Trading Foreign Exchange Triplets
TTrading Foreign Exchange Triplets (cid:73)
Forthcoming, SIAM J. Financial Mathematics ´Alvaro Cartea a , Sebastian Jaimungal b , Tianyi Jia b a Mathematical Institute, University of Oxford, Oxford, UKOxford-Man Institute of Quantitative Finance, Oxford, UK b Department of Statistical Sciences, University of Toronto, Toronto, Canada
Abstract
We develop the optimal trading strategy for a foreign exchange (FX) broker who must liqui-date a large position in an illiquid currency pair. To maximize revenues, the broker considerstrading in a currency triplet which consists of the illiquid pair and two other liquid currencypairs. The liquid pairs in the triplet are chosen so that one of the pairs is redundant. Thebroker is risk-neutral and accounts for model ambiguity in the FX rates to make her strategyrobust to model misspecification. When the broker is ambiguity neutral (averse) the tradingstrategy in each pair is independent (dependent) of the inventory in the other two pairs in thetriplet. We employ simulations to illustrate how the robust strategies perform. For a rangeof ambiguity aversion parameters, we find the mean Profit and Loss (P&L) of the strategyincreases and the standard deviation of the P&L decreases as ambiguity aversion increases.
Keywords:
Foreign Exchange, Currency Pairs, Optimal Liquidation, Execution, InventoryAversion, Ambiguity Aversion
1. Introduction
Trading activity in the foreign exchange (FX) market is vast. On average, the dailyturnover is in excess of 5 trillion USD, where approximately a third of this turnover is in theFX spot market, and the remainder is in FX derivatives: forwards, swaps, and options (see (cid:73)
SJ would like to acknowledge the support of the Natural Sciences and Engineering Research Councilof Canada (NSERC), [funding reference numbers RGPIN-2018-05705 and RGPAS-2018-522715]. We thankseminar participants at the Fields Institute, PIMS Summer School in Mathematical Finance, LABEX–Paris,SIAM (Austin 2017), Dublin City University, Technical University Berlin, Humboldt University of Berlin.This work first appeared on https://ssrn.com/abstract=3054656 . Email addresses: [email protected] ( ´Alvaro Cartea), [email protected] (Sebastian Jaimungal), [email protected] (Tianyi Jia) a r X i v : . [ q -f i n . T R ] A p r IS (2016)). All currencies are traded in the FX spot market, but most of the turnover is ina handful of pairs (USD/EUR, USD/JPY, USD/GBP, USD/AUD, USD/CAD). The marketquotes a rate for those who buy (bid) or sell (ask) the currency pair, where the differencebetween the ask and the bid prices is the quoted spread and the average of the bid and askprices is the mid-exchange rate. A currency pair is traded by simultaneously taking a longposition in one currency and a short position in the other currency of the pair. The conversionrate between the two currencies is given by the bid (resp. ask) if the investor is going long(resp. short) the currency pair.In this paper we show how a broker liquidates a large position in an illiquid currency pair.One approach is to trade only the pair the broker aims to liquidate. An alternative is for thebroker to trade in three currency pairs, one of which is the illiquid pair and the additional pairsare such that the three pairs are formed by combinations of only three currencies. Therefore,by no-arbitrage, one of the currency pairs can be replicated by taking positions in the othertwo pairs, and hence the dynamics of the three currency pairs exhibit strong co-movements.Ideally, at least one of the two additional pairs is very liquid (heavily traded), so this “triangle”dependence between the pairs is considered by the broker to devise liquidation strategies thatoffset the illiquidity in one pair with the two other more liquid pairs.There is no central Exchange framework in FX, so brokers trade in electronic communi-cation networks (ECNs) with multiple liquidity providers and via other channels with theirown pool of clients, see Cartea et al. (2015) and Oomen (2017). We refer to the broker’strading activities on all ECNs as trading in the ‘Exchange’. The broker can control her ownliquidity taking orders sent to the Exchange, but has little or no control on the arrival rateof the orders of her clients.The broker is risk-neutral and acknowledges that her model of mid-exchange rates of thecurrency pairs, which is represented by a reference measure P , may be misspecified. She dealswith this model uncertainty, also referred to as ambiguity aversion, by considering alternativemodels when she develops the optimal execution strategy, see Cartea et al. (2017) and Carteaet al. (2016). These alternative models are characterized by probability measures that areabsolutely continuous with respect to the reference model P . The decision to reject thereference measure is based on a penalty that the broker incurs if she adopts an alternativemodel. The magnitude of the penalty depends on the broker’s degree of ambiguity aversionand is based on a measure of the distance between the reference and the alternative measure.The broker solves an optimal control problem where the objective is to liquidate a largeposition in an illiquid pair, and does by submitting liquidity taking orders to the Exchangein all three currency pairs. These orders have a temporary impact in the quote currency ofthe pair. That is, when the broker executes trades in the Exchange she receives worse ratesthan the quoted mid-exchange rate. The broker simultaneously fills orders from her pool ofclients. These orders are filled at the mid-exchange rate and clients pay a service fee to the2roker, where the size of the fee is proportional to the size of the order in the currency pair.We show that when the broker is ambiguity neutral, i.e., fully trusts the reference model,the optimal trading strategy in each currency pair is independent of the level of inventory inthe other two pairs. Thus, the broker’s strategy does not employ the other two currency pairsof the triplet to optimally execute a position in an illiquid pair. On the other hand, whenthe broker is ambiguity averse, the inventory position in each pair affects the trading strategyin the other two pairs. As the level of ambiguity aversion increases, the speed at which thestrategy builds positions in the liquid pairs of the triplet increases.We also demonstrate that the ambiguity averse broker makes her model robust to mis-specification by adopting a candidate measure where the drifts of the mid-exchange rates aredifferent from those of the reference model. Specifically, when the broker’s initial position inthe illiquid pair is short (resp. long), the broker devises a strategy where the drift in the mid-exchange rates of the currency pairs in the triplet, relative to the model under the referencemeasure P , is larger (resp. smaller). This affects the speed of trading in the robust model,relative to that of the reference model, because the expected growth in the mid-exchangerate of the currency pair affects the value of the inventory. For example, if a currency pair isexpected to appreciate, then, everything else being equal, it is optimal to increase the positionin that pair.We use simulations, based on parameters calibrated to FX data, to illustrate the perfor-mance of the strategy. Our results show that when the broker makes her model robust tomisspecification, the mean Profit and Loss (P&L) of the strategy increases and the standarddeviation of the P&L decreases.To the best of our knowledge this is the first paper that shows how FX brokers manage largepositions in currency pairs. Our framework can be extended to FX trading algorithms to makemarkets and those designed to take speculative positions in currency pairs with strong co-dependence. Although the literature on algorithmic trading in the FX market is scant, thereis a large body of work that looks at optimal liquidation, and other algorithmic strategiesin equity markets. For comprehensive treatments of algorithmic trading and microstructureissues in equity markets see Lehalle and Laruelle (2013), Cartea et al. (2015), Gu´eant (2016),Abergel et al. (2016).The remainder of the paper proceeds as follows. Section 2 presents the broker’s referencemodel for the currency pairs and discusses the broker’s cashflows from dealing currency pairswith her pool of clients and from executing trades in the FX Exchange. Section 3 solves thecontrol problem for the ambiguity neutral broker. Section 4 shows how the broker introducesmodel ambiguity to make the model robust to misspecification and solves the control problemof the ambiguity averse broker and provides the optimal speed to trade the currency pairsin the triplet. Section 5 illustrates the performance of the strategy and Section 6 concludes.3 21 X t Y t Z t Figure 1: Currencies and exchange rates for pairs in the FX triplet. The quote currency (the start of thearrow) is sold at the rate shown on the link, and one unit of the base currency (the end of the arrow) purchased.For example, when the broker goes long the currency pair (2 ,
1) in a frictionless market, she buys one unit ofthe base currency 2 and sells X t units of quote currency 1. Finally, some proofs are collected in the Appendix.
