Cross impact in derivative markets
CCross impact in derivative markets
Mehdi Tomas , Iacopo Mastromatteo , and Michael Benzaquen
LadHyX UMR CNRS 7646, Ecole Polytechnique, 91128 Palaiseau Cedex, France CMAP UMR CNRS 7641, Ecole Polytechnique, 91128 Palaiseau Cedex, France Chair of Econophysics & Complex Systems, Ecole Polytechnique, 91128 Palaiseau Cedex, France Capital Fund Management, 23-25, Rue de l’Université 75007 Paris, France
February 8, 2021
Abstract
We introduce a linear cross-impact framework in a setting in which the price of some given financial instruments( derivatives ) is a deterministic function of one or more, possibly tradeable, stochastic factors ( underlying ). We show that aparticular cross-impact model, the multivariate Kyle model, prevents arbitrage and aggregates (potentially non-stationary)traded order flows on derivatives into (roughly stationary) liquidity pools aggregating order flows traded on both derivativesand underlying. Using E-Mini futures and options along with VIX futures, we provide empirical evidence that the priceformation process from order flows on derivatives is driven by cross-impact and confirm that the simple Kyle cross-impactmodel is successful at capturing parsimoniously such empirical phenomenology. Our framework may be used in practicefor estimating execution costs, in particular hedging costs.
Contents a r X i v : . [ q -f i n . T R ] F e b Data 17
Understanding the relation that connects trade imbalances and price changes is arguably one of the main goals of marketmicrostructure theory. By leveraging a data deluge that increased both quantity and quality of financial data over the past20 years, many empirical results have shed light on market impact (see e.g. [2, 4, 17]). Further, it lead to an adjustment ofthe theoretical framework underpinning the foundations of market microstructure [7, 9], and it drove a more data-drivencommunity to build novel settings in which to accommodate these findings (see e.g. [4, 15]). On the other hand, onlysparse attention has been given in the literature to the related problem of establishing a relation between order flowsand price changes in a multi-instrument setting ( cross-impact ). Compared to its univariate counterpart, cross-impact ismuch harder to characterize empirically, due to the larger number of degrees of freedom involved, and the consequentlysmaller signal-to-noise ratio [8, 12, 14, 19]. The challenge of cross-impact modeling also raises from genuinely new effectsthat only appear in the multivariate setting, such as the possible presence of cross-sectional arbitrages. Accounting forthese problems requires building some dedicated theoretical infrastructure [1, 11], and it led us to investigate in [16] theprinciples by which a cross-impact model should abide in order to be free from basic inconsistencies and arbitrages.This works aims precisely at illustrating the consequences of such a principled approach in a limit case that exposesboth the flaws of inconsistently formulated models and the benefits of having strong theoretical guarantees on the behaviorof a cross-impact model. In fact, we focus our study on cross-impact in a universe of instruments that comprises potentiallyilliquid derivatives. This is the prototypical case in which a univariate approach to modeling impact and liquidity fallsshort, due to two main reasons respectively related to prices and liquidity . Indeed, in an efficient market the prices ofderivatives should be locked by non-arbitrage, and hence they are not expected to respond to trades independently of theirunderlying. Regarding liquidity , it is intuitive that in presence of many strongly correlated and individually illiquid financialinstruments (e.g. options), it is necessary to aggregate the liquidity of multiple products into common liquidity pools inorder to have a satisfactory description of the price response, which would otherwise appear anomalously strong. Bothpoints indicate that in order to have a viable model to describe impact on derivative markets in presence of fragmentedliquidity it is unavoidable to take a multivariate perspective on the system, namely one that is able to single out the relevantliquidity factors.Our approach allows to map this problem onto a dimensionality reduction one, showing that one can replace a highdimensional space of instruments (say, option surfaces) with a lower dimensional representation of underlying factors(spot, implied volatility), which are the only degrees of freedom allowed to respond to order flow imbalances.It is worth emphasizing at this point that, although other models for cross-impact of derivatives (in particular foroptions) have already appeared in the literature, our focus is different in several respects. A first stream of works aims atthe characterization of non-arbitrage properties in cross-impact models [1], aiming at defining necessary conditions thatshould be satisfied in the general case. Here, we study a specific instance of such class of models, which leads to a veryrich phenomenology that cannot be fully appreciated in a completely generic setting. A second series of works focuseson the implications that some specific cross-impact models have in the context of option replication and hedging [3, 10],due to the fact that in presence of cross-impact non-linear effects arise, and potentially dangerous feedback loops emerge.Finally, in [18] the emphasis is given to the empirical determination of cross-impact; one of its findings of interest to us isthe identification of impact along a non-trivial underlying factor (level of implied volatility) that does not mechanicallycorrespond to any individual option, and yet emerges from the aggregation of the whole volatility surface. An importantcontribution that pushes this approach even further is [13], which indicates that other factors (skew of the volatility surface)are also necessary to accurately describe cross-impact on options. Even though the last two references strongly relateto our work, there are several differences that characterize our approach. First, we try to infer from data the rules to beused to aggregate liquidities, rather than postulating them. Second, our approach can easily accommodate multi-factorcross-impact models, whereas in such references only one factor at the time was considered. Third and last, the theoreticalfoundations our our approach are strongly grounded in the classical market microstructure literature [5, 12], and inparticular in [6], which provides a solid micro-foundation of our model (see also [16]).The paper is organized as follows. In Section 2 we introduce the notations used. Section 3 presents our modelingframework. Section 4 provides illustrative examples of its applications. Section 5 presents the empirical results of cross-impact on options. In Section 6 we conclude on the contributions of the paper, open questions, and directions for futurework. 2
Notations
