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This chapter delves into the intricate world of signature trading strategies, focusing on how algorithmic traders can transform predictive signals into effective trading strategies. It begins by examining the diverse approaches of financial market participants, contrasting long-term investors with fast algorithmic traders who rely on sophisticated models and vast computational resources. The text then explores the complex interplay between signal and noise in financial markets, highlighting the challenges traders face in distinguishing true signals from noisy market orders. The core of the chapter is dedicated to the design and development of trading strategies, emphasizing the importance of extracting alpha and managing execution and risk. It discusses various optimization techniques, including classical financial optimization problems like portfolio optimization and optimal trading and execution. The chapter also introduces the concept of path-dependent optimization, which allows traders to continuously update their positions based on new information. A notable feature of this text is its focus on signature trading strategies, which represent trading strategies as linear functionals applied to the signature of past paths. This approach enables traders to capture path-dependent behaviors and obtain closed-form solutions that are easy to compute and analyze. The chapter concludes with a detailed discussion on the advantages of signature trading strategies, making it a comprehensive guide for professionals looking to enhance their trading strategies.
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Abstract
In this chapter, the primary objective is to recall the concept of a trading strategy and to extend this concept to the setting of signatures. We discuss how one can extend classical methods of optimal trading by tackling stochastic control problems through the lens of signatures, in order to incorporate path-dependencies. First introduced in (Signatures in machine learning and finance. PhD thesis, University of Oxford, 2020), the notion of a signature trading strategy allows to solve path-dependent optimization problems in a simple and explicit way. We first outline the key concepts of signature trading and highlight its associated benefits. Additionally, we present a solution to the mean-variance criterion and illustrate the advantages of signature-based methods using intuitive examples. Finally, we explore how signature trading strategies can be applied to obtain approximate solutions to optimal execution problems.
1 Introduction
1.1 Background and Motivation
Financial Market Participants
There are many different types of participants in financial markets, each with different objectives and preferences, deploying various methods and strategies depending on their reasons for trading at any given time. Certain investors, such as mutual funds, pension funds, and endowments, typically engage in longer-term investments without the need for frequent rebalancing of their positions. These firms generally opt for more classical portfolio construction techniques. On the other end of the scale are fast algorithmic traders, including hedge funds and proprietary trading firms, who utilise quantitative methods to capitalise on short-term market inefficiencies, with more frequent, dynamic rebalancing of their positions. These participants rely heavily on sophisticated algorithms, machine learning models, and vast computational resources to execute trades at speeds measured in microseconds. They also employ risk models capable of managing a variety of more complex risks dynamically. The dichotomy between these market participants illustrates the diverse approaches to financial optimization, where slower asset managers aim for long-term stability and growth, while fast algorithmic traders pursue rapid gains through advanced quantitative techniques. In this chapter of the book, we are more focused on the latter, traders who generally already have access to predictive signals and want to transform those signals into a trading strategy while exploiting any path dependencies and temporal structures that may exist.
Modelling Signal and Noise
In any market setting, the interplay between buyers and sellers dictates the current price of any asset, reflecting the transactions that have taken place in that asset over some past time horizon. This price is influenced by the information available to market participants, encompassing intrinsic asset value, reactions to order flow, news, and broader economic, political, or geographic events. This information then causes market participants to buy or sell in reaction to it, creating chain reactions and feedback loops within the price. For example, if there is a flurry of buyers within a market, other participants may believe that the market has positive information (that they do not have) and also decide to buy, driving the price even higher. True signal emanates from diverse sources, arriving at different speeds as market participants react in various ways. Additionally, market orders may be placed by institutions that are driven by factors such as capital rebalancing or option flow hedging, which occur randomly without necessarily implying an expectation of future price movements. These “noisy” orders in the market are made without access to any exogenous signal to inform when or how that asset should be bought or sold, adding extra layers of noise on top of any true signal that may be present in asset price movements. Information within the market is also noisy itself, since it can be subjective (either positive or negative based on one’s opinion). Consequently, the relationship between signal and noise in finance is complex and takes a variety of non-linear forms and varying speeds, giving rise to quantitative traders tasked with slicing through the layers of noise by deploying mathematical tools to model the behavior of asset prices.
1.2 Trading Strategies
Transforming a Signal into a Strategy
The design and development of trading strategies is a critical element of finance and has been the subject of extensive research over recent decades. Creating a trading strategy can often be divided into two areas: extracting alpha (expected excess returns) and managing the associated execution and risk. Methods of extracting alpha (or an edge) are done in a modelling phase, perhaps achieved through statistical techniques, in order to capture the signal within the noisy price dynamics mentioned previously. The exact modelling setup chosen is dependent on the underlying asset that is being traded, as well as the trade frequency and the investment horizon. Meanwhile, the choice of execution and allocation of the associated risk generally depends on your choice of model and objective criterion. Conventionally, choosing an optimal strategy with respect to some risk objective is done in an optimization phase, where various methods are used to transform the signal-asset dynamics in a way that maximizes the chosen utility function of the strategy’s PnL, given the underlying model. A pioneering example of financial optimization is modern portfolio theory, introduced in Markowitz’s seminal work [18] and then extended to continuous time in Merton’s portfolio problem (both of which we discuss further in Sect. 1.3.1). These simple theories base portfolio allocation on investors’ risk and return preferences, resulting in a well-diversified portfolio.
As illustrated in Fig. 1, the range of possible model choices is extensive. This includes various methods of feature engineering (selecting predictive exogenous variables), return prediction via statistical techniques, and the selection of probabilistic models. The choice of optimization technique is then tailored to the chosen utility function and model.
Fig. 1
Standard blueprint for obtaining an optimal trading strategy
Portfolio optimization frameworks help investors decide how to allocate capital across a universe of assets, this is typically large and can be in the number of hundreds. Typically, such strategies are self-financing, meaning they operate with a fixed initial wealth, without additional capital injections over time. This assumption reflects the priority of growing the portfolio’s value over the long term, rather than on capturing short-term market movements. In such self-financing portfolio strategies, the control (decision) variable is the portfolio’s weights, representing the proportion of total capital allocated to each asset. These weights must sum to one, ensuring the portfolio remains fully invested without the need for external funds. This framework is especially suitable for portfolios that do not require frequent rebalancing, as excessive trading incurs transaction costs that can quickly erode returns, especially for portfolios with a large number of assets. For rebalancing to be justified, any expected excess returns (or alpha) must be substantial enough to outweigh these costs, making infrequent adjustments generally more practical.
Markowitz Portfolio Framework (1952)
The classical Markowitz mean-variance problem [18] that led to Modern Portfolio Theory (MPT), is considered the foundational approach to portfolio optimization and provides a simple and useful benchmark. The Markowitz model is a one-period framework where an investor maintains a constant allocation \(w = (w_1, \dots , w_d)\) across d underlying assets at an initial time, say \(t=0\), until some time in future \(t=T\). In this framework we assume that the returns of each asset are normally distributed, where the expected return of each asset from now until time T is given as \(\boldsymbol {\mu } = (\mu _1, \dots , \mu _d)^\top \) and the corresponding covariance of the returns is given in a covariance matrix \(\Sigma \). The portfolio expected PnL (return), and variance of PnL at time T, is given as
The goal is to maximize expected return (change in wealth) for a specified level of risk (variance). If there are no weight constraints, the optimal solution is straightforward:
When a weight constraint is imposed (e.g., weights summing to 1), the optimization problem can still be solved using quadratic programming, reducing to a linear system by applying a Lagrangian and first-order conditions. We can observe that this is among the simplest portfolio optimization models: it assumes normally distributed returns and uses static weights that remain fixed over the investment period. The objective function that we are considering is quadratic and hence convex, leaving a closed form, intuitive and easy to compute solution.
Merton’s Portfolio Problem (1969)
Building on the success of Modern Portfolio Theory and utility maximization approaches like the Kelly criterion (1956), Merton extended portfolio optimization to continuous time, allowing for dynamic rebalancing over both finite and infinite time horizons. In Merton’s framework, the price of the underlying asset follows the stochastic differential equation (SDE):
where \((W_t)_{t\geq 0}\) is a standard one-dimensional Brownian motion on the filtered probability space \((\Omega , \mathcal {F}, \mathbb {F} = (\mathcal {F}_t)_{t \geq 0}, \mathbb {P})\). We denote the fraction of wealth of the strategy time t as \(\pi _t\), where the remainder of the investors wealth is invested in a risk-free asset, hence self-financing. Assuming we are trading over a short enough time horizon such that interest rates are negligible and there is no consumption of wealth, the PnL of the strategy at time T is given by:
where the instantaneous gain/loss at time t is given as \(dV_t = \pi _t dX_t\). We note here that the PnL V is simply the wealth of the process at time T minus the initial wealth at time 0, and this is the control process that we are able to influence with \(\pi \). In stochastic control problems, our goal is to solve problems of the form
where g is a running reward function and F is the terminal reward function, which are utility functions that are generally non-decreasing and concave. This extension allows for continuous rebalancing of the portfolio in order to maximize our given utility function.
