Equations and Shape of the Optimal Band Strategy
EEquations and Shape of the Optimal Band Strategy
Joachim de Lataillade, Ayman ChaoukiCapital Fund Management23 rue de l’Université, 75007 Paris, FranceMarch 18, 2020
Abstract
We consider the problem of the optimal trading strategy in the presence of a price predictor,linear trading costs and a quadratic risk control. The solution is known to be a band system,a policy that induces a no-trading zone in the positions space. Using a path-integral methodintroduced in a previous work, we give equations for the upper and lower edges of this band, andsolve them explicitly in the case of an Ornstein-Uhlenbeck predictor. We then explore the shapeof this solution and derive its asymptotic behavior for large values of the predictor, withoutrequiring trading costs to be small.
Price returns on financial markets are by nature very difficult to predict, and the goal of statisticalarbitrage is to find small but significant predictive patterns in all available data. However, froma practitioner’s perspective, the prediction of the price is only an ingredient in the building of atrading system: controlling the risk taken by this system, and avoiding high costs when trading,are crucial elements of success.In the present paper, we focus on the optimisation of trading in a specific case: we considerthe single-asset case, where the risk is controlled through a penalty on the square of the exposure(or position) on that asset, and with a linear cost of trading of the form Γ | Q | , where Q is thequantity bought or sold at a given time. Because of the relation between costs and market impactmodels [TLD +
11, DBMB14], quadratic or at least superlinear models of costs are often consid-ered [DL07, GP09]. Linear transaction costs are nonetheless relevant when considering market andbrokerage fees, or costs for crossing the bid-ask spreads, as they become dominant for small tradingamounts.Systems with linear (aka. proportional) transaction costs have been considered on many occa-sions in the literature [DN90, SS94, Con86, MS11, Mar12], with a focus on particular on the limitof small transaction costs [MKRS17, LMKW14, RBdL + a r X i v : . [ q -f i n . M F ] M a r aper, using a method first introduced in [dLDPB12], which infers the limit of a no-trading zone bystudying the possible future paths of the predictor when starting from this limit, we end up witha much more explicit solution for the upper and lower edges of the band. This allows in particularto derive new asymptotic results, which do not require costs to be small. In particular, we derive:i) the asymmetry of the band when the predictor becomes large ii) the asymptotic size of the bandand iii) the position of the band around zero when trading costs become large.The content of the paper is as follows: after having formalized the problem we want to solve,we show why the shape of its optimal solution is necessarily a band (as we are not aware of anysuch proof already existing in the literature for this exact problem), and then extend the techniquesintroduced in [dLDPB12] to derive path-integral equations for the upper and lower edges of the band.We then restrict ourselves to the case of a predictor following an Ornstein-Uhlenbeck dynamics andobtain explicit solutions in this case, for which we can derive the asymptotic behavior as a function ofthe predictor’s value. Finally, we run numerical estimations of our analytical formulas and comparethe resulting policy against a system with a constant and symmetric band. The problem we address in this paper is to find the optimal strategy for a trader in the presenceof a predictor, a quadratic risk penalty and a linear cost term. This means we want to find at anymoment the optimal position π t , given: • A predictor of the future price returns, following a random process ( p t ) t , which generates again p t · π t . • A risk penalty for holding a position: λπ t . • A cost penalty for trading: Γ | π t − π t − | .We require the predictor to be a Markovian process, independent of time t , and unbounded: ∀ q ∀ p, ∃ (cid:15) q,p > s.t. P ( p t +1 > q | p t = p ) > (cid:15) q,p