2. Model Setup
FX traders buy and sell currency pairs. Trading in a pair of currencies involves thesimultaneous sale of one currency and purchase of another currency and the price at whichthis transaction is done is the mid-exchange rate plus (minus) half the quoted spread whenlonging (shorting) the pair. In this paper we focus on a triplet of currency pairs that links theexchange rates of three currencies. We denote the three currencies by { , , } , and denotethe mid-exchange rates of the currency pairs by X t , Y t , and Z t , where t denotes time. Here X t is the mid-exchange rate for the pair (2 , Y t is the mid-exchange rate for the pair (3 , Z t is the mid-exchange rate for the pair (2 , ,
1) in a frictionless market (i.e., no feesand zero spread), she buys one unit of currency 2, known as the base currency in the pair,and sells X t units of currency 1, known as the quote currency in the pair. Similarly, the costof one unit of currency 3 is Y t units of currency 1; and the cost of one unit of currency 2 is Z t units of currency 3. By no-arbitrage, the mid-exchange rates for the reverse direction of thepairs, i.e., for (1 , , , /X t , 1 /Y t , 1 /Z t , respectively. See Figure1. One of the three currency pairs is redundant because it can be obtained by simultaneouslyexecuting transactions in the other two pairs. For example, buying one unit of the pair (2 , X t units of the pair (1 , , Z t = X t /Y t . (1)We focus on a triplet with a redundant currency pair because brokers who need to unwindpositions in highly illiquid currency pairs may rely on the triplet to enhance the financial4erformance of the liquidation strategy. For example, assume that a broker needs to unwinda position in one illiquid pair only. One approach is to devise an optimal liquidation strategybased on exclusively trading the illiquid pair. An alternative is to devise a liquidation strategywhere the broker also trades in two other pairs which are more liquid. These two additionalpairs are chosen to form, in conjunction with the illiquid pair, a triangular triplet. Thedynamics of the three pairs exhibit strong co-dependence, and the broker may offset some ofthe costs arising from the illiquidity in one pair by taking positions in the other, more liquid,pairs. To devise the optimal trading strategy, the broker assumes a model for the dynamics of thetriplet. This model is described by the completed probability space (Ω , F = {F t } t ∈ [0 ,T ] , P ),where F is the natural filtration generated by the processes X = ( X t ) t ∈ [0 ,T ] and Y = ( Y t ) t ∈ [0 ,T ] ,which satisfy the stochastic differential equations (SDEs) dX t = µ x X t dt + σ x X t dW xt , (2a) dY t = µ y Y t dt + σ y Y t dW yt , (2b)where σ x > σ y > µ x , µ y are constants, W x = ( W xt ) t ∈ [0 ,T ] and W y = ( W yt ) t ∈ [0 ,T ] are P –Brownian motions with correlation ρ ∈ [ − , Z = ( Z t ) t ∈ [0 ,T ] are implied by the no-arbitrage relationship (1) and are given by the SDE dZ t = µ z Z t dt + σ x Z t dW xt − σ y Z t dW yt , where µ z = µ x − µ y + σ y − ρ σ x σ y . (2c)The exchange rates of the triplet satisfy identity (1) if there are no trading fees, the quotedspread is zero (i.e., best bid and best ask prices in the LOB are the same), and the market isarbitrage-free. On the other hand, if the product of the three exchange rates is greater thanone, there is an arbitrage opportunity, see Fenn et al. (2009). For example, assume that Z isthe exchange rate for the pair EUR/USD, Y is for the pair USD/CHF, and X is for the pairEUR/CHF, and assume that Z bidt Y bidt /X askt = f , where f >
1. If a trader initially holdsone Euro, the arbitrage consists of: (i) with one Euro purchase Z bidt USD, (ii) which buys Y bidt units CFH, (iii) and sell this position for 1 /X askt Euros. Clearly, this chain of transactionswill ‘convert’ one Euro into f Euros with no risk.We could assume that the exchange rate Z is as (2c) plus noise so that the product ofthe three rates is a stochastic process which is most of the time below 1 and very seldomabove 1. However, to keep the model as simple as possible, we assume that the product ofthe mid-exchange rates of the currency pairs of the triplet is 1 for all t .5 .2. Speed of trading, order flow, and inventory We assume the broker uses the Exchange exclusively to take liquidity (i.e., does not provideliquidity to the Exchange), and provides liquidity to her pool of clients by filling the orders ofclients. Thus, the broker controls the speed at which she trades aggressively in the Exchange,but cannot control the rate at which she fills the liquidity taking orders of her clients. Thebroker trades in the triplet of currencies over the time window [0 , T ] and aims at holding zeroinventory in all pairs by the terminal date T . We denote by ν = ( ν t ) t ∈ [0 ,T ] the vector-valuedprocess of execution speeds in the Exchange, for the pairs x, y, z , and write ν t = ( ν xt , ν yt , ν zt ) (cid:124) ,where (cid:124) is the transpose operator.The vector-valued process Q ν = ( Q ν t ) t ∈ [0 ,T ] represents the controlled inventory in eachcurrency pair of the triplet, which results from trading in the Exchange and from tradingwith her own pool of clients. That is, Q ν tracks the units of base currency in each currencypair – for modelling purposes, it is convenient to use “inventory of currency pair” insteadof “units of base currency” to track trades in different currency pairs that share a commonbase currency. At t = 0, the broker’s initial holding in the currency pairs is denoted by Q ν = ( Q x, ν , Q y, ν , Q z, ν ) (cid:124) . To understand our notation (in a frictionless market) if the firstentry of the inventory triplet is Q x = 100, then the broker is short 100 × X units of currency1 and long 100 units of currency 2. Similarly, if Q x = −
100 the broker is long 100 × X unitsof currency 1 and short 100 units of currency 2.When the broker’s initial inventory is Q ν , her initial portfolio of currency pairs consists of Q x + Q z units of currency 2, and Q y units of currency 3. Then, the value of the portfolio incurrency 1 is Q x × X + Q y × Y + Q z × Z × Y in a frictionless market, and in the absence ofarbitrage, the value of the portfolio is Q x × X + Q y × Y + Q z × X . As mentioned previously,two inventory processes, Q x, ν t and Q z, ν t , track the units of the base currency 2 in the pairs(2 ,
1) and (2 , t the strategy is to execute in the Exchange the amount ν xt ∆ t > , Q x, ν t +∆ t = Q x, ν t − ν xt ∆ t . If we further assume that over the time interval ( t, t + ∆ t ] the mid-exchangerate X t does not change, then the effect of the execution strategy is to reduce by: ν xt ∆ t × X t the amount the broker is short in currency 1, and ν xt ∆ t the amount the broker is long incurrency 2.While the broker’s target is to hold zero units in the base currency of all currency pairs bythe terminal date, i.e., Q ν T = , she continues to fill buy and sell orders of the clients in thethree pairs, and this affects the broker’s inventory. We assume the liquidity taking orders of theclients arrive according to independent compound Poisson processes. Let O k, ± = ( O k, ± t ) t ∈ [0 ,T ] k ∈ { x, y, z } . Specifically, O k, ± t = N k, ± t (cid:88) n =1 ξ k, ± n , (3)where market sell is denoted by + and market buy is denoted by − . Here, { ξ k, ± n } n =1 , ,... i.i.d. ∼ F k, ± , where F k, ± is the distribution function, with support on [0 , ∞ ), of the volume of theclients’ buy and sell orders in currency pair k . The counting processes N k, ± = ( N k, ± t ) t ∈ [0 ,T ] are independent Poisson processes with intensities λ k, ± , and represent the number of market(buy and sell) orders received by the broker in each currency pair through time. The Poissonprocesses, the size of the trades, and the Brownian motions W x,y are all mutually independent.All distribution functions are under the reference measure P . At this point, we extend thecompleted filtered probability space (Ω , F = {F t } t ∈ [0 ,T ] , P ), so that henceforth, F t representsthe natural filtration generated by ( W x , W y , O x, ± , O y, ± , O z, ± ).For example, when the broker fills a client’s market sell order of volume ξ x, + and there isno brokerage service fee, the broker’s inventory position in the pair x increases by ξ x, + ; thatis, the broker has shorted an additional ξ x, + × X t units of currency 1 and longed an additional ξ x, + units of currency 2.In all, the broker’s inventory is affected by the controlled executions sent to the Exchange,and the clients’ filled market orders. The dynamics of the inventory in each pair are given by Q k, ν T = Q k, ν t − (cid:90) Tt ν ku du − (cid:90) Tt (cid:90) R r J k, − ( du, dr ) + (cid:90) Tt (cid:90) R r J k, + ( du, dr ) , (4)where J k, ± is the jump measure of O ± k,t , with intensity measure λ k, ± F k, ± ( dr ) du . The broker’s trades in the Exchange receive temporary price impact. When the brokersends executions to the Exchange, she enters a position in the quote currency at a value thatis worse than the quoted mid-exchange rate. For example, if at time t the broker sends tothe Exchange a sell order for r units of the currency pair (2 , r × X t units of the quote currency 1 (in exchange of r units of the base currency 2), she receives r × X t (1 − a x r ) units of currency 1 (in exchange of r units of currency 2). The parameter a x ≥ , X t , Y t , Z t . For example, if a client sends a7ell order of r units in the pair (2 ,
1) the client receives the rate X t (1 − c x r ) r , which is worsethan the mid-exchange rate X t if c x >
0. We refer to this spread as a brokerage service fee.
At every instant in time the broker’s cash position is marked-to-market in currency 1and the cumulative value is denoted by X ν = ( X ν ) t ∈ [0 ,T ] . This cash position results fromthe market making activity with her clients, the liquidity taking orders she sends to the FXExchange, and is given by X ν T = X ν t + (cid:90) Tt (cid:104) (cid:98) X u ν xu + (cid:98) Y u ν yu + (cid:98) Z u ν zu (cid:105) du (5a)+ (cid:90) Tt (cid:90) R + (cid:101) X + u r J x, − ( du, dr ) − (cid:90) Tt (cid:90) R + (cid:101) X − u r J x, + ( du, dr ) (5b)+ (cid:90) Tt (cid:90) R + (cid:101) Y + u r J y, − ( du, dr ) − (cid:90) Tt (cid:90) R + (cid:101) Y − u r J y, + ( du, dr ) (5c)+ (cid:90) Tt (cid:90) R + (cid:101) Z + u r J z, − ( du, dr ) − (cid:90) Tt (cid:90) R + (cid:101) Z − u r J z, + ( du, dr ) , (5d)where (cid:98) X t = X t (1 − a x ν xt ) , (cid:101) X ± t = X t (1 ± c ∓ x r ) , (6a) (cid:98) Y t = Y t (1 − a y ν yt ) , (cid:101) Y ± t = Y t (1 ± c ∓ y r ) , (6b) (cid:98) Z t = Y t Z t (1 − a z ν zt ) , (cid:101) Z ± t = Y t Z t (1 ± c ∓ z r ) . (6c)Recall that a k ≥ c ± k ≥ , , , ν x ispositive, the broker is selling the pair (2 , ,
1) andmarked-to-market in the quote currency 1. 8 . Optimal trading in triplet of currency pairs
In this section we pose and solve the broker’s control problem when the objective is toliquidate an inventory position in the currency pairs by the terminal date T , and the brokeris confident about the reference model P . The broker maximizes terminal expected wealthresulting from trading in the Exchange and dealing currency pairs with her clients. Theresults in this section are used as a benchmark to compare how the broker’s strategy changeswhen the broker acknowledges that the reference model P is misspecified (see Section 4). The broker’s performance criterion is H ν ( t, X , p , q ) = E P t, X , p , q (cid:20) X ν T + (cid:96) (( X T , Y T ) , Q ν T ) (cid:21) , (7)where p = ( x, y ), q = ( q x , q y , q z ), (cid:96) ( p , q ) = x q x (1 − α x q x ) + y q y (1 − α y q y ) + x q z (1 − α z q z ) , α x,y,z ≥ . (8)The terminal payment (cid:96) ( p , q ) denotes the cash proceeds from liquidating any terminal inven-tory in the Exchange minus a penalty paid by the broker, marked-to-market in currency 1.The terminal payment includes a penalty of the form α k q k , which consists of fees stemmingfrom crossing the spread and a non-financial penalty included by the broker to tweak the liq-uidation strategy. For example, in the absence of order flow from clients, in the limit α k → ∞ the optimal strategy guarantees full liquidation by the terminal date. The state variable z is not included as an argument in the performance criterion because by no-arbitrage it is aredundant state variable. Finally, note that the broker is risk-neutral because the perfor-mance criterion (7) optimises expected wealth, i.e., the broker is not sensitive to the risk inthe outcome of terminal wealth.The broker’s value function is H ( t, X , p , q ) = sup ν ∈V H ν ( t, X , p , q ) (9)where the set of admissible controls is V = (cid:26) ν : ν is F -predictable, and E P (cid:20) (cid:90) T ( ν s ) ds (cid:21) < ∞ (cid:27) . (10)To solve the optimal control problem in (9), the dynamic programming principle holds,and the value function is the unique solution of the Hamilton-Jacobi-Bellman (HJB) equation0 = (cid:18) ∂ t + L p + sup ν L ν (cid:19) H + (cid:88) k ∈{ x,y,z } ,i = ± (cid:90) R + ∆ k, − it, X , p , q ,r H λ k,i F k,i ( dr ) , (11a)9ith terminal condition H ( T, X , p , q ) = X + x q x (1 − α x q x ) + y q y (1 − α y q y ) + x q z (1 − α z q z ) , (11b)where the various operators are defined as follows: L p H = µ x x ∂ x H + µ y y ∂ y H + σ x x ∂ xx H + σ y y ∂ yy H + ρ σ x σ y x y ∂ xy H , (12a) L ν H = (cid:88) k ∈{ x,y,z } (cid:20) − a k (cid:98) k ∂ X H ( ν k ) + ( (cid:98) k ∂ X H − ∂ q k H ) ν k (cid:21) (12b)∆ k, + t, X , p , q ,r H ( t, X , p , q ) = H ( t, X + ˜ k + r, p , q − r k ) − H ( t, X , p , q ) , (12c)∆ k, − t, X , p , q ,r H ( t, X , p , q ) = H ( t, X − ˜ k − r, p , q + r k ) − H ( t, X , p , q ) , (12d) (cid:98) k = k { k ∈{ x,y }} + x { k = z } , x = (1 , , (cid:124) , y = (0 , , (cid:124) , z = (0 , , (cid:124) , and ˜ k ± ∈ { ˜ x, ˜ y, ˜ z } denotes the mid-exchange rates defined in (6). Proposition 1.