Stochastic Portfolio Theory
Stochastic Portfolio Theory (SPT), introduced by Robert Fernholz [8], is a framework for analyzing and constructing portfolios in a continuous-time, market-relative setting. Unlike traditional approaches that focus on absolute returns, SPT models the dynamics of portfolio weights and asset market capitalizations over time, aiming to achieve robust portfolio growth relative to a benchmark. A key feature of SPT is that it does not rely on specific distributional assumptions about asset returns, and no requirement of alpha or expectation on the underlying drift. For a given universe of assets \(X^1, \dots , X^n\), each of which is assigned a market weight\(\mu _i(t)\), satisfying \(\sum _{i=1}^d \mu _i(t) = 1\), where the market weights will evolve over time as the asset prices change. The portfolio weights are denoted as \(\pi (t) = (\pi _1(t), \dots , \pi _d(t))\) and the portfolio value\(V(t)\) evolves according to
This theory is well-suited for longer term portfolios that aim to maximize long term growth, due to the multiplicative nature of the portfolio value, making it more analagous to the Kelly framework. Intuitively, if we assumed that our stock dynamics follow a geometric Brownian motion with static coefficients, then the optimal proportion of our wealth to invest should also be static in time. Therefore, when trading a martingale (and assuming you have no extra information), there is no temporal structure or alpha to exploit, and hence no path-dependent characteristics to model. However, the reality of financial markets is that volatility is not constant and is in fact stochastic and path-dependent, causing asset returns to be non-normal and fat-tailed over a fixed time horizon. Recently, an excellent use case of Stochastic Portfolio Theory was developed by Cuchiero and Möller in [6], in which the portfolio weights are parameterised as path-dependent functionals. These strategies are relatively general, and includes a particularly relevant case where the functional is modeled as a linear functional on the signature, an approach aligned with the topics of this chapter. This is particularly powerful in financial markets, since by using stochastic volatility and path-dependencies, you are able to better minimise risk and maximize growth over the longer term.
1.3.2 Optimal Trading and Execution
Alternatively, some traders and quantitative funds trade intraday at a higher frequency, and use market information (signals) to inform their decision making. These kind of financial market participant are generally not capital constrained, meaning the notion of being fully invested (weights summing to one) does not make sense here. Rather than having a fixed amount of cash to invest, they are instead constrained by liquidity, volatility and transaction costs (market impact). It is common for trading strategies to be allocated a specific daily dollar volatility, meaning that they are constrained with how much their PnL can vary from day to day for a given average exposure of one dollar. These traders often only handle trading strategies with a much smaller universe of assets as opposed to those in the portfolio optimization case seen previously. With this, there are two main optimization tasks that are required to efficiently trade based on the information that they have. The first of which is finding the optimal amount of inventory (or position) to hold at a given time, such that it maximizes the expected PnL, subject to a terminal time variance constraint. Secondly, once the desired position in known, they are tasked with how to transition from the current position to the desired position, while minimizing the costs and risk associated with doing so. We discuss further the mean-variance optimization for the optimal position in Sect. 3 and the optimal execution case in Sect. 4.
1.4 Path-Dependent Optimization
In this chapter, we are concerned with finding optimal systematic and dynamic trading strategies such that the trader continuously updates their position as new information filters in. This is done with no discretion and is defined as a function of the market state, which continuously updates through time, leading to a new position for each time t. Throughout, we denote \(T \in \mathbb {R}\) the terminal time. Let \((\Omega , \mathcal {F}, (\mathcal {F}_t)_{t \in [0,T]}, \mathbb {P})\) be a filtered probability space. Furthermore, denote by \(X=(X_t)_{t \in [0,T]}\) a non-negative \(\mathbb {R}^d\)-valued stochastic process satisfying \(X_0^m=1\) for \(m \in \{1,\dots ,d\}\). We are interested in finding an optimal, predictable dynamic trading strategy \((\xi )_{t \in [0,T]}\) that maximizes the expected utility of the PnL
We can see that this optimization is within the stochastic control framework in (2), while in this chapter we do not consider a running reward function, although this is a natural extension. The optimal strategy \(\xi \) is a function of the market state (i.e. filtration \(\mathcal {F}\) at time t) and its optimality is with respect to the constraints and utility function that the trader chooses. \(V_T^\xi \) is the terminal value of the trading strategy (i.e. the PnL),
In order to first fix the dynamics of the underlying asset price X, we usually need to construct a model. In the classical stochastic control frameworks, this may be done by modelling the asset price and the trader’s wealth via a diffusion process such as (1) and using dynamic programming to solve the Hamilton-Jacobi-Bellman (HJB) equation. However for quantitative traders, it is most likely that they are trading under the presence of exogenous information f to enrich the filtration \(\mathcal {F}\) (and hence the model of the dynamics of X) and so can estimate its coefficients through statistical techniques and forecasts. For example, in a one-period mean-variance optimization, we require estimates of the mean and variance of the underlying asset returns, which will need to be calibrated via some form of prediction or modelling. Therefore, a large number of models tend to fall into the predict-then-optimize framework, where assumptions are made on the asset returns/signals that are input into any prediction, leaving the final solution exposed to residual errors that are fat tailed and have non-trivial autocorrelation structure, eventually becoming compounded in the optimization phase, leading to large positions in assets.
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Common objective criteria focus on a single trade-by-trade optimization basis, overlooking the potential path that the trading strategy will take. However, in practice signals and asset prices can have strong temporal dependencies and so subsequent trading strategy positions will inherit autocorrelation structure, which we can use to our advantage to exploit through increasing our expected returns, but also by recycling risks in a pathwise manner. As a motivating example in Fig. 2, we consider the comparison between a strategy that is mean-reverting (has autocorrelation) vs one that doesn’t. Both strategies yield identical daily PnL distributions (and hence Sharpe ratio), but have different distributions at a future time due to the temporal structure within the strategy over time.
Fig. 2
Comparison of the PnL profile through time of a mean-reverting strategy vs a non mean-reverting strategy
The main difference from the stochastic control methods, is that we aim to work in a “model-free” sense, so we are not fixing a type of stochastic process of our underlying asset or any information (signal) that we may have. We instead choose a class of functions to model the trading strategy itself, this means we can bypass direct return prediction, resulting in an end-to-end optimization framework and thereby mitigating the accumulation of errors in the two-stage approach. Why would we do this? Since we want to optimize over a large class of stochastic processes, for example
where the SDE coefficients are dependent on the history of the path. For example if the drift is path-dependent, there is temporal structure that is exploitable as \(\alpha \) and if there is path-dependent volatility, then we are able to manage our risk dynamically. The general form for a path-dependent trading strategy is
which is a path-dependent criterion, that is maximized by a path-dependent function \(\phi ^*\). Like most function estimation/optimization, we require a class of functions to parameterise and then solve the optimization by finding the optimal parameters. In this chapter, we parameterise this class of functionals as linear functionals on the signature. In this chapter we also focus on optimising under a dynamic mean-variance criterion. The general dynamic mean-variance optimization can be framed as
is the strategy PnL at some future time T and \(\lambda \) is a risk aversion parameter that can be solved for to obtain a specific level of variance. Many classical mean-variance techniques either require a one-period optimization, heavy modelling assumptions or require numerical techniques. However, by using signatures, we are able to recover a closed form solution when observing the class of signature trading strategies.
2 Signature Trading Strategies
First introduced in the work of Perez et al. in [1, 14, 17], Signature Trading Strategies were adopted to capture a large class of trading strategies by representing them as a linear functional applied to the signature of the past path. Naturally, this makes sense, since in Sect. 1.1 we discussed how signal takes complex path-dependent forms. We can think of a trading strategy as a decision made using the knowledge of the current state (e.g. the previous price path, plus some exogenous trading signal); in other words as a function from path space to some decision process. This inevitably raises the question—can the class of linear functionals on the signature be seen as a rich enough class of maps that represent such trading strategies? We will show that this is possible to due to signatures’ capability to approximate continuous functions on paths. Also perhaps crucially, the Signature Trading framework does not impose the restriction that the underlying asset or signal be Markovian, allowing the Sig-Trader to capture auto-correlation and mean-reverting behaviours within the process that perhaps some more classical methods are not able to exploit. This enables us to incorporate path-dependencies into our trading decisions, while still obtaining a closed-form solution, that is easy to compute and simple to analyse. Throughout the remainder of this chapter there are some notations that we will frequently use and for the avoidance of doubt (since there is often different notation for the same mathematical objects), we list them below.
