Journal of Risk

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Risk-averse dynamic arbitrage in illiquid markets

Somayeh Moazeni, Ricardo A. Collado and Andy Zhang

  • To rule out dynamic arbitrage, trades adapted to the price process filtration should be searched.
  • The concept of risk-averse dynamic arbitrage using a time-consistent dynamic risk measure is introduced.
  • Sufficient conditions are established to certify no-dynamic arbitrage by only searching in the space of $\mathcal F_0$-measurable admissible round-trip trades.

Arguments on the existence of dynamic arbitrage and price manipulation strategies are often invoked to guide the modeling of price impacts of large trades. We revisit the concept of dynamic arbitrage in illiquid markets in the presence of time-varying stochastic price impact functions and a broad class of market price dynamics. We first establish a sufficient condition under which searching in the space of F0-measurable admissible round-trip trades is enough to attain a no-dynamic arbitrage characterization. This result both simplifies the identification of price impact structures that rule out dynamic arbitrage and supports analyses in some of the existing literature, where its assessment has been limited to searching in a set of F0-measurable round-trip trades. For time-varying stochastic linear price impact functions, we show that this condition is necessary and sufficient for the absence of dynamic arbitrage. The present quantitative analysis implies that a trader’s opinion concerning the existence of dynamic arbitrage opportunities for a price impact model depends on their belief about expected future price changes and expected future price impacts, which can be revised over time due to the collection of new information. This motivates us to let the existence of such arbitrage opportunities depend not only on the trader’s belief about expected price movements but also on their risk attitude. We thus introduce the concept of risk-averse dynamic arbitrage using a general time-consistent dynamic risk measure and a risk-aversion threshold level. Similar sufficient conditions are studied under which searching in the space of static round-trip trades enables us to conclude on no-risk-averse dynamic arbitrage.

1 Introduction

Illiquid financial markets are characterized by the material cost of security execution within a short time frame and its effects on asset prices and returns (see, for example, Amihud and Mendelson 2010; Amihud et al 2013; Foucault et al 2013). Limited liquidity is characterized by the correlation between an incoming order and the subsequent price change, referred to as price impact. Since this is an indirect cost, price impact is often difficult to measure and model (Almgren 2010; Moazeni et al 2010, 2013). Arguments about market efficiency, and the consequent absence of trading strategies that enable the manipulation of assets prices in a financially beneficial way, can then be invoked to guide modeling price impact; for a broad discussion on price manipulation, see, for example, Kyle and Viswanathan (2008).

Various concepts of arbitrage have been studied to address the existence of price manipulation trading strategies and, consequently, price impact model specifications. Huberman and Stanzl (2004) build on the concept of a δ-arbitrage opportunity, proposed by Ledoit (1995) as a portfolio with a Sharpe measure strictly greater than δ. Here, the value of δ is determined a priori by the economist.11Bernardo and Ledoit (2000) generalized the concept of δ-arbitrage using a gain–loss ratio and referred to an investment opportunity with the infinite gain–loss ratio as an arbitrage. For normally distributed returns, their gain–loss ratio constraint is equivalent to a Sharpe ratio restriction. Huberman and Stanzl (2004) refer to a finite sequence of admissible trades, starting with a zero initial holding and ending at a zero position, as a round-trip trade. They extend the notion of δ-arbitrage to the stronger concept of quasi-arbitrage, defined as a sequence of round-trip trades that produces infinite expected execution payoffs with an infinite Sharpe ratio. A quasi-arbitrage is a δ-arbitrage. Employing this arbitrage definition, Huberman and Stanzl (2004) show that only permanent price impact functions, linear in amount traded, can rule out the availability of quasi-arbitrage. Gatheral (2010) studies (bounded) price manipulation trading strategies as a sequence of round-trip trades with a strictly negative expected execution cost or, equivalently, a strictly positive expected execution payoff. He refers to the existence of a price manipulation trading strategy as dynamic arbitrage and argues that permanent price impact must be linear in volume traded in order to rule out dynamic arbitrage.22We note that this definition is not fully consistent with the common concept of dynamic arbitrage, defined as an arbitrage opportunity that involves trading instruments in the future, contingent on market states (Allingham 1991; Wilmott 1998; Kondor 2009). Fruth et al (2014) investigate the occurrence of dynamic arbitrage for a limit order book model with linear price impact functions under different parameter assumptions. Gueant (2014) shows that a specific class of nonlinear price impact functions does not admit dynamic arbitrage. A limit order book model with a nonlinear price impact function that excludes the availability of dynamic arbitrage is studied in Alfonsi and Schied (2010). All of these concepts belong to the class of pseudo-arbitrage (as opposed to pure arbitrage) in the sense that they can be profitable but are not risk-free (Damodaran 2012).