The DPE (11a) admits the solution H ( t, X , p , q ) = X + x ( q x + q z ) + y q y − h x ( t ) x − h y ( t ) y − h x ( t ) x q x − h y ( t ) y q y − h z ( t ) x q z − h x ( t ) x ( q x ) − h y ( t ) y ( q y ) − h z ( t ) x ( q z ) , (13) where h x,y,z ( t ) , h x,y,z ( t ) , h x,y ( t ) are deterministic functions of time given explicitly in (57) , (59) , (61) . For a proof see Appendix A.
Theorem 2. If | µ (cid:98) k | < | α k /a k | , then the value function of problem (9) is given by (13) andthe controls ν ∗ are optimal and are given componentwise by ν k, ∗ t = a k (cid:16) h k ( t ) + 2 h k ( t ) Q k, ν ∗ t (cid:17) , k ∈ { x, y, z } , (14) where h k , ( t ) are provided in (57) and (59) . For a proof see Appendix B. From the above theorem, one sees that the optimal strategy doesnot have any cross effects. That is, the speed of execution in one pair is independent of theinventory held in the other two pairs. 10s a result of Theorem 2, when Q x = 0, Q y = 0, Q z (cid:54) = 0, µ x = 0, µ y = 0, λ x, ± = 0and λ x, ± = 0, the ambiguity neutral broker will only trade currency pair (2 ,
3) – recallthat in our model, an ambiguity neutral broker is equivalent to a risk-neutral one. Anotherobservation is that the optimal trading strategy does not depend on mid-exchange rates andis a deterministic function of time if there are no orders from clients. However, below weshow that the ambiguity averse broker will always trade the three currency pairs of the tripletregardless of the initial inventory and drift of the mid-exchange rates.The following proposition shows the execution strategy as the terminal date approachesand the terminal liquidation penalty is arbitrarily large.
Proposition 3. (i) When λ k, ± = 0 , the optimal trading strategy remains admissible in thelimit as the terminal inventory penalty parameter α k → + ∞ (for k ∈ { x, y, z } ). Moreover,near the end of the trading horizon, we have lim α k → + ∞ ν k, ∗ t = ( T − t ) − Q k, ν ∗ t + o ( T − t ) , (15) which results in complete liquidation of all inventories by T .(ii) When the arrival rate of the orders from clients is positive, i.e., λ k, ± > , the optimaltrading strategy is not admissible in the limit as α k → + ∞ (for k ∈ { x, y, z } ). For a proof see Appendix C.
4. Ambiguity Aversion on the Mid-Exchange Rates
The broker assumes that the dynamics of the three currency pairs are given by the referencemeasure P , see SDEs (2a)-(2c). However, the broker is not fully confident about the referencemodel for the mid-exchange rates, so she considers alternative measures to make the modelrobust to misspecification, see Cartea et al. (2017). This ambiguity about model choice, ormodel uncertainty, has an effect on the optimal strategy employed by the broker when tradingin the three currency pairs. We incorporate this ambiguity about model choice in two steps.First, we characterize alternative measures that describe the mid-exchange rate dynamics,and then we determine how the broker decides between employing the reference measure P or one of the alternative measures. 11he broker considers candidate measures Q that are equivalent to P and characterized bythe Radon-Nikodym derivative d Q ( κ ) d P = exp (cid:26) (cid:90) T κ (cid:124) u ρ − κ u du + (cid:90) T κ (cid:124) u ρ − d W κ u (cid:27) , (16)where ρ = (cid:20) ρρ (cid:21) , and κ = (( κ xt , κ yt ) (cid:124) ) t ∈ [0 ,T ] is a two-dimensional F -adapted process.We denote by Q the class of alternative measures Q = (cid:26) Q ( κ ) | κ is F − adapted and E P (cid:104) exp (cid:110) (cid:82) T κ (cid:124) u ρ − κ u du (cid:111)(cid:105) < ∞ (cid:27) , and write dX t = X t ( µ x dt + σ x κ xt dt + σ x dW x, κ t ) , (17a) dY t = Y t ( µ y dt + σ y κ yt dt + σ y dW y, κ t ) , (17b) dZ t = Z t (( µ z + σ x κ xt − σ y κ yt ) dt + σ x dW x, κ t − σ y dW y, κ t ) , (17c)where W x,y, κ = ( W x,y, κ t ) t ∈ [0 ,T ] are Q ( κ )-Brownian motions, and [ W x, κ , W y, κ ] t = ρ t , ρ ∈ [ − , t to T , i.e., H t,T ( Q | P ) = E Q t (cid:20) log (cid:18)(cid:18) d Q d P (cid:19) T (cid:30) (cid:18) d Q d P (cid:19) t (cid:19)(cid:21) . (18)Note that H t,T ( Q ( κ ) | P ) ≥ H t,T ( Q ( κ ) | P ) = 0 if and only if κ = , i.e., if and only if Q ( κ ) = P almost surely. As before, the broker’s aim is to liquidate the position in the currency pairs by the terminaldate T while maximizing expected terminal wealth and considering candidate models thatare penalized using relative entropy. The broker’s performance criterion is H ν ( t, X , p , q ) = inf Q ∈ Q (cid:110) E Q t, X , p , q [ X ν T + (cid:96) (( X T , Y T ) , Q ν T ) ] + ϕ H t,T ( Q | P ) (cid:111) , (19)where (cid:96) ( p , q ) is as in (8) and recall that p = ( x, y ), q = ( q x , q y , q z ).Here the parameter ϕ is a non-negative constant that represents the broker’s degree of am-biguity aversion. If the broker is confident about the reference measure P , then the ambiguityaversion parameter ϕ is small and any deviation from the reference model is very costly. In12he extreme ϕ →
0, the broker is very confident about the reference measure, so she chooses P because the penalty that results from rejecting the reference measure is too high. Thus,when ϕ →
0, the broker is ambiguity neutral and the problem is equivalent to solving thevalue function in (9) discussed above when the broker is risk-neutral.On the other hand, if the broker is very ambiguous about the reference model, consideringalternative models results in a very small penalty. In the extreme ϕ → ∞ , deviations fromthe reference model are costless, so the broker considers the worst case scenario.One can also assume that the broker is risk-averse and employs the performance criterion H ν ( t, X , p , q ) = E P t, X , p , q (cid:20) U ( X ν T + (cid:96) (( X T , Y T ) , Q ν T )) (cid:21) , where, as above, p = ( x, y ), q = ( q x , q y , q z ) and (cid:96) ( p , q ) is as in (8). Here, U ( · ) is thebroker’s utility function, which is concave. If the utility function is linear, we obtain therisk-neutral performance criterion in (9) discussed above. In general, when U is concave,the broker’s problem becomes more difficult to solve. We note, however, that under certainassumptions, ambiguity aversion and exponential utilities are equivalent, see e.g., Schweizer(2010) for details. The broker’s value function is H ( t, X , p , q ) = sup ν ∈V H ν ( t, X , p , q ) , (20)where the set of admissible controls is as in (10). To solve the optimal control problem in(20), the dynamic programming principle holds, and the value function is the unique viscositysolution of the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation0 = (cid:18) ∂ t + L p + sup ν L ν + inf κ L κ (cid:19) H + (cid:88) k ∈{ x,y,z } ,i = ± (cid:90) R + ∆ k, − it, X , p , q ,r H λ k,i F k,i ( dr ) , (21a)with terminal condition H ( T, X , p , q ) = X + x q x (1 − α x q x ) + y q y (1 − α y q y ) + x q z (1 − α z q z ) , (21b)where L p , L ν , and ∆ k, ± t, X , p , q ,r are given in (12a)–(12d), L κ H = κ (cid:124) D H + 12 ϕ κ (cid:124) ρ − κ , and D H = ( σ x x ∂ x H , σ y y ∂ y H ) (cid:124) . In the following proposition we show that the optimal trading speed does not depend onthe cash position. 13 roposition 4.