Notation
\(T((\mathbb {R}^d))\)
extended tensor algebra over \(\mathbb {R}^d\)
\(T^{(N)}(\mathbb {R}^d)\)
truncated tensor algebra over \(\mathbb {R}^d\) of order \(N \in \mathbb {N}\)
\(T((\mathbb {R}^d)^*)\)
dual space of the tensor algebra
\(C^{p-var}\)
the space of paths with bounded p-variation
\(G^{[p]}(\mathbb {R}^d)\)
the space of geometric p-rough paths
X
continuous underlying tradeable asset process
\(\hat {X}\)
time-augmented process of X, \((t, X\))
\(\hat {X}^{LL}\)
Hoff lead-lag process of \(\hat {X}\)
\(\mathbb {X}^n\)
n-th term of the signature of X
\(\mathbb {X}^{\leq M}\)
signature of X truncated at order N
\(\mathbb {X}^{< \infty }\)
full signature of X
\(\alpha \)
un-tradeable exogenous signal
\(\hat {Z}\)
market signal process \((t, X_t, \alpha _t)\)
\(\mathcal {Z}^\alpha \)
space of all market signal trajectories
\(\hat {\mathbb {Z}}^{LL, < \infty }\)
full signature of the Hoff lead-lag process \(\hat {Z}^{LL}\)
w
a word corresponding to a multi-index of the tensor algebra
f\((m)\)
an index mapping from one word to another
wv
concatenation of two words \({\mathbf {w}}\) and \({\mathbf {v}}\)
\(\mathcal {A}_d\)
alphabet of \(d \in \mathbb {N}\) letters
\(\mathcal {W}(\mathcal {A}_d)\)
vector space of all words with letters in the alphabet of \(\mathcal {A}_d\)
Diagram of a simple electrical circuit with a battery symbol on the left, connected to two vertical lines representing a resistor. The circuit is open, indicating no current flow.
shuffle product
\(\mathcal {T}^Z\)
space of continuous trading strategies with respect to \(Z =(t, X, \alpha )\)
\(\mathcal {T}^Z_{\text{sig}}\)
space of linear signature trading strategies with respect to Z
\(\mathcal {S}^Z_{\text{sig}}\)
space of linear signature trading speeds with respect to Z
\(\ell \)
linear functional as an element in the dual space of the tensor algebra
\(\xi \)
an \(\mathbb {R}^d\)-valued trading strategy process
Let \(\hat {X}:[0,1] \to \mathbb {R}^{d+1}\) be the continuous time-augmented process of X, discretely sampled at \(t=t_0,\dots ,t_{2N} \in [0,1]\). The Hoff lead-lag transformed path is defined as the piecewise linear interpolation \(\hat {X}^{LL}:[0,1] \to \mathbb {R}^{2(d+1)}\) such that
There also exists a more intuitive and straightforward definition of a lead-lag process, however we decide not to opt for this version and instead consider the so-called Hoff process [13], due to its fundamental properties in the case when we want to calculate the PnL of our trading strategy (Theorem 2.14). This is due to the powerful, non-trivial result of Theorem 2.13, proven in [9] that states that the Itô integral of a function of a process X against itself, can be recovered via the components of the Hoff lead-lag transformation.
Definition 2.3 (Linear Functional on the Tensor Algebra)
Note that there is a natural pairing between the extended tensor algebra \(T((\mathbb {R}^d))\) and its dual space \(T((\mathbb {R}^d)^*)\), by which we denote
for \(\ell \in T((\mathbb {R}^d)^*)\), \(\mathbb {X} \in T((\mathbb {R}^d)).\) We make clear that there exists a canonical isomorphism \((\mathbb {R}^d)^* \cong \mathbb {R}^d\) through the mapping \((\mathbb {R}^d)^* \ni \langle \ell , \cdot \rangle \mapsto \ell \in \mathbb {R}^d\).
Definition 2.4 (Shuffle Product Property)
Let \(\mathbb {X}^{< \infty }_{0,T} \in G^{[p]}(\mathbb {R}^d)\) be a geometric rough path and let \(\ell _1, \ell _2 \in T((\mathbb {R}^d)^*)\) be elements of the dual space of the tensor algebra, then
The image displays a mathematical formula involving sequences and operations. The formula is: [ langle ell_1, X_{0,T}^{<infty} rangle langle ell_2, X_{0,T}^{<infty} rangle = langle ell_1 bigsqcup ell_2, X_{0,T}^{<infty} rangle ] Key symbols include: - ell_1 and ell_2: sequence elements. - X_{0,T}^{<infty}: a sequence or set notation. - bigsqcup: a union or join operation. - langle cdot rangle: angle brackets indicating a pairing or inner product.
We can define the shuffle product
The image displays a mathematical notation representing a function. The function is denoted by a symbol resembling a square with a horizontal line on top, mapping from the Cartesian product of mathcal{W}(A_d) times mathcal{W}(A_d) to mathcal{W}(A_d) .
inductively by
The image contains two mathematical expressions involving a binary operation denoted by the symbol ??. The first expression is: [ text{ua} bigsqcup bigsqcup text{vb} = (text{u} bigsqcup bigsqcup text{v})text{a} + (text{ua} bigsqcup bigsqcup text{v})text{b} ] The second expression is: [ text{w} bigsqcup bigsqcup emptyset = emptyset bigsqcup bigsqcup text{w} = text{w} ] These expressions illustrate properties of the operation ??, including distributive and identity-like properties.
for all words \({\mathbf {u}}, {\mathbf {v}}\) and letters \({\mathbf {a}}, {\mathbf {b}} \in \mathcal {W}(\mathcal {A}_d)\).
Example 2.5
Let \({\mathbf {w}} = {\mathbf {12}}, {\mathbf {v}} = {\mathbf {34}}\), then
The image shows a symbol composed of the letters "W", "L", "L", and "V" in a stylized serif font, arranged horizontally.
is given by
The image displays a mathematical formula involving permutations of the numbers 1, 2, 3, and 4. The formula is expressed as: [ 12 parallel 34 = 1234 + 1324 + 1243 + 1342 + 3124 + 3142 + 3412 ] The symbol parallel is used to denote a specific operation or relation between the numbers.
Remark 2.6
We make extensive use of the fact that polynomials of linear functionals can be expressed as shuffle products of the linear functionals themselves.
Definition 2.7 (Trading Signal)
We will treat \(\alpha _t\) to be an un-tradable exogenous trading signal. This can be seen of as a prediction of a particular asset return (for a given future time-horizon) or perhaps a critical piece of data that we believe is informative about the future price movement of the asset that we are trading. We may often interchange the use of the words signal, alpha and factor all to mean the same thing.
Definition 2.8 (Market Signal Process)
Let \(X=(X_t)_{t \in [0,T]}\) be a d-dimensional tradable underlying asset process and \(\{\alpha ^i \}_{i=1}^N\) are N un-tradable exogenous trading signals. Then we define the (time-augmented) market signal process as the \((1+d+N)\)-dimensional process
that induces the natural filtered probability space \((\Omega , \mathcal {F}^{Z}, (\mathcal {F}^{Z}_t)_{t \in [0,T]}, \mathbb {P})\), where the filtration of X satisfies \(\mathcal {F}^X \subseteq \mathcal {F}^Z\) such that X is driven by \(\{\alpha ^i \}_{i=1}^N\). Throughout the remainder of this paper, we maintain such assumptions on \(\hat {Z}\). We define as the space of all market signal trajectories for given assets X and signals \(\alpha \),
Let \(\hat {Z}\) be a market signal process as defined in Definition 2.8. Let \(\xi = (\xi )_{t \in [0,T]}\) be an adapted, \(\mathcal {F}^Z_t\)-predictable and integrable strategy such that \(\int ^T_0 \xi _s^2 ds < \infty \). If the strategy \(\xi \) is then a function of the market state, that is \(\xi _t = \phi (\hat {Z}_{0,t})\), a continuous function of the previous market signal trajectory up to time t. We can extend \(\xi \) to be a signature trading strategy, such that
A natural question to ask is if this class of strategies \(\mathcal {T}^Z_{\text{sig}}\) is reasonable or if in fact it is a large enough class of trading strategies to consider. In fact, we can show that the class of signature trading strategies is dense in the space of continuous trading strategies on compact sets of paths.
Lemma 2.10
Let\(K \subset C^{1-var}([0,T],\mathbb {R}^d) \subset \mathcal {Z}^{f}\)be a compact set of market signal trajectories. Then for any exogenous signature trading strategy\(\xi = \phi (\hat {Z}_{0,T})\)that acts on paths in K and for every\(\epsilon > 0\), there exists a linear functional\(\ell \in T((\mathbb {R}^{N+d+1})^*)\), such that
where the choice of suitable candidate topology is discussed in [5].
Proof
We can see that this result follows from the universal approximation of continuous functions on paths by linear functionals acting on the signature. □
Remark 2.11
If instead we wanted to trade endogenously without any trading signals, we would simply take the market signals to be the null-process \(\emptyset \), such that \(\hat {X} = \hat {Z}\), the results would still hold.
Now that we have defined the framework required in order to trade a signature trading strategy embedded with market signals, we can state a key result in order to combine the importance of the Hoff process in Theorem 2.13, that allows us to explicitly state the PnL of the exogenous sig-trader without the need for an integral at all.