Throughout the paper, we write scalars in roman lower cases, vectors in bold lower cases and matrices in roman uppercases. The set of n by n real-valued square matrices is denoted by M n ( (cid:82) ), the set of orthogonal matrices by O ( n ), the set ofreal non-singular n by n matrices by GL n ( (cid:82) ), the set of real n by n symmetric positive semi-definite matrices by S + n ( (cid:82) ),and the set of real n by n symmetric positive definite matrices by S ++ n ( (cid:82) ). Further, given a matrix A in M n ( (cid:82) ), A (cid:62) denotesits transpose. Given A in S + n ( (cid:82) ), we write A for a matrix such that A ( A ) (cid:62) = A and (cid:112) A for the matrix square root,the unique positive semi-definite symmetric matrix such that ( (cid:112) A ) = A . We write ker( M ) for the null space of a matrix M ∈ M n , Π V for the projector on a linear subspace of V ∈ (cid:82) n and ¯ Π V = (cid:73) − Π V for the orthogonal projector. Finally, given avector v ∈ (cid:82) n , we write v = ( v ,..., v n ) and diag( v ) for the diagonal matrix with diagonal components the components of v .All stochastic processes in the text are defined on a probability space ( Ω , F ,( F t ) t ∈ (cid:82) , (cid:80) ) and will be adapted to thefiltration ( F t ) t ∈ (cid:82) unless stated otherwise. Standard Brownian motions are defined with respect to the probability measure (cid:80) .All stochastic differential equations introduced will be assumed to have a unique strong solution and correspondingly thefunctions appearing in these equations will be assumed to be sufficiently regular for this to be true. We denote by (cid:69) theexpectation with respect to the probability measure (cid:80) and (cid:69) t [...] will denote the conditional expectation (cid:69) [... | F t ]. We consider a universe comprising two classes of financial instruments, that we will refer to as underlying and derivatives .With some abuse of language, the notion of underlying will apply to a set of N stochastic processes, that might describe indis-tinctly a set of tradable financial instruments or an ensemble of stochastic factors. The prices of these N instruments will bedenoted p t = ( p t ,..., p Nt ). We define as derivatives a set of M instruments, whose prices P ( p t , t ) = ( P ( p t , t ),..., P M ( p t , t ))are deterministic functions of the underlying price process p t .We assume that impact is linear in the traded order flows and we denote by q t = ( q t ,..., q Nt ) the stochastic process cor-responding to the net traded order flows on the underlying and by Q t = ( Q t ,..., Q Mt ) the stochastic process correspondingto the net traded order flows on derivatives. Such flows are not associated to any specific agent, and rather denote aggregatemarket order flow. Since only the non-predictable component of order flows contributes to impact, up to replacing q t by q t − (cid:69) t − [ q t ] and Q t by Q t − (cid:69) t − [ Q t ] in the following, we assume that (cid:69) t − [ q t ] = (cid:69) t − [ Q t ] = We assume that the dynamics of the underlying ared p t = µ p ( p t , t )d t + G p ( p t , t )d w t + Λ pq ( p t , t )d q t + Λ pQ ( p t , t )d Q t , (1)where w is a standard N -dimensional Brownian motion, µ p : (cid:82) N × (cid:82) → (cid:82) N is the drift, G p : (cid:82) N × (cid:82) → GL N ( (cid:82) ) is thediffusion matrix, Λ pq : (cid:82) N × (cid:82) → M N , N ( (cid:82) ) is the underlying-underlying cross-impact matrix and Λ pQ : (cid:82) N × (cid:82) → M N , M ( (cid:82) )is the underlying-derivative cross-impact matrix. The underlying dynamics of Equation (1) could be enriched by jumpprocesses, but for the sake of simplicity we chose a simple continuous framework which covers several well-known modelsof derivatives pricing.Since derivative prices are deterministic functions of the underlying, without loss of generality, the derivative dynamicscan be written as d P t = µ P ( p t , t )d t + G P ( p t , t )d w t + Λ Pq ( p t , t )d q t + Λ PQ ( p t , t )d Q t , (2)where µ P : (cid:82) N × (cid:82) → (cid:82) M is the derivative drift, G P : (cid:82) N × (cid:82) → M M , N ( (cid:82) ) is the derivative diffusion matrix, Λ Pq : (cid:82) N × (cid:82) → M M , N ( (cid:82) ) is the derivative-underlying cross-impact matrix and Λ PQ : (cid:82) N × (cid:82) → M N , M ( (cid:82) ) is the derivative-derivativecross-impact matrix.For convenience cross-impact matrices can be compactly rearranged into a single matrix, Λ , which we refer to as thecross-impact matrix since it describes the cross-impact of the complete system Λ ( p t , t ) : = (cid:181) Λ pq Λ pQ Λ Pq Λ PQ (cid:182) ( p t , t ). (3)3imilarly drift and diffusion terms can be grouped as µ ( p t , t ) = ( µ p ( p t , t ), µ P ( p t , t )) G ( p t , t ) = ( G p ( p t , t ), G P ( p t , t )). (4)Obviously, not all the terms appearing in Equations (3) and (4) are independent, given that the function P ( p t , t ) fixesthe derivative prices as a function of the underlying. In Sec. 3.3.1 we will discuss the conditions required to prevent arbitrage.We also draw the reader’s attention to the fact that one may directly obtain an equation for the derivatives’ prices byapplying Ito’s formula and using the underlying dynamics of Equation (1). This would yield expressions for Λ Pq and Λ PQ asa function of Λ pq and Λ pQ . However, not all cross-impact models give expressions of Λ Pq and Λ PQ consistent with Ito’sformula and it is thus more convenient to choose a cross-impact model which, by construction, yields this result instead ofimposing it a priori . Our choice of cross-impact model is discussed in the next section. The impact model we propose involves two parameters, the return covariance matrix and the order flow covariance matrix.The underlying-underlying return covariance matrix Σ pp : (cid:82) N × (cid:82) → S + N ( (cid:82) ) is defined as Σ pp ( p t , t )d t : = (cid:69) t [d p t d p (cid:62) t ] − (cid:69) t [d p t ] (cid:69) t [d p (cid:62) t ], (5)and we similarly denote Σ pP , Σ Pp = Σ (cid:62) pP , Σ PP for the underlying-derivative, derivative-underlying and derivative-derivativereturn covariance matrices. Naturally, since derivative prices are deterministic function of the underlying, these matricesare all related to Σ pp . We denote by Ω qq : (cid:82) N × (cid:82) → S + N ( (cid:82) ) the underlying-underlying order flow covariance matrix Ω qq ( p t , t )d t : = (cid:69) t [d q t d q (cid:62) t ] − (cid:69) t [d q t ] (cid:69) t [d q (cid:62) t ], (6)and we denote Ω qQ , Ω Qq = Ω (cid:62) qQ and Ω QQ for the underlying-derivative, derivative-underlying and derivative-derivativeorder flow covariances. Contrary to return covariance matrices, there are no constraints betweeen these order flowcovariance matrices and Ω qq . The covariance structure of returns and flows for the whole system can be arrangedcompactly as Σ ( p t , t ) = (cid:181) Σ pp Σ pP Σ (cid:62) pP Σ PP (cid:182) ( p t , t ) Ω ( p t , t ) = (cid:181) Ω qq Ω qQ Ω (cid:62) qQ Ω QQ (cid:182) ( p t , t ).The impact model that we propose, first derived in [5], has been analyzed in this context in [6, 16], where it was referred asthe Kyle cross-impact model. The model prescribes using a Λ of the form Λ = (cid:112) Y Λ kyle ( Σ , Ω ) : = (cid:112) Y ( Ω − ) (cid:62) (cid:112) ( Ω ) (cid:62) ΣΩ Ω − , (7)where we have omitted the dependence on ( p t , t ) for convenience, and where we have introduced Λ kyle : S + N + M ( (cid:82) ) × S ++ N + M ( (cid:82) ) → M N + M , N + M ( (cid:82) ). We introduced 0 < Y < The Kyle cross-impact model (i) ensures derivatives are priced consistently with the underlying dynamics, (ii) reduces thedynamics of the system to a classic SDE when flow degrees of freedom are integrated inside volatility terms, (iii) allows fordimensionality reduction, (iv) properly aggregates liquid and illiquid instruments and (v) does not need to know whichinstruments belong to the set of underlyings and which ones are derivatives. We show these properties below and discusstheir implications.