Write the value function as H ( t, X , p , q ) = X + H( t, p , q ) . (22) Then, from (22) and the HJBI (21) , the function H satisfies ∂ t + L p + G + G ) H + (cid:88) k ∈{ x,y,z } ,i = ± (cid:90) R + (cid:16) i ˜ k i + ∆ k, − it, X , p , q ,r H (cid:17) λ k,i F k,i ( dr ) , (23a) subject to the terminal condition H( T, p , q ) = x q x (1 − α x q x ) + y q y (1 − α y q y ) + x q z (1 − α z q z ) , (23b) and the operators G and G act as follows: G H = (cid:88) k ∈{ x,y,z } ( (cid:98) k − ∂ q k H) a k (cid:98) k , (24a) G H = − ϕ (cid:18) σ x x ( ∂ x H) + σ y y ( ∂ y H) + 2 ρ σ x σ y x y ∂ x H ∂ y H (cid:19) . (24b) Furthermore, the optimal speed of trading in feedback form is ν k, ∗ = a k (cid:98) k ( (cid:98) k − ∂ q k H) , (25) and the optimal measure in feedback form is κ ∗ = − ϕ ρρ σ x x ∂ x H σ y y ∂ y H . (26) Proof.
Substitute (22) into (21a) and maximize the term L ν H to obtain the optimal tradingspeed (25). As this term is quadratic in ν (with negative quadratic coefficient), it is trivialto verify that the supremum ν ∗ satisfies the first order condition (FOC) in (25). Similarly,we obtain (26) by the FOC of the L κ H term in (21a), which is quadratic in κ with positivecoefficients for ( κ x ) and ( κ y ) , thus we obtain the minimizer κ ∗ . Then, by substituting (22),(25), and (26) into (21) results in (23). (cid:3) .2. Expansion with respect to the ambiguity parameter ϕ The HJBI (21) is nonlinear and we cannot obtain a solution in closed-form. We employperturbation methods, similar to those used by Lorig and Sircar (2016) and Fouque et al.(2017) in portfolio optimization problems, to approximate the value function with the expan-sion H ( t, X , p , q ) = X + H ( t, p , q ) + ϕ H ( t, p , q ) + ε ( t, p , q ) . (27)By construction, we anticipate that ε is o ( ϕ ) (see discussion after Proposition 8), and in the fol-lowing propositions we provide closed-form solutions for the terms H ( t, p , q ) and H ( t, p , q ). Proposition 5.
In the limit ϕ ↓ , the value function of the ambiguity averse broker H ( t, X , p , q ) →X + H ( t, p , q ) , where H ( t, p , q ) = x ( q x + q z ) + y q y − h x ( t ) x − h y ( t ) y − h x ( t ) x q x − h y ( t ) y q y − h z ( t ) x q z − h x ( t ) x ( q x ) − h y ( t ) y ( q y ) − h z ( t ) x ( q z ) , (28) h x,y ( t ) = h x,y ( t ) , h x,y,z ( t ) = h x,y,z ( t ) , h x,y,z ( t ) = h x,y,z ( t ) are deterministic functions, and h x,y ( t ) , h x,y,z ( t ) , h x,y,z ( t ) are in the appendix, see (57) , (59) , (61) . Proof.
It is straightforward to see from Proposition 4, that as ϕ ↓
0, (23) reduces to thePDE for the ambiguity neutral case in (11), and the result follows from Proposition 1.
Proposition 6.
Let H ( t, X , p , q ) be as in the expansion (27) , then H ( t, p , q ) satisfies ∂ t + L p ) H − (cid:88) k ∈{ x,y,z } (cid:98) k − ∂ qk H a k (cid:98) k ∂ q k H + (cid:88) k ∈{ x,y,z } ,i = ± (cid:90) R + ∆ k, − it, X , p , q ,r H λ k,i F k,i ( dr ) − D H (cid:124) ρ D H , (29) with terminal condition H ( T, p , q ) = 0 . Proof.
Substitute expansion (27) into (21), expand to first order in ϕ , use the fact that H satisfies (11), and equate the result to zero to obtain (29). (cid:3) H , the equation in (29) is a linear PIDE. Thus we use a Feynman-Kacprobabilistic representation to write the solution of H : H ( t, p , q ) = (cid:90) Tt E t, p , q (cid:2)(cid:0) − D H (cid:124) ρ D H (cid:1) ( u, X u , Y u , Q u ) (cid:3) du . (30)The term D H can be computed explicitly from (28), and here Q t = ( Q xt , Q yt , Q zt ) (cid:124) areindependent auxiliary processes that satisfy the SDEs d Q kt = β k ( t, Q kt ) dt − (cid:90) R rJ k, − ( dr, dt ) + (cid:90) R rJ k, + ( dr, dt ) , k ∈ { x, y, z } , (31)with the function β k ( t, q ) = − a k (cid:0) h k ( t ) + 2 h k ( t ) q (cid:1) . Based on the form of (28) and (30), we propose the ansatz H ( t, p , q ) = H ( t, q x , q z ) x + H ( t, q y ) y + H ( t, q x , q y , q z ) x y . (32)Insert (32) into (29), collect terms proportional to x , y , x y , and equate each term to zeroto obtain expressions for H , H , H . Proposition 7.
We have the following representation for H , H , H : H ( t, q x , q z ) = − (cid:90) Tt e (2 µ x + σ x ) ( u − t ) 12 σ x E t,q x ,q z (cid:34) (cid:18) ∂ x H ( u, Q xu , Q zu ) (cid:19) (cid:35) du , (33a) H ( t, q y ) = − (cid:90) Tt e (2 µ y + σ y ) ( u − t ) 12 σ y E t,q y (cid:34) (cid:18) ∂ y H ( u, Q yu ) (cid:19) (cid:35) du , (33b) H ( t, q ) = − (cid:90) Tt e ( µ x + µ y + ρ σ x σ y ) ( u − t ) × ρ σ x σ y E t,q x ,q y ,q z (cid:20) (cid:18) ∂ x H ( u, Q xu , Q zu ) (cid:19) (cid:18) ∂ y H ( u, Q yu ) (cid:19)(cid:21) du . (33c) Proof.
Substitute (32) into (29) and equate x terms to zero, then H ( t, q x , q z ) satisfies0 = ∂ t H + (2 µ x + σ x ) H − (cid:88) k ∈{ x,z } ( x − ∂ q k H )2 a k x ∂ q k H + (cid:88) k ∈{ x,z } ,i = ± (cid:90) R + ∆ k, − it, X , p , q ,r H λ k,i F k,i ( dr ) − σ x ( ∂ x H ) , (34)with terminal condition H ( T, q x , q z ) = 0. Then, use the Feynman-Kac Theorem to show that(33a) is a solution to the equation above. Similarly, we equate to zero the terms proportionalto y and x y and obtain the functions H and H . (cid:3) roposition 8. Let H ( t, X , p , q ) be as in the expansion (27) , then the residual function ε ( t, p , q ) satisfies the PIDE ∂ t + L p + (cid:88) i =1 B i ) ε + (cid:88) k ∈{ x,y,z } ,i = ± (cid:90) R + (cid:16) ∆ k, − it, X , p , q ,r ε (cid:17) λ k,i F k,i ( dr ) + f ε ( t, p , q ) , (35) with terminal condition ε ( T, p , q ) = 0 , and where the operators are defined as B ε = (cid:88) k ∈{ x,y,z } a k (cid:98) k ( ∂ q k ε ) , (36a) B ε = − (cid:88) k ∈{ x,y,z } a k (cid:98) k (cid:16)(cid:98) k − ∂ q k ( H + ϕ H ) (cid:17) ∂ q k ε , (36b) B ε = − ϕ [ D ε ] (cid:124) ρ D ε , (36c) B ε = − ϕ [ D ( H + ϕ H )] (cid:124) ρ D ε , (36d) f ε ( t, p , q ) = ϕ (cid:88) k ∈{ x,y,z } a k (cid:98) k ( ∂ q k H ) − D H (cid:124) ρ D H − ϕ D H (cid:124) ρ D H . (36e) Proof.
