform, such as, for example, a personal computer (PC).

The method for dynamic RFQ pricing using reinforcement learning and symbolic regression in order to obtain a pseudo-optimal nonlinear state-feedback controller for tracking a metric with increased interpretability may be implemented by a Dynamic Request For Quotation Pricing (DRFQP) device 202. The DRFQP device 202 may be the same or similar to the computer system 102 as described with respect to FIG. 1. The DRFQP device 202 may store one or more applications that can include executable instructions that, when executed by the DRFQP device 202, cause the DRFQP device 202 to perform actions, such as to transmit, receive, or otherwise process network messages, for example, and to perform other actions described and illustrated below with reference to the figures. The application(s) may be implemented as modules or components of other applications. Further, the application(s) can be implemented as operating system extensions, modules, plugins, or the like.

Even further, the application(s) may be operative in a cloud-based computing environment. The application(s) may be executed within or as virtual machine(s) or virtual server(s) that may be managed in a cloud-based computing environment. Also, the application(s), and even the DRFQP device 202 itself, may be located in virtual server(s) running in a cloud-based computing environment rather than being tied to one or more specific physical network computing devices. Also, the application(s) may be running in one or more virtual machines (VMs) executing on the DRFQP device 202. Additionally, in one or more embodiments of this technology, virtual machine(s) running on the DRFQP device 202 may be managed or supervised by a hypervisor.

In the network environment 200 of FIG. 2, the DRFQP device 202 is coupled to a plurality of server devices 204(1)-204(n) that hosts a plurality of databases 206(1)-206(n), and also to a plurality of client devices 208(1)-208(n) via communication network(s) 210. A communication interface of the DRFQP device 202, such as the network interface 114 of the computer system 102 of FIG. 1, operatively couples and communicates between the DRFQP device 202, the server devices 204(1)-204(n), and/or the client devices 208(1)-208(n), which are all coupled together by the communication network(s) 210, although other types and/or numbers of communication networks or systems with other types and/or numbers of connections and/or configurations to other devices and/or elements may also be used.

The communication network(s) 210 may be the same or similar to the network 122 as described with respect to FIG. 1, although the DRFQP device 202, the server devices 204(1)-204(n), and/or the client devices 208(1)-208(n) may be coupled together via other topologies. Additionally, the network environment 200 may include other network devices such as one or more routers and/or switches, for example, which are well known in the art and thus will not be described herein. This technology provides a number of advantages including methods, non-transitory computer readable media, and DRFQP devices that efficiently implement a method for dynamic RFQ pricing using reinforcement learning and symbolic regression in order to obtain a pseudo-optimal nonlinear state-feedback controller for tracking a metric with increased interpretability.

By way of example only, the communication network(s) 210 may include local area network(s) (LAN(s)) or wide area network(s) (WAN(s)), and can use TCP/IP over Ethernet and industry-standard protocols, although other types and/or numbers of protocols and/or communication networks may be used. The communication network(s) 210 in this example may employ any suitable interface mechanisms and network communication technologies including, for example, teletraffic in any suitable form (e.g., voice, modem, and the like), Public Switched Telephone Network (PSTNs), Ethernet-based Packet Data Networks (PDNs), combinations thereof, and the like.

The DRFQP device 202 may be a standalone device or integrated with one or more other devices or apparatuses, such as one or more of the server devices 204(1)-204(n), for example. In one particular example, the DRFQP device 202 may include or be hosted by one of the server devices 204(1)-204(n), and other arrangements are also possible. Moreover, one or more of the devices of the DRFQP device 202 may be in a same or a different communication network including one or more public, private, or cloud networks, for example.