In order for the model to be self-consistent (i.e., for the derivative to be efficiently priced at all times), one should make surethat the dynamics that we have postulated in Equation (2) are consistent with the one obtained by applying Ito’s formula tothe dynamics of the underlying via the relation P ( p t , t ) . This requirement prescribes that the following conditions shouldhold. 4 iffusion In order for the diffusion dynamics to be consistent, one should have G P ( p t , t ) = Ξ ( p t , t ) G p ( p t , t ), (8)where Ξ : (cid:82) N × (cid:82) → M M , N ( (cid:82) ) is a sensitivity matrix defined as Ξ i j ( p t , t ) = ∂ P i ∂ p j ( p t , t ). Impact
By the same token, the cross-impact matrix should satisfy the condition Λ ( p t , t ) = (cid:181) Λ pq Λ pQ ΞΛ pq ΞΛ pQ (cid:182) ( p t , t ), (9)indicating that the price response of the derivative should be completely fixed by the one on the underlying. Drift
Finally, the drift term should also match the one obtained via Ito’s formula, so that µ P ( p t , t ) = Θ ( p t , t ) + Ξ ( p t , t ) µ p ( p t , t ) + (cid:88) jk χ jk ( p t , t )( Σ pp ( p t , t )) jk , (10)where Θ i ( p t , t ) = ∂ P i ∂ t ( p t , t ) χ i , jk ( p t , t ) = ∂ P i ∂ p j ∂ p k ( p t , t ),are sensitivities, of nature similar to Ξ defined above.We will assume in the following that the conditions for drift and diffusion hold by construction, taking Equations (8)and (10) as definitions of respectively G P and µ P . On the other hand, since Equation (9) is not guaranteed a priori underour choice of the cross-impact model, we will be required to prove that our model satisfies Equation (9) above, and thusthat our construction is consistent.In order to do so, we need to discuss a crucial property of the Kyle model, that is referred in [16] as strong fragmentationinvariance . With strong fragmenation invariance, we denote the fact that for any (cid:59) ⊂ V ⊆ ker( Σ ) we have Π V Λ kyle ( Σ , Ω ) = Λ kyle ( Σ , Ω ) Π V = Λ kyle ( Σ , ¯ Π V Ω ¯ Π V ) = Λ kyle ( Σ , Ω ). (13)Equation (11) shows that linear combinations of instruments with constant price (zero volatility) are not impacted by theorder flow. In our setting, this guarantees that derivatives are always priced efficiently even in the presence of order flowpressure. Equation (12) maintains that order flow pressure on non-fluctuating modes should not be able to influencethe price of any other combination of products. This prevents pushing the price of fluctuating instruments by tradingzero-volatility (free) linear combination of instruments. Finally, Equation (13) shows that the way non-fluctuating modesare traded has no influence the price of any instrument.In our context, strong fragmentation invariance and symmetry imply that Λ ( p t , t ) = (cid:181) Λ pq Λ pq Ξ (cid:62) ΞΛ pq ΞΛ pq Ξ (cid:62) (cid:182) ( p t , t ). (14)Thus the Kyle model satisfies the condition of Equation (9) above, thus guaranteeing that (along with the conditions ofEquations (8) and (10) which we impose and are independent of the impact model) the derivative is efficiently priced.At this moment, it is worth emphasizing a central point of our approach: we are postulating that derivatives areefficiently priced at all times despite the presence of finite liquidity. This hides the implicit assumption that some marketactors are able to arbitrage away the spread between underlying and derivatives very quickly, and that those marketdirections are effectively frictionless for such actors. Inefficiencies can be implemented in this framework by relaxing theassumption that P t = P ( p t , t ), rather assuming that derivative prices mean-revert with a finite velocity to their theoreticalvalue. On long enough time scales, we expect the empirical estimate for Σ to nevertheless be close to the value given byno-arbitrage so that these considerations can be ignored. For pedagogical clarity, hereafter we will stick to the idealizedcase in which derivatives are efficiently priced. 5 .3.2 Consistency of frictionless and impacted dynamics We now want to show that the dynamics with impact are consistent with the frictionless dynamics that would be observedwhen disregarding the flows degrees of freedom. The term frictionless in this context does not indicate absence of impact,but rather identifies the effective system in which the degrees of freedom related to flows ( q t and Q t ) are integrated insidevolatility contributions.To prove this, we first remark that the Kyle model is the only linear, symmetric positive definite impact model that isconsistent with the covariance structure of the system (see [16]), namely the fact that Y Σ ( p t , t ) = Λ ( p t , t ) Ω ( p t , t ) Λ (cid:62) ( p t , t ). (15)This property is not surprising, given ubiquity of the inconspicous equilibrium property in theoretical frameworks inwhich market efficiency is achieved by rational agents that attempt to forecast future returns [5, 6]. In these frameworks,predictions are optimal when Equation (15) is verified. The prefactor Y expresses that a fraction Y of the total covariance isdue to trading activity while the remainder comes from shocks unrelated to the order flow.Now, let us relate a frictionless dynamics to the one that we have postulated in our model thanks to the covarianceconsistency condition (Equation (15)). To do so, consider the dynamics without impactd ˜ p t = ˜ µ p ( ˜ p t , t )d t + ˜ G p ( ˜ p t , t )d ˜ w t (16)d ˜ P t = ˜ µ P ( ˜ p t , t )d t + ˜ G P ( ˜ p t , t )d ˜ w t (17)where ˜ w is a standard Brownian motion, ˜ µ p : (cid:82) N × (cid:82) → (cid:82) N is the frictionless underlying drift, ˜ µ P : (cid:82) N × (cid:82) → (cid:82) M is thefrictionless derivative drift, ˜ G p : (cid:82) N × (cid:82) → GL N ( (cid:82) ) and ˜ G P : (cid:82) N × (cid:82) → M M , N ( (cid:82) ) are the frictionless diffusion matrices. Thenit is simple to verify that thanks to Equation (15) one can recover the equivalence between frictionless and impacteddynamics if G = (cid:112) − Y ˜ G and µ = ˜ µ . In intuitive terms, our price dynamics reduces to the vanilla stochastic evolution ofderivatives and underlying for any observer that does not possess information about the order flows. This stems from theGaussian nature of both order flow processes and underlying increments, along with the covariance consistency condition(Equation (15)) that is implicitly contained in the definition of the Kyle model. The strong fragmentation property introduced in Sec. 3.3.1 has another important implication in our setting: by imposingthe structure of the impact matrix to the form of Equation (14), it explicitly shows that rank( Λ ) = rank( Λ pq ) = N . This hasthe consequence of making explicit that no more than N distinct liquidity pools are necessary in