Price Impact on Term Structure
PPrice Impact on Term Structure
Damiano Brigo, Federico Graceffa, and Eyal NeumanDepartment of Mathematics, Imperial College London23rd November 2020
Abstract
We introduce a first theory of price impact in presence of an interest-rates termstructure. We explain how one can formulate instantaneous and transient price impacton bonds with different maturities, including a cross price impact that is endogenous tothe term structure. We connect the introduced impact to classic no-arbitrage theory forinterest rate markets, showing that impact can be embedded in the pricing measure andthat no-arbitrage can be preserved. We present pricing examples in presence of priceimpact and numerical examples of how impact changes the shape of the term structure.Finally, to show that our approach is applicable we solve an optimal execution problemin interest rate markets with the type of price impact we developed in the paper.
Key words:
Price impact, term structure models, fixed-income market impact, crossimpact, impacted risk-neutral measure, impacted yield curve, optimal execution, impactedbond price, impacted Eurodollar futures price.
Contents a r X i v : . [ q -f i n . T R ] N ov Examples 21
The main aim of this work is to present a combined theory of the term structure of interestrates and of price impact, with applications to optimal execution. This objective entails theinclusion of a specific type of cross price impact that is specific to fixed income.Term structure modeling with a view to derivatives valuation and hedging has beendeveloped over several decades. For the purposes of this work and what we might callthe classic theory we refer to the monographs by Bjork [7], Brigo and Mercurio [11] andFilipovic [15]. After the crisis that started in 2007, the gap between two of the rates thatwere used as benchmarks for risk free rates, namely interbank and overnight rates, widenedconsiderably, peaking in October 2008, following the defaults of several financial institutionsin the space of one month [12]. This highlighted the fact that interbank rates could nolonger be used as benchmarks for risk free rates and neither could they be used to derivezero-coupon curves that were not contaminated by credit and liquidity risk. This led tothe necessity to model multiple interest rate curves, treating interbank rates as risky ratesaffected by credit and liquidity risk and adopting overnight based rates as new risk freerates. This multiple-curve literature has been developed mostly by practitioners. Withoutdoing justice to the multi-curve literature in our limited space, here we mention briefly themonograph by Henrard [23] and Bormetti et al. [8] for multiple curves in conjunction withvaluation adjustments and credit risk. Further recent developments include the presenceof negative interest rates in many currencies, see for example the BIS report [32], and theongoing project of eliminating current interbank rates like the London Interbank Offer Rate(LIBOR), replacing them with new types of risk free rates inspired by overnight rates. Thisputs the multiple-curve area in a state of uncertainty, while negative rates prompted themainstream resurgence of Gaussian models that were previously justified only in very specialeconomies exhibiting negative rates, such as for example Switzerland in the seventies.Given the state of market uncertainty on benchmark interest rates, products and markets,we will not consider these recent developments in this work, except for allowing for negativerates in our formulation. We are interested in developing a combination of term structuremodeling and price impact in the classic theory of interest rates. We are confident thatmultiple curves and further discussion of negative rates, if still present in the market after2eforms and updated central bank policies, can be incorporated in further work after theclassic theory has been developed.The financial crisis of 2008-2009 have also raised concerns about the inventories keptby intermediaries. Regulators and policy makers took advantage of two main regulatorychanges (Reg NMS in the US and MiFID in Europe) and enforced more transparency onthe transactions and hence on market participants positions, which pushed the tradingprocesses toward electronic platforms [26]. Simultaneously, consumers and producers offinancial products asked for less complexity and more transparency.This tremendous pressure on the business habits of the financial system, shifted it froma customized and high margins industry, in which intermediaries could keep large (andpotentially risky) inventories, to a mass market industry where logistics have a centralrole. As a result, investment banks nowadays unwind their risks as fast as possible. Inthe context of small margins and high velocity of position changes, trading costs are ofparamount importance. A major factor of the trading costs is the price impact: the fasterthe trading rate, the more the buying or selling pressure will move the price in a detrimentalway.Academic efforts to quantify and reduce the transaction costs of large trades trace backto the seminal papers of Almgren and Chriss [3] and Bertsimas and Lo [6]. In both modelsone large market participant (for instance an asset manager or a bank) would like to buy orsell a large amount of shares or contracts during a specified duration. The cost minimizationproblem turned out to be quite involved, due to multiple constraints on the trading strategies.On one hand, the price impact demands to trade slowly, or at least at a pace which takesinto account the available liquidity (see [4] and references therein). On the other hand,traders have an incentive to trade rapidly, because they do not want to carry the risk of anadverse price move far away from their decision price. The importance of optimal trading inthe industry generated a lot of variations for the initial mean-variance minimization of thetrading costs (see [26, 14, 18] for details). These type of problems are usually formulated asoptimisation problems in the context of stochastic control where the agent tries to minimizethe transaction costs which result by the price impact and to reduce the risk associated withholding the assets for too long (see e.g. [25, 19, 28]).In this spirit, we will follow the term structure theory as developed by Bjork [7], modi-fying it to allow for the inclusion of price impact. We will start from the simplest possibledynamics, namely one factor short rate models, and extend it later to instantaneous forwardrate models. We will introduce price impact formulated on zero-coupon bonds, since theyare a possible choice of building blocks for the term structure. We will later connect thiswith impact on coupon bearing bonds that are more commonly traded.We assume that an agent who is executing a large order of bonds is creating two types ofprice impact which are extensively used in the literature. The first one is an instantaneous(or temporary) price impact, which affects the asset price only while trading, and fadesaway immediately after. This type of price impact occurs due to the fact that a large buy(sell) trade consumes the liquidity