The plurality of server devices 204(1)-204(n) may be the same or similar to the computer system 102 or the computer device 120 as described with respect to FIG. 1, including any features or combination of features described with respect thereto. For example, any of the server devices 204(1)-204(n) may include, among other features, one or more processors, a memory, and a communication interface, which are coupled together by a bus or other communication link, although other numbers and/or types of network devices may be used. The server devices 204(1)-204(n) in this example may process requests received from the DRFQP device 202 via the communication network(s) 210 according to the HTTP-based and/or JavaScript Object Notation (JSON) protocol, for example, although other protocols may also be used.

The server devices 204(1)-204(n) may be hardware or software or may represent a system with multiple servers in a pool, which may include internal or external networks. The server devices 204(1)-204(n) hosts the databases 206(1)-206(n) that are configured to store historical RFQ pricing data and data that relates to tracking metrics for optimizing RFQ pricing.

Although the server devices 204(1)-204(n) are illustrated as single devices, one or more actions of each of the server devices 204(1)-204(n) may be distributed across one or more distinct network computing devices that together comprise one or more of the server devices 204(1)-204(n). Moreover, the server devices 204(1)-204(n) are not limited to a particular configuration. Thus, the server devices 204(1)-204(n) may contain a plurality of network computing devices that operate using a master/slave approach, whereby one of the network computing devices of the server devices 204(1)-204(n) operates to manage and/or otherwise coordinate operations of the other network computing devices.

The server devices 204(1)-204(n) may operate as a plurality of network computing devices within a cluster architecture, a peer-to peer architecture, virtual machines, or within a cloud architecture, for example. Thus, the technology disclosed herein is not to be construed as being limited to a single environment and other configurations and architectures are also envisaged.

The plurality of client devices 208(1)-208(n) may also be the same or similar to the computer system 102 or the computer device 120 as described with respect to FIG. 1, including any features or combination of features described with respect thereto. For example, the client devices 208(1)-208(n) in this example may include any type of computing device that can interact with the DRFQP device 202 via communication network(s) 210. Accordingly, the client devices 208(1)-208(n) may be mobile computing devices, desktop computing devices, laptop computing devices, tablet computing devices, virtual machines (including cloud-based computers), or the like, that host chat, e-mail, or voice-to-text applications, for example. In an exemplary embodiment, at least one client device 208 is a wireless mobile communication device, i.e., a smart phone.

The client devices 208(1)-208(n) may run interface applications, such as standard web browsers or standalone client applications, which may provide an interface to communicate with the DRFQP device 202 via the communication network(s) 210 in order to communicate user requests and information. The client devices 208(1)-208(n) may further include, among other features, a display device, such as a display screen or touchscreen, and/or an input device, such as a keyboard, for example.

Although the exemplary network environment 200 with the DRFQP device 202, the server devices 204(1)-204(n), the client devices 208(1)-208(n), and the communication network(s) 210 are described and illustrated herein, other types and/or numbers of systems, devices, components, and/or elements in other topologies may be used. It is to be understood that the systems of the examples described herein are for exemplary purposes, as many variations of the specific hardware and software used to implement the examples are possible, as will be appreciated by those skilled in the relevant art(s).

One or more of the devices depicted in the network environment 200, such as the DRFQP device 202, the server devices 204(1)-204(n), or the client devices 208(1)-208(n), for example, may be configured to operate as virtual instances on the same physical machine. In other words, one or more of the DRFQP device 202, the server devices 204(1)-204(n), or the client devices 208(1)-208(n) may operate on the same physical device rather than as separate devices communicating through communication network(s) 210. Additionally, there may be more or fewer DRFQP devices 202, server devices 204(1)-204(n), or client devices 208(1)-208(n) than illustrated in FIG. 2.

In addition, two or more computing systems or devices may be substituted for any one of the systems or devices in any example. Accordingly, principles and advantages of distributed processing, such as redundancy and replication also may be implemented, as desired, to increase the robustness and performance of the devices and systems of the examples. The examples may also be implemented on computer system(s) that extend across any suitable network using any suitable interface mechanisms and traffic technologies, including by way of example only teletraffic in any suitable form (e.g., voice and modem), wireless traffic networks, cellular traffic networks, Packet Data Networks (PDNs), the Internet, intranets, and combinations thereof.