Substitute expansion (27) into (21) and collect powers of ϕ , and obtain (35) because H satisfies (11) and H satisfies (29). (cid:3) We observe that (35) is a semilinear PIDE, and B ε = sup ν L ν ε , B ε = inf κ L κ ε , where L ν ε = (cid:88) k ∈{ x,y,z } (cid:20) − a k (cid:98) k ( ν k ) − ( ∂ q k ε ) ν k (cid:21) , L κ ε = κ (cid:124) D ε + 12 ϕ κ (cid:124) ρ − κ , and the extremes are achieved at ν k, ∗ = a k (cid:98) k ( − ∂ q k ε ) and κ ∗ = − ϕ (cid:20) ρρ (cid:21) (cid:20) σ x x ( ∂ x ε ) σ y y ( ∂ y ε ) (cid:21) . (37)The PIDE in (35) is the HJBI equation of the stochastic control problem: ε ( t, p , q ) = sup ν inf κ E Q ( κ ) t, p , q (cid:20)(cid:90) Tt g ε ( p u , ν u ) + 12 ϕ ( κ u ) (cid:124) ρ − κ u + f ε ( u, p u , q u ) du (cid:21) , (38)17here g ε ( p , ν ) = (cid:88) k ∈{ x,y,z } ( − a k ) (cid:98) k ( ν k ) , (39)and p u = ( X u , Y u ), q u = ( q xu , q yu , q zu ) are auxiliary processes that satisfy the SDEs: d X t = X t ( µ x dt + σ x κ xt dt + σ x dW x, κ t ) + (cid:98) µ x ( t, p t , q t ) dt , (40) d Y t = Y t ( µ y dt + σ y κ yt dt + σ y dW y, κ t ) + (cid:98) µ y ( t, p t , q t ) dt , (41) d q kt = (cid:98) β k ( t, p t , q t , ν kt ) dt − (cid:90) R rJ k, − ( dr, dt ) + (cid:90) R rJ k, + ( dr, dt ) , k ∈ { x, y, z } , (42)with the functions H ϕ = H + ϕ H , (cid:98) µ x ( t, p , q ) = − ϕ (cid:0) σ x x ∂ x H ϕ + ρ σ x σ y x y ∂ y H ϕ (cid:1) ( t, p , q ) , (cid:98) µ y ( t, p , q ) = − ϕ (cid:0) σ y y ∂ y H ϕ + ρ σ x σ y x y ∂ x H ϕ (cid:1) ( t, p , q ) , (cid:98) β k ( t, p , q , ν k ) = − a k (cid:98) k (cid:16)(cid:98) k − ∂ q k H ϕ ( t, p , q ) (cid:17) − ν k , and initial conditions X , Y , Q x , Q y , Q z .Suppose the optimal processes ( ν ∗ t , κ ∗ t , p ∗ u , q ∗ u ) t ∈ [0 ,T ] are admissible, then (38) is indeed ε ( t, p , q ) = E Q ( κ ∗ ) t, p , q (cid:20)(cid:90) Tt g ε ( p ∗ u , ν ∗ u ) + 12 ϕ ( κ ∗ u ) (cid:124) ρ − κ ∗ u + f ε ( u, p ∗ u , q ∗ u ) du (cid:21) . (43)In addition, here we assume ε is O ( ϕ r ) for some r >
0. Then, because of the initialdata, we have ( p ∗ t ) t ∈ [0 ,T ] and ( q ∗ t ) t ∈ [0 ,T ] are O (1). By (37) we have ( ν ∗ t ) t ∈ [0 ,T ] is O ( ϕ r ) and( κ ∗ t ) t ∈ [0 ,T ] is O ( ϕ r ). By (39) and (36e), the first, second, and third terms of the integrandin (43) are O ( ϕ r ), O ( ϕ r ), and O ( ϕ ), respectively. Thus, the term ε should be O ( ϕ ),which is consistent with our assumption that O ( ϕ r ) for some r > ν k, ∗ = a k (cid:98) k (cid:104) ( (cid:98) k − ∂ q k H ) − ϕ ∂ q k H − ∂ q k ε (cid:105) , (44)so the optimal measure that arises from ambiguity aversion (26) is characterized by κ ∗ = − ϕ (cid:20) ρρ (cid:21) (cid:20) σ x x ( ∂ x H + ϕ ∂ x H + ∂ x ε ) σ y y ( ∂ y H + ϕ ∂ y H + ∂ y ε ) (cid:21) . (45)Thus, we approximate the optimal controls ν ∗ and κ ∗ with˜ ν k, ∗ = a k (cid:98) k (cid:104) ( (cid:98) k − ∂ q k H ) − ϕ ∂ q k H (cid:105) and ˜ κ ∗ = − ϕ (cid:20) ρρ (cid:21) (cid:20) σ x x ( ∂ x H + ϕ ∂ x H ) σ y y ( ∂ y H + ϕ ∂ y H ) (cid:21) . (46)18e observe that the speeds of trading in (46) exhibit cross effects. The coefficient ofambiguity aversion parameter ϕ depends on the price and inventory of each pair in the triplet.In the limit ϕ ↓ Proposition 9.
If the broker does not fill any orders from her clients, i.e., λ k, ± = 0 , thenthe trading strategy in (46) remains admissible in the limit α k → + ∞ (for k ∈ { x, y, z } ).Moreover, close to the end of the trading horizon, the trading speed can be expressed as lim α k → + ∞ ,k ∈{ x,y,z } ˜ ν k, ∗ t = Q k, ˜ ν ∗ t T − t (cid:16) ϕ σ (cid:98) k C (cid:98) k ( X t , Y t , Q x, ˜ ν ∗ t , Q y, ˜ ν ∗ t , Q z, ˜ ν ∗ t ) (cid:17) + o ( T − t ) , (47) where C x ( x, y, q x , q y , q z ) = σ x (cid:2) a x x ( q x ) + a z x ( q z ) (cid:3) + ρ σ y a y y ( q y ) ,C y ( x, y, q x , q y , q z ) = ρ σ x (cid:2) a x x ( q x ) + a z x ( q z ) (cid:3) + σ y a y y ( q y ) , (48) and C x + C y ≥ for any x , y , q x , q y , q z , and recall that (cid:98) k = k { k ∈{ x,y }} + x { k = z } . For a proof see Appendix D.
5. Performance of strategy
In this section we employ simulations to show the performance of the strategy, where weemploy the trading speed in (46). We choose the following triplet of currency pairs: (NZD,USD), (AUD, USD), (NZD, AUD). In our notation, USD is currency 1, NZD is currency 2,and AUD is currency 3. Therefore, we denote the mid-exchange rate for the three pairs by X t , Y t , and Z t respectively. Historically, in this triplet, the pair (AUD, USD) is the mostliquid, followed by (NZD, USD), and the least liquid pair of the triplet is (NZD, AUD).The broker trades in lots of currency pairs. Each lot consists of 10 currency pairs. Theinitial lot position is Q x = 0 , Q y = 0 , Q z = 200, the trading window is T = 1 hour and werun 10,000 simulations. 19 x × − σ y × − σ z × − ρ µ x × − µ y × − µ z × − X NZD,USD Y AUD,USD Z NZD,AUD ˜ P .
70 1 .
56 1 .
02 0.78 -6.491659 -3.155159 -3.299338 0.7459 0.7678 0.9715 P .
70 1 .
56 1 .
08 0.78 0 0 0.003650 0.7459 0.7678 0.9715
Table 1: Parameter estimates and reference measure
Statistical measure.
We assume that under the (true) statistical measure, denoted by ˜ P ,the dynamics of the pairs x , y , z satisfy the SDEs dX t = ˜ µ x X t dt + ˜ σ x X t d (cid:102) W xt , (49) dY t = ˜ µ y Y t dt + ˜ σ y Y t d (cid:102) W yt , (50) dZ t = ˜ µ z Z t dt + ˜ σ x Z t d (cid:102) W xt − ˜ σ y Z t d (cid:102) W yt , (51)where, by no-arbitrage, ˜ µ z = ˜ µ x − ˜ µ y + ˜ σ y − ˜ ρ ˜ σ x ˜ σ y , (52)and where (cid:102) W x = ( (cid:102) W xt ) t ∈ [0 ,T ] and (cid:102) W y = ( (cid:102) W yt ) t ∈ [0 ,T ] are ˜ P -Brownian motions. Moreover, d [ (cid:102) W x , (cid:102) W y ] t = ˜ ρ dt . Here, ˜ µ x , ˜ µ y , ˜ ρ , ˜ σ x >
0, ˜ σ y > Parameter estimates and reference measure.
We use FX data from HistData.com toobtain the parameters of the mid-exchange rate dynamics under statistical measure: ˜ µ k , ˜ σ k , k = { x, y, z } , and ˜ ρ . We employ data between the trading hours 00:00 am to 13:59 pmEaster Standard time (without daylight savings adjustment), on the 8th of September, 2016.The first row of Table 1 shows the maximum likelihood estimates of parameters used tosimulate mid-exchange rates under the statistical measure ˜ P . The second row of the tableshows the parameters used by the broker under the reference measure P . These parametersare obtained by assuming that µ k = 0, σ k = ˜ σ k , k = { x, y } , ρ = ˜ ρ , and the parameters µ z , σ z are determined by the no-arbitrage condition (1). The time unit is hours. Other model parameters.
We do not have access to limit order book data and marketorder activity. These data are useful to estimate the impact of orders on exchange rates, andto estimate the arrival rate and size of liquidity taking orders – see Gould et al. (2016) for arecent study of order flow in FX markets. Thus, for illustrative purposes we assume that thetemporary impact on the mid-exchange rate for the three pairs is a x = 5 × − , a y = 1 × − , a z = 1 × − . Here we assume that the most liquid pair is that with the deepest limit order book (i.e.,everything else being equal, it shows the smallest temporary price impact), followed by thesecond most liquid, and finally, the illiquid pair shows the shallowest book, i.e., highesttemporary price impact.Brokerage service fees are given by c ± x = a x , c ± y = a y , c ± z = a z , ime q A U D ; U S D ; $ -60-40-200 Time q N Z D ; U S D ; $ -40-20020 Time q N Z D ; A U D ; $ Figure 2: Inventory paths for currency pairs, ϕ = 0 .
1. The solid line is the mean of all simulated paths. and the terminal liquidation penalty parameters are α k = a k × .We assume that the sizes of clients’ orders are exponentially distributed with mean θ ± .The arrival rates and expected sizes of the clients’ orders are given by λ x, ± = 60 , λ y, ± = 90 , λ z, ± = 6 ; θ x, ± = 2 , θ y, ± = 1 , θ z, ± = 10 . Figure 2 shows three inventory paths for each pair and shows the mean inventory path(solid black line) for the 10,000 simulations for each pair when the broker’s degree of ambiguityaversion is ϕ = 0 .