Establishing the absence of an arbitrage opportunity involves solving a multistage stochastic optimization problem. Given the uncertainty in future security prices and the time requirement for implementing round-trip trades, the optimization should be over the set of admissible trading strategies that are contingent on market states. More precisely, the search is over the set of trading strategies that are t-measurable, where t is the sigma algebra generated by stochastic processes in the model. For example, to verify the existence of dynamic arbitrage studied in Gatheral (2010), one needs to maximize the expected execution payoff over the set of round-trip policies. Restricting our search to the smaller set of 0-measurable round-trip trades may yield misleading conclusions about the absence of dynamic arbitrage, since maximizing the expected execution payoff over the set of 0-measurable trading strategies may result in a negative optimal objective value. However, there may exist some t-measurable round-trip trade with a positive expected execution payoff. Still, some of the existing literature, such as Gueant (2014), explicitly searches in the set of 0-measurable trades (static strategies) to prove the absence of dynamic arbitrage. This motivated us to explore the impact of the market adaptability assumption of the price manipulation strategies on a trader’s belief regarding the absence of dynamic arbitrage under different risk attitudes.

In this paper, we consider two general classes of market price dynamics that can exhibit a broad range of stochastic market price processes. Following the discussion in Fruth et al (2014) that liquidity is not time independent, we let price impacts be time varying and stochastic. For this general setting, we first establish a sufficient condition under which searching in the space of 0-measurable admissible round-trip trades is enough to obtain a no-dynamic arbitrage characterization. More precisely, this sufficient condition involves the knowledge that, at the initial time, the expected market price change conditioned on the information set at any time t is zero. Here, no assumption on the linearity or nonlinearity of the permanent or temporary price impact functions is made.

This result simplifies the effort of verifying dynamic arbitrage and consequently characterizing the structure of price impact functions. We then show that for linear time-varying stochastic price impact functions this condition is both necessary and sufficient for no-dynamic arbitrage. This statement is valid for the two general classes of market price dynamics, which accommodate a broad variety of stochastic price processes. Our analysis implies that a trader’s opinion on the existence of dynamic arbitrage opportunities for a price impact model depends on their belief regarding expected future price changes and expected future price impacts, which can be updated over time as new observations are collected and more accurate estimations of conditional expectations are attained. In fact, these estimations can be different from those computed at the beginning of the trading time horizon. This motivated us to let the existence of such arbitrage opportunities depend on not only the investor’s belief concerning expected price movements but also the trader’s risk attitude.

We extend the concept of dynamic arbitrage of Gatheral (2010) to risk-averse dynamic arbitrage in a dynamic setting using time-consistent dynamic risk measures (Detlefsen and Scandolo 2005; Ben-Tal and Teboulle 2007; Föllmer and Weber 2015). A dynamic risk measure is a sequence of conditional risk measures {ρt,T}t=1T, each of which takes into account the information available at the time of risk assessment and adapts to the underlying filtration (Ruszczyński and Shapiro 2006a, 2006b; Ruszczyński 2010; Acciaio and Penner 2011). For more details on dynamic risk measures, the reader is referred to Cvitanić and Karatzas (1999), Ogryczak and Ruszczyński (1999, 2001), Riedel (2004), Boda and Filar (2006), Cheridito et al (2006), Frittelli and Scandolo (2006), Ruszczyński and Shapiro (2006a, 2006b), Klöppel and Schweizer (2007), Bion-Nadal (2008), Ruszczyński (2010) and Acciaio and Penner (2011). The essence of time consistency is to guarantee that a trading policy preferred from a future perspective is also preferred from the trader’s current point in time, and that, consequently, decisions are not contradictory over time. This property is crucial to the development of a dynamic theory of risk-aversion and to obtain recursive evaluations avoiding backtracking; see Defourny et al (2008) and Rudloff et al (2014) for examples of pathological cases leading into backtracking that risk-averse dynamic systems face in the absence of time consistency. Time consistency enables us to use the dynamic programming principle to find trading strategies with certain risk levels (Boda and Filar 2006; Shapiro 2009; Ruszczyński 2010; Rudloff et al 2014). Examples of consistent dynamic measures of risk include those obtained from recursive evaluations of conditional value-at-risk (CVaR) or recursive evaluations of conditional mean upper semideviation of order r (2014, Chapter 6).

Given a time-consistent dynamic convex risk measure {ρt,T}t=1T and a threshold level δ, we define a risk-averse dynamic arbitrage as a round-trip trading policy π such that ρ1,T(C1π,,CTπ)<δ. Here, Ctπ refers to the execution cost of trading nπ shares over (t-1,t] indicated by the policy π. The pseudo-arbitrage concept of risk-averse dynamic arbitrage is an extension of the (risk-neutral) dynamic arbitrage of Gatheral (2010), in which ρ()=?[] and δ=0. In addition, as will be discussed in Section 4, δ-arbitrage with the gain semi-standard deviation ratio can be cast as risk-averse dynamic arbitrage. The dynamic programming optimality equation can be employed to solve the optimization problem involved in the search for risk-averse dynamic arbitrage opportunities. We establish sufficient conditions under which searching in the set of static round-trip trades can prove the absence of risk-averse dynamic arbitrage.