The optimal policy can then be defined explicitely as the function π (cid:63) ( π, p ) given by:argmax π (cid:63) : R → R lim T →∞ E (cid:34) T T (cid:88) t =1 p t π t − λπ t − Γ | π t − π t − | (cid:12)(cid:12)(cid:12)(cid:12) π = 0 and π t = π (cid:63) ( π t − , p t ) ∀ t > (cid:35) Note that without loss of generality we can rescale all the positions by a constant factor, so wewill fix the value λ = 1 / . This allows to see the value p of the predictor itself as a position: it isthe position which maximizes the instantaneous gain g p ( π ) = p · π − π , sometimes called the idealposition .Finally, we will frequently use the function V ( π, p ) to indicate the future gains and losses if wechoose to stay in position π for a value p of the predictor (and then trade optimally): V ( π, p ) = E (cid:34) T (cid:88) t =1 p t π t − π t − Γ | π t − π t − | (cid:12)(cid:12)(cid:12)(cid:12) p = p , π = π = π and π t = π (cid:63) ( π t − , p t ) ∀ t > (cid:35)
2n theory V should be indexed by T , but in practice we will assume this T to be large enough sothat it does not really intervene in the results. We have then, for any π and p : π (cid:63) ( π, p ) = argmax π (cid:48) (cid:2) V ( π (cid:48) , p ) − Γ | π (cid:48) − π | (cid:3) By expansion of its first term, V also satisfies the equation: V ( π, p ) = pπ − π + (cid:90) (cid:2) V ( π (cid:63) ( π, p (cid:48) ) , p (cid:48) ) − Γ | π (cid:63) ( π, p (cid:48) ) − π | (cid:3) P ( p (cid:48) | p ) d p (cid:48) with P ( p (cid:48) | p ) = P ( p t +1 = p (cid:48) | p t = p ) . It is well-known folklore in the literature [MKRS17] that the optimal strategy in this context will bea band , also known as a DT-NT-DT (Discrete-Trading / No-Trading / Discrete-Trading) policy:it is the system described on Figure 1: • To each value p of the predictor are associated two positions (cid:96) ( p ) and u ( p ) , such that (cid:96) ( p ) ≤ p ≤ u ( p ) : these two positions determine a “band” around the predictor. • If the current position is inside the band for the current predictor p t , the optimal policy is todo nothing: π t = π t − . • If the current position is above (resp. below) the band, the optimal policy is to trade directlytowards it: π t = u ( p t ) (resp. π t = (cid:96) ( p t ) ). pLU Discrete TradingDiscrete TradingNo Trading ππ π Figure 1: Behavior of the band strategy (aka. DT-NT-DT).This policy is highly sparse on trades, which is coherent with the L constraint of the costpenalty. However we are not aware of any formal justification in the literature for the optimalityof that system, so in this section we would like to provide some arguments in that direction. Theproof will be made in four parts: 3. The function V is concave in π : ∂ V∂π < .2. For a given p , the no-trading zone { π ∈ R | π (cid:63) ( π, p ) = π } is convex, so it is a segment.3. When we are outside of the no-trading zone, we always trade towards the edge of it.4. The predictor p is always inside the no-trading zone. V is concave in π Let us consider a given position π and a fixed p . Setting current time at zero, we consider a path ( p t ) t ≥ for the future evolution of the predictor, and we call τ the first time in the future where π (cid:63) ( π, p τ ) = π (cid:54) = π : τ will be the first moment where we do a trade.If we call δV the component of V ( π, p ) coming from this particular future path, we have: δV = τ (cid:88) t =0 ( p t π − π ) − Γ | π − π | + V ( π , p τ ) so that: ∂ δV∂π = ∂ ∂π τ (cid:88) t =0 ( p t π − π )= − τ By summing over all possible future paths, we obtain that the second derivative along π is indeednegative. Consider three positions π < π < π for a given p , suppose that π , π are in the no-trading zonewhereas π is not. Then π (cid:63) ( π , p ) = π = π + δπ with δπ (cid:54) = 0 .Suppose that δπ > . Then V ( π , p ) − V ( π , p ) > Γ · δπ . By the mean value theorem thereexists π ∈ [ π , π ] such that ∂V∂π ( π , p ) = V ( π , p ) − V ( π , p ) δπ Since ∂ V∂π < everywhere, we would have ∂V∂π ( π , p ) > Γ . So, close enough around π , it would beworth trading: π could not belong to the non-trading zone.Of course we can apply the same argument if δπ < by using π instead of π . So for any p ,the no-trading zone is a convex set on R , hence a segment [ (cid:96) ( p ) , u ( p )] . First, we prove that if π (cid:63) ( π , p ) = π then π (cid:63) ( π , p ) = π : after a trade, we always end up in theno-trading zone. Indeed, if we had π (cid:63) ( π , p ) = π (cid:54) = π then we would have: V ( π , p ) − V ( π , p ) = V ( π , p ) − V ( π , p ) + V ( π , p ) − V ( π , p ) > Γ | π − π | + Γ | π − π | > Γ | π − π |