2.2 Signature Trading Strategy PnL
Definition 2.12 (Trading Strategy PnL)
Let \(\hat {Z} = (t,X_t, \alpha _t)\) be a market signal process with X a tradable underlying process and \(\alpha \) an untradable signal process. If \(\xi \in \mathcal {T}^Z_{\text{sig}}\) is a linear signature trading strategy such that \(\xi _t^m = \langle \ell _m, \hat {\mathbb {Z}}_{0,t} \rangle \), for each asset \(m=1,\dots ,d\), then we define the PnL of the signature trading strategy between time 0 and time T as
where the integral is understood in the Itô sense.
It is crucial to distinguish the difference between the Itô integral used in this definition versus the Stratonovich integral in the definition of the signature. If that integral was in fact an Itô integral then we could describe the above definition of PnL directly in terms of the add-time signature \(\hat {\mathbb {Z}}_{0,t}\), however this is unfortunately not the case! So in order to develop a more friendly version of \(V_T\), we require the following theorem involving the Hoff lead-lag process as seen in Definition 2.1.
Theorem 2.13 (Recovery of Itô Integral Using the Hoff Process, (Theorem 5.1, [9]))
Let\(X=(X_t)_{t \in [0,T]}\)be a stochastic process on the filtered probability space\((\Omega , \mathcal {F}, (\mathcal {F}_t)_{t \in [0,T]}, \mathbb {P})\). Suppose we observe piecewise smooth streams of X that are discretely sampled over a sequence of times\(\{t_i\}^N_{i=0}\). Let\(\hat {X}^{LL} = (\hat {X}_{t_i}^{\mathit{\mbox{lag}}}, \hat {X}_{t_i}^{\mathit{\mbox{lead}}})^{2N}_{i=1}\)be the associated observed Hoff lead-lag transform of\(\hat {X}\)as defined in Definition2.1. Let\(\phi = (\phi ^1, \dots , \phi ^d)\)be continuous functions on paths as described in Lemma2.10, then we have that
The basic idea of the proof is that the areas between the lead and lag components of the Hoff lead-lag process (captured by \(\hat {X}^{LL}\)), introduce a correction factor in the stochastic integral limit, consequently allowing us to recover the Itô, not Stratonovich, integral. □
This result allows us to show that applying a function to the observed lagged path, and integrating against the observed leading path, we recover the true Itô integral against the stochastic process X as the mesh size goes to zero. In this sense, we see that we are able to approximate asymptotically the true Itô integral as instead an integral of the observed, Hoff lead-lag stream. A natural next step is to consider the case where our strategy \(\xi \) is in fact a signature trading strategy, i.e. \(\xi _t^m = \langle \ell _m , \hat {\mathbb {Z}}_{0,t} \rangle \) for each asset \(m=1, \dots , d\). Moreover, how can we find an expression for the PnL \(V_T\) in (7), in the Itô integral sense, using Theorem 2.13?
Theorem 2.14 (PnL of a d-Asset Signature Trading Strategy Under Exogenous Signal, [11, Theorem 2.11])
Let\(\hat {Z} := (t, X_t, \alpha _t)_{t \in [0,T]}\)be the market signal process, where X is a d-dimensional tradable stochastic process and\(\alpha \)is a N-dimensional un-tradable signal process. Let\(\ell _1, \dots , \ell _d \in T((\mathbb {R}^{N+d+1})^*)\)and define our trading strategy as\(\xi _t^m = \langle \ell _m, \hat {\mathbb {Z}}_{0,t}^{< \infty } \rangle \)for\(m=1,\dots , d\). Then, we have that the PnL of the strategy between time 0 and time T can be represented as
where\({\mathbf {f}}(m): \{1, \dots , d\} \to \mathcal {W}(\mathcal {A}_{2(N+d+1)})\)is a shift operator which is defined as\({\mathbf {f}}(m) = \pi (e^*_{m+N+d+1})\)where\(\pi :T((\mathbb {R}^{2(N+d+1)})^*) \to \mathcal {W}(\mathcal {A}_{2(N+d+1)})\)the canonical isomorphism between the dual space\(T((\mathbb {R}^{2(N+d+1)})^*)\)and the space of all words\(\mathcal {W}(\mathcal {A}_{2(N+d+1)})\).
Now, due to the linearity of expectation, we are able to represent the expected PnL at time T as the following:
where \(\mathbb {E}( \hat {\mathbb {Z}}^{LL,<\infty }_{0,T})\) is the expected signature of the Hoff lead-lag transformation of \(\hat {Z}\).
Remark 2.15
Note that \({\mathbf {f}}(m): \{1,\dots ,d\} \to \mathcal {W}(\mathcal {A}_{2(N+d+1)})\) is simply just the shift operator that allows any multi-index \((j_1, \dots , j_n) \in \left \{1,\dots ,d \right \}^n\) for the original signature terms to be transformed into a multi-index for the signature terms of the \(2(N+d+1)\)-dimensional time-augmented lead-lag process. Let us consider the case of when we have two assets \(X = (X^1, X^2)\), one signal \(\alpha \), such that \(d=2\), \(N=1\). Then we define the 4-dimensional market signal process \(\hat {Z}:= (t,X^1, X^2,\alpha )\) and its corresponding Hoff lead-lag process \(\hat {Z}^{LL}\) will have dimension \(2(N+d+1) = 8\). Given that the assets correspond to \(m=1,2\), then we define the shift operator for each asset as
$$\displaystyle \begin{aligned} {\mathbf{f}}(1) = e^*_{1 + N + d + 1} = {\mathbf{5}} \\ {\mathbf{f}}(2) = e^*_{2 + N + d + 1} = {{\mathbf{6}}} \end{aligned} $$
This simply means that the linear function \(\ell \) is then applied to the correct expected lead-lag signature terms. By correct, we mean the dimensions of the lead-lag process that correspond to the lead part of the integrator \(dX_t^{m,\mbox{lead}}\) in the Itô integral of (8). To show this further, the dimensions of the lead-lag process are as follows:
The image displays a mathematical expression with variables and subscripts. It includes two groups of terms, each enclosed by braces. The first group is labeled with m = 1, 2 and consists of t_{lag}, X_{1,lag}, X_{2,lag}, alpha_{1,lag} . The second group is labeled with f(1) = 5, f(2) = 6 and consists of t_{lead}, X_{1,lead}, X_{2,lead}, alpha_{1,lead} .
More intuition on the shift operator will be provided in Example 3.3.
This result allows us to express any integral of a linear functional on the signature of a time-augmented market signal path as a newly defined linear functional on the time-augmented lead-lag market signal path instead.
Lemma 2.17 (Variance of the PnL of a Signature Trading Strategy Under Exogenous Signal, [11, Lemma 2.12])
Let X be a d-dimensional tradable stochastic process and let\(\alpha \)be a N-dimensional un-tradable signal process. Define\(\hat {Z}_t := (t, X_t, \alpha _t)\)as the market signal process and\(\ell _1, \dots , \ell _d \in T((\mathbb {R}^{N+d+1})^*)\)and define our trading strategy as\( \langle \ell _m, \hat {\mathbb {Z}}_{0,s}^{< \infty } \rangle \)for\(m=1,\dots , d\). Let the expected PnL of the strategy between time 0 and time T be defined as in (10). Then the Variance of the PnL at time T is defined as
The image shows a mathematical formula for the variance of V_T . It is expressed as: [ text{Var}(V_T) = sum_{m=1}^{d} sum_{=1}^{d} langle ell_m f(m) perp !!! perp ell_n f(), mathbb{E}(hat{Z}^{LL}_{0,T}) rangle - langle ell_m f(m), mathbb{E}(hat{Z}^{LL}_{0,T}) rangle langle ell_n f(), mathbb{E}(hat{Z}^{LL}_{0,T}) rangle. ] The formula includes summations, inner products, and expectations, with symbols such as ell, f, and hat{Z}^{LL}_{0,T}.
where
Diagram of a simple electrical circuit with a battery symbol on the left, connected to two vertical lines representing a resistor. The circuit is open, indicating no current flow.
This results follows simply from Theorem 2.14 and the fact that variance of a random variable is defined as \(\mbox{Var}(V_T) = \mathbb {E} (V_T^2) - (\mathbb {E}(V_T))^2\), where \(\mathbb {E}(V_T)\) is defined in (10) and
The image displays a mathematical formula involving expected values and summations. The formula is: [ mathbb{E}(V_T^2) = sum_{m=1}^{d} sum_{=1}^{d} langle ell_m f(m) bigsqcup ell_n f(), mathbb{E}(hat{Z}_0^{L,L,T}) rangle ] Key elements include the expected value symbol mathbb{E}, summation symbols sum, and various mathematical functions and operations.
In this section we derive an explicit and concise expression for the optimal signature trading strategy in the presence of exogenous market signals considering a path-dependent extension of the classical mean-variance criterion, as seen in [11]. Using the trading strategy expected PnL and variance of PnL as defined in Sect. 2.2, we show that the path-dependent mean-variance optimization problem is convex in the weights of the linear functionals \(\ell _1, \dots , \ell _d\).