In summary, this paper contributes to the existing literature on price impact modeling and arbitrage by highlighting the importance of searching in the space of nonanticipatory round-trip trades and traders’ beliefs regarding future conditional expectations as well as introducing the concept of risk-averse dynamic arbitrage.

This paper is organized as follows. Models for price impact and price dynamics are presented in Section 2. The optimization involved in the assessment of risk-neutral dynamic arbitrage is discussed in Section 3. Section 4 introduces the concept of risk-averse dynamic arbitrage, where a sufficient condition for no-risk-averse dynamic arbitrage certification based on static optimization is also established. Conclusions and directions for future research are summarized in Section 5.

2 Market price and price impacts

In a financial market with imperfect liquidity, trading affects asset prices. The relationship between price changes and incoming orders is referred to as price impact, which can be used as an illiquidity measure (Amihud et al 2013). There are short-term effects of inventory on prices, called temporary or transitory price impacts, which only affect the revenue of the trader who initiated the trade. In addition, there are permanent effects on market prices and expected return flows, referred to as permanent price impacts. Both of these impacts are often functions of the volume of assets traded.

Let the m-vector Pt denote the unit market price of a portfolio of m assets at time t. The deterministic initial market price is denoted by P0. We let the price update function be

  Pt=?t(Pt-1)-g~t(nt),   (2.1)

where Pt-1 refers to the (nonnegative) market price before placing the order at time t, and ?t(Pt-1) denotes the market price at time t if the trader had not placed the order nt. Here, nt is the amount traded at time t, and the function g~t() captures the permanent price impact of the trader’s order. A positive nt,i denotes selling the ith asset, and a negative nt,i implies that the ith asset is bought over (t-1,t]. In addition to the effect on the market price, the decision maker’s trade induces a temporary price impact (slippage) on the execution price. The m-vector execution price P~t per share is given by

  P~t=Pt-1-h~t(nt),   (2.2)

where h~t() is the uncertain temporary price impact. We note that the price impact functions (particularly temporary price impact) can be functions of the trading rate, instead of the volume traded, in which case the correction denominators are captured by g~t and h~t. Both g~t(nt) and h~t(nt) have the same sign as nt. It is common in the literature to let the price impact functions be deterministic and constant over time, ie, g~t(nt)=g(nt) and h~t(nt)=h(nt) for some deterministic functions g and h. Fruth et al (2014) argue that liquidity has seasonal patterns and allows for time-dependent deterministic price impacts. Following Fruth et al (2014), and for the generality of our discussion in this paper, we let the price impact functions be time varying and stochastic.

Examples of the functional ?t(Pt-1) in (2.1) include the following additive and multiplicative models:

  ?t(Pt-1) =~t+Pt-1,   (2.3)
  ?t(Pt-1) =(Im+Diag(~t))Pt-1,   (2.4)

where the m-vector ~t is independent of the price Pt-1 and represents a wide range of stochastic processes, such as diffusion, jump diffusion, Hawkes processes, etc, for the market price evolution. In (2.4), Diag(~t) is a diagonal matrix with ~t as its diagonal, while Im denotes the m×m identity matrix.

Price models (2.1) and (2.2) with price dynamics (2.3) or (2.4) provide us with a rich setting to exhibit a variety of models frequently used in the literature on portfolio liquidation and price impact modeling (see, for example, Almgren and Chriss 2000; Huberman and Stanzl 2004; Almgren et al 2005; Huberman and Stanzl 2005; Moazeni et al 2010, 2013, 2016; Foucault et al 2013; Amihud et al 2013; Gueant 2014). For instance, Bertsimas and Lo (1998), Almgren and Chriss (2000) and Almgren et al (2005) consider the additive model (2.3) with

  ~t:=ΣZ~t,   (2.5)

where the components of the l-vector Z~t are independent standard normals and Σ is an m×l volatility matrix of the asset returns. Hence, ?t(Pt-1)=Pt-1+ΣZt. Moazeni et al (2013) study both (2.3) and (2.4) with a jump-diffusion model as described below:

  ~t:=α+J~t+ΣZ~t,   (2.6)

where the jump term J~t is a summation of two compound Poisson processes capturing the price impacts of other concurrent buy-or-sell large trades. Here, the time-independent deterministic α can be interpreted as the expected price change due to small trades. Gatheral (2010) considers model (2.3), where t follows an arithmetic random walk whose drift is the decayed accumulation of price impacts of previous trades. One may apply the multiplicative model (2.4), where ~t is defined by the market beta β and Jensen’s alpha α (Jensen 1968) of the m-assets, as can be seen below:

  ~t:=αt+rtf?m+βt(r~tM-rtf)+ΣtZ~t,   (2.7)

where βt=(βt,1,,βt,m)T and αt=(αt,1,,αt,m)T. It is expected that αt is 0 (see Fama and French 2004). Here, rtf indicates the (deterministic) risk-free interest rate, r~tM is the market return and ?m denotes the m-vector of all ones.