4o that, starting from π , it would be better to jump to π than to π .Moreover, this trade is always towards the edge of the band: indeed, if we have π < π < π and π (cid:63) ( π , p ) = π then V ( π , p ) − V ( π , p ) − Γ | π − π | > V ( π , p ) − V ( π , p ) − Γ | π − π | (otherwise we would better jump to π than π ), so: V ( π , p ) − V ( π , p ) > Γ | π − π | − Γ | π − π | > Γ | π − π | so π is not in the no-trading zone. p ∈ [ (cid:96) ( p ) , u ( p )] The position p is the maximum of the function g p ( π ) = p · π − π . By definition of V , for any π we have: V ( π, p ) = g p ( π ) + (cid:90) (cid:0) V ( π p (cid:48) , p (cid:48) ) − Γ | π p (cid:48) − π | (cid:1) P ( p (cid:48) | p ) d p (cid:48) with π p (cid:48) = π (cid:63) ( π, p (cid:48) ) , so V ( π, p ) − Γ | π − p | ≤ g p ( p ) + (cid:90) (cid:0) V ( π p (cid:48) , p (cid:48) ) − Γ | π p (cid:48) − π | − Γ | π − p | (cid:1) P ( p (cid:48) | p ) d p (cid:48) ≤ g p ( p ) + (cid:90) (cid:0) V ( π p (cid:48) , p (cid:48) ) − Γ | π p (cid:48) − p | (cid:1) P ( p (cid:48) | p ) d p (cid:48) ≤ V ( p, p ) so that π (cid:63) ( p, p ) = p : the predictor is always inside the no-trading zone.Now that we have established the shape of the optimal strategy, we will derive the explicitequations for the values of u ( p ) and (cid:96) ( p ) . As already said, some equations of this sort alreadyappear in [MS11], but here we will provide more explicit solutions that will allow to calculate inSection 5.2 the asymptotic behavior in p . As in [dLDPB12], we will rely on an analysis of the optimal behavior when the position is close tothe non-trading zone in order to establish the equations for the band. However, since this time wehave two parameters to determine (the two edges of the band), we need to find a system of twoequations.Let us consider a value p for the predictor, we will note u = u ( p ) and (cid:96) = (cid:96) ( p ) . We alsointroduce p as the value of the predictor for which (cid:96) is the upper edge : u ( p ) = (cid:96) ( p ) .We suppose that the current position ( t = 0 ) is at (cid:96) , and consider two cases:i) The current value of the predictor is p , and we wonder if it is worth buying an infinitesimalquantity δπ . 5 p p Γ LU −2 Γ φφ φ
12 1 ’’ φ Figure 2: Configurations giving rise to the equations for the band.ii) The current value of the predictor is p and we wonder if it is worth selling an infinitesimalquantity δπ .In each case we will consider the different future paths taken by the predictor, keeping in mindthat our future behaviour is the optimal one (stay inside the band or trade towards it). The situationis summarised on Figure 2. Note that we do not need to look at what happens after we exit theband, because the optimal position will not depend anymore on what we did at t = 0 . Note alsothat, because the predictor dynamics is unbounded, the paths that stay inside the band foreverhave a null contribution when we integrate over all paths, so we can safely ignore them.Let us consider first the case i). If we buy δπ starting from position (cid:96) then we are inside theband, and we will stay there as long as: • either the predictor becomes larger than p (path φ ), • or it becomes smaller than p (path φ ).Compared to the case where we stayed at (cid:96) without buying, we will not have suffered anyadditional cost if the predictor follow the path φ , whereas we will have paid · δπ in the case ofthe path φ (because we paid linear costs when buying δπ , and then again by selling it when thepredictor goes below p ). We denote by δ C this potential additional cost.Now, in terms of gains, the difference between both situations is simply δ G = T φ (cid:88) t =0 φ ( t ) · δπ for φ ∈ { φ , φ } , where T φ is the first time where φ ( t ) > p or φ ( t ) < p .And finally, in terms of risk, the difference is δ R = T φ (cid:88) t =0 (cid:18) · ( (cid:96) + δπ ) − · (cid:96) (cid:19) = T φ · (cid:96) · δπ + O ( δπ ) for φ ∈ { φ , φ } . 