Denote\(T \in \mathbb {N}\)the terminal time. Let X be a d-dimensional tradable stochastic process and let\(\alpha \)be a N-dimensional un-tradable signal process. Define\(\hat {Z}_t := (t, X_t, \alpha _t)\)as the market signal process. Define a signature trading strategy\(\xi _t^m\)through a linear functional on the signature of the market signal process, i.e.\(\xi _t^m = \langle \ell _m, \hat {\mathbb {Z}}_{0,t} \rangle \). Then, for a given truncation level M, the mean-variance optimal signature trading strategy\(\ell ^* = (\ell _1^*,\dots ,\ell _d^*),\)\(\ell _m^* \in T^{(M)}((\mathbb {R}^{N+d+1})^*)\), satisfies
and the “Signature PnL covariances” as the\(d \cdot \vert \mathcal {W}_{N+d+1}^M \vert \times d \cdot \vert \mathcal {W}_{N+d+1}^M \vert \)matrix\(\Sigma ^{\mathit{\mbox{sig}}}\)as
The image displays a mathematical formula involving summation and various functions. The formula is: [ Sigma^{text{sig}}_{text{wf}(m), text{vf}()} = leftlangle text{wf}(m) bigsqcup text{vf}(), mathbb{E}(hat{mathbb{Z}}^{LL}_{0,T}, < infty) rightrangle ] Key elements include the summation symbol with superscript "sig" and subscripts "wf(m)" and "vf()", the functions "wf(m)" and "vf()" combined with the disjoint union symbol, and an expectation operator involving a hat notation over "Z" with subscripts and superscripts.
and we are interested in finding saddle points \(\ell _1^*, \dots , \ell _d^* \in T^{(M)}((\mathbb {R}^{N+d+1})^*), \lambda ^0 \in \mathbb {R}\) that satisfy \(\nabla \mathcal {L}(\ell _1^*, \dots , \ell _d^*, \lambda ) = 0\). For this purpose recall Theorem 2.14, where the expected terminal PnL is given by (10). Furthermore, we can decompose the variance of the terminal PnL, given in Eqs. (11) and (12). Next, we can compute for each asset \(m \in \{1,\dots ,d\}\) and each word \(\mathbf {w} \in \mathcal {W}_{N+d+1}^M\), the gradients of (10), (11), and (12) with respect to \(\langle \ell _m, e_{\mathbf {w}} \rangle \)
The image contains three mathematical expressions involving derivatives and expectations. The first expression is: [ frac{partial mathbb{E}(V_T)}{partial (ell_m, e_w)} = langle text{wf}(m), mathbb{E}(hat{Z}^{LL}_{0,T}, <infty) rangle ] The second expression is: [ frac{partial mathbb{E}(V_T^2)}{partial (ell_m, e_w)} = 2 sum_{=1}^{d} sum_{v in W^M_{+d+1}} langle ell_n, e_v | text{wf}(m) perp text{vf}(), mathbb{E}(hat{Z}^{LL}_{0,T}, <infty) rangle ] The third expression is similar to the second, with a slight variation: [ frac{partial mathbb{E}(V_T^2)}{partial (ell_m, e_w)} = 2 sum_{=1}^{d} sum_{v in W^M_{+d+1}} langle ell_n, e_v | text{wf}(m), mathbb{E}(hat{Z}^{LL}_{0,T}, <infty) rangle text{vf}(), mathbb{E}(hat{Z}^{LL}_{0,T}, <infty) ] These expressions involve mathematical symbols such as derivatives, expectations, summations, and inner products.
Using these computed gradients, we can compute the gradient of the Lagrangian with respect to \(\langle \ell _m, e_{\mathbf {w}} \rangle \)
The image displays a mathematical formula involving summation and vector functions. The formula is: [ Sigma^{text{sig}}_{text{wf}(m), text{vf}()} = langle text{wf}(m) perp text{vf}(), mathbb{E}(hat{Z}^{LL}_{0,T}, <infty) rangle - langle text{wf}(m), mathbb{E}(hat{Z}^{LL}_{0,T}, <infty) rangle (text{vf}(), mathbb{E}(hat{Z}^{LL}_{0,T}, <infty)) ] Key elements include the summation symbol Sigma, vector functions wf(m) and vf(), expectation mathbb{E}, and a perpendicular symbol perp.
for all \(\mathbf {w},\mathbf {v} \in \mathcal {W}_{N+d+1}^M\) and \(m, n \in \{1,\dots ,d\}\).
Using the first order conditions, observe that we obtain the system of linear equations
equations and \(d_M\) unknowns. Hence, assuming that \(\Sigma ^{\mbox{sig}}\) is invertible, we can solve (17) and obtain for \(m \in \{1,\dots ,d \}, \mathbf {w} \in \mathcal {W}^M_{N+d+1}\)
where we assumed by complementary slackness (KKT) that \(\lambda \neq 0\). We can then substitute (18) into the variance constraint to obtain two solutions for \(\lambda \)
The image displays a mathematical formula involving summations and products. The formula is: [ lambda_{pm} = pm frac{1}{2sqrt{Delta}} left( sum_{m=1}^{d} sum_{=1}^{d} sum_{w, v in W_{+d+1}^{M}} langle ell_{m}, e_{w} rangle langle ell_{}, e_{v} rangle Sigma_{text{wf}(m), text{vf}()}^{text{sig}} right)^{frac{1}{2}} ] [ = pm frac{1}{2sqrt{Delta}} left( sum_{m=1}^{d} sum_{=1}^{d} sum_{w, v in W_{+d+1}^{M}} ((Sigma^{text{sig}})^{-1} mu^{text{sig}})_{text{wf}(m)} ((Sigma^{text{sig}})^{-1} mu^{text{sig}})_{text{vf}()} times Sigma_{text{wf}(m), text{vf}()}^{text{sig}} right)^{frac{1}{2}} ] The formula includes Greek letters such as lambda (lambda), sigma (Sigma), and mu (mu), and involves complex mathematical operations.
for \(m \in \{ 1, \dots , d \}, \mathbf {w} \in \mathcal {W}^M_{N+d+1}\). □
Remark 3.2
The matrix \(\Sigma ^{\mbox{sig}}\) and vector \(\mu ^{\mbox{sig}}\) are just placeholders for different terms and combinations of \(\mathbb {E} (\hat {\mathbb {Z}}^{LL,<\infty }_{0,T})\). Hence, all we need in order to know what our optimal functional is, is the expected Hoff lead-lag signature, and this is straightforward to compute for reasonable orders of truncation and number of assets and signals. The following example aims to provide some practical intuition behind how the optimal strategy is calculated and the relation between the signature and words of the tensor algebra.
Example 3.3
Let us consider the case of when we have two assets \(X = (X^1, X^2)\), one signal \(\alpha \), such that \(d=2\), \(N=1\). Then we define the 4-dimensional market signal process \(\hat {Z}:= (t,X^1, X^2,\alpha )\). Let us fix the truncation level to be \(M=2\). The 2nd level truncated signature \( \mathbb {Z}_{0,t}^{\leq 2}\) has \(\vert \mathcal {W}^2_{4} \vert = 21\) terms, namely the associated words are:
Hence, the optimal linear signature trading strategy will correspond to two linear functionals (one for each asset) \(\ell _1, \ell _2\), each of length 21, defined as
Hence, \(\mu ^{\mbox{sig}}\) will be a \(d \cdot \vert \mathcal {W}^M_{N+d+1} \vert = 2 \times 21 = 42\) length vector, containing elements of the expected lead-lag signature of order 3.
Recall that the shift operator \({\mathbf {f}}(m)\) is defined for each asset \(m=1,2\) as \({\mathbf {f}}(1) = {\mathbf {5}}, {\mathbf {f}}(2) = {\mathbf {6}}\), which correspond to the 5th and 6th dimensions of the lead-lag process. Hence, we see that \(\mu ^{\mbox{sig}}_1\) contains the expected lead-lag signature terms corresponding to the index of words
Now, we consider the \(42 \times 42\) matrix \(\Sigma ^{\mbox{sig}}\), defined element-wise as
The image displays a mathematical formula involving summation and various functions. The formula is: [ sum_{text{wf}(m), text{vf}()}^{text{sig}} = leftlangle text{wf}(m) perp text{vf}(), mathbb{E} left( hat{Z}_{0,T}^{LL, <infty} right) rightrangle - leftlangle text{wf}(m), mathbb{E} left( hat{Z}_{0,T}^{LL, <infty} right) rightrangle leftlangle text{vf}(), mathbb{E} left( hat{Z}_{0,T}^{LL, <infty} right) rightrangle ] Key elements include the summation symbol, functions wf(m) and vf(), the expectation operator mathbb{E}, and the notation hat{Z}_{0,T}^{LL, <infty}.
for all \(\mathbf {w},\mathbf {v} \in \mathcal {W}_{4}^2\) and \(m, n =1,2\).