which is available in the market by “walking through”the first few price levels of the limit order book (see e.g. [3] and Chapter 6.3 of [14]).However, empirical studies have shown that price impact also has a transient effect. A shortof liquidity due to a large trade creates an imbalance between supply and demand, which3n turn pushes the price in a detrimental direction. This effect decays within a short timeperiod after each trade (see [4] and [30]). Execution in presence of transient price impact wasstudied extensively in the context of optimal control problems (see e.g. [17, 16, 2, 29, 5]).In this work we incorporate these two types of price impact models into a bonds tradingframework, as was done in [29] for equities.Applying these price impact models to the term structure will be challenging. At everypoint in time the term structure of interest rates is a high dimensional object, or even aninfinite dimensional one when considering all possible maturities for interest rates or zero-coupon bonds at a given time. This is a unique feature of term structure modeling, wheredifferently from FX or equity modeling for example, we model a whole curve dynamics ratherthan a point dynamics. We can expect that trading a bond with a specific maturity mayimpact the price of bonds with different maturities on the same curve . In this sense, the crossprice impact is endogenous to the same underlying asset, differently from what happens inother markets. We will investigate how price impact interacts with no-arbitrage dynamics,and we will encapsulate the effect of price impact in the definition of a new no-arbitragepricing measure embedding impact itself. This will be done by extending the market priceof risk to an impacted version embedding the bond price impact speed. The impacted zero-coupon bond dynamics will then be written as the unimpacted bond dynamics but under adifferent measure. We will also introduce an impacted physical measure that could be usefulfor risk management and risk analysis. Finally, we will illustrate our theory by proposingan application to optimal execution.We will not limit ourselves to short rate models. We will also see how in the Heath-Jarrow-Morton model the no-arbitrage drift condition for instantaneous forward rate dy-namics can be maintained under price impact by resorting to the modified pricing measure.The paper is structured as follow. In section 2 we propose a roadmap for readers who arenot familiar with at least one of the two areas of the paper, namely term structure modelingand market impact. In Section 3 we introduce the short rate models setup and the maintheoretical results. In particular, we introduce price impact for zero-coupon bonds, and welook at the impacted market price of risk, absence of arbitrage and the impacted risk neutralmeasure. We define the impacted yield curve and extend impact to coupon bearing bonds.We further show how to use the impact setup in a HJM framework. Section 4 features afew examples including valuation of impacted Eurodollar futures with the Hull and Whitemodel. Section 5 presents some numerical results illustrating how the yield curve behavesunder impact. Finally, we introduce a result on optimal execution with impacted bonds inSection 6. Sections 7 and 8 include proofs that have not been included in the main text.
This paper combines areas that are typically disjointed in the literature, such as term struc-ture modeling and price impact/optimal execution. It is possible that most readers will befamiliar with one area but not the other one. We expect to find few readers who may readthe whole paper without effort. We therefore provide a roadmap for the reader who wishesto have a quick understanding of what the paper deals with and how it is organized, with a4articular focus on where finding what and what needs to be read more carefully dependingon the reader’s background.
Main section of the paper.
Section 3 is the section where both the term structuremodel is introduced and the impact model is postulated and developed in detail, linking thedevelopments to no-arbitrage pricing. The reader should start ideally from here.
Interest rate and bond dynamics under different measures.
In Section 3.1we introduce our initial assumption on the interest rate dynamics, assuming it is a one-dimensional short rate model dynamics. This will be relaxed later with a Heath-Jarrow-Morton setting in Section 3.6 but the paper arguments and innovations are best appreciatedin a short-rate setting. We start postulating a diffusion dynamics for the short rate r under the physical measure, giving then also the dynamics of the bond price P under thesame measure. We introduce the market price of risk leading to the risk neutral measuredynamics for the zero-coupon bond and show that, as classically done in Bjork with no-arbitrage assumptions related to self-financing and locally risk free portfolios, it does notdepend on the bond maturity. With this market price of risk we then derive the dynamicsof the short rate r under the risk neutral measure. All this is standard theory that the termstructure expert will find immediate and may want to read quickly. For readers who are moreinto price impact but relatively unfamiliar with term structure modeling, we recommend acareful reading of this introductory part. Inventory and impact.
We define the trader’s inventory X , the execution speed v andthe instantaneous and transient impacts, leading to the impacted bond price ˜ P in Section 3.1.This part will be easier to the market microstructure expert but, given that applications ofprice impact to full term structure modeling are not known, we recommend to read carefullythis part given the nuances involved in developing an impact theory under term structure.We then show how it is possible to incorporate the effect of price impact in a new definitionof impacted market price of risk, leading to an impacted risk neutral measure. Impacted risk neutral and physical probability measures.
This measure is in-troduced in Section 3.2. Under this measure, the impacted zero-coupon bond dynamics isthe same as the risk neutral dynamics of the unimpacted bond provided one replaces theBrownian motion under the original risk neutral measure with a Brownian motion underthe new impacted measure. The impacted risk neutral measure allows us to define an im-pacted physical measure as well, and again the bond price dynamics under the impactedphysical measure is the same as the bond price dynamics under the original physical meas-ure, provided that the Brownian motion is replaced with a Brownian motion under the newmeasure. In this setting, we can show that the impacted zero-coupon bond is the impactedrisk neutral expectation of the stochastic discount factor. In a way, the impacted risk neutralmeasure behaves indeed as much as possible as a risk neutral measure in presence of impact.One of the main purposes of introducing the risk neutral measure in classic derivativespricing is the idea that the existence of this measure implies absence of arbitrage. Thereader might then wonder whether we can say the same of the impacted risk measure. Thisis dealt with in the Section 3.3.