6e now need to integrate over all possible paths: for a finite path φ : [0 , n ] → R , we note: T φ = nφ b = φ (0) φ e = φ ( n ) P ( φ | p ) = P ( p z = φ ( z ) , z ∈ [0 , n ] | p = p ) (cid:90) z F ( φ ( z )) d z = n − (cid:88) i =0 F ( φ ( i )) Then it is worth buying δπ at t = 0 if, and only if: φ e ≥ p ∨ φ e ≤ p (cid:90) φ b = p p <φ ( z )
7. A difference in risk equal to: δ R = T φ (cid:88) t =0 (cid:18) · ( (cid:96) − δπ ) − · (cid:96) (cid:19) = − T φ · (cid:96) · δπ + O ( δπ ) So it is indeed worth selling δπ if, and only if: δπ · φ e ≥ p ∨ φ e ≤ p (cid:90) φ b = p p <φ ( z )
8n the next section we will consider a continuous dynamics for the predictor, in which case eachterm in the above equations is equal to zero by definition (except P ( p ) which goes to ), and theequations become trivial. This is the classical issue of evaluating a continuous stochastic systemclose to a boundary, and this is solved by requiring the equalities above to be true around p and p up to first-order expansion : G (cid:48) ( p ) − (cid:96) · R (cid:48) ( p ) − P (cid:48) ( p ) = 0 (3) G (cid:48) ( p ) − (cid:96) · R (cid:48) ( p ) − P (cid:48) ( p ) = 0 (4) Let us now consider the case where the dynamics of the predictor ( p t ) t is given by a discreteOrnstein-Uhlenbeck process: p t +1 − p t = − ε · p t + β · ξ t (5)where ( ξ t ) t ∈ R is a set of independent N (0 , Gaussian random variables.In what follows, contrary to [dLDPB12], we will only consider the continuous limit: β (cid:28) Γ (no single-step jump in the predictor is significant compared to the costs). The dynamics of thepredictor can then be written in a more continuous form:d p = − εp d t + β d X t (6)where ( X t ) t is a Wiener process. Now that the dynamics of the predictor is fixed, we can calculate the functions G , R and P , and solveEquations (3) and (4). To make the reasonings easier to follow, we will redefine them temporarilyas functions of two variables: G ( p, t ) = G ( p ) , R ( p, t ) = R ( p ) and P ( p, t ) = P ( p ) .To calculate G , we can make use of It¯o’s lemma with Equation (6):d G = ∂ G ∂t d t + ∂ G ∂p d p + 12 β ∂ G ∂p d t d G = (cid:18) ∂ G ∂t − εp ∂ G ∂p + 12 β ∂ G ∂p (cid:19) d t + β ∂ G ∂p d X t Let us now consider the operator (cid:104)·(cid:105) d X which integrates over all possible values for d X t : bydefinition of G , we can write, for p ∈ [ p , p ] , G ( p, t ) = p d t + (cid:104) G ( p + d p, t + d t ) (cid:105) d X so that (cid:104) d G (cid:105) d X = − p d t . Since we also have (cid:104) d X t (cid:105) d X = 0 and ∂ G / ∂t = 0 , it gives β ∂ G ∂p − εp ∂ G ∂p = − p One can understand this by considering only one discrete, infinitesimal step starting from p or p , followed bya continuous dynamics. G ( p ) = 0 and G ( p ) = 0 . This is the Kolmogorov backward equation of the system for the gain term.This equation can be solved as: G ( p ) = 1 ε (cid:18) p − p − p − p I (cid:90) pp e ax d x (cid:19) with a = εβ and I = (cid:90) p p e ax d x A similar reasoning can be applied to find the Kolmogorov backward equation for R : β ∂ R ∂p − εp ∂ R ∂p = − with initial conditions R ( p ) = R ( p ) = 0 .Its solution is: R ( p ) = 2 aKε (cid:18) I (cid:90) pp e ax d x − K (cid:90) pp e ax (cid:20)(cid:90) xp e − ay d y (cid:21) d x (cid:19) with K = (cid:90) (cid:90) p (cid:54) x (cid:54) y (cid:54) p e a ( x − y ) d x d y And finally, the equation for P is: β ∂ P ∂p − εp ∂ P ∂p = 0 with initial conditions P ( p ) = 0 and P ( p ) = 1 .Its solution is: P ( p ) = 1 I (cid:90) pp e ax d x Plugging the functions above into Equations (3) and (4) (with unknown p and (cid:96) ), we obtain: (cid:40) Ie − ap − ( p − p ) − a(cid:96) · K − ε = 0 Ie − ap − ( p − p ) − a(cid:96) · K + 2 a(cid:96) · IJ − ε = 0 with J = (cid:90) p p e − ax d x By simply solving this system of two equations, we finally end up with the result:10 roposition 1.