To verify the absence of dynamic arbitrage using a set of T discrete trades over [0,T], and assuming the market price dynamics (2.1) started at the initial price P0 under execution price model (2.2), one faces solving the following multistage stochastic dynamic optimization problem to maximize the total amount received at the end of T trades over the set of all admissible round-trip trades:

  V(P0):=maxn1,,nT?,ntist-1-measurable?(t=1TntTP~t)   (2.8)

such that t=1Tnt=0. Here, ?m is the set of admissible round-trip trades, and the information at time t is given by the sigma-algebra t-1={Pt-1,xt-1}, where xt-1 is the portfolio position before time t.

Given price impact models g~t and h~t, a nonpositive value V(P0)0 provides us with a certificate for the absence of dynamic arbitrage, ie, there exists no price manipulation strategy with a strictly positive expected execution payoff. In contrast, V(P0)>0 ensures the presence of dynamic arbitrage.

Depending on the number of assets in the portfolio, the modeling assumptions on the price impact functions, and the underlying market and execution prices, solving problem (2.8) can be a challenging task due to the curse of dimensionality (see, for example, Bertsekas and Tsitsiklis 1996; Powell 2011). Alternatively, one may choose to make an approximation by restricting the search for the price manipulation strategies to a smaller set of admissible round-trip strategies – such as only 0-measurable trades – and solving the following, computationally cheaper, optimization problem:

  V^(P0):=maxn1,,nT?,ntis0-measurable?(t=1TntTP~t)   (2.9)

such that t=1Tnt=0. This is a one-stage numerical optimization problem that can be solved using mathematical programming techniques (see, for example, Nocedal and Wright 2006).

In general, V^(P0)V(P0). Therefore, while V^(P0)>0 implies that a dynamic arbitrage opportunity exists, V^(P0)0 cannot certify the absence of dynamic arbitrage possibilities. Therefore, limiting the search to the set of 0-measurable round-trip trades may mislead us about identifying price impact structures that rule out dynamic arbitrage. Gueant (2014) solves (2.9) and concludes that a class of nonlinear permanent price impact functions can also exclude dynamic arbitrage. In the next section, we shed some light on the validity of this choice, discussing the difference between the solution sets of these two problems and their suitability in proving no-dynamic arbitrage.

3 Absence of (risk-neutral) dynamic arbitrage

We first present a sufficient condition under which searching in the set of 0-measurable round-trip trades can ensure the absence or presence of dynamic arbitrage. This result is valid for general time-dependent stochastic price impact functions and market price dynamics.

Proposition 3.1.

Let the market price dynamics and the execution price model be given by (2.1) and (2.2), respectively. Assume that the price impact functions g~t() and h~t() satisfy

  ?[h~t(nt)t-1]=?[h~t(nt)xt-1],  
  ?[g~t(nt)t-1]=?[g~t(nt)xt-1].   (3.1)

Let

  ?[?t(Pt-1)Pt-1]=Pt-1.   (3.2)

Then, V^(P0)=V(P0), ie, the absence of an 0-measurable round-trip trade with positive expected execution cost provides us with a certificate for the absence of dynamic arbitrage.

A proof of Proposition 3.1 is given in Appendix B (available online).33Proposition 3.1 can be seen as an extension of Moazeni et al (2013), Theorem 3.1 to general time-dependent stochastic price impact functions. In Theorem 3.1, the price impact functions are assumed to be deterministic, time-independent linear functions of the volume traded.

For both market price dynamics (2.3) and (2.4), (3.2) is reduced to ?t[~t]=0. Therefore, using the tower property of expectation and the equalities

  t=lTnt=xl-1andt=1Tnt=x0=0,  

the numerical optimization problem (2.9) is given by

  V^(P0) =maxn1,,nT?,ntis0-measurable(-t=1T-1xtT?[g~t(nt)]-t=1TntT?[h~t(nt)])  
    =-minn1,,nT?,ntis0-measurablet=1T{xtT(?[g~t(xt-1-xt)-h~t(xt-1-xt)])  
                       +xt-1T?[h~t(xt-1-xt)]}.  