Derivatives pricing and absence of arbitrage.
First, we show that the existence ofan impacted equivalent pricing measure implies that the impacted model is arbitrage free.5t then follows that we can extend our result for zero-coupon bonds, namely we can priceinterest-rate derivatives under impact by taking the impacted risk neutral expectation oftheir discounted cash flows. We present an example involving Eurodollar futures.An important part of the theory of price impact for interest rates is what we might call endogenous cross-impact . With this we point to the impact of a zero-coupon bond witha given maturity on the price of a zero-coupon bond with a different maturity on the samecurve . Exogenous cross impact would be the impact of a bond in one curve on the bondon another curve, but this is not what we deal with here. The theory of endogenous crossimpact is developed in Section 3.4. This has to be read very carefully by all readers, as itrepresents a very peculiar aspect of any theory of price impact applied to a term structure.In term structure modeling the underlying of derivatives is a whole curve, so price impacthas to be postulated also for one part of the curve over other parts. This is not a problemin markets where assets are points rather than curves, like stocks or FX rates.
Coupon bonds.
The theory developed up to this point of the paper deals with zero-coupon bonds. However, most traded bonds are coupon bonds, so it is necessary to developa theory of price impact for coupon bearing bonds. This is done in Section 3.5. First,by looking at prices as expectations of discounted cash flows, we deduce that the price ofimpacted coupon bearing bond is simply obtained by the unimpacted one, by replacing allunimpacted zero-coupon bonds in its expression with impacted ones. We find a way toconnect the impact on the coupon bearing bond to the impact of single zero-coupon bonds,and find assumptions under which the impacted dynamics may even be the same.Finally, as explained earlier, Section 3.6 goes into quite some detail to show that thetheory developed so far can be equally formulated in the
Heath-Jarrow-Morton (HJM)framework , using instantaneous forward rates to model the term structure. It is well knownto interest rate experts that the no-arbitrage condition in the HJM framework is expressedby requiring that the drift in the forward rate dynamics is a specific transformation of thedynamics volatility. We show that the same relationship between drift and volatility holdsfor impacted instantaneous forward rates under the impact-adjusted risk neutral measure.The classic HJM theory then carries over to the impacted case. This part may be helpfulto readers who wish to see our framework work outside the short rate models setting, butif the reader is not too interested in the specific interest rate model used, she can skip thispart and still get all the ideas on impacted term structure modeling.
Pricing examples.
These are presented in Section 4, including pricing impactedEurodollar Futures with Gaussian short rate models. This section is interesting for readerswho wish to see how the model can be effectively applied to price interest rate derivativesunder price impact.
Numerical results.
Section 5 presents numerical examples of impacted yield curves,showing how the term structure changes once price impact is included. We analyze theinterplay between the bond pull-to-par and cross-price impact parameters. This can be agood section for readers who wish to develop some intuition on how price impact may changethe shape of the term structure. 6 ptimal execution with impacted bonds.
For the reader wondering how this theoryof price impact can be included in classic problems such as optimal execution, we solve theoptimal execution problem in a specific setting. This is presented in Section 6. The exampleis rather technical and the solution involves a system of forward-backward SDEs, showingthat optimal execution with term structure models can be technically demanding, and thisis recommended only for readers who are interested in optimal execution.Concluding, the best way to read the paper is sequentially, skipping the parts that arenot of interest and looking more carefully at areas outside the specific expertise of the reader.We tried to provide a guide to what can be skipped and on the different emphasis for differentareas of expertise above.
We introduce our initial assumption on the interest rate dynamics, assuming it is a one-dimensional short rate model dynamics. This will be relaxed later with a Heath-Jarrow-Morton setting in Section 3.6.Let us fix a maturity
T > and let (Ω , F , ( F t ) t ≥ , P ) be a filtered probability spacesatisfying the usual conditions, on which there is a standard ( F t ) t ∈ [0 ,T ] -Brownian motion W P . We consider the following dynamics of the short rate under the real world measure P dr ( t ) = µ ( t, r ( t )) dt + σ ( t, r ( t )) dW P ( t ) , (3.1)where µ ( t, r ) , σ ( t, r ) are given real valued functions, assumed to be regular enough to ensurethe SDE has a unique strong solution. For example one can assume that both µ and σ areLipschitz continuous in the r coordinate, and has at most linear growth in r uniformly in t ∈ [0 , T ] . We moreover assume that σ ( t, r ( t )) is P -a.s. strictly positive for any t > .Assume that the dynamics of a zero-coupon bond with maturity at time T , under thereal world measure, is given by dP ( t, T ) = µ T ( t, r ( t )) dt + σ T ( t, r ( t )) dW P ( t ) , (3.2)with µ T , σ T depending on the maturity T and regular enough as in (3.1). Typically, onemight assume the price process of the T -bond to be of the form P ( t, T ) = F ( t, r ( t ); T ) for some function F smooth in three variables. Then, under suitable assumptions (seeAssumption 3.2 in Chapter 3.2 of [7]) one can define for any finite maturity T > thestochastic process λ ( t ) = µ T ( t, r ( t )) − r ( t ) P ( t, T ) σ T ( t, r ( t )) , (3.3)and show that λ may depend on r but it does not actually depend on T . Such process iscalled market price of risk . Provided the Novikov condition holds, this process can be used7o define a change of measure from the real world measure P to the risk neutral measure Q : d Q d P = exp (cid:18)(cid:90) t λ ( s ) dW P ( t ) − (cid:90) t λ ( s ) ds (cid:19) . (3.4)The dynamics of the short rate under Q becomes dr ( t ) = [ µ ( t, r ( t )) − λ ( t ) σ ( t, r ( t ))] dt + σ ( t, r ( t )) dW Q ( t ) . Due to the lack of completeness of the bond market, the model will be specified once thestochastic process λ is defined. Our strategy for establishing a mathematical framework thatencompasses both risk neutral pricing and price impact in the context of interest rates de-rivatives consists, first of all, in specifying the dynamics for an impacted bond with maturity T under the real world measure P .We consider a trader with an initial position of x T > zero-coupon bonds with maturity T . Let < τ ≤ T denote some finite deterministic time horizon. The number of bonds thetrader holds at time t ∈ [0 , τ ] is given by X T ( t ) = x T − (cid:90) t v T ( s ) ds. (3.5)where the function v T denotes the trader’s selling rate, which takes negative values in caseof a buy strategy. In what follows we assume that v T = { v T ( t ) } ≤ t ≤ τ is progressivelymeasurable and has a P -a.s. bounded derivative (in the t -variable), that is, there exists M > such that sup ≤ t ≤ τ + | ∂ t v T ( t ) | < M, P − a . s ., (3.6)where ≤ t ≤ τ . After the trading stops, we assume that v T ( t ) = 0 . We denote the class ofsuch trading speeds as A T .The main idea the assumption of the differentiability of v T is that the overall impact weadd to the zero-coupon bond should affect the drift only (see (3.12)). Moreover due to priceimpact effect, we allow bond price which are large than for some time intervals but we doneed to control their upper bound.We consider a price impact model with both transient and instantaneous impact, whichis a slight generalization of the model which was considered in [29]. The impacted bondprice is therefore given by ˜ P ( t, T ) = P ( t, T ) − l ( t, T ) v T ( t ) − K ( t, T )Υ vT ( t ) . (3.7)Here Υ vT represents the transient impact effect and it has the form Υ vT ( t ) := ye − ρt + γ (cid:90) t e − ρ ( t − s ) v T ( s ) ds, (3.8)where y, ρ and γ are positive constants. The term v T ( t ) in (3.7) represents the instantaneousprice impact, where we absorb in the function l any constants that should factor it. Lastly,8 and K are differentiable functions with respect to both variables ( t, T ) which take positivevalues on ≤ t < T and for any ≤ τ < T we have inf ≤ t ≤ τ l ( t, T ) > . (3.9)Moreover we assume that sup ≤ t ≤ τ | ∂ t l ( t, T ) | < ∞ , lim t → T l ( t, T ) = 0 , sup ≤ t ≤ τ | ∂ t K ( t, T ) | < ∞ , lim t → T K ( t, T ) = 0 . (3.10)While the assumption on boundedness of the derivatives of functions K and l arise fromtechnical reasons which has similar motivation as the reason for (3.6), the assumptions onthe behaviour at expiration is meant to enforce the boundary condition on the price of theimpacted bond at expiration, which is ˜ P ( T, T ) = 1 . Note that K and l are time-dependentversions of the parameters λ, k in [29]. A prominent example of such functions is l ( t, T ) = κ (cid:18) − tT (cid:19) α , K ( t, T ) = (cid:18) − tT (cid:19) β , for some constants α, β ≥ and κ > .We define for convenience the overall price impact: I T ( t ) := l ( t, T ) v T ( t ) + K ( t, T )Υ vT ( t ) . (3.11)Then, since v T is in A T we can rewrite (3.7) as follows: d ˜ P ( t, T ) = dP ( t, T ) − J T ( t ) dt, ˜ P ( T, T ) = 1 , (3.12)with J T ( t ) := ∂ t I T ( t )= ∂ t l ( t, T ) v T ( t ) + l ( t, T ) ∂ t v T ( t ) + ∂ t K ( t, T )Υ vT ( t ) + K ( t, T )[ − ρ Υ vT ( t ) + v T ( t )] . (3.13)Our model so far describes how trading a T -bond affects its price. Next, we show theexistence of an impacted market price of risk process which will be a generalization of (3.3).Using this process we will define an equivalent martingale measure, under which bonds andderivatives prices can be computed. Such a measure will be called an impacted risk neutralmeasure . It is important to remark that, as in the classical case, this change of measure willbe unique for all bond maturities.Before stating the main theorem of this section, let us first introduce a few importantdefinitions. Definition 3.1 (Impacted portfolio) . Let ˆ T < + ∞ be some finite time horizon. An im-pacted portfolio is a ( n + 1) − dimensional, bounded progressively measurable process ˜ h = ˜ h t ) t ∈ [0 , ˆ T ] with ˜ h t = (˜ h t , ˜ h t , . . . , ˜ h nt ) , where ˜ h it represents the number of shares in the im-pacted bond ˜ P ( t, T i ) held in the portfolio at time t . The value at time t of such a portfolio ˜ h is defined as ˜ V ( t ) ≡ ˜ V ( t, ˜ h ) := n (cid:88) i =0 ˜ h i ( t ) ˜ P ( t, T i ) . Definition 3.2 (Self-financing) . Let ˆ T < + ∞ be some finite time horizon. and let ˜ h be animpacted portfolio as in Definition 3.1. We say that ˜ h is self-financing if its value ˜ V is suchthat d ˜ V ( t, ˜ h ) = n (cid:88) i =0 ˜ h i ( t ) d ˜ P ( t, T i ) , for all ≤ t ≤ ˆ T . (3.14)
Definition 3.3 (Locally risk free) . Let ˜ h be an impacted portfolio as in Definition 3.1 andlet ˜ V be its value. Let also α = ( α t ) t ∈ [0 , ˆ T ] be an adapted process. We say that ˜ h is locallyrisk-free if, for almost all t , d ˜ V ( t ) = α ( t ) ˜ V ( t ) = ⇒ α ( t ) = r ( t ) , where r ( t ) is the risk-free interest rate introduced in (3.1) . Here is the main result of this section.