For a predictor whose dynamics is governed by Equation (5), the lower edge of theband associated to a value p = p of the predictor is: (cid:96) ( p ) = e − ap − e − ap a · J (7) where p is given, as a function of p , by: p − p = 2Γ ε − Ie − ap + K · ( e − ap − e − ap ) J (8) with: a = εβ , I = (cid:90) p p e ax d x , J = (cid:90) p p e − ax d x , K = (cid:90) (cid:90) p (cid:54) x (cid:54) y (cid:54) p e a ( x − y ) d x d y Similarly, the upper edge of the band associated to a value p = p of the predictor is: u ( p ) = e − ap − e − ap a · J (9) where p is given, as a function of p , by Equation (8). All the parameters of the problem can actually be factorized in Proposition 1: indeed, if we set q = p √ a , q = p √ a and (cid:96) (cid:63) = (cid:96) √ a , then the result can be rewritten as: (cid:96) (cid:63) ( q ) = F ( q , q ) with q given by: G ( q , q ) = 2Γ ε / β (10)where F ( q , q ) = e − q − e − q (cid:82) q q e − x d xG ( q , q ) = q − q + e − q (cid:90) q q e x d x − e − q − e − q (cid:82) q q e − x d x · (cid:90) (cid:90) q (cid:54) x (cid:54) y (cid:54) q e x − y d x d y As explained in [dLDPB12], up to a factor √ , βε − / is the standard deviation σ p of thepredictor and βε − / its integrated average gain (taking into account its autocorrelation). So thefactor Γ /βε − / from Equation (10) is a very natural scale for the problem, since it compares theaverage total gain coming from the predictor to the cost of a trade. The rescaling q = p √ a = p/βε − / is also easy to interpret: it is just a rescaling of the predictor by its standard deviation σ p (multiplied by √ ).So, after normalisation of the predictor, the edges of the band are only determined by thepredictor’s value and the ratio Γ /βε − / . 11 .2 Asymptotic shape of the band Now that we have the explicit solutions for (cid:96) ( p ) and u ( p ) through Proposition 1, we can look attheir asymptotic behavior when the predictor p takes very large or very small values.The questions we are interested in are the following: • How does the size of the band evolve with large / small values of p ? • How is the symmetry of the band around the predictor affected in those limits? p → If p = 0 , the symmetry of the system is straightforward: u (0) = − (cid:96) (0) . Now, using the notation p = p for simplicity, Equation (10) becomes: G (0 , p √ a ) = 2Γ ε / β As we would like to consider the limit p → , this requires Γ to be small, more precisely: Γ (cid:28) βε − / In this limit, one has then: I = p + a · p + O ( p ) J = p − a · p + O ( p ) K = p O ( p ) So Equation (8) becomes, to the main order in p : − p = 2Γ ε − p − a · p + p · ap p − a · p which leads to: p = − · Γ β Equation (7) then gives: (cid:96) = ap a ( p − a · p ) = p O ( p ) So we obtain (cid:96) (0) = − (cid:114) · Γ β and u (0) = (cid:114) · Γ β Those are the limits found in [MS11] as well as in [dLDPB12] in the case of small linear costs.See in particular [Rog04] where an explanation is provided for the appearance of a / exponent onthe parameter Γ . 12 .2.2 Case where p → + ∞ (continuous case) We now consider the limit p , p → + ∞ . First, let us recall that: (cid:90) p e αx d x ∼ p →∞ e αp αp + C te for any α , positive or negative.This leads to: I ∼ − e ap ap (1 − η · p p ) J ∼ − e ap ap (1 − η · p p ) with η = e a ( p − p ) and also: K ∼ (cid:90) p p e ax · (cid:32) e − ap ap − e − ax ax (cid:33) d x ∼ e − ap ap (cid:90) p p e ax d x − (cid:90) p p d x ax ∼ − a p (1 − η · p p ) − a ln p p Plugging everything into Equation (8), we obtain, to first order in p and p : p − p = 2Γ ε + 12 ap (1 − η · p p ) − p · − η − η · p p (cid:18) ln p p + 12 ap (1 − η · p p ) (cid:19) We can then assume that p − p (cid:28) p and η (cid:28) , so we have: p − p = 2Γ ε − p · p − p p We set B = p − p , to get: B = 2Γ ε + B · (1 − Bp ) so: B = (cid:112) ε · p The equation for the lower edge of the band gives: (cid:96) ∼ p · − η − η · p p ∼ p So, in this limit, the band becomes completely asymmetric : the upper edge is equal to thevalue of the predictor. Consequently, B = p − p is in fact the size of the band, which grows asthe square-root of the predictor . 