When the round-trip trade {nt=0}t=1T is an admissible strategy for (2.9), V^(P0)0. Therefore, under the assumptions in Proposition 3.1, price impact functions g~t and h~t imply no-dynamic arbitrage (V(P0)0) if and only if

  minx0,x1,,xT?x,x0=0,xT=0t=1T{xtTkt(xt-1-xt)+xt-1Tht(xt-1-xt)}=0,   (3.3)

where kt(x):=?[g~t(x)-h~t(x)], ht(x):=?[h~t(x)] and ?xm is a feasible set for positions xt, corresponding to the admissibility set ? for amounts traded nt.

Example 3.2.

Consider a simple case in which T=2. Therefore, for any trading strategy {nt?}t=1,2 where n1+n2=0, we have

  ?[n1TP~1+n2TP~2P0] =?[n1TP0-n1Th~1(n1)+n2T?1(P0)-n2Tg~1(n1)-n2Th~2(n2)P0]  
    =n1T(P0-?[?1(P0)P0])+n1T?[-h~1(n1)+h~2(-n1)+g~1(n1)P0].  

Hence, when ?[?1(P0)P0]=P0, a dynamic arbitrage opportunity exists if and only if the second term is positive for some n1, ie, the total temporary price impact cost is expected to be recovered through the impact on the market price. This can hold even for nonlinear price impact functions.

The nonlinear price impact function in Gueant (2014),

  g~t(nt)=f(|xt|)nt=f(|xt-1-nt|)nt,   (3.4)

for some deterministic function f, satisfies (3.1). In addition, Gueant (2014) models the market price dynamics by an arithmetic Brownian motion with zero drift, which satisfies (3.2). Hence, Proposition 3.1 implies that for the setting in Gueant (2014) it is sufficient to search in the set of 0-measurable round-trip trades to verify the existence of dynamic arbitrage.

Linear price impact functions have been frequently used in the literature (see, for example, Bertsimas and Lo 1998; Almgren and Chriss 2000; Alfonsi and Schied 2010; Obizhaeva and Wang 2013). Given time-varying stochastic matrixes G~t and H~t, whose distributions do not depend on the market price Pt-1, the linear price impact functions

  h~t(nt)=H~tntandg~t(nt)=G~tnt   (3.5)

also satisfy (3.1). Therefore, according to Proposition 3.1, (3.2) implies that searching in the set of 0-measurable round-trip trades is enough to certify no-dynamic arbitrage. In fact, for such linear price impact functions, (3.2) yields a stronger result: for linear price impact model (3.5), (3.2) implies the absence of dynamic arbitrage. This is more formally summarized in the following corollary.

Corollary 3.3.

Let assumptions (3.1) and (3.2) in Proposition 3.1 hold. In addition, assume that the price impact functions are linear as in (3.5) and the objective function of (2.9) is concave. Then, no dynamic arbitrage opportunities exist.

This result can be derived via Proposition 3.1, based on which it is sufficient to search in the set of 0-measurable round-trip trades and to directly solve the numerical optimization problem (2.9) or the minimization (3.3). For linear price impact functions and when ?=m, (3.3) is reduced to an unconstrained quadratic programming problem (see Moazeni et al 2010) with an optimal value that equals zero, ie, V(P0)=0. This consequently guarantees the absence of dynamic arbitrage. We leave it for future research to apply the problem formulation (3.3) and the result for linear price impact functions to prove no-dynamic arbitrage for nonlinear price impact functions that can be bounded by linear or piecewise linear functions.

Alternatively, Corollary 3.3 can be proved directly as in part (b) of Propositions A.1 and A.2 in Appendix A (available online). Part (a) of Propositions A.1 and A.2 shows that the best expected execution payoff of a round-trip trade equals V(P0)=E1 for the market price dynamics model (2.3) and is given by V(P0)=P0TA1P0 for the multiplicative model (2.4), where the matrixes E1 and A1 are defined in Appendix A. Part (b) of Propositions A.1 and A.2 shows that, when (3.2) holds, E10 and A10, and consequently V(P0)0, ie, there is no dynamic round-trip trade with a strictly positive expected execution payoff. This result is general and does not rely on any assumption regarding the stochastic model of ~t; hence, it is applicable to a broad range of market price model specifications.

Next, we show that for linear price impact functions, (3.2) not only suffices to ensure no-dynamic arbitrage but is also necessary. This is formally stated in the following proposition.

Proposition 3.4.

Assume T2 and ?[?t(Pt-1)Pt-1]Pt-1 for some t in {1,,T-1} (or equivalently ?[~tt-1]0 for some t). Consider the linear price impact model (3.5) and let the market price dynamics be

  1. (i)

    as in ( 2.3 ), while the matrixes {Qt}t=2T – defined in (A.15) in the appendix – are positive definite, or

  2. (ii)

    as in ( 2.4 ), while the matrixes {Qt}t=2T – defined in (A.1) in the appendix – are positive definite, and

      ?m?mT+cov[~tt-1]0.   (3.6)

Then, dynamic arbitrage exists, ie, there exists an t-measurable round-trip trade with a strictly positive expected execution payoff.