Theorem 3.4 (Impacted market price of risk) . Let ˆ T < + ∞ be some finite time horizon andlet T := (0 , ˆ T ] . Let J T be the impact density defined in (3.13) . Given an impacted portfolio ˜ h as in (3.14) , we assume that it is self-financing and locally risk-free, as in Definitions (3.2) and (3.3) , respectively. Then, there exists a progressively measurable stochastic process ˜ λ ( t ) such that ˜ λ ( t ) = µ T i ( t, r ( t )) − r ( t ) ˜ P ( t, T i ) − J T i ( t ) σ T i ( t, r ( t )) , t ≥ , (3.15) for each maturity T i , i = 1 , .., n , with ˜ λ depending on the short rate r but not on T i . The proof of Theorem 3.4 is given in Section 7.
Remark 3.5 (Self-financing in presence of price impact) . In presence of price impact it isof course not obvious that the self-financing condition should hold. Adjusted self-financingconditions have been proposed, for instance, by Carmona and Webster [13]. We notice that,in their work, the adjustment consists of two parts: the covariation between the inventoryand the price process, and the bid-ask spread. In our work we will assume the inventory is afinite variation process and that the bid ask spread is negligible, thereby obtaining the classicself-financing condition.
Remark 3.6 (Intrinsic price impact) . From Theorem 3.4 it follows that endogenous crossprice impact naturally emerges in our framework. Indeed, once an agent trades a bond withmaturity T , the process ˜ λ is uniquely determined. Note that ˜ λ does not depend on thematurity. For any bond with maturity T ∈ T , which is not traded, we have J T ≡ butby (3.15) , the price ˜ P ( t, T ) will be affected by the trade on the bond with maturity T . Weremark that, by endogenous , we mean that the bonds with different maturities T and T are thought of as belonging to the same curve . If we were to discuss multiple interest ratecurves, then exogenous cross price impact should be taken into account as well. .2 Impacted risk-neutral measure We previously introduced two measures: the real world measure P and the classical riskneutral measure Q , as defined in (3.4). Now we use the result of Theorem 3.4 to define athird measure, which we call impacted risk neutral measure and denote by ˜ Q . This is definedas follows: d ˜ Q d P = exp (cid:26)(cid:90) t ˜ λ ( s ) dW P ( s ) − (cid:90) t ˜ λ ( s ) ds (cid:27) . (3.16)The well posedness of ˜ Q can be checked via the Novikov condition. It might be useful torecall that the usual approach does not consist in determining the conditions on µ T , σ T underwhich the Novikov condition is fulfilled. Rather, one chooses a specific short rate model tobegin with. Then, one can specify the market price of risk process, exploiting the fact thatit depends on t and r , but not on T . For example, in the case of Vasicek model, the marketprice of risk is assumed to be λ ( t ) = λr ( t ) , for some constant λ . At this point, Novikovcondition can be checked much more easily. Since we proved that ˜ λ depends on t and r only,we can assume the two processes to have the same structure and follow the same idea. Inthe case of Vasicek model, for example, we can assume ˜ λ ( t ) = ˜ λr ( t ) , for some constant ˜ λ incorporating the impact. Consequently, determining the existence and well-posedness of ˜ Q is fundamentally equivalent to determining the existence and well-posedness of Q .The Girsanov change of measure from the classical risk neutral measure to the impactedone is given by d ˜ Q d Q = d ˜ Q d P d P d Q . with d P d Q = exp (cid:26) − (cid:90) t λ ( s ) dW Q ( s ) − (cid:90) t λ ( s ) ds (cid:27) . where λ was defined in (3.3). Hence, d ˜ Q d Q = exp (cid:26)(cid:90) t ˜ λ ( s ) dW P ( s ) − (cid:90) t ˜ λ ( s ) ds + (cid:90) t λ ( s ) dW Q ( s ) − (cid:90) t λ ( s ) ds (cid:27) . Since W P ( t ) := W Q ( t ) + (cid:82) t λ ( s ) ds is a Brownian motion under the measure P , we have d ˜ Q d Q = exp (cid:26)(cid:90) t (˜ λ ( s ) − λ ( s )) dW Q ( s ) − (cid:90) t (cid:16) λ ( s ) + ˜ λ ( s ) − λ ( s )˜ λ ( s ) (cid:17) ds (cid:27) . In other words, W ˜ Q ( t ) := W Q ( t ) − (cid:90) t (˜ λ ( s ) − λ ( s )) ds, is a Brownian motion under the measure ˜ Q . It is then straightforward to notice that theimpacted zero-coupon bond under the impacted measure ˜ Q will be described by the dynamics d ˜ P ( t, T ) = r ( t ) ˜ P ( t, T ) dt + σ T ( t, r ( t )) dW ˜ Q ( t ) . (3.17)11e further remark that, in principle, we could start by defining a new measure ˜ P to get ridof the additional drift due to impact. Just rewrite the dynamics of the impacted zero-couponbond as d ˜ P ( t, T ) = µ T ( t, r ( t )) dt + σ T ( t, r ( t )) (cid:18) J T ( t ) σ T ( t, r ( t )) dt + dW P ( t ) (cid:19) . This suggests to define d ˜ P d P = exp (cid:40)(cid:90) t J T ( s ) σ T ( s, r ( s )) dW P ( s ) − (cid:90) t (cid:18) J T ( s ) σ T ( s, r ( s )) (cid:19) ds (cid:41) . The impacted bond under this measure would follow the dynamics d ˜ P ( t, T ) = µ T ( t, r ( t )) dt + σ T ( t, r ( t )) dW ˜ P ( t ) . (3.18)At this point, ˜ Q can be defined from ˜ P by using the classical market price of risk λ ( t ) . Inother words, d ˜ Q d ˜ P = d ˜ Q d Q d Q d ˜ P = d ˜ P d P d Q d ˜ P = d Q d P . Putting everything together, we have the following commuting diagram