Each element of this matrix represents a covariance term between the PnL attributions that we discussed previously. For example, let us observe an arbitrary element of \(\Sigma ^{\mbox{sig}}\). Let \(\mathbf {w} = \mathbf {01}, \mathbf {v} = \mathbf {23}, m=1, n=2\). Then \(\mathbf {wf}(m) = {\mathbf {015}}, \mathbf {vf}(n) = \mathbf {236}\) and the corresponding element in the matrix is given as
The image displays a mathematical formula involving summation and expected value notation. The formula is: [ Sigma_{015,236}^{text{sig}} = left( frac{015 perp !!!! perp 236, mathbb{E} hat{Z}_{alpha, T}^{LL, <delta}}{015, mathbb{E} hat{Z}_{alpha, T}^{LL, <3}} right) - left( frac{015, mathbb{E} hat{Z}_{alpha, T}^{LL, <3}}{236, mathbb{E} hat{Z}_{alpha, T}^{LL, <3}} right) ] The formula includes Greek letters and mathematical symbols such as summation (Sigma), expected value (mathbb{E}), and a hat symbol (hat{}) indicating an estimate or approximation.
where
The image shows a sequence of numbers and symbols: "015 ? 236". The symbol in the middle resembles a horizontal bracket or a bottom half of a rectangle.
is a sum of 20 different words in \(\mathcal {W}_4^6\). We refer the reader also to Example 2.5 for another example of the shuffle product. It is evident that, while our linear functional is only applied to the second order truncated signature, we require the sixth order signature in order to compute the covariance matrix, which can cause a computational bottleneck in practice.
3.2 Examples
3.2.1 Lead-Lag and Pairs Trading
The true advantage that data-driven methods have over classical parametric frameworks is that they can exploit the inefficiencies present in financial time series data, without specifying the explicit dynamics that they are trying to capture. In this example, we try to understand the multivariate relationship between two assets that track each other, perhaps due to the fact that the intrinsic value of these two assets are extremely similar. There are many such examples in finance, perhaps they are two adjacent interest rates or futures contracts, or perhaps they are just two stocks that happen to be very similar to one another. The fundamental idea is that the fair value of these two assets should be similar or the same as one another, and hence they will move in the same direction. If the asset prices move apart from one another for whatever reason, then this can represent a mispricing, and one that a trader would like to exploit, causing the prices to converge back to one another. If one of the assets always moves before the other, then we could see this as a lead-lag relationship, otherwise if there is no clear leader or lagger then this can be seen as a general pairs trading strategy, where one aims to capitalise on divergences in the two asset prices. The sentiment is that while both assets have their own dynamics, the difference (spread) between the prices of the two assets should hold some predictability on future (co-)movements. Classical literature focuses on modelling this relationship using a mean-reverting process such as an Ornstein-Uhlenbeck process. The strategy should then contain a buy signal when the spread falls below some threshold and a sell signal when the spread is above the threshold, in the anticipation that this spread should converge back to the threshold. In this experiment we take two assets such that
where X is a standard Brownian motion with zero drift and volatility \(\sigma ^X\). Y however is modelled as a mean-reverting process where its drift is proportional to the spread between X and Y . That is, X is the leading process and Y is a lagging process. We can imagine that we are a mid-frequency quantitative trader who wants to exploit this relationship and trades/rebalances every 30 minutes, and wants to maximize their expected PnL and minimise the variance of the PnL at the end of the day (perhaps due to margin/capital constraints being defined daily), just like in our objective in (3). In a 9-hour trading day, we would see this essentially gives us 16 trading opportunities. We can clearly see in this toy example that the only exploitable alpha within the framework is the temporal dependence through mean-reversion in Y since there is no long term drift in X or Y .
If a trader was to deploy a static buy and hold strategy here, the PnL would be zero on average, however, for higher order Sig-Traders, it is possible to exploit the mean-reversion dynamics. In fact, this example demonstrates how the above-mentioned alpha is self-contained within the 1st level of the signature (which is the increment of the path, or the ‘drift’) and so orders of truncation of \(M \geq 2\) do not contain any more predictive power on excess returns than that of \(M=1\). However, on the right hand side (RHS) of Fig. 3, the signature efficient frontier illustrates that when trading with a daily look-forward horizon, the ratio of return to variance of daily PnL is greater as we increase the order of the Sig-Trader. This is due to the higher levels of the signature capturing non-linear path-dependencies that can help reduce variance in the PnL distribution at the end of the day. We relate back to Fig. 2 to demonstrate that in fact higher order Sig-Traders are able to construct mean-reverting strategies that naturally limit drawdowns, which are an inherently path-dependent characteristic.
Fig. 3
Weekly PnL distribution (LHS) and the signature efficient frontier comparing different orders of truncation (RHS)
In this synthetic example, we compare the Sig-Trader strategy to that of the original factor model. The generic factor model is set up as a supervised linear regression on future returns, as a function of the current signature,
where r is the 2-dimensional asset returns, B is a \(2 \times N\) matrix of factor coefficients, \(\hat {\mathbb {Z}}_{0,t}\) is a vector of the signature values of the market signal process and \(\varepsilon \) is a vector of the 2 assets’ (unexplained) residuals returns. We can see that this model uses the same input as the Sig-Trader (the signature up to time t), with the same tools (applying a linear function). The key distinguishment however is that the factor model is optimal for maximising the mean-variance PnL profile of 30-minute portfolio returns, which ignores any pathwise dynamics of the strategy itself, throughout the course of the day. This is very important, since in fact, volatility/volume profiles are not constant over the course of a trading day, and hence risks are very different depending on the time of day. Figure 4 demonstrates the difference in nature between the Sig-Trader and the factor model. In the order 1 case, as described above, the exploitable alpha in terms of returns are captured by the 1st level of the signature and so we see a similar position profile for the Sig-Trader and the factor model. However, when the factor model tends to go more short (or long), the higher order Sig-Traders, e.g. order 3, tend to reduce its position, since this will be more beneficial to minimising the PnL at the end of the trading day. We can think of this behaviour as being similar to applying a sigmoid function to your position in search of robustness, or applying a stop-loss to avoid becoming too leveraged - which are common practices. By incorporating path-dependencies in the strategy, we have access to a more robust and intuitive extension to classic factor models.
Fig. 4
Comparison between the Markowitz Order 3 factor model positions and the corresponding Sig Trader positions, for orders 1,2,3
As discussed in Sect. 1, obtaining an optimal trading strategy is always with respect to the market scenario we are trading within. While mean-variance optimization is regarded as perhaps the most common and intuitive of utility preferences, various other objectives and circumstances may influence trading decisions, contingent upon personal utility and preferences. In addition to individual preferences, it’s crucial to consider characteristics of market microstructure when trading, particularly for large investors or when dealing with illiquid assets, since the market becomes sensitive to any trade we make. This means that by actively trading, we are able to fundamentally alter the price process that we are trying to model, incurring a feedback loop. We explore such market microstructure problems in this section and present the results of Kalsi et al. in [14], in which they obtain approximate solutions to the problem of optimal execution within the class of signature trading strategies.
4.1 Optimal Execution Problem
The classical optimal execution problem concerns a trader who wants to transition from one position (or level of inventory) \(Q_0\) to another \(Q_T\) over a specified time horizon. If the trader were to immediately post a market order in the limit order book (LOB), insufficient liquidity within the book might result in execution at a less favorable price, often referred to as market impact. To mitigate this impact, it may be prudent to instead split the parent order up into smaller child orders and execute them gradually since they will have much less market impact, however this approach exposes the trader to potential adverse market movements during the execution period. Consequently, the trader faces the dilemma of trading as swiftly as possible to minimize uncertainty regarding future market movements while simultaneously minimizing market impact. As seen in the blueprint of Fig. 1, in order to solve this type of optimal trading problem, we require a model for the asset midprice dynamics as well as a way of modelling market impact, including the execution price. Once the modeling phase is complete, the next step is to formulate an objective function aligned with the trader’s preferences. The goal is typically to maximize total wealth at the terminal time, which depends on the prices executed between time 0 and T, as well as the final execution at the terminal time. Optionally, a penalty may be imposed on any remaining inventory held at time T as this is generally undesirable.
Signature Trading Speeds
In previous sections we modelled the position (holding) of the underlying asset at time t by \(\xi _t\), as a signature trading strategy. In execution problems we often model our strategy as a trading speed, which is defined as the change in our position (inventory) per unit time.
Definition 4.1 (Signature Trading Speed)
Let \(\hat {Z}\) be a market signal process as defined in Definition 2.8 where \(d=1\) and we have access to N exogenous signals. We denote \(\mathcal {S}^Z\) as the space of trading speeds with respect to market signal process Z, defined as the space of all continuous adapted processes with respect to the filtration generated by \(\hat {Z}\). Similarly to Definition 2.9, we define the space of signature trading speeds\(\mathcal {S}^Z_{\mathrm {sig}}\) as
Modelling price impact from a large market order is a well-studied topic of mathematical finance and there exists many formulations in the literature. Generally the execution price \(P_t^\theta \) of the strategy \(\theta \) at time t is modelled as
where X is the unperturbed midprice if the trader does not trade the underlying asset. We observe that the market impact \(I(\theta _{0,t})\) is a function of the path of past trading speeds, which in our framework is a linear function of the signature of the past path of the market signal process, that is to say
where \(g^\ell \) is a linear functional that models the market impact I as a function of the past market signal trajectory. There are several ways discussed in the literature to model market impact, derived from empirical observations, which include but not limited to:
1.