order to compute impact.In particular, we will show that it is enough to estimate ¯ Π V Ω ¯ Π V instead of the (potential) rank N + M matrix Ω in order tofully determine the cross-impact matrix, where with V we denote the subspace spanning the direction P t − P ( p t , t ), whichobviously belongs to ker( Σ ) if we assume that derivatives are consistently priced. In intuitive terms, it is only the liquidityin the direction of the fluctuating degrees of freedom that contributes to the overall liquidity pool, whereas any trading inthe direction of the mispricing has no effect on the system.To show this, remark that Equation (14) allows us to express the full cross-impact matrix Λ once an expression for thelower-rank Λ pq object is available. We will thus derive the expression of Λ pq . Writing ˆ p and ˆ P for the underlying andderivative impact component of the price, one has d ˆ P t = Ξ d ˆ p t and (cid:181) d ˆ p t d ˆ P t (cid:182) = (cid:181) I Ξ I (cid:182)(cid:181) d ˆ p t (cid:182) = (cid:181) I Ξ I (cid:182)(cid:181) Λ pq
00 0 (cid:182)(cid:181) I Ξ (cid:62) I (cid:182)(cid:181) d q t d Q t (cid:182) ,so that the impacted underlying price is d ˆ p t = Λ pq (d q t + Ξ (cid:62) d Q t ).Since Λ is symmetric and positive-definite, one can easily show that Λ pq is symmetric and positive-definite. Furthermore, Λ pq is covariance-consistent and Y Σ pp = Λ pq Ω Ξ Λ (cid:62) pq ,where we have introduced the aggregated order flow covariance matrix Ω Ξ : = (cid:69) t [(d q t + Ξ (cid:62) d Q t )(d q t + Ξ (cid:62) d Q t ) (cid:62) ] = Ω qq + Ξ (cid:62) Ω QQ Ξ + Ξ (cid:62) Ω Qq + Ω qQ Ξ .6herefore, since these properties imply that Λ pq is the Kyle cross-impact model with return covariance Σ pp and order flowcovariance Ω Ξ one has Λ pq = (cid:112) Y Λ kyle ( Σ pp , Ω Ξ ) = (cid:112) Y ( Ω − Ξ ) (cid:62) (cid:113) ( Ω Ξ ) (cid:62) Σ pp Ω Ξ Ω − Ξ .This property is very useful in practice since it gives a recipe for computing Λ pq . It also tells us that the aggregated orderflow covariance matrix combines direct trading of underlying and indirect trading through derivatives exposures to theunderlying. In particular, even if an underlying is not tradeable, as long as derivatives which are exposed to that factor aretradeable its aggregated liquidity will be non-zero. The Kyle-model is also cross-stable (see [16]), meaning that trading an illiquid combination of products cannot move liquidproducts by a disproportionate amount. To make this property explicit, let us introduce a subset of illiquid instruments W ,on which we assume to measure a modified covariance matrix Ω (cid:178) : = ( ¯ Π W + (cid:178) Π W ) Ω ( ¯ Π W + (cid:178) Π W ),where Π W = (cid:73) − ¯ Π W is a projector on the space W . The covariance matrix above is the one we would have observed ifliquidities of instruments belonging to W were multiplied by (cid:178) . Then, for any ( Σ , Ω ) ∈ ( S + n ( (cid:82) ) × S ++ n ( (cid:82) )), the Kyle modelsatisfies ¯ Π W Λ kyle ( Σ , Ω (cid:178) ) Π W = (cid:178) → O (1) (18)¯ Π W Λ kyle ( Σ , Ω (cid:178) ) ¯ Π W → (cid:178) → ¯ Π W Λ kyle (cid:161) ¯ Π W Σ ¯ Π W , ¯ Π W Ω ¯ Π W (cid:162) ¯ Π W . (19)Equation (18) indicates that volumes traded on illiquid products cannot move by a disproportionate (diverging in (cid:178) ) theprice of liquid instruments, whereas Equation (19) indicates that the impact of the flows of liquid instruments is notaffected by the presence of other, illiquid, instruments.These properties allow one to include non-tradeable instruments (such as implied volatilities) within the definition ofthe underlying because even though the quantities Ω − appearing in Equation (7) are divergent whenever an illiquidproduct appears, this property implies that the divergence is restricted to the subspace of zero-liquidity products. This isconvenient also for cases in which liquidity is fragmented in a large set of products that might display weak liquidity (e.g.,out-of-the money options) as this property guarantees that such behavior won’t induce spurious features on the overallliquidity pool of the system. Throughout our approach, we have split instruments between underlyings and derivatives. However, a compelling propertyof the chosen cross-impact model is that Equation (7) does not need to know which is which in the universe of instruments.This may be convenient when dealing with a large number of instruments where it would be time-consuming to makeexplicit which instruments are derivatives and the associated sensitivity matrix Ξ . To illustrate the flexibility of our setup and the usefulness in practice of the properties of the Kyle cross-impact model, wediscuss examples in increasing complexity below.
For our first example, we consider a universe of N = M = p t and a futures con-tract expiring at a later time T , quoting a price P ( p t , t ). By assuming a constant, continuously compounded, deterministicinterest rate r one has P ( p t , t ) = e r ( T − t ) p t .In this case Ξ ( p t , t ) = ∂ p P ( p t , t ) = e r ( T − t ) and Equation (14) yields Λ ( p t , t ) = (cid:112) Y p t σ ( p t , t ) ω ( p t , t ) (cid:181) e r ( T − t ) e r ( T − t ) e r ( T − t ) (cid:182) ,7here σ ( p t , t ) : = (cid:69) t [ p t ] − (cid:69) t [ p t ] p t ω ( p t , t ) : = (1, e r ( T − t ) ) (cid:62) Ω ( p t , t )(1, e r ( T − t ) ).The meaning of this formula is rather simple: when dealing with a spot and a future market, there is a single relevantliquidity pool. Such liquidity pool should mix the flow traded on the future and the one traded on the spot. Volumes tradedon the futures market should be properly adjusted for the interest rate. We now consider a system with a single underlying N = M derivatives. The underlying is a spot with price p t , whereasthe derivatives are a set of European call or put options labeled by i = M , differing for either their strike or theirmaturity. We assume the price p t follows the usual log-normal dynamics, with risk-free rate r and implied volatility σ d p t = r p t d t + σ p t d w t .Then, with the usual notation for the Black-Scholes ∆ , we have Ξ i ( p t , t ) = ∂ p P i ( p t , t ) : = ∆ it ( p t , t ),and, writing ∆ : = ( ∂ p P ( p t , t ), ··· , ∂ p P M ( p t , t )), Equation (14) yields Λ ( p t , t ) = (cid:112) Y p t σω ( p t , t ) (cid:181) ∆ (cid:62) ∆ ∆∆ (cid:62) (cid:182) ( p t , t ),where ω ( p t , t ) = (1, ∆ ( p t , t )) (cid:62) Ω ( p t , t )(1, ∆ ( p t , t )).Thus, as in the previous example, there is a single liquidity pool, with volumes traded on options adjusted for the options’ ∆ . Volume traded on deep in-the-money options ( ∆ i ≈
1) contribute to the overall liquidity pool as if it was the spot itselfthat was traded, whereas deeply out-of-the-money options ( ∆ i ≈
0) give negligible contributions.