1. In the figure, the first picture corresponds to the very liquid pair (AUD,USD), followed by the picture for the liquid pair (NZD,USD), and the last picture is for theilliquid pair. We first discuss the mean inventory paths and then discuss individual paths.At the beginning of the trading window, the mean inventory paths for the very liquid andliquid pairs show how the strategy builds a short position in both pairs while the positionin the illiquid pair is gradually liquidated. The initial short positions in the very liquid andilliquid pairs are built to compensate for the broker’s initial long position in the illiquid pair.At some point the (mean) strategy reverses the direction of trading in the two liquid pairsbecause the short positions must be closed by the terminal time. As expected, the strategyrelies more on the very liquid currency pair because trades in this pair have the smallestadverse impact in the quote currency received by the broker.The sample inventory paths in Figure 2 show how individual simulations deviate from themean behavior of the strategy as a result from the broker filling clients’ orders. For example,the red dash-dot line in the middle picture shows that early on, the strategy changes fromselling the pair (NZD,USD) to purchasing it – instead of building a short position in thepair (NZD, USD) as shown in the mean path. For that path we also see that the inventory q NZD,USD crosses zero a number of times before reaching the terminal date. This optimalbehavior becomes clear by looking at the speeds of trading and the broker’s activity with herpool of clients, both of which we discuss below.21 ime A U D ; U S D ; $ -2000200400 Time N Z D ; U S D ; $ -2000200400 Time N Z D ; A U D ; $ -2000200400600800 Figure 3: Trading speed for each currency pair, ϕ = 0 .
1. Solid line represents the mean trading speed overall paths.
Time O A U D ; U S D ; ! Time O N Z D ; U S D ; ! Time O N Z D ; A U D ; ! Figure 4: Buy order flow from broker’s clients, ϕ = 0 .
1. Solid line is mean over all paths.
Figure 3 shows the optimal speeds of trading for the triplet. As above, the solid black linerepresents the average over 10,000 simulations. Recall that if the continuous rate of trading ν k is positive, then the strategy is selling the pair k . As already discussed, the strategy reliesmore on trading the most liquid pair (AUD,USD) because it exhibits the lowest temporaryimpact on the mid-exchange rate.A number of factors determines the speed at which the strategy trades the pairs; one ofwhich is the exogenous order flow from clients. Every time the broker fills a client’s order, theinventory in that pair jumps, and this causes the optimal speed of trading in the three pairsto change. There are cases in which the size of the client’s order is large enough to reversethe direction of the broker’s trading in that pair, i.e., the speed ν k changes sign.Figure 4 shows buy order flow (in the interest of space we do not show sell order flow) fromthe pool of clients which are filled by the broker. At the beginning of the trading window,the broker quickly fills around 20 lots of the currency pair (NZD,USD), which is above theaverage fills shown by the solid line in the second picture of the figure. Thus, the broker’sinventory q NZD,USD drops very quickly and the speed at which the broker sells that pair inthe Exchange also decreases.The broker’s degree of ambiguity aversion also affects the speeds of trading. When thebroker is ambiguous to the drift of the mid-exchange rates, the trading strategy assumes that22he dynamics of the rates are given by (17a), (17b), (17c), which for convenience we repeathere dX t = X t (cid:16) σ x κ x, ∗ t dt + σ x dW x, κ ∗ t (cid:17) ,dY t = Y t (cid:16) σ y κ y, ∗ t dt + σ y dW y, κ ∗ t (cid:17) ,dZ t = Z t (cid:16) ( µ z + σ z κ z, ∗ t ) dt + σ z dW z, κ ∗ t (cid:17) , where κ z, ∗ t = σ x κ x, ∗ t − σ y κ y, ∗ t (cid:112) σ x + σ y − ρ σ x σ y and dW z, κ ∗ t = (cid:16) σ x dW x, κ ∗ t − σ y dW y, κ ∗ t (cid:17)(cid:112) σ x + σ y − ρ σ x σ y . (53)In Figure 5 we show the drift adjustments κ x, ∗ t , κ y, ∗ t , κ z, ∗ t that stem from ambiguity aversion.For each mid-exchange rate, the black solid line shows the mean of the drift adjustment forall the simulations. For the three pairs, the mean of κ k, ∗ is always negative and graduallyincreases to zero by the end of the trading window. When the drift adjustment is negative,the strategy assumes that the mid-exchange rate is decreasing (on average), and therefore itis optimal to sell the pair. Thus, everything else being equal, the effect of ambiguity aversionis to increase the speed of trading.To gain insights into the effect of ambiguity aversion on the broker’s optimal strategywe also discuss the results of the strategy if, everything else being equal, the broker’s initialinventory is Q x = 0 , Q y = 0 , Q z = − Q x = 0 , Q y = 0 , Q z =200 is as that shown in Figure 5, but with the opposite sign. That is, for all three pairs, themean adjustment κ k, ∗ is positive and gradually decays to zero as the strategy approaches theterminal date.Therefore, when the broker’s initial position in the illiquid pair is short (resp. long), theambiguity averse broker devises a strategy where the drift in the mid-exchange rates, relativeto the model under the reference measure P , is larger (resp. smaller). In the two casesdiscussed above ( Q z = −
200 and Q z = 200), the drift of the pairs x and y is zero underthe reference measure and the drift adjustment under the optimal measure is positive (resp.negative) if the initial position in the illiquid pair is short (resp. long). Hence, ambiguityaversion increases the speed at which the strategy enters positions in the liquid pairs ofthe triplet. The magnitude of the increase in the speed is linear in the ambiguity aversionparameter ϕ . 23 ime A U D ; U S D ; $ -0.02-0.01 0 0.01 Time N Z D ; U S D ; $ -0.03-0.02-0.0100.01 Time N Z D ; A U D ; $ -0.015 -0.01-0.005 0 0.005 Figure 5: Drift adjustments as result of ambiguity aversion, ϕ = 0 .
1. Solid line is mean over all paths.
The left panel in Figure 6 shows the profit and loss (P&L) of the strategy when the brokertrades in the triplet (gray-fill histogram) or trades only the illiquid pair (white-fill histogram).For comparative purposes, when the broker trades the triplet of currency pairs, we assumethat she also trades with her clients, so λ x, ± = 60, λ y, ± = 90, λ z, ± = 6, and ϕ = 0 .
1. Whenthe broker only trades the illiquid pair we employ the parameters λ x, ± = 0, λ y, ± = 0, λ z, ± = 6,and ϕ = 0. The mean P&L of the strategy that trades in the triplet is higher (solid verticalline) than the P&L of the strategy when only the illiquid pair is traded (dash vertical line)– the difference is approximately 88 .
88 USD per lot of currency 1. The mean P&L when thebroker only trades the illiquid pair is 7 . × , and when the broker trades in all the pairsthe mean P&L is 7 . × .The right panel in Figure 6 shows the mean P&L against the standard deviation of theP&L of the strategy for ϕ ∈ [0 ,
50] and λ x, ± = 60, λ y, ± = 90, λ z, ± = 6. The lowest meanP&L, which also corresponds to the highest standard deviation of the P&L, corresponds to ϕ = 0, i.e., when the broker does not doubt the reference model. As the value of the ambiguityaversion parameter ϕ increases, the mean P&L increases and the standard deviation of P&Ldecreases up to a point ( ϕ = 36) after which the mean P&L starts to decrease. Thus, there isa range of the ambiguity aversion parameter where increasing the broker’s ambiguity aversionimproves the tradeoff between mean and standard deviation of the profitability of the strategy.To investigate the additional value that stems from the opportunity to trade two liquidcurrency pairs, we assume the broker does not trade with her clients, i.e., λ x, ± = λ y, ± = λ z, ± = 0, and define the relative improvement of P&L as∆ P nL % =
P nL ϕ − P nL P nL , where P nL and P nL ϕ are the P&L with ϕ = 0 and ϕ >
0, respectively. We compute a Recall that if ϕ > S ∆ PnL % = E ˜ P [∆ P nL %] (cid:113) V ar ˜ P [∆ P nL %] . The left-hand picture of Figure 7 shows log(1 + S ∆ PnL % ) against log(1 + ϕ ) for ϕ ∈ [0 , ϕ = 0, we see that ∆ P nL % ≡
0, i.e., the broker does not doubt the reference modeland only trades in the illiquid pair. As the level of ambiguity aversion increases, the Sharperatio increases and then decreases. Thus, there is a range of ϕ where increasing the broker’sambiguity aversion improves the tradeoff between mean and standard deviation of the valueof trading liquid pairs relative to trading the illiquid pair alone – the maximum Sharpe ratiois 0 . ϕ = 16.The right-hand picture of Figure 7 shows the probability that relative P&L improvementis greater than x , i.e. P ∆ PnL % ( x ) = ˜ P (∆ P nL % > x × − ). When ϕ = 0, the relative P&Limprovement has a point probability mass of 1 at 0. The dotted curve is for ϕ = 0 .
1, andthe solid curve is for ϕ = 16. For both curves, the probability of positive relative P&Limprovement is around 60%. The ∆ P nL % has 50 th -percentiles equal to 0.01% and 0.03% for ϕ = 0 . ϕ = 16, respectively. Thus, when the broker trades liquid pairs, she achieves(compared with the results obtained when the broker only trades the illiquid pair) a higherP&L of 74 .
59 USD ( ϕ = 0 .
1) and 223 .
77 USD ( ϕ = 16) in half of the 10,000 trade simulations. In addition to Proposition 9, to illustrate the effect of the terminal liquidation penaltyon: inventory, trading speed, and drift adjustment, we perform additional simulations. Weassume that the broker does not trade with her pool of clients ( λ k, ± = 0) and employs thefollowing terminal liquidation penalties: α k = 1 × a k , α k = 2 . × a k , and α k = 10 × a k , k ∈ { x, y, z } .Figure 8 shows average inventory paths for each currency pair when ϕ = 0 . Q x = Q y = 0, Q z = 200. As expected, as the value of the terminal liquidationpenalty decreases, the strategy first sells and then buys the two liquid pairs (AUD, USD),(NZD, USD), and sells the pair (NZD, AUD) throughout the trading period. Finally, whenthe value of the liquidation penalty is sufficiently low, it is optimal for the strategy to havenon-zero inventories at the end of trading of the trading horizon.Figure 9 shows the average of the trading speed for each currency pair. As the value ofthe liquidation penalty parameter decreases we observe the following: the broker is willingto hold more inventory; the sell (buy) speed increases (decreases) in approximately the first25second) half of the trading period for the liquid pairs (AUD,USD) and (NZD,USD); and thesell speed decreases for the illiquid pair (NZD, AUD).Figure 10 shows the average drift adjustment, which is a result of the broker’s ambiguityaversion, for each currency pair for various levels of the terminal penalty parameter. Observethat at the beginning of the trading horizon, the impact of the terminal liquidation penaltyon the drift adjustment is very small. As time evolves, the effect of the terminal penaltydiffers for each currency pair. For example, the mean drift adjustment for the pair (NZD,AUD) is negative during the entire trading horizon, so it is always optimal to sell. Recall thatthe initial inventory in the pair (NZD, AUD) is 200 lots, so the impact of the negative driftadjustment on the strategy is to sell quicker than otherwise.Figure 11 shows the P&L of a strategy when α k = 1 × a k (white-fill histogram) and α k = 10 × a k (gray-fill histogram), and λ k, ± = 0, k ∈ { x, y, z } . The mean and standarddeviation of the P&L for the strategies where the terminal liquidation penalty is low are7 . × and 501 .