The DRFQP device 202 is described and illustrated in FIG. 3 as including a dynamic request for quotation (RFQ) pricing module 302, although it may include other rules, policies, modules, databases, or applications, for example. As will be described below, the dynamic RFQ pricing module 302 is configured to implement a method for dynamic RFQ pricing using reinforcement learning and symbolic regression in order to obtain a pseudo-optimal nonlinear state-feedback controller for tracking a metric with increased interpretability.

An exemplary process 300 for implementing a mechanism for dynamic RFQ pricing using reinforcement learning and symbolic regression in order to obtain a pseudo-optimal nonlinear state-feedback controller for tracking a metric with increased interpretability by utilizing the network environment of FIG. 2 is illustrated as being executed in FIG. 3. Specifically, a first client device 208(1) and a second client device 208(2) are illustrated as being in communication with DRFQP device 202. In this regard, the first client device 208(1) and the second client device 208(2) may be “clients” of the DRFQP device 202 and are described herein as such. Nevertheless, it is to be known and understood that the first client device 208(1) and/or the second client device 208(2) need not necessarily be “clients” of the DRFQP device 202, or any entity described in association therewith herein. Any additional or alternative relationship may exist between either or both of the first client device 208(1) and the second client device 208(2) and the DRFQP device 202, or no relationship may exist.

Further, DRFQP device 202 is illustrated as being able to access a historical RFQ pricing data repository 206(1) and a tracking metrics for RFQ pricing database 206(2). The dynamic RFQ pricing module 302 may be configured to access these databases for implementing a method for dynamic RFQ pricing using reinforcement learning and symbolic regression in order to obtain a pseudo-optimal nonlinear state-feedback controller for tracking a metric with increased interpretability.

The first client device 208(1) may be, for example, a smart phone. Of course, the first client device 208(1) may be any additional device described herein. The second client device 208(2) may be, for example, a personal computer (PC). Of course, the second client device 208(2) may also be any additional device described herein.

The process may be executed via the communication network(s) 210, which may comprise plural networks as described above. For example, in an exemplary embodiment, either or both of the first client device 208(1) and the second client device 208(2) may communicate with the DRFQP device 202 via broadband or cellular communication. Of course, these embodiments are merely exemplary and are not limiting or exhaustive.

Upon being started, the dynamic RFQ pricing module 302 executes a process for dynamic RFQ pricing using reinforcement learning and symbolic regression in order to obtain a pseudo-optimal nonlinear state-feedback controller for tracking a metric with increased interpretability. An exemplary process for dynamic RFQ pricing using reinforcement learning and symbolic regression in order to obtain a pseudo-optimal nonlinear state-feedback controller for tracking a metric with increased interpretability is generally indicated at flowchart 400 in FIG. 4.

In process 400 of FIG. 4, at step S402, the dynamic RFQ pricing module 302 receives first information that relates to a bid price for a financial instrument and second information that relates to an ask price for the financial instrument. In an exemplary embodiment, the financial instrument may include any one or more of a stock that relates to an entity, such as a corporation or other type of commercial concern; a bond that relates to the entity; an option that relates to the entity; and/or a derivative financial instrument that relates to the entity.

At step S404, the dynamic RFQ pricing module 302 selects a metric to be used in conjunction with a determination of an RFQ price with respect to the financial instrument. In an exemplary embodiment, the metric may include any one or more of a hit ratio, a performance ratio, an expected profit, an expected market share, a risk level, and/or any other suitable type of metric.

At step S406, the dynamic RFQ pricing module 302 formulates a Markov Decision Process (MDP) that relates to a determination of the RFQ price, and then uses the MDP to define an RFQ pricing controller with respect to the metric selected in step S404. In an exemplary embodiment, the MDP is designed to minimize a tracking error that corresponds to a difference between an observed value of the metric and a target value of the metric.