Temporary market impact. We let \(g^\ell = \lambda \ell \), with \(\lambda >0\), then
which is a linear temporary market impact studied in [2, 4, 16].
2.
Permanent market impact. In [2‐4], a permanent market impact given by \(\int ^t_0 \theta _s ds\) is considered. If we set \(g^\ell = \ell \mathbf {0}\) as a concatenation of the linear functional \(\ell \) with the word \(\mathbf {0}\) which is the designated time index of the process \((t,X_t,f_t)\). Therefore the market impact is defined as
Transient market impact. In [7, 12], the authors considered a transient market impact that is given by \(\int ^t_0 K(t-s) \theta _s ds\) where \(K(x) := \exp (-\rho x)\) for \(\rho >0\) is an exponential time kernel. We can therefore find a linear functional \(g^\ell \) such that the market impact is well approximated, i.e.
In the previous example of (20), we had specifically separated the market impact from the midprice but since the midprice is one of the channels of the market signal process, it becomes a specific case of (21) where \(f^\theta = {\mathbf {1}} + \boldsymbol {\emptyset } - g^\theta \).
Optimal Execution Formulation
Suppose a trader at time 0 is in the presence of \(Q_0\) inventory and is hoping to liquidate all of their position by time T. The optimal execution task aims to strike an equilibrium between trading fast enough so as to avoid uncertain future price moves, while minimising their price impact, caused by market orders that are too large relative to the limit order book. The objective criterion is as follows:
The image displays a mathematical formula related to financial modeling. The formula is: [ V(theta) = underbrace{W_T^theta}_{text{accumulated cash}} + underbrace{Q_T^theta (P_T^theta - alpha Q_T^theta)}_{text{terminal execution}} - underbrace{phi int_0^T (Q_s^theta)^2 , ds}_{text{running inventory penalty}} ] The formula includes Greek letters theta, alpha, and phi, and involves an integral from 0 to T. The terms are annotated to indicate their financial significance.
(22)
where the wealth accumulated at timetcorresponding to the trading speed\(\theta \), is defined by
The first term of the objective function is clearly our accumulated wealth from liquidating our assets, which is influenced by our trade order size plus market movements, hence the balance that we are trying to strike. The second term refers to the terminal execution if we arrive at time T with remaining inventory, here \(\alpha \geq 0\) is the terminal liquidation penalty parameter (the larger \(\alpha \) then the larger the penalty for holding inventory at the terminal time since this would result in a greater market impact). The final term is a running inventory penalty, since we want to trade as quick as possible in order to transition into the new position. Therefore, the goal of the optimization is to find the signature trading speed, \(\theta ^*\), that maximizes the objective criterion \(V(\theta )\) defined in (22), i.e. we aim to solve
We will show that, while the optimization problem in (23) is nonlinear in the underlying midprice process, since we are parameterising \(P, X,\) and \(\theta \) as linear functions on the signature, then so too do W and Q. Due to the shuffle product property of the signature (Definition 2.4), we are able to re-write the value functions as a new linear functional on the signature of the market signal process. In (23) there are 4 quantities of interest (\(W, Q, \int ^t_0 Q^2_s ds, Q(P-\alpha Q)\)) that we want to compute and then combine, in order to represent the objective as one main linear functional.
Lemma 4.2
Let\(\theta \in \mathcal {S}_{\mathrm {sig}}^Z\)be a signature trading speed given by\(\langle \ell , \hat {\mathbb {Z}}_{0,t}^{<\infty } \rangle \), hence\(\ell \)is the linear functional that represents our trading speed that we are solving for. Recall that the word associated to the constant linear functional is\(\boldsymbol {\emptyset }\), the word associated to the time dimension is\(\mathbf {0}\)and the word associated with the midprice process is\({\mathbf {1}}\). Then the following statements hold:
1.
The wealth process if we trade according to\(\ell \):
The image displays a mathematical formula involving integrals and various symbols. The formula is: [ W_T^ell = int_0^T P_s^ell langle ell, hat{Z}_{0,s}^{<infty} rangle ds = int_0^t (X_s - langle g^ell, hat{Z}_{0,s}^{<infty} rangle) langle ell, hat{Z}_{0,s}^{<infty} rangle ds ] [ = int_0^T leftlangle (1 + varnothing - g^ell) bigsqcup ell, hat{Z}_{0,s}^{<infty} rightrangle ds ] [ = leftlangle left[(1 + varnothing - g^ell) bigsqcup ell right] 0, hat{Z}_{0,T}^{<infty} rightrangle ] The formula includes Greek letters, integrals, and various mathematical symbols.
2.
The inventory process if we trade according to\(\ell \):
The running inventory penalty if we trade according to\(\ell \):
The image displays a mathematical equation involving integrals and various symbols. The equation is: [ int_{0}^{T} (Q_{s}^{ell})^{2} , ds = int_{0}^{T} leftlangle (Q_{0} emptyset - ell 0) bigsqcup_{L^{2}}, hat{Z}_{0,s}^{<infty} rightrangle , ds ] [ = leftlangle left( (Q_{0} emptyset - ell 0) bigsqcup_{L^{2}} right) 0, hat{Z}_{0,T}^{<infty} rightrangle ] The equation includes integrals from 0 to T, Greek letters, and mathematical symbols such as angle brackets, empty set, and hat notation.
4.
The terminal execution if we trade according to\(\ell \):
The image displays a complex mathematical equation involving various symbols and operations. The equation is: [ mathcal{L}_T(P_T^ell - alpha Q_T^ell) = (Q_0 emptyset - ell_0, hat{Z}_{0,T}^{<infty}) lambda (1 + emptyset - g^ell - alpha(Q_0 emptyset - ell_0), hat{Z}_{0,T}^{<infty}) ] [ = left((Q_0 emptyset - ell_0) bigsqcup (1 + emptyset - g^ell) - alpha(Q_0 emptyset - ell_0) bigsqcup_2, hat{Z}_{0,T}^{<infty}right) ] The equation includes Greek letters such as ell and alpha, mathematical symbols like emptyset (empty set), lambda (lambda), and operations such as bigsqcup (coproduct).
Therefore, we can now state the overall optimal execution problem under the class of signature trading speeds.
Theorem 4.3
Let\(\theta \in \mathcal {S}_{\mathrm {sig}}^Z\)be a signature trading speed given by\(\langle \ell , \hat {\mathbb {Z}}_{0,t}^{<\infty } \rangle \), then the value function in (22) can be written as
The image displays a mathematical formula involving various symbols and operations. The formula is: [ V(theta) = left[ left( 1 + varnothing - g^{ell} right) bigsqcup ell right] circ - phi left( Q_0 varnothing - ell 0 right) circ - alpha left( Q_0 varnothing - ell 0 right) bigsqcup ell^2, ] [ + (Q_0 varnothing - ell 0) bigsqcup (1 + varnothing - g^{ell}), hat{Z}_{0,T}^{<infty} rangle. ] The formula includes Greek letters such as theta, varnothing, and phi, and mathematical symbols like bigsqcup, circ, and hat{}.
Hence, the optimal execution problem that we are trying to solve for\(\theta ^*\), is reduced to finding a linear functional\(\ell ^*\)that satisfies
The image contains a complex mathematical expression involving various symbols and operations. The expression is as follows: [ sup_{ell in mathcal{T}((mathbb{R}^{+2})^*)} leftlangle left[ (1 + varnothing - g^ell) bigsqcup ell right] varnothing - phi left( (Q_0 varnothing - ell 0) bigsqcup ell^2 right) varnothing - alpha (Q_0 varnothing - ell 0) bigsqcup ell^2, right. ] [ left. + (Q_0 varnothing - ell 0) bigsqcup (1 + varnothing - g^ell), mathbb{E} left[ hat{Z}_{0,T}^{<infty} right] rightrangle. ] The expression includes Greek letters such as varnothing, phi, and alpha, mathematical operations like supremum (sup), and various set and function notations.
Recall, that in the more general case, when we are under the presence of exogenous signals, we can incorporate the market impact function\(g^\ell \)directly within a new linear functional that is dependent on the exogenous signals. In this specific case we have\(f^\ell = {\mathbf {1}} + \boldsymbol {\emptyset } - g^\ell \), however we can in fact model this as any linear functional\(f^\ell \), that is we solve the optimization
The image displays a mathematical expression involving a supremum function. The formula is: [ sup_{ell in mathcal{T}((mathbb{R}^{+2})^*)} leftlangle (f^ell bigsqcup ell) odot 0 - phi left( (Q_0 odot ell 0) bigsqcup ell 0 right) odot alpha (Q_0 odot ell 0) bigsqcup ell 0^2, right. ] [ left. + (Q_0 odot ell 0) bigsqcup ell f^ell, mathbb{E} left[ hat{Z}_{0,T}^{<infty} right] rightrangle. ] The expression includes Greek letters such as phi and alpha, mathematical symbols like sup, mathbb{R}, and mathbb{E}, and operations such as bigsqcup and odot.