96 USD respectively. When the terminal liquidation penalty is large,the mean and standard deviation of the P&L per lot are 7 . × and 496 .
05 USD,respectively. The average cost of executing large inventories at time T is 2 . × USD. Finally, we investigate the behavior of the optimal strategy when the value of the ambiguityaversion parameter is high. We let ϕ ∈ { . , , } and assume that the broker ensures fullliquidation by the end of the terminal horizon, so α k = 10 × a k , k ∈ { x, y, z } and λ k, ± = 0, k ∈ { x, y, z } , i.e., the broker does not trade with her pool of clients. Below, in Figures 12,13, 14, the black, blue, and red solid lines represent cases of ϕ = 0 .
1, 1, 10, respectively.Figure 12 shows the average inventory paths for the three currency pairs. Figure 13 showsaverage trading speeds in the currency pairs. Finally, Figure 14 shows the average of the driftadjustment for each currency pair.
6. Conclusions
We showed how a broker executes a large position in an illiquid foreign exchange currencypair. The broker’s strategy considers taking positions on two other more liquid currency pairs.These additional pairs are chosen to form a triplet where, by no arbitrage, one of the pairs We compute the costs of unwinding terminal inventory as follows. Let Q kT represent the outstandinginventory at the terminal time T , and let ∆ T be the time interval to unwind Q kT . The contribution to theP&L of unwinding the inventory is (cid:88) k ∈{ x,y,z } Q kT × max (cid:16) (cid:98) K T (cid:16) − a k Q kT ∆ T (cid:17) , (cid:17) , (54)where (cid:98) K T = X T { k ∈{ x,z }} + Y T { k = y } and ∆ T = 10 − hours, which is the time step we use in the simulations. nL of Strategy F r e q u e n c y Std. PnL of Strategy M e a n P n L o f S t r a t e g y ' increasing Figure 6: Left panel: P&L of strategy trading in triplet ( ϕ = 0 . λ x, ± = 60, λ y, ± = 90, λ z, ± = 6) or inilliquid pair only ( ϕ = 0, λ x, ± = 0, λ y, ± = 0, λ z, ± = 6). Right panel: mean and standard deviation of P&Lfor ϕ ∈ [0 , λ x, ± = 60, λ y, ± = 90, λ z, ± = 6. log (1 + ' ) ; ' [0 ; l o g ( + S " P n L % ) x -0.2 0 0.2 P " P n L % ( x ) Figure 7: Left panel: Sharpe ratio of relative P&L improvement of strategy trading in triplet for a range of ϕ . Right panel: The probability of relative P&L improvement greater than a given value x in percentage, thedotted line is for ϕ = 0 . ϕ = 16. becomes redundant.When the broker is ambiguity neutral, i.e., fully trusts the reference model, the optimalstrategy in each currency pair is independent of the inventory held in the other two pairs.Thus, the ambiguity neutral broker’s strategy is to liquidate each pair independently from theother pairs. On the contrary, when the broker makes her model robust to misspecification,we show that the optimal trading strategy in each currency pair is affected by the inventoryholdings in the other two pairs.We use simulations to compare the tradeoff between mean and standard deviation of theProfit and Loss (P&L) of the optimal liquidation strategy. We show that when the brokermakes her model robust to model misspecification the mean P&L of the strategy increases andthe standard deviation of the P&L decreases (for a range of ambiguity aversion parameter).27 ime q A U D ; U S D ; $ -80-60-40-20 0 Time q N Z D ; U S D ; $ -40-20 0 20 Time q N Z D ; A U D ; $ Figure 8: Average inventory over all simulated paths for currency pairs, ϕ = 0 . λ k, ± = 0, k ∈ { x, y, z } . Theblue, red, black solid lines are for α k = 1 × a k , α k = 2 . × a k , and α k = 10 × a k , k ∈ { x, y, z } , respectively. Time A U D ; U S D ; $ Time N Z D ; U S D ; $ -200 0 200 400 Time N Z D ; A U D ; $ -200 0 200 400 Figure 9: Average trading speed over all simulated paths for each currency pair, ϕ = 0 . λ k, ± = 0, k ∈{ x, y, z } . The blue, red, black solid lines represent α k = 1 × a k , α k = 2 . × a k , and α k = 10 × a k , k ∈ { x, y, z } , respectively. Appendices
A. Proof of Proposition 1
Proof.
We show that H can be solved analytically and it takes the form (13). First, thesupremum in (11a) is obtained point-wise by ν k, ∗ = (cid:98) k ∂ X H − ∂ q k H a k (cid:98) k ∂ X H , for k ∈ { x, y, z } and (cid:98) k = k { k ∈{ x,y }} + x { k = z } . Substituting the form (13) into the above leads to ν k, ∗ = a k ( h k ( t ) + 2 h k ( t ) q k ) . (55)Substituting the feedback control (55) and (13) into (11a) we obtain a new equation which isclosed under the form of the solution (13). Collecting the x ( q x ) , y ( q y ) and x ( q z ) terms,we find the ODEs − ∂ t h k + a k ( h k ) − µ (cid:98) k h k = 0 , (56)28 ime A U D ; U S D ; $ -0.02-0.01 0 0.01 Time N Z D ; U S D ; $ -0.03-0.02-0.01 0 0.01 Time N Z D ; A U D ; $ -0.015 -0.01-0.005 0 0.005 Figure 10: Average drift adjustments over all simulated paths as result of ambiguity aversion, ϕ = 0 .
1, andno trades with broker’s clients, λ k, ± = 0, k ∈ { x, y, z } . The blue, red, black solid lines are for α k = 1 × a k , α k = 2 . × a k , and α k = 10 × a k , k ∈ { x, y, z } , respectively. PnL of Strategy F r e q u e n c y Figure 11: P&L of strategy trading in triplet with ϕ = 0 . λ k, ± = 0, k ∈ { x, y, z } . The plot with blackedge white fill is P&L for α k = 1 × a k ; and the plot with gray edge and fill is P&L for α k = 10 × a k , where k ∈ { x, y, z } . with terminal conditions h k ( T ) = α k . These ODEs have the explicit solutions h k ( t ) = (cid:0) − υ k e − µ (cid:98) k ( T − t ) (cid:1) − a k µ (cid:98) k , µ (cid:98) k (cid:54) = 0 , (cid:0) α − k + a − k ( T − t ) (cid:1) − , µ (cid:98) k = 0 , (57)with constant υ k := (1 − a k α k µ (cid:98) k ).Next, collecting x q x , y q y and x q z terms, and setting θ k, ± := (cid:82) R + r F k, ± ( dr ), we have − ∂ t h k + µ (cid:98) k (1 − h k ) + a k h k h k − γ k − h k = 0 , (58)with terminal conditions h k ( T ) = 0, and γ k − := ( λ k, + θ k, + − λ k, − θ k, − ). These equations admitthe explicit solutions h k ( t ) = (cid:0) − υ k e − µ (cid:98) k ( T − t ) (cid:1) − (cid:8) (1 − a k γ k − ) µ (cid:98) k ( t − T ) + υ k (1 − e − µ (cid:98) k ( T − t ) ) (cid:9) , µ (cid:98) k (cid:54) = 0 , γ k − (cid:0) α − k + a − k ( T − t ) (cid:1) − ( T − t ) , µ (cid:98) k = 0 . (59)29 ime q A U D ; U S D ; $ -100 -50 0 Time q N Z D ; U S D ; $ -100 -50 0 Time q N Z D ; A U D ; $ Figure 12: Average inventory over all simulated paths for three currency pairs, α k = 10 × a k , λ k, ± = 0, k ∈ { x, y, z } . The red, blue, black solid lines are for ϕ = 10, ϕ = 1, and ϕ = 0 . Time A U D ; U S D ; $ -400-200 0 200 400 Time N Z D ; U S D ; $ -200 0 200 400 Time N Z D ; A U D ; $ Figure 13: Average trading speed over all simulated paths for each currency pair, α k = 10 × a k , λ k, ± = 0, k ∈ { x, y, z } . The red, blue, black solid lines are for ϕ = 10, ϕ = 1, and ϕ = 0 . ime A U D ; U S D ; $ -0.02-0.01 0 0.01 0.02 Time N Z D ; U S D ; $ -0.03-0.02-0.01 0 0.01 Time N Z D ; A U D ; $ -0.04-0.03-0.02-0.01 0 Figure 14: Average drift adjustments over all simulated paths as result of ambiguity aversion for all threecurrency pairs, α k = 10 × a k , λ k, ± = 0, k ∈ { x, y, z } . The red, blue, black solid lines are for ϕ = 10, ϕ = 1,and ϕ = 0 . Finally, collecting x and y terms, and let η k, ± := (cid:82) R + r F k, ± ( dr ), we have0 = − ( ∂ t + µ x ) h x + ψ x + ψ z − γ x − h x ( t ) − γ z − h z ( t ) − δ x h x ( t ) − δ z h z ( t )+ a x ( h x ) + a z ( h z ) , (60a)0 = − ( ∂ t + µ y ) h y + ψ y − γ y − h y ( t ) − δ y h y ( t ) + a y ( h y ) , (60b)with terminal conditions h x ( T ) = h y ( T ) = 0, and constants δ k := ( λ k, + η k, + + λ k, − η k, − ) and ψ k := ( c − k λ k, − η k, − + c + k λ k, + η k, + ). These ODEs can be solved by quadratures and are givenby h x ( t ) = (cid:88) (cid:96) ∈{ x,z } (cid:8) h (cid:96) ( t ) + h (cid:96) ( t ) + h (cid:96) ( t ) + h (cid:96) ( t ) (cid:9) , (61a) h y ( t ) = h y ( t ) + h y ( t ) + h y ( t ) + h y ( t ) , (61b)where, for all µ (cid:98) k h k ( t ) = − (cid:90) Tt e µ (cid:98) k ( u − t ) ψ k du , h k ( t ) = (cid:90) Tt e µ (cid:98) k ( u − t ) γ k − h k ( u ) du , h k ( t ) = (cid:90) Tt e µ (cid:98) k ( u − t ) δ k h k ( u ) du , h k ( t ) = − (cid:90) Tt e µ (cid:98) k ( u − t ) 14 a k (cid:0) h k ( u ) (cid:1) du . (62) (cid:3) B. Proof of Theorem 2
