Trading Strategies Generated by
Pathdependent Functionals of Market Weights
Abstract
Almost twenty years ago, E.R. Fernholz introduced portfolio generating functions which can be used to construct a variety of portfolios, solely in the terms of the individual companies’ market weights. I. Karatzas and J. Ruf recently developed another methodology for the functional construction of portfolios, which leads to very simple conditions for strong relative arbitrage with respect to the market. In this paper, both of these notions of functional portfolio generation are generalized in a pathwise, probabilityfree setting; portfolio generating functions are substituted by pathdependent functionals, which involve the current market weights, as well as additional boundedvariation functions of past and present market weights. This generalization leads to a wider class of functionallygenerated portfolios than was heretofore possible, and yields improved conditions for outperforming the market portfolio over suitable timehorizons.
Keywords and Phrases: Stochastic portfolios, pathwise functional Itô formula, trading strategies, functional generation, regular functionals, strong relative arbitrage.
1 Introduction
The concept of ‘functionally generated portfolios’ was introduced by Fernholz (1999, 2002) and has been one of the essential components of stochastic portfolio theory; see Fernholz and Karatzas (2009) for an overview. Portfolios generated by appropriate functions of the individual companies’ market weights have wealth dynamics which can be expressed solely in terms of these weights, and do not involve any stochastic integration. Constructing such portfolios does not require any statistical estimation of parameters, or any optimization. Completely observable quantities such as the current values of ‘market weights’, whose temporal evolution is modeled in terms of continuous semimartingales, are the only ingredients for building these portfolios. Once this structure has been discerned, the mathematics underpinning its construction involves just a simple application of Itô’s rule. Then the goal is to find such portfolios that outperform a reference portfolio, for example, the market portfolio.
Karatzas and Ruf (2017) recently found a new way for the functional generation of trading strategies, which they call ‘additive generation’ as opposed to Fernholz’s ‘multiplicative generation’ of portfolios. This new methodology weakens the assumptions on the market model: asset prices and market weights are continuous semimartingales, and trading strategies are constructed from ‘regular’ functions of the semimartingales without the help of stochastic calculus. Trading strategies generated in this additive way require simpler conditions for strong relative arbitrage with respect to the market over appropriate time horizons; see also Fernholz et al. (2018).
Along a different, but related, development, Föllmer (1981) has shown that Itô calculus can be developed ‘path by path’, without any probability structure. Then Dupire (2009), and Cont and Fournié (2010, 2013) introduced an associated functional Itô calculus. This new type of stochastic calculus extends the classical Itô formula to functionals depending on the entire history of the path, not only on its current value. Versions of this pathwise Itô calculus have recently been applied to various fields in mathematical finance, in particular, to stochastic portfolio theory. Schied et al. (2016) developed pathdependent functional version of Fernholz’s ‘master formula’ for portfolios which are generated multiplicatively by functionals of the entire past evolution of the market weights.
In this paper, we generalize both additive and multiplicative functional generation of trading strategies in two ways. First, we use pathwise functional Itô calculus to show that one can construct trading strategies additively and multiplicatively from a functional depending on the entire history of market weights, in a manner completely devoid of probability considerations. The only analytic structure we impose is that the market weights admit continuous covariations in a pathwise sense. The usage of this advanced Itô calculus enables us to construct trading strategies which depend not only on the current market weights, but also on the past market weights. Secondly, we admit generating functionals that depend on an additional argument of finite variation. Introducing a new argument other than the market weights gives extra flexibility in constructing portfolios; see Strong (2014), Schied et al. (2016), Ruf and Xie (2018). We present various types of additional argument, to the effect that a variety of new trading strategies can be generated from a functional depending on it; these strategies yield new sufficient conditions for strong relative arbitrage with respect to the market portfolio.
We also present a new sufficient condition for strong relative arbitrage via additively generated trading strategies. The existing sufficient condition in Karatzas and Ruf (2017) requires the generating function to be ‘Lyapunov’, or the corresponding ‘Gamma functional’ to be nondecreasing. In contrast, the new sufficient condition in this paper depends on the intrinsic nondecreasing structure of the generating function itself. This new condition shows that trading strategies outperforming the market portfolio can be generated from a much richer collection of functions, or of functionals depending on the market weights and on an additional argument of finite variation. We give some interesting examples of such additively generated trading strategies and empirical analysis of them.
This paper is organized as follows: Section 2 presents the pathwise functional Itô calculus that will be needed for our purposes. Section 3 first defines trading strategies and regular functionals, then discusses how to generate trading strategies from regular functionals in ways both additive and multiplicative. Section 4 gives sufficient conditions for such trading strategies to generate strong relative arbitrage with respect to the market. Section 5 builds trading strategies depending on current and past values of market weights, in contexts where the use of the pathwise functional Itô calculus is essential. Section 6 gives some examples of trading strategies additively generated from entropic functionals and corresponding sufficient conditions for strong arbitrage. Section 7 contains empirical results of portfolios discussed in Section 6. Finally, Section 8 concludes.
2 Pathwise functional Itô calculus
In what follows, we let be a valued continuous function, representing a dimensional vector of assets whose values change over time, and is the value of the ^{th} asset at time . As in Schied et al. (2016), we require that the components of admit continuous covariations in the pathwise sense with respect to a given, refining sequence of partitions of . The sequence is such that each partition satisfies , for each , as well as , and the mesh size of decreases to zero on each compact interval as . We fix such a sequence of partitions for the remainder of this paper. We denote by the successor of a given in , i.e.,
(2.1) 
With this notation, we define the pathwise covariation of and on the interval , denoted by for any , as the limit of the sequence
We assume that this limit is finite, and that the resulting mapping is continuous. We also define the pathwise quadratic variation of by as usual.
Here, we need to emphasize that the existence of the pathwise covariations and quadratic variations for the components of depends heavily on the choice of the sequence of partitions. Example 5.3.2. in Cont (2016), and the arguments following this example, illustrate this fact. Also, we note that the existence of the pathwise covariations and quadratic variations is required for Itô’s formula to hold in a pathwise sense.
We will consider only the finite timehorizon case, so we fix for the remainder of this paper. For an open subset of a Euclidean space, we denote by the space of right continuous valued functions with left limits on . As usual, we denote by the space of continuous valued functions; whereas stands for the space of those functions in which are of bounded variation, and stands for the space of continuous functions in .
Let be an open set of , and an open subset of . Here, is the dimension of our vector function mentioned earlier, and is the dimension for some additional vector function of finite variation on compact timeintervals. For in , and for in , we denote the distance
Next, we state a pathwise functional Itô formula, as well as some definitions and notation based on Appendix of Schied et al. (2016).
2.1 Nonanticipative functional
A functional is called nonanticipative, if holds for each . Here we denote by the function, or “path”, stopped at , for any given path defined on . We use the term “path” instead of function, to emphasize that it represents an evolution of value which changes over time. We present some concepts regarding a nonanticipative functional , acting on a space of such paths, as follows:

is called leftcontinuous if, for any given and for any given , there exists such that, for all and with , we have

is called boundednesspreserving if, for any given compact sets and , there exists a constant such that
holds for all .
2.2 Horizonal and vertical derivatives
We present now two notions of differentiability for a given nonanticipative functional . Intuitively, the first of these notions, horizontal differentiation, concerns the arguments that correspond to functions of bounded variation; we set here . The second notion, vertical differentiation, concerns the arguments that correspond to general, right continuous functions with left limits.
Definition 2.1 (Horizontal derivative).
A nonanticipative functional is called horizontally differentiable if, for each , the limits
exist and are finite. We use here the convention . Then, the horizontal derivative of at is given by the dimensional vector
and the map
defines a nonanticipative vectorvalued functional called the horizontal derivative of , with the convention .
We note that our definition of horizontal derivative is based on the lefthand limit as in Schied et al. (2016), and is different from that of Dupire (2009) and Cont and Fournié (2010). Thus, only the past evolution of the underlying path is relevant for this definition, while no assumptions on future values are imposed.
Now let us fix , and , and define the vertical perturbation of the stopped path , as the rightcontinuous path obtained by shifting the value of the path stopped at by the amount on the interval , i.e.,
Definition 2.2 (Vertical derivative).
A nonanticipative functional is called vertically differentiable at if the map is differentiable at . In this case, the i^{th} vertical partial derivative of the functional at is defined as
Here is the i^{th} unit vector in , and the last equality holds because is nonanticipative. The corresponding gradient is denoted by
and is called the vertical derivative of at . If is vertically differentiable at every triple in , then the mapping
defines a nonanticipative functional with values in , called the vertical derivative of .
We can iterate the operations of horizontal and vertical differentiation, to define higherorder horizontal and vertical derivatives as long as the functional admits horizontal and vertical derivatives. In particular, we define the mixed vertical derivatives
Definition 2.3.
We denote by the set of all nonanticipative functionals which satisfy the following conditions.

is continuous at fixed times , uniformly in over compact intervals. That is, for all and , there exists such that for all with , we have

is jtimes horizontally differentiable and ktimes vertically differentiable.