Building on our previous example, we want to focus on the same class of instruments (one spot and a strip of M Europeanoptions) in the case in which it is necessary to add an implied volatility term to the Black-Scholes formula to obtain theirprice. One is required to use a pricing formula P i ( p t , ˆ σ it , t ) in order to accurately describe the price of the options, whereˆ σ it is an implied volatility .In order to reduce the dimensionality of the M implied volatilities ˆ σ it , we assume that they lie on a low dimensionalsurface, so that we are allowed to write ˆ σ it = F i ( ς t ),where ς t = ( ς t ,..., ς Qt ) is a set of volatility factors that completely describe the volatility surface through a set of M functions( F i ( ς )) Mi = . Note that with some abuse of notation, we will often write P i ( p t , ˆ F i ( ς t ), t ) = P i ( p t , ς t , t ),and we will employ a similar notation for other functions of the implied volatility ˆ σ it . We then have an underlyingconsisting of N = + Q instrument, of which only one is tradeable (the spot), and where the other Q factors correspond tonon-tradeable volatility factors. The sensitivities of the system in this case correspond to Ξ i t ( p t , ς t , t ) = ∆ it ( p t , ς t , t ) : = ∂ P i ( p t , ς t , t ) ∂ p t Ξ i ( q + t ( p t , ˆ σ it , t ) = ∂ P i ( p t , ˆ σ it , t ) ∂ ˆ σ it ∂ F i ( ς t ) ∂ς q : = V it ( p t , ς t , t ) β iq ( p t , ς t , t ) = : Υ iq ,8here q = Q and where, as it is customary in the literature on option pricing, we have introduced the vega V it ( p t , ˆ σ it , t ) = ∂ P it ( p t , ˆ σ it , t ) ∂ ˆ σ it ,and the sensitivities of the volatility surface to ς t β iq ( p t , ς t , t ) = ∂ F i ( ς t ) ∂ς q .For convenience, we write the sensitivity matrix in block Ξ = (cid:161) ∆ | Ξ · | ··· | Ξ · Q (cid:162) = : (cid:161) ∆ | Υ (cid:162) .At this point, the most general expression for the cross-impact matrix that we can write is Λ ( p t , ς t , t ) = Λ pp Λ p ς Λ pp ∆ (cid:62) + Λ p ς Υ (cid:62) Λ (cid:62) p ς Λ ςς Λ ςς Υ (cid:62) + Λ (cid:62) p ς ∆ (cid:62) ∆ Λ pp + ΥΛ (cid:62) p ς ΥΛ ςς + ∆ Λ p ς ∆ Λ pp ∆ (cid:62) + ΥΛ ςς Υ (cid:62) + ∆ Λ p ς Υ (cid:62) + ( ∆ Λ p ς Υ (cid:62) ) (cid:62) . Single-factor model
It is instructive to understand the behavior of the system in the case in which Q =
1, and the volatilitysurface is parametrized by a single level factor: ˆ σ it ( ς t ) = F ( ς t ) = ς t ,so that one has Ξ i ( p t , ς t , t ) = V it ( p t , ς t , t ).In this case the rank two cross-impact matrix can be written as Λ ( p t , ς t , t ) = Λ pp Λ p ς Λ pp ∆ (cid:62) + Λ p ς V (cid:62) Λ p ς Λ ςς Λ ςς V (cid:62) + Λ p ς ∆ (cid:62) ∆ Λ pp + V Λ p ς V Λ ςς + ∆ Λ p ς Λ pp ∆∆ (cid:62) + Λ ςς VV (cid:62) + Λ p ς ( ∆ V (cid:62) + V (cid:62) ∆ ) .In the case where the ∆ order flow and V order flow are not correlated, i.e. writing ∆ c : = (1,0, ∂ P ∂ p , ··· , ∂ P M ∂ p ), V c : = (0,1, ∂ P ∂ς , ··· , ∂ P M ∂ς ) we have ∆ (cid:62) c Ω V c =
0, we can obtain the expression of the kyle cross-impact matrix for the underly-ing: (cid:181) Λ pp Λ p ς Λ p ς Λ ςς (cid:182) ( p t , ς t , t ) = (cid:113) σ ω ∆ + ξ ω V + σξρω ∆ ω V (cid:195) σ + ω V ω ∆ σξ (cid:112) − ρ σξρσξρ ξ + ω ∆ ω V σξ (cid:112) − ρ (cid:33) ( p t , ς t , t ), (20)where ω ∆ : = ∆ (cid:62) c Ω ∆ c is the delta-aggregated liquidity, ω V : = V (cid:62) c Ω V c is the vega-aggregated liquidity, σ ( p t , ς t , t ) : = (cid:69) t [ p t ] − (cid:69) t [ p t ] is the spot volatility, ξ ( p t , ς t , t ) : = (cid:69) t [ ς t ] − (cid:69) t [ ς t ] is the volatility of volatility and ρ is the spot-vol correlation. We now illustrate Section 3 with an empirical analysis of cross-impact on derivatives markets. Section 5.1 describes theuniverse of instruments and the chosen derivative modeling. Section 5.2 shows the empirical observables Σ pp and Ω Ξ usedin Section 5.3 to compute the resulting cross-impact matrix Λ pp . Finally, Section 5.4 stress-tests the fit of cross-impactmodels and Section 5.5 examines non-parametric evidence of cross-impact. The universe of instruments is made up of (i) the front-month E-mini future, (ii) the two front-month VIX futures, (iii) a setof M − M derivatives. We model the implied volatility surface with Q volatility factors, as discussed in Section 4.3. We aggregate returns and order flows on a time window of five minutes, so that9 p t is approximated by the price change on the timeframe of five minutes and (d q t ,d Q t ) is approximated by the signedtraded order flow during this time window. To avoid confusion, we write δ p t , δ q t and δ Q t for these empirical quantities inthe rest of this section. Prices and order flows for these instruments are taken from trades and quotes data and more detailon the dataset is provided in Appendix A.We consider a linear approximation of the implied volatility surface with volatility factors, so that using the notationsof Section 4.3, we have F i ( ς ) = (cid:80) Qq = β iq ς q where i = M . To fit surfaces, we choose M = level factor. The second factor corresponds to the skew of the implied volatilitysurface, referred to as the skew factor hereafter. The third factor explains the term structure of the implied volatility, hencethe name the term factor in the following. T i
20 40 60 80100120 i i + i Level factor T i
20 40 60 80100120 i i + i Skew factor T i
20 40 60 80100120 i i + i Term factor
Figure 1: Effect of the different volatility factors on the implied volatility surface.