In an exemplary embodiment, the use of the MDP to define the RFQ pricing controller includes applying a predetermined neural network to a predetermined set of policy parameters that relate to the metric. In an exemplary embodiment, the method may further include performing a symbolic regression process in order to obtain an algebraic approximation of the predetermined neural network. In an exemplary embodiment, the use of the MDP to define the RFQ pricing controller is performed within a Monte-Carlo simulation of the dynamic RFQ pricing operation. In an alternative exemplary embodiment, instead of applying a neural network to the set of policy parameters, the use of the MDP to define the RFQ pricing controller may include using other types of parametric function approximation tools, such as, for example, decision trees. In this aspect, the “model class” defining the policy to which the set of policy parameters corresponds may include nonlinear functions, which contribute to the property that the RFQ pricing controller may be a nonlinear controller.

At step S408, the dynamic RFQ pricing module 302 applies a reinforcement learning algorithm in order to tune a behavior of the RFC pricing controller. In an exemplary embodiment, the Monte-Carlo simulation implements an ability to tune and evaluate the RFQ pricing controller for robustness against prediction error by introducing random noise in outputs of the RFQ pricing controller based on samples from a predetermined Gaussian noise function. In addition, the Monte-Carlo simulation may also implement an ability to tune and evaluate the RFQ pricing controller for constraints in an actuation of the RFQ pricing controller by introducing an upper bound and a lower bound to a range of outputs of the RFQ pricing controller. In an exemplary embodiment, the reinforcement learning algorithm is not applied every time that the RFQ pricing controller is queried to produce a control output, but is instead applied from time to time depending on a variety of factors, such as any factor that affects the behavior of the market as a dynamic system, e.g., news, market regimes, pricing trends, changes in competitors or liquidity, etc.

At step S410, the dynamic RFQ pricing module 302 uses each of the RFQ pricing controller, the first information that relates to the bid price of the financial instrument, and the second information that relates to the ask price of the financial instrument to determine an RFQ price. In an exemplary embodiment, the as-determined RFQ price is optimized with respect to the tracking error of the selected metric as a result of the use of the MDP to define the RFQ pricing controller and the tuning of the RFQ pricing controller as implemented by the Monte Carlo simulation.

The present inventive concept arises within the domain of quantitative research that is performed in order to develop quantitative tools and models to assist traders in financial markets. Determining pricing for RFQs in an automated and systematic way allows traders to respond to RFQs more efficiently and consistently in a way that aligns with the goals of the traders, as specified by the value of the metric that the traders would like to track.

There are several challenges when devising a RFQ pricing system that effectively tracks the value of a metric with low tracking error. A first challenge relates to robustness to a bias and a variance of a prediction model. The prediction model aims to predict the effect of quoting a price for an RFQ. Inevitably, there will be some bias and variance in such predictions. Designing a controller that is robust with respect to prediction model bias and variance is a challenge for the practical use of the system.

A second challenge relates to determining optimal parameters for the controller. In an exemplary embodiment, once a control architecture has been devised, the parameters that determine the behavior of the controller must be tuned for optimality against the objective of the controller.

A third challenge relates to incorporation of constraints to the controller. In an exemplary embodiment, the system must enable the optimization of the controller in situations when the control outputs are constrained within a given region or in a given direction.

Research question: In an exemplary embodiment, the broad research question to be addressed by the present inventive concept is: Given a prediction model that can predict what the quote price corresponding to a metric is, and a target value for that metric, how can a controller be devised to minimize the tracking error with respect to the metric when combined with a prediction model for RFQ pricing?

In an exemplary embodiment, the two solutions proposed include: 1) Proportional-Integral-Derivative (PID) Control with Bayesian Optimization to determine the optimal linear state-feedback controller for tracking a given metric; and 2) Reinforcement Learning (RL) with optional Symbolic Regression to determine a pseudo-optimal nonlinear state-feedback controller for tracking a given metric.