P Q ˜ P ˜ Q λ ˜ λ ˜ λ − λ ˜ λ − λλ By (3.17) and usual arguments it follows that discounted impacted traded prices, that is { ˜ P ( · , T ) /B ( t ) } t ≥ , are martingales for any ≤ T ≤ ˆ T under ˜ Q . Here B is the usual moneymarket account at time t given by B ( t ) = e (cid:82) t r ( s ) ds . (3.19)We therefore have ˜ P ( t, T ) B ( t ) = E ˜ Q (cid:34) ˜ P ( T, T ) B ( T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) . (3.20)Multiplying both sides by B ( t ) and exploiting the boundary condition ˜ P ( T, T ) = 1 , weobtain the fundamental equation ˜ P ( t, T ) = E ˜ Q (cid:104) e − (cid:82) Tt r ( s ) ds (cid:12)(cid:12) F t (cid:105) . (3.21) Remark 3.7 (Interpretation impacted real world measure) . From (3.18) we observe that,financially speaking, under the impacted real world measure ˜ P , impacted bond dynamics ˜ P ( · , T ) has the same dynamics as the classical bond (without price impact modeling) in (3.2) under P . In particular we have that ˜ P ( T, T ) = 1 for all maturities T . .3 Applications to pricing of interest rate derivatives We start this section by remarking that the notion of arbitrage we use in our work is theclassical one (see e.g. Harrison and Kreps [20] or Harrison and Pliska [21]), adjusted withimpacted portfolios.
Definition 3.8 (Arbitrage portfolio) . An arbitrage portfolio is an impacted self-financingportfolio ˜ h such that its corresponding value process ˜ V satisfies1. ˜ V (0) = 0 .2. ˜ V ( T ) ≥ P -a.s.3. P ( ˜ V ( T ) > > Using this definition of arbitrage we show that the first fundamental theorem of assetpricing holds in our setting.
Theorem 3.9 (Absence of arbitrage) . Assume that there exists an impacted equivalentmartingale measure ˜ Q as in (3.16) . Then, our impacted model is arbitrage free. The proof of Theorem 3.9 is given in Section 7.A key consequence Theorem 3.9 is that our term structure model with price impact isindeed free of arbitrage. This allows to price interest rate derivatives by taking the expect-ation of discounted payoffs under the impacted risk neutral measure ˜ Q . As a benchmarkexample, we consider the price of an impacted Eurodollar future. In the classical context, aEurodollar-futures contract provides its owner with the payoff (see Chapter 13.12 of [11]) N (1 − L ( S, T )) , where N denotes the notional and L ( S, T ) is the LIBOR rate, defined as (see Chapter 1 of[11], Definition 1.2.4) L ( S, T ) := 1 − P ( S, T ) τ ( S, T ) P ( S, T ) , (3.22)with τ ( S, T ) denoting the year fraction between S and T . Motivated by this, we introducethe impacted counterpart of the LIBOR rate in (3.22), i.e. ˜ L ( S, T ) := 1 − ˜ P ( S, T ) τ ( S, T ) ˜ P ( S, T ) , (3.23)with τ defined as above and the impacted zero-coupon bond in place of the classical one.This new rate ˜ L is interpreted as the simply-compounded rate which is consistent with theimpacted bond. This corresponds to the classical LIBOR rate which is the constant rate atwhich one needs to invest P ( t, T ) units of currency at time t in order to get an amount ofone unit of currency at maturity T . Then, the fair price of an impacted Eurodollar futureat time t is (see [11], Chapter 13, eq. (13.19)) ˜ C t = E ˜ Q t [ N (1 − ˜ L ( S, T ))] , = N (cid:18) τ ( S, T ) − τ ( S, T ) E ˜ Q t (cid:20) P ( S, T ) (cid:21)(cid:19) , (3.24)13here the discounting was left out due to continuous rebalancing (see again Chapter 13.12 of[11]). We will demonstrate in Section 4 how such expectation can be computed analyticallyprovided the short rate model is simple enough as in Vasicek and Hull-White models. Remark 3.10 (Linear and nonlinear pricing equations) . Our success in retaining analyticaltractability and linearity in the pricing equation may look surprising at first. In the con-text of equities, pricing derivatives in presence of price impact typically leads to nonlinearPDEs. This, in turn, motivated the study of super-replicating strategies and the so-calledgamma constrained strategies. Several works provide also necessary and sufficient condi-tions ensuring the parabolicity of the pricing equation, hence the existence and uniqueness ofa self-financing, perfectly replicating strategy. We refer, for example, to Abergel and Loeper[1], Bourchard, Loeper et al. [9, 10] and Loeper [27]. The point we would like to stress here isthat the nonlinearity of the pricing equation is a consequence of the trading strategy having adiffusion term, or a consequence of the presence of transaction costs. In other words, underthe assumption that trading strategies have bounded variation and no transaction costs arepresent, the pricing PDE becomes linear again. Hence, our work is actually in agreement towhat can be found in the context of equities.