(24)
Numerically Solving the Optimal Execution Problem
While in Sect. 3 we were able to obtain a closed form solution for \(\ell ^*\), in this problem we require numerical techniques to solve the optimization. We also previously required a notion of the Itô integral in our definition of the PnL of a trading strategy in order to capture the predictability of such a trading strategy. However, when working with trading speeds, we define our objective slightly differently since we are not directly to model every price movement, but instead focus on modelling the market impact and hence executable price, P. This leads us to instead only require the estimation of the signature of the market signal process and not the Hoff lead-lag process for the fitting procedure, which significantly reduces the bottleneck in computation. The main building blocks for numerically solving the problem are:
1.
Pre-determine via a well-studied market impact model or empirically fit for a linear functional \(f^\theta \) that satisfies \(P_t^\theta = \langle f^\theta , \hat {\mathbb {Z}}_{0,t}^{<\infty } \rangle \).
2.
Estimate the expected signature of the market signal process, \(\mathbb {E} \left [ \hat {\mathbb {Z}}_{0,T}^{<\infty } \right ]\).
3.
Numerically solve the optimization problem (24). If we consider linear permanent or temporary market impact, the value function becomes quadratic in the weights of \(\ell \) and we can simply find the unique global maximum.
4.3 Numerical Examples
Geometric Brownian Motion
If the underlying asset paths behave as a geometric Brownian motion, the optimal execution solution can be obtained in closed form. The optimal execution strategy in this setting is a time-weighted average price (TWAP) strategy. When the signature trading model is fitted using geometric Brownian motion paths, we observe the same closed form solution, as demonstrated in Fig. 5, showing that the signature trading framework agrees with classical methods. Kalsi, Lyons and Perez in [14] also conduct experiments that incorporate exogenous information, such as order-flow, into the midprice dynamics, allowing to estimate the optimal trading speed when the closed form solution is unknown, outperforming the Almgren-Chriss benchmark. This is particularly useful since the execution algorithm is also dependent on how much liquidity is in the market, which leads to solutions such as a Volume Weighted Average Price (VWAP) strategy, rather than time weighted. Further discussion and details of other variants including VWAP can be found in [3]. Figure 5 also demonstrates how increasing the parameter \(\phi \) delivers a more urgent execution program. The choice of \(\phi \) is dependent on many things, including the speed of your signal. If you believe the price is about to decrease in the next second, then you need to sell out of your position much quicker than if the price won’t decrease for another 30 seconds into the future, hence this is a parameter that may be tuned at the discretion of the trader.
Fig. 5
Optimal inventory, trading speed and wealth trajectories for the optimal execution program when we are trading a Markovian Brownian motion, for different urgency parameters \(\phi \)
As we have seen in Sect. 3, quantitative traders often have alpha that they can exploit, perhaps through the form of mean-reversion. However, there is a trade off since if we trade too quickly, we will incur a large market impact, but if we trade too slowly, we will be exposed to the risk of missing the price change that we were trying to position for. In Fig. 6, we see the PnL (including market impact/trading costs) of a trading strategy, depends on the speed of mean-reversion. As the mean-reversion speed increases, this is good since the signal becomes less noisy, meaning our signal to noise ratio is higher. However, there becomes a point at which the mean-reversion can be too fast since we are unable to trade it without incurring too large costs. This implies that there exists a sweet spot, in this case a mean-reversion of \(\alpha =0.05\).
Fig. 6
PnL of a trading strategy, depending on the speed of mean-reversion (LHS). An example of the change in inventory when trading with a mean-reverting alpha signal (RHS)
Throughout this chapter, we used the package signatory [15], alongside PyTorch for calculating and performing functionality related to tensors and the signature transform. We also performed some computations with iisignature integrated in numpy. The GitHub repository for this work can be obtained in [10]. Here, we provide an extensive overview and implementation of Signature Trading, both the mean-variance solution and optimal execution solution. We provide examples and functionality in both signatory/PyTorch, as well as iisignature/numpy, for whichever the user prefers.
The signature trading method can be completely self-contained in the sense that it does not require inputted assumptions such as expected returns or covariances and that they are inferred along with characteristics of the whole process within the algorithm itself. For both the mean-variance solution and the optimal execution strategy, once a linear functional \(\ell \) is obtained from past data samples it is straightforward to unravel this into an implementable trading strategy characterised by the number of units to buy/sell at each time point \(t \in [0,T]\). Since the strategy is dynamic, as new data arrives the Sig-Trader will continuously compute the signature and update their position accordingly.
Whilst Sig-Trading is data driven and does not require direct probabilistic assumptions on the underlying model, just like most frameworks, there is versatility when deploying Sig-Trading in practice as there does still remain quantities that need reliably estimating in the fitting procedure. In the mean-variance framework, the expected lead-lag signature can be noisy when calibrated through time and so we leave any questions of robustness (with respect to the fitting procedure) to future ongoing extensions of the Signature Trading project, as there remains question marks over how well defined such a solution is, especially in the context of rapidly changing regimes such as in financial markets. The most straightforward method to estimate the expected signature is a Monte Carlo approach by calculating the empirical expected lead-lag signature. However, there remains a multitude of alternative ways to approach this for some given sample/fitting data, in order to ensure robustness out of sample. This chapter does not directly discuss these approaches but we do point out that any trader’s favourite statistical estimation and training procedures can work in this setting, for example it is possible to weight more to recent fitting data which may be incorporated via a rolling fitted model. Hence, we leave such statistical procedures directly to the discretion of the user and instead provide a framework in which such techniques can be ensembled.
5 Conclusion
Path dependencies, such as non-Markovian data structures or time series exhibiting temporal correlation, are common phenomena in financial data. Momentum and mean-reversion (of assets or signals) are two of the most fundamental features of time series data that a trader can exploit, and these features are inherently dependent on the entire path. However, many traditional techniques used for portfolio optimization are too inflexible to handle such structural complexities in the data and signals. Signature trading strategies, on the other hand, offer a versatile representation of investment strategies and have proven to be powerful tools in pricing, hedging, and optimal execution.
We observe that the advantages of signature trading strategies can, in fact, be extended to more general portfolio optimization problems, as they encompass many common trading styles used in practice (including the aforementioned momentum and mean-reversion). Specifically, in this chapter, we obtained solutions to two of the most fundamental stochastic control problems in finance: the mean-variance criterion and the optimal execution problem. In summary, signature trading provides an alternative to machine learning methods in its ability to handle path-dependent, nonlinear dynamics and signals, while also extending classical solutions in financial stochastic control problems. Overall, these solutions offer more intuition than ML-based methods and provide greater discretion in the fitting process than classical methods, offering a flexible framework that effectively addresses many challenges encountered when working with financial data.
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
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We can define the shuffle product
for all words \({\mathbf {u}}, {\mathbf {v}}\) and letters \({\mathbf {a}}, {\mathbf {b}} \in \mathcal {W}(\mathcal {A}_d)\).
□
Using these computed gradients, we can compute the gradient of the Lagrangian with respect to \(\langle \ell _m, e_{\mathbf {w}} \rangle \) where we define \(\mu ^{\mbox{sig}} = (\mu ^{\mbox{sig}}_1, \dots , \mu ^{\mbox{sig}}_d)^\top \) as and the matrix \(\Sigma ^{\mbox{sig}}\) as
for all \(\mathbf {w},\mathbf {v} \in \mathcal {W}_{N+d+1}^M\) and \(m, n \in \{1,\dots ,d\}\).
Hence, we obtain the solution for \(m \in \{ 1, \dots , d \}, \mathbf {w} \in \mathcal {W}^M_{N+d+1}\). □
for all \(\mathbf {w},\mathbf {v} \in \mathcal {W}_{4}^2\) and \(m, n =1,2\).
where
where the wealth accumulated at timetcorresponding to the trading speed\(\theta \), is defined by and the remaining inventory at timetis defined by The first term of the objective function is clearly our accumulated wealth from liquidating our assets, which is influenced by our trade order size plus market movements, hence the balance that we are trying to strike. The second term refers to the terminal execution if we arrive at time T with remaining inventory, here \(\alpha \geq 0\) is the terminal liquidation penalty parameter (the larger \(\alpha \) then the larger the penalty for holding inventory at the terminal time since this would result in a greater market impact). The final term is a running inventory penalty, since we want to trade as quick as possible in order to transition into the new position. Therefore, the goal of the optimization is to find the signature trading speed, \(\theta ^*\), that maximizes the objective criterion \(V(\theta )\) defined in (22), i.e. we aim to solve 