Both approaches assume availability of a prediction model to ultimately produce RFQ pricing. Both approaches can be optimized for robustness under bias and variance of prediction model, as well as optimization under constraints on the output of the controller.

FIG. 5 is a diagram 500 that illustrates a comparison between a baseline approach and two proposed methods for dynamic RFQ pricing, according to an exemplary embodiment. A descriptive diagram of the systems proposed, compared against a baseline approach which uses a rule-based controller determined by human subject matter expertise, is illustrated in FIG. 5.

In an exemplary embodiment, the main techniques used to solve the problem of obtaining a controller suitable for the RFQ pricing problem include classical control theory tools such as PID control, as well as machine learning tools such as Bayesian Optimization, Reinforcement Learning, and multi-objective evolutionary algorithms for Symbolic Regression.

Solution 1—PID Control with Bayesian Optimization: The PID controller is determined by the control law as follows:

u
⁡
(
t
)

=

K
p

⁢

e
⁡
(
t
)

+

K
i

⁢

∫

e
⁡
(
t
)

⁢
dt

+

K
d

⁢

d
⁢
e

d
⁢
t

In the control law as expressed above, the output of the controller at time t is u(t), the tracking error is e(t) and is defined as the difference between the measured value and the target value of the metric at time t, and the parameters determining the behavior of the controller are K_p, K_i, K_d. It is noted that these parameters may also be referred to as gains, and that they respectively correspond to the proportional, integral, and derivative gains. For RFQ pricing, the final output of the controller at any time is equal to u(t)+P_target, where P_targetis the target value of the metric being tracked, and the error e(t)=P_observed−P_target, where P_observedis the observed value of the metric being tracked. This is a linear controller because the input to the controller is the error P_observed−P_target, which means the controller is agnostic to the actual magnitude of P_observedor P_targetas it operates on the difference between the two, providing the same behavior for different values of P_observedand P_targetwhich yield the same error P_observed−P_target.

In an exemplary embodiment, the proportional term controls the response of the controller to the instantaneous tracking error, yielding higher control outputs as corrections to higher instantaneous tracking errors. The integral term controls the response of the controller to the accumulated tracking error over past time, yielding higher control outputs if there is a persistent tracking error over time. The derivative term controls the response of the controller to the rate of change of the tracking error, yielding higher control output if the rate of change of the tracking error increases.

In an exemplary embodiment, the optimal gains of the PID controller can be obtained using Bayesian Optimization within a Monte-Carlo simulation, where the objective is to determine the value of the proportional, integral, and derivative gains which yield minimum tracking error within the Monte-Carlo simulations. Bayesian Optimization allows for global optimization of a black-box system without any assumptions on its functional form, and is thus suitable for optimizing the performance of the controller as evaluated in the Monte-Carlo simulation. Formally, the goal is to find the location x∈ [Image Omitted] ⁿcorresponding to the global minimum of a function ƒ:ⁿ→. In an exemplary embodiment, the function ƒ represents a measure of the performance of the controller as a function of controller gains. In an exemplary embodiment, and without loss of generality, the function being evaluated is the standard error of the tracking error of the controller against the target value of the metric across the Monte-Carlo samples at a given timestep, integrated over a period of time.

The methodology used for Bayesian Optimization includes modeling the function determining the performance of the controller in the Monte-Carlo simulation for different values of controller parameters, i.e., control gains, using a Gaussian Process (GP). GPs are formally defined as stochastic processes where any finite subcollection of random variables has a multivariate Gaussian distribution. Using a mean function m(·) and covariance function, k(·,·), a GP is denoted as h(·)~ [Image Omitted] P(m(·), k(·,·)). The Bayesian Optimization problem then becomes a GP regression problem, which can be solved using the following iterative algorithm:

Bayesian Optimization using GP Regression.