A proof of Proposition 3.4 is given in Appendix C (available online). Hence, for the linear price impact model (3.5), there is no-dynamic arbitrage if and only if (3.2) holds. Note that the statement in Proposition 3.4 does not rely on a specific assumption on the market price dynamics; hence, it is applicable to a broad range of stochastic models for ~t.

From Proposition 3.4, ?t[~t]0 at some time period implies the existence of a t-measurable round-trip trade. When ?0[~t]0, a static round-trip trade with a strictly positive expected benefit also exists. This can be formally investigated by limiting the trading activity to only three time steps. In this case, the maximization problem (2.9) is reduced to the following quadratic optimization problem on the 2m-vectors n:=(n1,n2),

  max(n1,n2)2m12nT(a+aT-(Θ+ΘT)-Θ+aT-ΘT+a-(Θ+ΘT))n-((?0[2~+1~]+?0[1~]?0[2~]).*P0(?0[Im+1~]?0[2~]).*P0)Tn,  

where Θ=defH+HT-G, a:=Diag(?0[2~])G and A.*B denotes the component-wise (Hadamard) product of the matrixes A and B. If the Hessian of this quadratic function is negative definite or is not negative semidefinite, a nonzero value of ?0[~1]?0[~2] implies that an 0-measurable round-trip trade with a strictly positive expected execution payoff exists. In the former case, the 0-measurable price manipulation strategy uniquely exists, while in the latter case there are infinitely many 0-measurable price manipulation strategies. A round-trip trade over three time steps with a strictly positive expected execution payoff implies a dynamic arbitrage opportunity for longer trading time horizons.

When the initial estimations of the expected price changes equal zero, ie, ?[~t0]=0 for all t, but after collecting more observations the trader estimates that ?[~tt-1]0, searching in the set of static round-trip trades will erroneously evince the absence of dynamic arbitrage, ie, the best expected execution payoff of static round-trip trades equals zero, while a dynamic round-trip trade with a strictly positive expected benefit does actually exist.

In summary, the existence of dynamic arbitrage is directly related to the trader’s estimation of the expected future price changes conditioned on the price or return distribution at time t. In addition, by collecting more observations, the trader’s belief regarding the expected value of ~t will change over time, perhaps becoming more accurate. Therefore, in order to certify that a price impact model does not permit dynamic arbitrage opportunities, one should search in the set of t-measurable round-trip trades and take into account updated estimations of conditional expectation of ~t over time. Limiting the search to the set of 0-measurable round-trip trades computed at time zero can mislead us to conclude the absence of dynamic arbitrage, when bounded dynamic price manipulation strategies do, in fact, exist.

Since a trader’s opinion on the existence of a dynamic arbitrage opportunity using round-trip trades depends on the trader’s belief about the expected future price changes, it seems natural to let the existence of such arbitrage opportunities depend on the trader’s risk attitude. This is the topic of Section 4, where we extend the concept of dynamic arbitrage in the presence of price impacts to the pseudo-arbitrage concept of risk-averse dynamic arbitrage given a dynamic risk measure.

4 Risk-averse dynamic arbitrage

We start by presenting the definition of dynamic risk measures and introducing our concept of risk-averse dynamic arbitrage in illiquid markets. In our presentation, we focus on execution costs or losses of trading strategies.

Let (Ω,,?) be our probability space with the filtration 0T, defined by the market price process, where 0={Ω,}. Define the spaces of execution costs t=p(Ω,t,?), p[1,+), t=0,,T, where the space 0 is associated with , and let t,T=t××T. Consider the adapted sequence of random variables Ctt-1, t=1,,T, where Ct is the stage-wise execution cost. The following definitions are from Ruszczyński (2010).

A conditional risk measure is a mapping ρt,T:t,Tt, 1tT, such that

  ρt,T(Z)ρt,T(W)for all Z,Wt,TsuchthatZW.   (4.1)

One can define a broader family of conditional-risk measures by setting

  ρτ,θ(Cτ,,Cθ)=ρτ,T(Cτ,,Cθ,0,,0),1τθT.  

Similarly, the one-step conditional risk measures ρt:t+1t, t=1,,T-1 are defined by

  ρt(Ct+1):=ρt,t+1(0,Ct+1).  

Let {ρt,T}t=1T-1 be a time-consistent conditional risk measure satisfying the following conditions:

  ρt,T(Ct,Ct+1,,CT) =Ct+ρt,T(0,Ct+1,,CT),   (4.2)
  ρt,T(0,,0) =0   (4.3)

for all Ct,T and all t=1,,T-1.