Since (13) is classical, we only need to check that the controls (14) are admissible. Tocheck that ν ∗ is admissible, it suffices to show that E P [ (cid:82) t ( Q k, ν ∗ u ) du ] < + ∞ because ν k, ∗ is31ffine in Q k, ν ∗ . It further suffices to show E P [( Q k, ν ∗ t ) ] < + ∞ on t ∈ [0 , T ].From (4), we have Q k, ν ∗ t = Q k, ν ∗ − (cid:90) t a k (cid:0) h k ( u ) + 2 h k ( u ) Q ν ∗ ,ku (cid:1) du − O k, − t + O k, + t . (63)Use the integrating factor π kt := e ak (cid:82) t h k ( s ) ds to find Q k, ν ∗ t = Q k, ν ∗ π k π kt − a k (cid:90) t π ku π kt h k ( u ) du + (cid:90) t π ku π kt ( dO k, + u − dO k, − u ) . (64)From (57) we see that h k ( t ) does not change sign on the interval t ∈ [0 , T ]. Next, if0 < µ (cid:98) k < α k /a k , then ∃ C > C >
0, s.t. h k ( t ) = (cid:0) C − C e − µ (cid:98) k ( T − t ) (cid:1) − . Hence, by explicitintegration, π kt = e C t (cid:0) C − C e − µ (cid:98) k ( T − t ) (cid:1) C , where C = 1 /a k C > C = C / ( C − C e − µ (cid:98) k T ) > C = C / ( C − C e − µ (cid:98) k T ) > C = − /a k µ (cid:98) k C < E P (cid:34)(cid:18)(cid:90) t π ku π kt dO k, ± u (cid:19) (cid:35) ≤ (cid:90) t (cid:16) π ku π kt (cid:17) λ k, ± θ k, ± du < ∞ , ∀ ≤ t ≤ T , (65)where the last inequality follows because C > C and therefore π kt remains bounded on thedomain t ∈ [0 , T ]. Similar arguments hold when − α k /a k < µ (cid:98) k < µ (cid:98) k = 0, from (57), we have h k ( t ) = (cid:0) α − k + a − k ( T − t ) (cid:1) − . Hence, by explicitintegration, π kt = (cid:0) α − k + a − k ( T − t ) (cid:1) − (cid:0) α − k + a − k T (cid:1) . We have the analogous inequality(65). Hence the result follows. (cid:3) C. Proof of Proposition 3
We proof the case µ (cid:98) k (cid:54) = 0 ( µ (cid:98) k = 0 is similar). From (57) and (59) we havelim α k → + ∞ h k ( t ) = (cid:0) − e − µ (cid:98) k ( T − t ) (cid:1) − a k µ (cid:98) k , and lim α k → + ∞ h k ( t ) = (cid:0) − e − µ (cid:98) k ( T − t ) (cid:1) − (1 − a k γ k − ) µ (cid:98) k ( t − T ) + 1 . Then lim t → T (cid:110)(cid:0) − e − µ (cid:98) k ( T − t ) (cid:1) − a k µ (cid:98) k (cid:111) ( T − t ) = a k , and lim t → T (cid:0) − e − µ (cid:98) k ( T − t ) (cid:1) − (1 − a k γ k − ) µ (cid:98) k ( t − T ) + 1 = 2 a k γ k − . α k → + ∞ and T − t (cid:28)
1, we obtainlim α k → + ∞ ν k, ∗ t = ( T − t ) − Q k, ν ∗ t + γ k − + o ( T − t ) , (66)recall γ k − = ( λ k, + θ k, + − λ k, − θ k, − ) and θ k, ± := (cid:82) R + r F k, ± ( dr ).The SDE of the inventory process is dQ k, ν ∗ u = − (cid:2) ( T − u ) − Q k, ν ∗ u + γ k − (cid:3) du + d ( O k, + u − O k, − u ) + o ( T − t ) , u ∈ [ t, T ] , (67)which has solution Q k, ν ∗ s = T − sT − t Q k, ν ∗ t + (cid:90) st T − sT − u (cid:2) d ( O k, + u − O k, − u ) − γ k − du (cid:3) + o ( T − t ) , s ∈ [ t, T ] . (68)As s → T the controlled inventory Q k, ν ∗ s vanishes. In other words, the strategy guaranteesthat the broker completely liquidates her position by maturity.If we substitute Q k, ν ∗ s given in (68) into (66), the speed of trading becomes ν k, ∗ s = Q k, ν ∗ t T − t + (cid:90) st T − u (cid:2) d ( O k, + u − O k, − u ) − γ k − du (cid:3) + γ k − T − s + o ( T − t ) , s ∈ [ t, T ] , (69)and if we assume that the expected order flow from clients γ k − (cid:54) = 0, it is easy to see that thebroker’s speed of trading is infinitely fast close to maturity. Let γ k − = 0. Then, the speed oftrading becomes ν k, ∗ s = Q k, ν ∗ t T − t + (cid:90) st T − u ( dO k, + u − λ k, + θ k, + du ) − (cid:90) st T − u ( O k, − u − λ k, − θ k, − du ) + o ( T − t ) . Note that E (cid:34)(cid:18)(cid:90) st T − u ( dO k, ± u − λ k, ± θ k, ± du ) (cid:19) (cid:35) = λ k, ± η k, ± (cid:18) T − s − T − t (cid:19) , and recall η k, ± := (cid:82) R + r F k, ± ( dr ) > λ k, ± = 0, we therefore have E (cid:104) (cid:82) Tt ( ν k, ∗ s ) ds (cid:105) < ∞ , and hence, in this case, the strategyis admissible, and from (68) we see that the broker completely liquidates her position by T .If λ k, ± >
0, we have E (cid:104) (cid:82) Tt ( ν k, ∗ s ) ds (cid:105) = ∞ , which is not an admissible strategy. (cid:3) D. Proof of Proposition 9
We prove the case µ (cid:98) k = 0, the proof is similar when µ (cid:98) k (cid:54) = 0. The control ˜ ν ∗ , given in (46)and repeated here for convenience is˜ ν k, ∗ = (cid:98) k − ∂ q k H a k (cid:98) k − ϕ ∂ q k H a k (cid:98) k , k ∈ { x, y, z } and (cid:98) k = k { k ∈{ x,y }} + x { k = z } . λ k, ± = 0, the function H in (28) simplifies to H ( t, x, y, q ) = x q x + y q y + x q z − h x ( t ) x ( q x ) − h y ( t ) y ( q y ) − h z ( t ) x ( q z ) . In the limit α k → ∞ the first term in ˜ ν k, ∗ , see (57), becomeslim α k → + ∞ (cid:98) k − ∂ q k H a k (cid:98) k = q k T − t . By (33a), (33b), (33c), we have H ( t, q x , q z ) = (cid:90) Tt − e (2 µ x + σ x ) ( u − t ) × σ x E t,q x ,q z (cid:34) (cid:18) Q xu + Q zu − h x ( u ) ( Q xu ) − h z ( u ) ( Q zu ) (cid:19) (cid:35) du ,H ( t, q y ) = (cid:90) Tt − e (2 µ y + σ y ) ( u − t ) σ y E t,q y (cid:34) (cid:18) Q yu − h y ( u ) ( Q yu ) (cid:19) (cid:35) du ,H ( t, q x , q y , q z ) = (cid:90) Tt − e ( µ x + µ y + ρ σ x σ y ) ( u − t ) × ρ σ x σ y E t,q x ,q y ,q z (cid:20) (cid:18) Q xu + Q zu − h x ( u ) ( Q xu ) − h z ( u ) ( Q zu ) (cid:19) × (cid:18) Q yu − h y ( u ) ( Q yu ) (cid:19)(cid:21) du , and by (31), (32), the auxiliary processes Q ku , k ∈ { x, y, z } are deterministic functions of u and satisfy d Q ku = − h k ( u ) a k Q ku du , u ∈ [ t, T ] , with initial condition Q kt = q k . The solution is Q ku = a k + α k ( T − u ) a k + α k ( T − t ) q k . After evaluating the integrals, taking derivatives with respect to q k , taking the limit α k → ∞ ,and in the limit t → T , the second term in ˜ ν k, ∗ becomeslim α k → + ∞ ,k ∈{ x,y,z } − ϕ ∂ q k H a k (cid:98) k = ϕ σ (cid:98) k q k T − t C (cid:98) k ( x, y, q x , q y , q z ) + o ( T − t ) , where C x ( x, y, q x , q y , q z ) = σ x (cid:2) a x x ( q x ) + a z x ( q z ) (cid:3) + ρ σ y a y y ( q y ) ,C y ( x, y, q x , q y , q z ) = ρ σ x (cid:2) a x x ( q x ) + a z x ( q z ) (cid:3) + σ y a y y ( q y ) . (cid:3) eferencesReferenceseferencesReferences