For t=1, 2, . . . , T repeat:

- 1) Find the next sampling point x_tby optimizing the acquisition function over the

GP
:

x
t

=

arg

max
x

u
⁡
(

x
⁢

❘
"[LeftBracketingBar]"

D

1
:

t
-
1

)

- 2) Obtain a possibly noisy sample y_t=ƒ(x_t)+∈_tfrom the objective function ƒ
- 3) Add the sample to previous samples _1:t={_1:t−1, (x_t, y_t)} and update the GP

In an exemplary embodiment, executing this algorithm with the Monte-Carlo simulation being executed to evaluate the value of ƒ, i.e., the performance of the controller, for different values of x, i.e. the PID controller gains, at each iteration, yields the controller parameters that globally minimize the tracking error of the controller used for RFQ pricing.

In an exemplary embodiment, using Bayesian Optimization to tune the gains of the PID controller yields the optimal linear controller for the RFQ pricing problem.

In an exemplary embodiment, the role of the Monte-Carlo simulation is to provide a simulation environment for tuning and testing the performance of the controller, where RFQs are received over time and they are executed according to a stochastic process. Importantly, the Monte-Carlo simulation can be used to evaluate the performance of the controller with two considerations.

The first consideration relates to tuning and evaluating for robustness against prediction error in the prediction model, for example by introducing random noise in the control outputs of the controller sampled from a Gaussian

𝒩
⁡
(

μ
e

,

σ
e
2

)

which represents the error introduced by imperfect prediction model. This is valuable for the practical use of the system as prediction model errors are commonplace for RFQ pricing and it is desirable for the controller to be robust against this.

The second consideration relates to tuning and evaluating for constraints in the actuation of the controller, for example, by setting upper and/or lower bounds to the range of controller outputs. Consequently, the tuning process can determine the optimal gains accounting for the controller constraints. This is valuable for the practical use of the system the capability to add constraints in the direction and range of pricing expands the utility of the system for automated RFQ pricing for a more diverse set of market scenarios and strategic needs for the market-maker.

Solution 2—Reinforcement Learning (RL) with Optional Symbolic Regression: In an exemplary embodiment, the controller, also referred to herein as policy in the context of RL terminology, is determined by a neural network π_θ parametrized by parameters θ, which is tuned using Reinforcement Learning. Optionally, a symbolic approximation {circumflex over (π)}≈π_θ may be obtained via symbolic regression, using a multi-objective evolutionary algorithm. This optional approximation may be valuable in cases when a simple algebraic expression is preferred over a neural network as a controller for RFQ pricing.

The Reinforcement Learning controller is obtained by formulating a Markov Decision Process (MDP) which represents the decision-making process the RFQ pricing controller aims to solve, i.e., determining the control outputs which minimize the tracking error of the controller to a given RFQ target metric. An MDP consists of a tuple ( [Image Omitted] , γ) where is the set of states called the state space, is the set of actions called the action space, is the transition dynamics of the process determining the evolution in states of the process as actions are taken, is the reward function representing the immediate reward obtained upon state transitions, and γ is a scalar discount factor which discounts the value of future rewards. In an exemplary embodiment, a possible MDP that can be formulated for the RFQ pricing problem includes the following: 1) The state space includes [realized value of metric at current time, target value of the metric]. 2) The action space is the desired deviation from the target value of the metric at a given time, i.e., equivalent to the way the PID controller output is used. In this regard, the action of the controller is a and the value used for RFQ pricing is α+P_target. The actions are sampled as α~π_θ(s) where s is the state at a given time. 3) The transition dynamics are given by the Monte-Carlo simulation, capturing stochastic dynamics across multiple samples. 4) The reward function is a function that provides increased value of rewards for lower tracking errors, i.e., the negative of a cost function of the performance of the controller. In an exemplary embodiment, the reward function is a Radial Basis Function on the tracking error of the controller against the desired value of the metric, given by φ(e(t))=φ(P_observed−P_target), where without loss of generality, φ(·) can be a Gaussian Radial Basis Function given by φ(e)=exp(−ε*e) for shape coefficient E. The role of the reward function is to provide larger values of reward for lower values of error e.