Starting from a historical implied volatility ˆ σ i , we show the modified implied volatility surface after with a small contribution from thefactor q : ˆ σ i + (cid:178)β iq . The original (non-modified) implied volatility surface is shown in light opacity for reference. Using Section 3.3.3, it suffices to compute the Kyle model associated to the underlying return covariance matrix Σ pp ( p t , t )and the aggregated order flow covariance matrix Ω Ξ ( p t , t ) to obtain the full cross-impact matrix. As we have approximatedthe behaviour of the system with 4 underlyings, these observables are 4 by 4 matrices. To estimate them, we make theadditional assumption that Σ pp and Ω Ξ are stationary and independent of p t . Figure 2 displays the underlying returncorrelation matrix (cid:37) pp : = diag( σ ) − Σ pp diag( σ ) − and the risk order flow covariance matrix Ω risk Ξ : = diag( σ ) Ω Ξ diag( σ )where σ = (( Σ pp ) , ··· ,( Σ NNpp ) ) is the underlying volatility.The traded risk (volatility times liquidity) is concentrated on the spot and level directions. This justifies approximatingcross-impact on options using solely spot and level underlyings, which we delve in more detail in Section 5.3. The tradedrisk in the skew direction is much smaller than all other directions and skew order flow is thus expected to contribute lessto cross-impact. The underlying return correlation matrix correlation matrix shows strong negative correlation betweenthe spot and level mode. This is a well-known stylised fact, sometimes referred to as the "leverage effect". This will playan important role in the form of the cross-impact model, as highlighted in Equation (24). Unsurprisingly, the correlationbetween spot and level order flow is much smaller, although still noticeable (around -0.15%). We can now use the empirical estimates of Σ pp and Ω Ξ from the previous section to compute the cross-impact matrix Λ .For comparison purposes, we also introduce other cross-impact models. The first cross-impact model used for comparison10 pot level skew termspotlevelskewterm 1.00 -0.88 -0.26 0.72-0.88 1.00 0.31 -0.91-0.26 0.31 1.00 -0.270.72 -0.91 -0.27 1.00 pp spot level skew term129.02 -4.85 0.19 -1.95-4.85 12.98 -0.23 4.330.19 -0.23 0.02 -0.08-1.95 4.33 -0.08 1.49 Figure 2: Empirical estimates of return correlation matrix (cid:37) and order flow covariance Ω . The return correlation matrix (cid:37) (left) and the order flow covariance matrix Ω (right) estimates on our dataset. The order flow is reportedin thousands of dollars of risk. is the Black-Scholes cross-impact model introduced in Section 4.2 which has a single underlying: the spot. It is defined as Λ bs ( p t , t ) : = σ (cid:113) ∆ (cid:62) c Ω ∆ c ∆ c ∆ (cid:62) c . (21)The Black-Scholes model coincides with the Kyle cross-impact model if all the liquidity is concentrated on the spot. Inparticular, this model is unable to account for changes in the volatility factors. We thus introduce the two-dimensionaldirect model Λ direct-2d which accounts for the spot and implied volatility factor but ignores cross-sectional effects, definedas Λ direct-2d ( p t , t ) : = σ (cid:113) ∆ (cid:62) c Ω ∆ c ∆ c ∆ (cid:62) c + ξ (cid:113) V (cid:62) c Ω V c V c V (cid:62) c . (22)To account for all underlyings without correcting for cross-sectional effects, we introduce the four-dimensional directmodel Λ direct-4d ( p t , t ) : = σ (cid:113) ∆ (cid:62) c Ω ∆ c ∆ c ∆ (cid:62) c + ξ (cid:113) V (cid:62) c Ω V c V c V (cid:62) c + Q + (cid:88) i = (cid:118)(cid:117)(cid:117)(cid:116) Σ iipp Ω ii Ξ Ξ · i ( p t , t )( Ξ · i ( p t , t )) (cid:62) . (23)Direct models ignore the off-diagonal structure of Σ pp and Ω Ξ . In particular they do not account for the leverage effect,which is an essential characteristic of the underlying return covariance matrix Σ pp . To fix this, we introduce the two-dimensional Kyle cross-impact model Λ which captures the two dominating underlyings of the system: the spot andlevel factor. Since Figure 2 shows that the delta and vega order flow correlation is small (around − ξω V (cid:191) σω ∆ ,we can use Equation (20) to obtain the approximation Λ ( p t , ς t , t ) ≈ σ (cid:113) ∆ (cid:62) c Ω ∆ c ∆ c ∆ (cid:62) c + ξ (cid:112) − ρ (cid:113) V (cid:62) c Ω V c V c V (cid:62) c + ξρ (cid:113) ∆ (cid:62) c Ω ∆ c ( V c ∆ (cid:62) c + ∆ c V (cid:62) c ). (24)The two-dimensional Kyle cross-impact model predicts that when trading options, one pushes the price in the amount ofnotional V traded divided by the typical V liquidity, which is compatible with findings from the meta-order study [18]. Theprefactor (cid:112) − ρ ≈ − Λ is shown in Figure 3. Compared to the two-dimensional Kylecross-impact model, it decouples the contribution of options on the level mode depending on the direction. This increasesthe explanatory power of the model, as shall be evidenced in Table 1. For practical applications, a good cross-impact model should explain realized price changes from order flows. Thus tocompare the models previously introduced, we now examine their explanatory power on empirical data. Given a realization11 pot level skew termspotlevelskewterm 0.04 -0.03 -0.01 0.03-0.03 0.27 0.06 -0.57-0.01 0.06 3.33 0.030.03 -0.57 0.03 1.53 Figure 3: Four dimensional Kyle cross-impact model on options.
We report the four dimensional Kyle model estimated using empirical estimates of the covariances of Figure 2. The cross-impact matrixare reported in units of risk and in basis points so that Λ i j encodes by how many basis points of volatility Asset i is pushed by tradingone dollar of risk on Asset j . of the underlying price process ( δ p t ) ≤ t ≤ T of length T , a corresponding series of predictions ( (cid:99) δ p t ) ≤ t ≤ T and a symmetricpositive semi-definite matrix M , we introduce the generalized R ( M ) error as R ( M ) : = − (cid:80) ≤ t ≤ T ( δ p t − (cid:99) δ p t ) (cid:62) M ( δ p t − (cid:99) δ p t ) (cid:80) ≤ t ≤ T δ p (cid:62) t M δ p t .The matrix M is used to examine a model’s predictive power for different portfolios. As the underlyings of our system arenatural directions to consider, we report the R ( M ) in Table 1 for Π (1,0,0,0) = : Π spot , Π (0,1,0,0) = : Π level , Π (0,0,1,0) = : Π skew and Π (0,0,0,1) = : Π term . Model Scores R ( Π spot ) R ( Π level ) R ( Π skew ) R ( Π term ) Λ bs ± − ± − ± − ± Λ direct-2d ± − ± − ± ± Λ direct-4d ± − ± − ± − ± Λ ± ± − ± ± Λ ± ± − ± ± Table 1: Scores of different cross-impact models.