and all its horizontal and vertical derivatives are leftcontinuous and boundednesspreserving.
2.3 Pathwise functional Itô formula
We are in a position now, to state the celebrated functional changeofvariable formula, in the form of Schied et al. (2016) or Schied and Voloshchenko (2016). The proof can be found in Schied and Voloshchenko (2016). As before, we let be a fixed refining sequence of partitions of , and suppose a continuous function admits finite covariations , along the sequence of partitions .
Theorem 2.4 (Pathwise functional Itô formula).
Let and be given functions, recall the notation of (2.1), define the th piecewiseconstant approximation of by
and let . Then, for any functional , the pathwise Itô integral along , namely
exists; and with , we have the expansion
3 Trading strategies generated by pathdependent functionals
As in the previous section, let be a valued continuous function which admits continuous covariations with respect to a refining sequence of partitions of and be an additional vector function of finite variation. For the purpose of this section, the components of will denote the value processes of tradable assets, and eventually stand for the vector of market weights in an equity market. At the same time, the components of will model the evolution of an observable, but nontradable quantity related to the market weights. We have the following definition of trading strategy with respect to the pair in the manner of Karatzas and Ruf (2017).
Definition 3.1 (Trading strategies).
For the pair of a dimensional function and an dimensional function , suppose that is a dimensional function with a representation
for a vector of nonanticipative functionals, for which we can define an integral with respect to ; we write , to express this. We shall say that is a trading strategy with respect to if it is ‘selffinanced’, in the sense that
(3.1) 
holds. Here and in what follows,
(3.2) 
denotes the value process of the strategy at time .
The interpretation here, is that stands for the “number of shares” invested in asset at time . If is the price of this asset, then is the dollar amount invested in asset at time , and is the total value of investment across all assets.
The preceding Itô formula in Theorem 2.4 suggests that integrands of the special form , for some nonanticipative functional , play a very important role for integrators that admit finite covariations , along an appropriate nested sequence of partitions. This gives rise to the following definition.
Definition 3.2 (Admissible trading strategy).
Let be a dimensional function in , and be an dimensional function in . A dimensional trading strategy in is called an admissible trading strategy for the pair , if there exists a nonanticipative functional in the space , such that
(3.3) 
If is an admissible trading strategy for , the last integral of (3.1) above can be viewed as either the usual vector Itô integral (when is a continuous vector semimartingale in a probabilistic setting), or as the pathwise Itô integral (in the context of our Theorem 2.4).
In the following, we will define a regular functional for the dimensional continuous function and the dimensional function in in a manner similar to that of Karatzas and Ruf (2017).
Definition 3.3 (Regular functional).
We say that a nonanticipative functional in is regular for the pair of a dimensional continuous function and a function , if the continuous function
(3.4) 
has finite variation on compact intervals of . Here, is the function of (3.3) right above, with components
(3.5) 
Remark 3.4.
In order to define a pathwise functional Itô integral and be able to use pathwise functional Itô formulas, we need a sufficiently smooth (in general, at least ) functional, and an integrand which can be cast in the form of a vertical derivative of this functional. Thus, thanks to the above definition, we can always apply the pathwise functional Itô formula (Theorem 2.4) to the functional as in Definition 3.3 above, and get another expression for the socalled “Gamma functional” in (3.4); namely,
(3.6) 
Here and are, respectively, the horizontal derivative and the secondorder vertical derivative of at .
The difference in Definition 3.3 here, with Definition 3.1 of Karatzas and Ruf (2017), should be noted and stressed. In Karatzas and Ruf (2017), the integrand need not be the form of ‘gradient’ of a regular function . Here, we need the special structure of (3.5) for the integrand; this is the “price to pay” for being able to work in a pathwise, probabilityfree setting, without having to invoke the theory of rough paths.
3.1 Trading strategies depending on market weights
We place ourselves from now onward in a frictionless equity market with a fixed number of companies. For an open set with , we also consider a vector of continuous functions where represents the capitalization of the company at time . Here we take and allow to vanish at some time for all , but assume that the total capitalization does not vanish at any time . Then we define another vector of continuous functions that consists of the companies’ relative market weights
(3.7) 
We also assume that the components of admit finite covariations , along a nested sequence of partitions of . In the following, we will consider only regular functionals of the form which depend on the vector of market weights and on some additional function . Examples of such functions appear in (4.3), (4.4).
Remark 3.5.
In order to simplify the expression of (3.6) and to do concave analysis in a manner analogous to that of Karatzas and Ruf (2017), we can make depend only on the function . For example, if we consider the Gibbs entropy function and set , elementary computations show that the first term on the righthand side of (3.6) vanishes and we obtain ; this coincides with Example 5.3 of Karatzas and Ruf (2017).
3.2 Additively generated trading strategies