Using the above action space, the controller obtained is a nonlinear controller, because the input to the controller is not the error P_observed−P_target, but rather the values of P_observedand P_targetseparately, which means the controller can adjust its behavior depending on the magnitude of P_observedand P_target. It is noted that this represents a difference with the PID controller, which is a linear controller. In this aspect, the policy itself may be a nonlinear function, which also enables the nonlinear behavior of the RL controller.

In an exemplary embodiment, the RL algorithm Proximal Policy Optimization (PPO) is used to tune the behavior of the RL controller or policy. The PPO objective includes policy parameter updates via the following:

θ

k
+
1

=

arg

max
θ

𝔼

s
,

a
∼

π

θ
k

[

L
⁡
(

s
,
a
,

θ
k

,
θ

)

]

where L is a clipped objective given by:

L
⁡
(

s
,
a
,

θ
k

,
θ

)

=

min
⁢

(

π
θ

(

a
⁢

❘
"[LeftBracketingBar]"

s

)

π

θ
k

(

a
⁢

❘
"[LeftBracketingBar]"

s

)

⁢

A

π

θ
k

(

s
,
a

)

,

g
⁡
(

ϵ
,

A

π

θ
k

(

s
,
a

)

)

)

g
⁢

(

ε
,
 

A

π

θ
k

⁢

(

s
,
 
a

)

)

=

{

(

1
+
ϵ

)

⁢
A

A
≥
0

(

1
-
ϵ

)

⁢
A

A
<
0

and where A corresponds to the advantage value as defined in RL, which can be the simple value baseline advantage or alternatives such as those provided by Generalized Advantage Estimation (GAE).

Using PPO, there are no general theoretical guarantees of global optimality for the learned policy. Therefore, with this approach, a pseudo-optimal nonlinear controller for the RFQ pricing problem is obtained.

The capability of the RL controller to respond to past information is determined by the choice of architecture for the policy network π_θ. If the architecture is a feedforward network, the policy will only be able to use present information to determine its control outputs. If the architecture is a recurrent network, the policy will be able to incorporate past information to determine its control output due to the presence of a hidden state. Alternatively, the use of past information can be incorporated by a different choice of state space definition for the MDP, where the state includes a concatenation of values of P_observedand P_targetfor past times.

In an exemplary embodiment, the MDP uses the Monte-Carlo simulation as dynamics of the MDP to solve the control problem. The role of the Monte-Carlo simulation is to provide a simulation environment for tuning and testing the performance of the controller, where RFQs are received over time and they are executed according to a stochastic process. Importantly, the Monte-Carlo simulation can be used to evaluate the performance of the controller with two considerations.

The first consideration relates to tuning and evaluating for robustness against prediction error in the prediction model, for example by introducing random noise in the control outputs of the controller sampled from a Gaussian

𝒩
⁡
(

μ
e

,

σ
e
2

)

which represents the error introduced by imperfect prediction model. This is valuable for the practical use of the system as prediction model errors are commonplace for RFQ pricing and it is desirable for the controller to be robust against this.

The second consideration relates to tuning and evaluating for constraints in the actuation of the controller, such as, for example, by setting upper and or lower bounds to the range of controller outputs. Consequently, the tuning process can determine the optimal gains accounting for the controller constraints. This is valuable for the practical use of the system the capability to add constraints in the direction and range of pricing expands the utility of the system for automated RFQ pricing for a more diverse set of market scenarios and strategic needs for the market-maker.

Optionally, Symbolic Regression may be used as a means to perform symbolic distillation of the neural network policy obtained via Reinforcement Learning, yielding a symbolic approximation {circumflex over (π)}≈π