13o, to summarize, the equations give: u ( p ) ∼ p →∞ p(cid:96) ( p ) ∼ p →∞ p − (cid:112) ε · p Taking a step back, the fact that the band becomes asymmetric and bigger for larger p can beunderstood intuitively: the ideal position π = p is the one maximising the instantaneous gain/riskterm, and the role of the band is to avoid incuring excessive costs by following this position exactlyat any moment. Now, when the predictor becomes large, it becomes extremely likely that it willrevert towards zero, considering its dynamics given by Equation (5). So: • If we are above the ideal position, it makes sense to trade towards it since we will maximizethe instantaneous gain/risk term while doing a trade that we are very likely to do anywayduring the next time steps; hence the asymmetry of the band. • If we are below the ideal position, any trade we do towards the predictor will give us animmediate reward in the gain/risk term, but this rewards will most likely be offset by the factthat we will have to trade back during the next time steps; hence the lower edge getting fareraway from the predictor, and the band increasing in size with the predictor’s value. p → + ∞ (discrete case) The results above apparently imply that the size of the band will grow indefinitely. . . But there isan important pitfall there: when we introduced the continuous Ornstein-Uhlenbeck dynamics, westated that no single-step jump in the predictor is significant compared to the costs .This hypothesis is in general guaranteed by the fact that β (cid:28) Γ , since σ p is of the order of β(cid:15) − / . But if we take the freedom to explore very large predictor’s values for p , then we will reachthe point where the decrease ε · p coming in the next time step through Equation (5) becomescomparable to the cost Γ . Then the continuity hypothesis is broken, and all our calculations aboveare not valid anymore.Fortunately, in this extreme limit, the size of the band can actually be inferred from intuitivearguments. Suppose we are at position π slightly below the optimal lower bound, the predictor’svalue p being extremely large (and positive). At the next time step the predictor will almostcertainly be below π , so any trade we do in the direction of the band will have to be revertedimmediately.For any buy trade q > , one has then: V ( π + q, p ) = p · ( π + q ) −
12 ( π + q ) − · q + V where V is independent of q . So the maximum is reached when: ∂V∂q = 0 ⇔ p − q − π −
2Γ = 0 ⇔ π + q = p − p , we obtain: u ( p ) ∼ p →∞ p(cid:96) ( p ) ∼ p →∞ p − so the band size converges to .To summarize, when p becomes large, the size of the optimal band first grows as a square-root,as long as the system stays continuous, until we reach a region where the cost of trading are dwarfedby the instantaneous reward of the gain-risk term, and the band size then saturates. This behavioris very reminiscent of what happened in [dLDPB12] to the value of the threshold when β grows. By inverting Equation (8), one can find numerically the values of the lower and the upper boundsfor a given value of p . Note that the process can be quite unstable since large exponential values areinvolved, so one needs to be careful when initializing the solver. This gives in the end the resultsshown on Figure 3, where the upper and lower edges of the band are shown as functions of p , fordifferent values of Γ - or, more precisely, as functions of p/σ p for different values of the parameter Γ /βε − / , since we want to comply with the universality of Equation (10).One can see several interesting results on these figures:1. The asymmetry of the band is clearly visible for all values of Γ .2. By contrast, the increase of the band size when p grows is much more apparent for large valuesof Γ .3. The width of the (necessarily symmetric) band around p = 0 seems to reach a maximum when Γ grows.The third point in particular is interesting and rather counter-intuitive, but well supported bythe equations: indeed, for large values of Γ it is pretty clear that we will have | p | (cid:29) σ p : the valueof the predictor that initiates a trade towards (cid:96) (0) has to be large in order to beat the costs. So,without solving Equation (8), we have: lim Γ → + ∞ J = (cid:90) −∞ e − ax d x = − (cid:114) πa and consequently: lim Γ → + ∞ u (0) − (cid:96) (0) = 2 √ π · a = (cid:114) π · σ p (11)This probably deserves a little bit of explanation: why would a no-trading band reach a maximalwidth when linear