In this section we discuss how trading a bond P ( t, T ) impacts the yield curve. For the sakeof analytical tractability, we will consider affine short-rate models, that is, those modelswhere bond prices are of the form P ( t, T ) = A ( t, T ) e − B ( t,T ) r ( t ) , ≤ t ≤ T, (3.25)for some deterministic, smooth functions A and B and r is given by (3.1). The remarkableproperty of these models is that they can be completely characterized as in the followingtheorem (see, e.g., Filipovic [15], Section 5.3, Brigo and Mercurio [11], Section 3.2.4, Bjork[7] Section 3.4 and references therein). Lemma 3.11 (Characterization affine short-rate models) . The short rate model (3.1) isaffine if and only if there exist deterministic, continuous functions a, α, b, β such that thediffusion and the drift terms in (3.1) are of the form σ ( t, r ) = a ( t ) + α ( t ) r,µ ( t, r ) = b ( t ) + β ( t ) r, and the functions A, B satisfy the following system of ODEs − ∂∂t ln A ( t, T ) = 12 a ( t ) B ( t, T ) − b ( t ) B ( t, T ) , A ( T, T ) = 1 ,∂∂t B ( t, T ) = 12 α ( t ) B ( t, T ) − β ( t ) B ( t, T ) − , B ( T, T ) = 0 , for all t ≤ T .
14s explained in [15], the functions a, α, b, β can be further specified by observing thatany non-degenerate short rate affine model, that is a model with σ ( t, r ) (cid:54) = 0 for all t > ,can be transformed, by means of an affine transformation, in two cases only, depending onwhether the state space of the short rate r is the whole real line R or only the positive part R + . In the first case, it must hold α ( t ) = 0 and a ( t ) ≥ , with b, β arbitrary. In the secondcase, it must hold a ( t ) = 0 , α ( t ) , b ( t ) ≥ and β arbitrary.Let ˆ T < + ∞ be some finite time horizon. The yield curve at a pre-trading time t (i.e.before price impact effects kick in) according to classical theory of interest rates is definedby Y ( t, T ) := P ( t, T ) − /T − , ≤ t ≤ t , (3.26)for all maturities ≤ T ≤ ˆ T . Next, we consider the impacted bond dynamics d ˜ P ( t, T ) = dP ( t, T ) − J T ( t ) dt, where J T was defined in (3.13). Recall that the dynamics of r ( t ) is given in (3.1). ApplyingIto’s formula on P ( t, T ) in (3.25) we get dP ( t, T ) = e − B ( t,T ) r ( t ) (cid:20) ∂A∂t − A ( t, T ) ∂B∂t r ( t ) + 12 A ( t, T ) B ( t, T ) σ ( t, r ( t ))+ − A ( t, T ) B ( t, T ) µ ( t, r ( t )) (cid:21) dt − σ ( t, r ( t )) B ( t, T ) A ( t, T ) e − B ( t,T ) r ( t ) dW P ( t ) . From this equation, we readily extract the drift and the diffusion of the zero-coupon bondwith maturity T : µ T ( t, r ( t )) := e − B ( t,T ) r ( t ) (cid:20) ∂A∂t − A ( t, T ) ∂B∂t r ( t ) + 12 A ( t, T ) B ( t, T ) σ ( t, r ( t )) − A ( t, T ) B ( t, T ) µ ( t, r ( t )) (cid:21) ,σ T ( t, r ( t )) := − σ ( t, r ) A ( t, T ) B ( t, T ) e − B ( t,T ) r ( t ) . (3.27)Next, we consider the effect of an agent trading on the bond with maturity T on a bondwhich is not traded by the agent with maturity S . We call this effect the endogenous cross-impact on the bond with maturity S . Recall that in this case the dynamics of the S -bondis given by d ˜ P ( t, S ) = µ S ( t, r ( t )) dt + σ S ( t, r ( t )) dW P ( t ) , (3.28)where the coefficients µ S and σ S are given by analogous formulas to (3.27). Since we aretrading the T -bond only, J S in (3.13) will be identically equal to zero. Hence, the definitionof the impacted market price of risk (3.15) implies the following relationship µ T ( t, r ( t )) − r ( t ) ˜ P ( t, T ) − J T ( t ) σ T ( t, r ( t )) = µ S ( t, r ( t )) − r ( t ) ˜ P ( t, S ) σ S ( t, r ( t )) . S -bond has to change in order to avoid arbitrage.That is, this equation describes the cross-price impact . Specifically we have µ S ( t, r ( t )) = σ S ( t, r ( t )) σ T ( t, r ( t )) (cid:104) µ T ( t, r ( t )) − r ( t ) ˜ P ( t, T ) − J T ( t ) (cid:105) + r ( t ) ˜ P ( t, S ) . Substituting this drift in (3.28) we get d ˜ P ( t, S ) = r ( t ) ˜ P ( t, S ) dt + σ S ( t, r ( t )) σ T ( t, r ( t )) (cid:104) µ T ( t, r ( t )) − r ( t ) ˜ P ( t, T ) − J T ( t ) (cid:105) dt + σ S ( t, r ( t )) dW P ( t ) . (3.29)Finally, we define the impacted yield curve for all t ≤ T ≤ ˆ T as follows: ˜ Y ( t, T ) := ˜ P ( t, T ) − /T − . (3.30) Remark 3.12 (Cross impacted bonds at maturity) . We have shown in (3.12) that accordingto our model ˜ P ( T, T ) = 1 . However, we should also ensure that all cross-impacted bondswith maturity S (cid:54) = T reach value at their maturities. This of course, would make the modelmuch more involved and we may lose tractability. It is worth recalling that the zero-coupon bond P ( t, T ) is rarely traded. In practice, its priceis derived using some bootstrapping procedure applied, for instance, to coupon