A sequence of conditional risk measures {ρt,T}t=1T-1 is called a dynamic risk measure. From now on, we only consider time-consistent dynamic risk measures, ie, a sequence of conditional risk measures satisfying properties (4.1)–(4.3). Since these dynamic risk measures are completely specified by the one-step conditional risk measures ρt, t=1,,T-1, we simplify notation by denoting them as {ρt}t=1T-1. A dynamic risk measure {ρt}t=1T-1 is convex (coherent) if its one-step conditional risk measures are convex (coherent) (see Ruszczyński 2010; Acciaio and Penner 2011). Recursive evaluations of CVaR or recursive evaluations of conditional mean upper semideviation of order r (Shapiro et al 2014, Chapter 6), ie,

  ρt(Ct+1):=?[Ct+1t]+κ(?[(Ct+1-?[Ct+1t])+rt])1/r,  

where r[1,) is a fixed parameter and (z)+=max{z,0}, are examples of consistent dynamic measures of risk. In the rest of this section, we assume that the dynamic risk measure {ρt}t=1T-1 is coherent.

Ruszczyński (2010) shows that the risk measure follows a recursive evaluation, namely

  ρt,T(Ct,,CT) =Ct+ρt(Ct+1+ρt+1(Ct+2++ρT-2(CT-1+ρT-1(CT)))),   (4.4)

where ρt, t=t,,T-1 is a one-step conditional risk measure. Moreover, if the one-step conditional risk measures are coherent (see Artzner et al (1999) and Shapiro et al (2014), Chapter 6 for an in-depth treatment of the subject), then the recursive evaluation is given by

  ρt,T(Ct,,CT)=ρt(ρt+1(ρT-2(ρT-1(Ct+Ct+1++CT-1+CT)))).   (4.5)

For a round-trip trading policy π, let Ctπ denote the execution cost of selling nπ shares over (t-1,t]. For a given coherent time-consistent dynamic risk measure {ρt}t=1T-1, the evaluation of the risk of losses in the sequence of execution costs given by the round-trip trading policy π is

  π: =ρ1,T(C1π,,CTπ)  
    =C1π+ρ1(C2π+ρ2(C3π++ρT-2(CT-1π+ρT-1(CTπ))))  
    =-(n1π)TP~1π  
      +ρ1(-(n2π)TP~2π+ρ2(-(n3π)TP~3π+  
         +ρT-2(-(nT-1π)TP~T-1π+ρT-1(-(nTπ)TP~Tπ)))).  

If {ρt}t=1T-1 is coherent, then (4.5) is reduced to

  π =ρ1(ρ2(ρT-2(ρT-1(t=1TCtπ))))  
    =ρ~(-t=1T(ntπ)TP~tπ),  

where the composite risk measure is ρ~:=ρ1ρT-1.

We are now ready to define the concept of risk-averse dynamic arbitrage.

Definition 4.1.

Given a time-consistent dynamic risk measure {ρt}t=1T-1 ({ρt}t?) and threshold level δ, the presence of a risk-averse dynamic arbitrage opportunity at threshold level δ refers to the existence of some round-trip trading policy π such that π<δ.

To verify such arbitrage opportunities, one can solve the following dynamic programming problem:

  V(P0):=minn1,,nT?,ntist-measurableρ~(-t=1T(ntπ)TP~tπ)   (4.6)

such that t=1Tntπ=0. Then, V(P0)δ implies no-risk-averse dynamic arbitrage.

When the search for round-trip trades is limited to the space of 0-measurable trades, one solves

  V^(P0):=minn1,,nT?,ntis0-measurableρ~(-t=1T(ntπ)TP~tπ)   (4.7)

such that t=1Tntπ=0. Problems (4.6) and (4.7) can be solved, for example, using the scenario decomposition method of Collado et al (2012).

Analogous to Proposition 3.1, below we provide a sufficient condition under which searching in the set of static round-trip trades is sufficient to conclude that risk-averse dynamic arbitrage does not exist.

We then proceed by establishing a sufficient condition under which searching in the set of 0-measurable round-trip trades is enough to obtain a certification for no-risk-averse dynamic arbitrage. Similar to the (risk-neutral) dynamic arbitrage discussed in the previous section, this sufficient condition implies no-risk-averse dynamic arbitrage for linear price impact functions.

Proposition 4.2.