costs become very large? The situation is in fact the following: • For high values of Γ /βε − / , one will have to wait for a very long time before seeing a predic-tor’s value which justifies to trade away from (ie. which "beats its costs").15 p l ( p ) u ( p )= p (a) Γ /βε − / = 0 . p l ( p ) u ( p )= p (b) Γ /βε − / = 0 . p l ( p ) u ( p )= p (c) Γ /βε − / = 1 p l ( p ) u ( p )= p (d) Γ /βε − / = 1 . Figure 3: Numerical estimations of the upper and lower edges of the band in the position vs.predictor space, for different values of Γ /βε − / . The x-axis and y-axis have been rescaled by σ p ,to make the resulting curves universal. • Consequently, when the predictor’s value is zero, there is no incentive to stay in a position farfrom it: we will suffer a loss due to the risk term while desperately waiting for the predictor tobeat its costs again. More specifically, if Γ grows by a factor k , the cost of trading is multipliedby k , whereas the waiting time before having a value of p that triggers a trade is increasedexponentially, and so will be the loss due to the risk penalty. • However, even if we trade, the optimal policy is not to trade directly towards zero: indeed,once close enough from zero, one can afford to wait a little bit to see whether the predictorbecomes positive or negative . If it becomes negative (and if our position is positive), we’ll This reasoning is interestingly reminiscent of an optimal liquidation problem with a predictor [LN19]: indeed,the high value of Γ means that one is only allowed to trade in one direction, but one can play with the value of thepredictor to decide when it is best to do the trades. Γ for larger values of p .Finally, we have compared the results we obtain through the equations with a simple grid-searchon a fixed and symmetric band system: u ( p ) = p + B/ and (cid:96) ( p ) = p − B/ , where B is optimizedfor any tuple Γ , β, ε by simply maximizing a PnL over a set of sample trajectories for the predictor.To compare the two systems, we ran 100 simulations of 50 000 time steps for each value of Γ (with β = ε = 0 . ) and looked at the PnL after risk and cost penalties. The results are shown onTable 1: as expected, the system induced by the equations outperforms significantly the constant andsymmetric band in all cases. In particular, in the case of high linear costs when Γ /βε − / = 0 . , thissystem is still able to generate some positive PnL whereas the more basic band avoids any tradingat all. Γ /βε − / Optimal Band Grid Search0.01 110.44 (0.75) 94.13 (0.73)0.1 67.85 (0.67) 65.17 (0.69)0.15 54.97 (0.63) 49.07 (0.67)0.2 45.11 (0.60) 32.98 (0.66)0.3 30.95 (0.53) 17.75 (0.55)0.5 14.81 (0.41) 0 (0)Table 1: Comparing simulation results between the optimal system given by Proposition 1 and aconstant-size, symmetric band found by grid-search. Number in parentheses indicate the statisticalerror calculated on the sample of simulations.
Conclusion
In this paper we have given explicit solutions for the optimal edges of the band in a system withlinear costs, quadratic risk control and an Ornstein-Uhlenbeck price predictor. This allows to studythe shape of this band precisely and to derive some asymptotic behaviors of interest. Furthermore,we have shown that the method of analyzing paths in a no-trading zone introduced in [dLDPB12]is a solid alternative to the explicit calculation of a value function, that may apply to other specificoptimization problems like mixing linear and non-linear costs [RBdL +
15] or the study of the multi-asset case [Mar12, EPB19].Another interesting direction to dig into would be to see how much of the present results canbe recovered through a more exploration-based approach, using modern machine-learning methodsto solve the problem. The reinforcement learning viewpoint presented in [CHS +
20] has been tried17n the context of the present work but, for the high values of Γ that we have been testing, we foundthe system to be too unstable to offer a strong benchmark against our analytical solution. Acknowledgements
We would like to thank Jean-Philippe Bouchaud and Stephen Hardiman for many fruitful interac-tions on the content of this article, as well as Johannes Muhle-Karbe for his help with academicreferences.
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