All scores were computed in-sample using the same data used for the calibration of the cross-impact models.
Looking at R ( Π spot ), we see that all models show similar scores on the spot, with cross-impact models being slightlybetter. Furthermore, there is no difference between Λ and Λ . This is consistent with the liquidity reported in Figure 6.Indeed, most of the liquidity is placed on the spot and the order flow traded on other factors is small in comparison.Models which only take into account the spot thus capture most of the order flow explanatory power. There is also asmall advantage in using order flow on the level mode since Λ and Λ score better, but using term and skew order flowprovides no improvement.While using solely spot liquidity to explain spot returns is a good approximation, R ( Π level ) shows the same is not truefor the level underlying. Indeed, only models with spot-level cross-impact are able to account for returns of the impliedvolatility level. Therefore, to explain price changes on the level underlying we need to use order flow traded on the spot.This is natural as most of the traded order flow is on the spot but there is a high negative correlation between spot and levelunderlying (see Figure 6). Unfortunately, all models fail to explain skew returns. We suspect this comes from the low signalto noise ratio and low liquidity (in risk terms) of the skew underlying (see Figure 6).On all metrics, Λ performs at least as well as Λ , which shows that the model is able to combine additional factorswithout suffering from noise. The additional underlying also help weigh trades appropriately on the implied volatilitysurface, which improves the R ( Π level ) score.Finally, we report the expected realized return conditional on the prediction of Λ in Figure 4. This shows that, skewaside, Λ provides a good fit for the realized returns of the different underlying as (cid:69) t [ δ p t | δ ˆ p t ] ≈ δ p t .12 .0 0.5 0.0 0.5 1.0 1.5 p [ p ]1.51.00.50.00.51.01.5[ p p ][ p ] spotlevelskewterm Figure 4: Predictions of the four-dimensional Kyle model on the main directions of the system.
We report the expected price change conditional on the predicted price change of the four-dimensional Kyle model for the four maindirections of the system: in red for the spot, blue for the level, green for the skew and purple for the term structure. Predicted pricechanges and conditional averages are both normalized by the standard deviation of price changes along the given direction.
Section 5.4 showed that only cross-impact models are able to explain returns for the level and term underlying. Aside fromthis explanatory power, this section tests their ability to explain other features of our data. To do so, we introduce the crossaggregate impact metric. The cross aggregate impact induced from the portfolio u ∈ (cid:82) N + M on the return of the portfolio v ∈ (cid:82) N + M is Agg u , v ( x ) : = (cid:69) t [ v (cid:62) ( δ p t , δ P t ) | u (cid:62) ( δ q t , δ Q t ) = x ].If returns are given by a linear cross-impact model Ψ and if we further assume ( δ q t , δ Q t ) is a zero-mean Gaussian, thenAgg u , v ( x ) = (cid:69) t [ v (cid:62) Ψ ( δ q t , δ Q t ) | u (cid:62) ( δ q t , δ Q t )] : = Agg Ψ u , v ( x ) = a Ψ x ,where the slope a Ψ depends on the cross-impact model Ψ and on the order flow covariance. Even in the absence ofcross-impact, the presence of order flow correlations between two portfolios u and v may lead to a non-zero cross aggregateimpact. Thus, to test whether there is cross-impact, we compare the empirically measured Agg u , v to the prediction Agg Ψ u , v for different cross-impact models Ψ . We differentiate models between those which have no off-diagonal contributions( Λ bs , Λ direct-4d , Λ direct-2d ) and thus ignore cross-impact and those that take it into account ( Λ , Λ ).We report Agg u , v in Figure 4 for different portfolios u , v described in Table 2. Diagonal plots show aggregate directimpact. As expected, buying the E-Mini increases, on average, the price of the E-Mini as shown by the u , v = spot plot (firstrow, first column). We see from the u , v = level plot (second row, second column) that buying options and VIX futuresincreases, on average, the implied volatility. Furthermore, buying options and VIX futures decreases, on average, the E-Miniprice as shown by the u = level, v = spot plot (first row, second column). This same plot, among others of Figure 4, showsthat direct models provide a poor fit for cross aggregate impact. This suggests that the cross aggregate impact can only beexplained by using a cross-impact model with off-diagonal elements, such as Λ . Further, the fit is noticeably better for Λ than Λ which highlights the importance of taking into account the skew and term factors. Let us summarize what we have achieved. Our main objective was to examine cross-impact when some instruments,derivatives, are deterministic functions of others, underlying. This posed modeling challenges on two fronts. First,13 ame Componentsspot VIX VIX optionsspot (1, 0, 0, 0, ··· ,0)level (0, β , β , β , ··· , β M )skew (0, β , β , β , ··· , β M )term (0, β , β , β , ··· , β M ) Table 2: Description of different directions used in this section. v = s p o t v = l e v e l v = s k e w u = spot v = t e r m u = level u = skew u = term Agg u , v ( x ) v x u = bs = direct 2d = direct 4d = = Agg u , v Figure 5: Normalized cross aggregate impact curves.