Let the price impact functions g~t() and h~t() satisfy the following conditions:

  ρT-1[xT-1Th~T(nT)T-1]=ρT-1[xT-1Th~T(nT)xT-1],   (4.8)
  ρt-1[(nt-xt-1)T?t(Pt-1)+ntTh~t(nt)+(xt-1-nt)Tg~t(nt)t-1]=ntTPt-1-xt-1TPt-1+νt-1(xt-1,nt)+ρt-1[ntTh~t(nt)+(xt-1-nt)Tg~t(nt)xt-1],   (4.9)

where νt-1(xt-1,nt) is a time-varying function of xt-1 and nt, constant with respect to the realization of Pt-1. Then, V^(P0)=V(P0) and the optimal round-trip policy of (4.6) is 0-measurable.

The equality V^(P0)=V(P0) implies that the absence of an 0-measurable round-trip trade with π<δ is sufficient to conclude no-risk-averse dynamic arbitrage. This statement is valid for both market price dynamics (2.3) and (2.4), and it does not rely on the linear assumption about price impact functions. A proof of Proposition 4.2 is given in Appendix B (available online).

When the time-dependent price impact functions h~t() and g~t() are deterministic with respect to the information set t-1 and do not depend on the market price, (4.9) is reduced to

  ρt-1[(nt-xt-1)T?t(Pt-1)t-1]=(nt-xt-1)TPt-1+νt-1(xt-1,nt).   (4.10)

This condition indicates that at time t-1, in the realized sequence of market prices and in the absence of orders, the difference between the risk of the next market price, ρt-1[(nt-xt-1)T?t(Pt-1)], and the present observed market price, (nt-xt-1)TPt-1, is equal to a value that depends on time stamp t but not to the realized market price level Pt-1. This characteristic is akin to the martingale property, extended to dynamic risk measures.

In particular, for the additive model (2.3), under the assumption that the time-dependent price impact functions h~t() and g~t() are deterministic with respect to the information set t-1, (4.10) with

  νt-1(xt-1,nt)=ρt-1[(nt-xt-1)T~tt-1]  

holds, ie,

  ρt-1[(nt-xt-1)T(~t+Pt-1)t-1] =ρ[(nt-xt-1)T~t+(nt-xt-1)TPt-1t-1]  
    =ρt-1[(nt-xt-1)T~tt-1]+(nt-xt-1)TPt-1  
    =(nt-xt-1)TPt-1+νt-1(xt-1,nt),  

where the second equality comes from the translation invariance property. Hence, (4.10) always holds for additive market price dynamics and any coherent risk measure. However, for multiplicative price dynamics (2.4), (4.10) can hold only for processes that satisfy the following equation:

  ρt-1[(nt-xt-1)TDiag(~t)Pt-1]=νt-1(xt-1,nt).   (4.11)

Therefore, in contrast to the classical (risk-neutral) dynamic arbitrage in which (3.2) yields the absence of dynamic arbitrage for the multiplicative price model and linear price impact functions, constant with respect to market price, (4.9) does not rule out risk-averse dynamic arbitrage for this setting.

5 Concluding remarks

We investigate the importance of nonanticipativity constraint on round-trip trades to be assessed for dynamic arbitrage in illiquid markets. In our study, we let price impact functions be time varying and stochastic. We also consider two general classes of price impact dynamics, which can exhibit a broad range of stochastic market price processes. We first provide a condition under which searching in the space of 0-measurable admissible round-trip trades is enough to attain a no-dynamic arbitrage characterization. This result sheds light on analyses in some of the existing literature, such as Gueant (2014), in which a no-dynamic arbitrage is concluded based solely on a search in the space of 0-measurable round-trip trades, instead of searches in the larger set of dynamic round-trip trades. We show that for linear price impact models this condition is both sufficient and necessary for the stronger statement that dynamic arbitrage does not exist. For nonlinear price impact models, this sufficient condition significantly simplifies the effort of searching for price manipulation strategies and dynamic arbitrage opportunities. In particular, we present a simple numerical minimization problem that can be employed for general time-varying stochastic nonlinear price impact models as well as both additive and multiplicative market price dynamics for no-dynamic arbitrage verification. The validity of the sufficient condition, however, involves a precise belief from the very beginning that the conditional expected market price change at any time step is zero. This motivates us to extend the concept of dynamic arbitrage to risk-averse dynamic arbitrage in order to incorporate traders’ risk attitudes toward decisions on the presence of arbitrage opportunities. Similar to our analysis for (risk-neutral) dynamic arbitrage, we prove some sufficient condition. This enables us to limit searching in the space of static round-trip trades to establish the absence of risk-averse dynamic arbitrage.

We leave it for future research to investigate classes of price impact models that rule out risk-averse dynamic arbitrage at a risk-aversion threshold level δ and for a given market price dynamics and a time-consistent dynamic risk measure. The investigation of further relationships between the proposed concept of risk-averse dynamic arbitrage and alternative notions of arbitrage is another interesting direction for future work.

Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.

Acknowledgements

The authors wish to thank the editor and referee for their valuable comments and suggestions.

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