We would like now to introduce an additivelygenerated trading strategy, starting from a regular functional. For this, we will need a result from Karatzas and Ruf (2017). For any given functional which is regular for the pair , where is the vector of market weights and an appropriate function in , we consider the vector with components
(3.8) 
as in (3.5) of the Definition 3.3, and the vector with components
(3.9) 
Here,
(3.10) 
is the “defect of selffinancibility” at time of the integrand in (3.8), and
(3.11) 
is the “defect of balance” at time of the regular functional . By analogy with Proposition 2.3 of Karatzas and Ruf (2017), the vector of (3.9), (3.8) defines a trading strategy with respect to .
Definition 3.6 (Additive generation).
Proposition 3.7.
We can think of in (3.4), (3.6) and (3.12), as expressing the “cumulative earnings” of the strategy around the “baseline” .
Proof.
The proof is exactly the same as the proof of Proposition 4.3 of Karatzas and Ruf (2017), if we change , there, into , . ∎
Remark 3.8.
3.3 Multiplicatively generated trading strategies
Next, we introduce the notion of multiplicatively generated trading strategy. We suppose that a functional is regular as in Definition 3.3 for the pair , where is the vector of market weights and is some additional function in , and that is locally bounded. This holds, for example, if is bounded away from zero. We consider the vector function defined by
(3.17) 
in the notation of (3.4), (3.8) for . The integral here is welldefined, as is assumed to be locally bounded. Moreover, we have , since from Definition 3.1, and the exponential process is again locally bounded. As before, we turn this into a trading strategy by setting
(3.18) 
in the manner of (3.9), and with , defined as in (3.10) and (3.11).
Definition 3.9 (Multiplicative generation).
Proposition 3.10.
Proof.
We follow the proof of Proposition 4.8 in Karatzas and Ruf (2017), using the pathwise functional Itô formula instead of the standard Itô formula. If we denote the exponential in (3.19) by , the pathwise functional Itô formula (Theorem 2.4) yields
Here, the second equality uses the expression in (3.6), and the last equality relies on Proposition 2.3 of Karatzas and Ruf (2017). Since (3.19) holds at time zero, it follows that (3.19) holds at any time between and . The justification of (3.20) is exactly the same with that of Proposition 4.8 in Karatzas and Ruf (2017). ∎
Remark 3.11.
4 Sufficient conditions for strong relative arbitrage
We consider the vector of market weights as in (3.7). For a given trading strategy with respect to the market weights , let us recall the value process from Definition 3.1. As we will always consider trading strategies with respect to market weights, we write instead of from now on. For some fixed , we say that is strong relative arbitrage with respect to the market over the timehorizon , if we have
(4.1) 
along with
(4.2) 
Remark 4.1.
The notion of strong relative arbitrage defined above does not depend on any probability measure and is slightly more strict than the existing definition of strong relative arbitrage. The classical definition involves an underlying filtered probability space, and posits that the market weights should be continuous, adapted stochastic processes on this space. Also, there are two types of classical arbitrage; relative arbitrage and ‘strong’ relative arbitrage as in Definition 4.1 of Fernholz et al. (2018). In this old definition, an underlying probability measure is essential in defining this ‘weak’ version of relative arbitrage. However, if we say that is strong relative arbitrage when (4.2) holds for ‘every’ realization of , instead of ‘almost sure’ realization of , the notion of strong relative arbitrage can be established without referring to any probability structure. Since we constructed trading strategies in a pathwise, probabilityfree setting, the ‘strong’ version of relative arbitrage is a more appropriate concept of arbitrage for this paper, and we adopt the above strict definition of strong relative arbitrage from now on.
The value process of a trading strategy generated functionally, either additively or multiplicatively, admits a quite simple representation in terms of the generating functional and the derived functional as in (3.12) and (3.19). This simple representation provides in turn nice sufficient conditions for strong relative arbitrage with respect to the market; for example, as in Theorem 5.1 and Theorem 5.2 of Karatzas and Ruf (2017). In this section, we find such conditions on trading strategies generated by a pathwise functional which depends not only on the vector of market weights , but also on an additional finitevariation process related to . We also give a new sufficient condition leading to strong relative arbitrage for additively generated trading strategies, which is different from Theorem 5.1 of Karatzas and Ruf (2017).
Until now, we have not specified the dimensional function , so it is time to consider some plausible candidates for this finite variation function. A first suitable candidate would be the dimensional vector
(4.3) 
of quadratic variation of market weights. We can also think about a more general candidate; namely, the valued covariation process of market weights. Here, is the notation for symmetric positive matrices, and we will use double bracket to distinguish this dimensional vector from (4.3): namely,
(4.4) 
The advantage of choosing as in (4.4), is that we can match the integrators of the two integrals in (3.6), and the resulting expression for can then be cast as one integral.
There are many other functions of finite variation which can be candidates for the process . We list some examples below:

The moving average of defined by

The running maximum of the market weights with the components , and the running minimum of the market weights with the components for .

The local time process of the continuous semimartingale at the origin, for . We call this process the “collision local time of order ” for the ranked market weights