We report the cross aggregate cross impact curves for the spot, level, skew and term structure directions. Aggregate traded volumes arenormalized by the typical deviations ω u : = (cid:69) t [( u (cid:62) ( δ q t , δ Q t )) ] and portfolio returns by the typical deviations σ v : = (cid:69) t [( v (cid:62) ( δ p t , δ P t )) ].Estimated cross aggregate impact Agg u , v is reported along with predicted cross aggregate impact Agg Ψ u , v for different choices of linearcross-impact models Ψ . derivative prices with impact must be locked-in by no-arbitrage. Second, the liquidity of derivative instruments may bevery small and heavily non-stationary.Leveraging the results of [16], we introduced the Kyle cross-impact model on derivatives. We showed in Section 3.3 thatthis model (i) prevents arbitrage, (ii) provides impact dynamics which can be factored to recover frictionless dynamics, (iii)aggregates traded order flow to a few liquidity pools, (iv) is well-behaved even if some instruments are highly illiquid, and(v) can be applied without specifying which instruments are derivatives. The Kyle cross-impact model is thus theoreticallysatisfying and practical for applications. This justifies its use compared to other cross-impact models (see [16] for examplesof other models).To stress-test our framework on empirical data, we used the front-month E-Mini future, E-Mini vanilla options and VIXfutures. Despite our simplistic approach to model implied volatility dynamics, we showed (see Figure 5 and Table 1) thatthe Kyle model better explains returns than models which ignored cross-impact, and that the Kyle model can be improvedby more precise modeling of the implied volatility surface. This points at the effectiveness of the proposed framework toaggregate liquidity and suggests more sophisticated implied volatility modeling may further improve results. Aside fromexplanatory power, cross-impact models are consistent with some empirical observations noted in [13, 18]. We providedevidence of cross-impact in derivative markets by studying price responses conditional on traded order flow and showingmodels which ignore cross-impact are unable to provide an adequate fit (see Figure 5).On both theoretical and empirical grounds, the simple static, linear framework presented here thus accounts forimportant properties of impact on derivative markets. The methodology is flexible and can be readily adapted to handle14omplex price dynamics and exotic derivatives. The cross-impact estimates may be used in practice for estimatingexecution costs, in particular hedging costs.While this work provides a first proof of concept which captures many features of cross-impact on derivatives, it failsto capture the auto-correlations of order flows. A multi-period cross-impact model, apart from providing more accuratedescriptions of cross-impact, would yield insight into the cross-impact of meta-orders, another topic left unexploredto this date and particularly difficult to measure on options. We leave this extension to future work since it raises newchallenges. For example, multi-period, cross-sectional arbitrages yield new constraints for the chosen cross-impact model.Furthermore, to remain tractable, such models need to find the proper way to aggregate traded order flow in liquidity pools. We warmly thank J.-P. Bouchaud, Z. Eisler and B. Toth and M. Rosenbaum for fruitful discussions. This research wasconducted within the
Econophysics & Complex Systems
Research Chair under the aegis of the
Fondation du Risque , a jointinitiative by the
Fondation de l’École polytechnique, l’École polytechnique and Capital Fund Management.
References [1] Aurélien Alfonsi, Florian Klöck, and Alexander Schied. Multivariate transient price impact and matrix-valued positivedefinite functions.
Mathematics of operations research , 41(3):914–934, 2016.[2] Robert Almgren, Chee Thum, Emmanuel Hauptmann, and Hong Li. Direct estimation of equity market impact.
Risk ,18(7):58–62, 2005.[3] Bruno Bouchard, Grégoire Loeper, and Yiyi Zou. Hedging of covered options with linear market impact and gammaconstraint.
SIAM Journal on Control and Optimization , 55(5):3319–3348, 2017.[4] Jean-Philippe Bouchaud, Julius Bonart, Jonathan Donier, and Martin Gould.
Trades, Quotes and Prices . CambridgeUniversity Press, 3 2018.[5] Jordi Caballe and Murugappa Krishnan. Imperfect competition in a multi-security market with risk neutrality.
Econometrica (1986-1998) , 62(3):695, 1994.[6] Luis Carlos Garcia del Molino, Iacopo Mastromatteo, Michael Benzaquen, and Jean-Philippe Bouchaud. The multi-variate kyle model: More is different.
SIAM Journal on Financial Mathematics , 11(2):327–357, 2020.[7] Lawrence R Glosten and Paul R Milgrom. Bid, ask and transaction prices in a specialist market with heterogeneouslyinformed traders.
Journal of financial economics , 14(1):71–100, 1985.[8] Joel Hasbrouck and Duane J Seppi. Common factors in prices, order flows, and liquidity.
Journal of financialEconomics , 59(3):383–411, 2001.[9] Albert S Kyle. Continuous auctions and insider trading.
Econometrica: Journal of the Econometric Society , pages1315–1335, 1985.[10] Gregoire Loeper and others. Option pricing with linear market impact and nonlinear Black–Scholes equations.
TheAnnals of Applied Probability , 28(5):2664–2726, 2018.[11] Iacopo Mastromatteo, Michael Benzaquen, Zoltán Eisler, and Jean-Philippe Bouchaud. Trading Lightly: Cross-Impactand Optimal Portfolio Execution. 2017.[12] Paolo Pasquariello and Clara Vega. Strategic cross-trading in the US stock market.
Review of Finance , 19(1):229–282,2015.[13] Emilio Said. How option hedging shapes market impact.
Available at SSRN 3470915 , 2019.[14] Michael Schneider and Fabrizio Lillo. Cross-impact and no-dynamic-arbitrage.
Quantitative Finance , 19(1):137–154,2019.[15] Damian Eduardo Taranto, Giacomo Bormetti, Jean-Philippe Bouchaud, Fabrizio Lillo, and Bence Tóth. Linear modelsfor the impact of order flow on prices. I. History dependent impact models.
Quantitative Finance , 18(6):903–915, 2018.1516] Mehdi Tomas, Iacopo Mastromatteo, and Michael Benzaquen. How to build a cross-impact model from first principles:Theoretical requirements and empirical results. arXiv preprint arXiv:2004.01624 , 2020.[17] Nicolo Torre. BARRA market Impact model handbook.
BARRA Inc., Berkeley , 1997.[18] Bence Tóth, Zoltán Eisler, and J-P Bouchaud. The square-root impact law also holds for option markets.
Wilmott ,2016(85):70–73, 2016.[19] Shanshan Wang, Rudi Schäfer, and Thomas Guhr. Price response in correlated financial markets: empirical results. arXiv preprint arXiv:1510.03205 , 2015. 16
Data
This section gives motivation about our choice of instruments, details on the data and methodology which were omitted inthe main text for conciseness.
Feb Mar Apr Jun Jul Aug Sep0.00.20.40.60.81.0 R i s k e x c h a n g e d ( r e l a t i v e t o t o t a l ) optionsVIX_0VIX_1VIX_2VIX_3VIX_4 Figure 6: Distribution of liquidity among VIX futures and options.
Choice of instruments
To stress-test our approach, we sought an actively traded derivative market with many derivatives.Thus we considered E-mini vanilla options and their underlying (both quoted on the CME), the front-month futurescontract. However, a large fraction of the traded risk in derivatives comes from VIX futures (quoted on the CBOE) as shownin Figure 6. The VIX index is computed using options with maturities between 23 and 37 days and is meant to track thelevel of the implied volatility for options expiring in one month. Thus, because of their liquidity and close relationship withthe implied volatility of options, order flow traded on VIX futures play an important role and should not be ignored.
Filtering instruments
Given the very large number of options quoted on the market, we kept options within a givenrange of strikes and maturities to limit the size of the data set.
Resolution and time frame
Our dataset contains the trades and quotes of all previously selected products, at the fiveminute time scale, from January 2019 to September 2019. This time frame was chosen because of the large level of noise onderivatives’ prices and the size of the data set which encumbered analysis. In a given five minute bin, signed trades wereaggregated on their volumes, so that we have the opening and closing prices of instruments along with the aggregatedsigned traded order flow. We considered hours where both options and their underlying are liquid, further removing 30minutes around opening and closing for stationarity purposes. Doing so, data ranges between 3PM and 8:30PM UTC.
Implied volatility and greeks