bonds. Inthe classical theory, coupon bonds are defined as B ( t, T ) = n (cid:88) i =1 c i P ( t, T i ) + N P ( t, T n ) , where N denotes the reimbursement notional, ( c i , T i ) ni =1 are the coupons and the maturitiesat which the coupons are paid, respectively. In order to determine an expression for theimpacted coupon bond, we start from its cash flow C ( t ) := n (cid:88) i =1 c i D ( t, T i ) + N D ( t, T n ) , where D ( t, T ) is the stochastic discount factor defined by D ( t, T ) := e − (cid:82) Tt r ( s ) ds , where r is given by (3.1). Then, we define the impacted coupon bond as the expectation ofthis cash flow under the impacted risk neutral measure ˜ Q (see (3.16)): ˜ B ( t, T ) := E ˜ Q [ C ( t )] . C immediately yields ˜ B ( t, T ) = n (cid:88) i =1 c i E ˜ Q [ D ( t, T i )] + N E ˜ Q [ D ( t, T n )]= n (cid:88) i =1 c i ˜ P ( t, T i ) + N ˜ P ( t, T n ) , (3.31)where ˜ P ( · , T i ) is the (directly) impacted price of a zero-coupon bond as defined in (3.7).Note that (3.31) gives the price of the coupon bonds in terms of impacted zero-couponbonds. Since zero-coupon bonds are not always traded we would like to get a direct pricingformula for coupon bonds. Let { v T i } ni =1 be admissible trading speeds on zero-coupon bondswith maturities { T i } ni =1 as defined in Section 3, that is V T i ∈ A T i for any i = 1 , ...n . From(3.7) and (3.31) we get ˜ B ( t, T ) = n (cid:88) i =1 c i ˜ P ( t, T i ) + N ˜ P ( t, T n )= B ( t, T ) − n (cid:88) i =1 c i l ( t, T i ) v T i ( t ) − N l ( t, T n ) v T n ( t ) − n (cid:88) i =1 c i K ( t, T i ) ye − ρt − N K ( t, T n ) ye − ρt − γ (cid:90) t e − ρ ( t − s ) (cid:32) n (cid:88) i =1 c i K ( t, T i ) v T i ( s ) + N K ( t, T n ) v T n ( s ) (cid:33) ds. Let us now assume that l = κK at all times and for all maturities, where κ > is a constant.Then, the impacted coupon bond dynamics can be written as ˜ B ( t, T ) = B ( t, T ) − ye − ρt (cid:34) n (cid:88) i =1 c i K ( t, T i ) + N K ( t, T n ) (cid:35) − (cid:90) t e − ρ ( t − s ) κδ ( s − t ) (cid:34) n (cid:88) i =1 c i K ( t, T i ) v T i ( s ) + N K ( t, T n ) v T n ( s ) (cid:35) ds − γ (cid:90) t e − ρ ( t − s ) (cid:34) n (cid:88) i =1 c i K ( t, T i ) v T i ( s ) + N K ( t, T n ) v T n ( s ) (cid:35) ds, (3.33)where δ denotes the Dirac delta. Notice that under this assumption the impacted zero-coupon bond dynamics defined in (3.7) boils down to ˜ P ( t, T ) = P ( t, T ) − K ( t, T ) (cid:20) ye − ρt + (cid:90) t e − ρ ( t − s ) v T ( s ) ( γ + κδ ( s − t )) ds (cid:21) . (3.34)This suggest we can define K B ( t, T ) := n (cid:88) i =1 c i K ( t, T i ) + N K ( t, T n ) . v B ( t, s ) := 1 K B ( t, T ) (cid:34) n (cid:88) i =1 c i K ( t, T i ) v T i ( s ) + N K ( t, T n ) v T n ( s ) (cid:35) . (3.35)Therefore, we obtain the following price impact model for the coupon bond: ˜ B ( t, T ) = B ( t, T ) − K B ( t, T ) (cid:20) ye − ρt + (cid:90) t e − ρ ( t − s ) v B ( t, s ) ( γ + κδ ( s − t )) ds (cid:21) . (3.36)Interestingly, under the simplifying assumption that the functions l and K are equal up tosome constant, we observe that the impacted zero-coupon bond ˜ P ( t, T ) in (3.34) and theimpacted coupon bond ˜ B ( t, T ) in (3.36) are described by the same kind of dynamics.This is particularly useful because, provided enough data on traded coupon bonds areavailable, one might attempt to use (3.35) and (3.36) to bootstrap the trading speeds v T i relative to the zero-coupon bonds. Using the price impact model (3.7), it would be thenpossible to price impacted zero-coupon bonds consistently with market data. Finally, usingthese impacted zero-coupon bonds as building blocks, it would be possible to price, consist-ently with market data, more complicated and less liquid impacted interest rate derivatives,as discussed in Section 3.3. In this section we turn our discussion to incorporating price impact into the Heath, Jarrowand Morton framework [22], in order to model the forward curve. Notice that this approach,although it may look different, has some common aspects to the framework developed inSection 3.1. We first add artificially a price impact term to the forward rate dynamics,which creates an impacted interest rate. This corresponds to adding price impact to zero-coupon bonds in Section 3.1. Then we will develop the connection between the price impactof zero-coupon bonds to the price impact term incorporated to the forward rate in orderto reveal the financial interpretation of the later. Note that both the zero-coupon bondsand the forward rate can be used as building blocks for the whole interest rates theory.We are therefore interested in showing the connection between the two in the presence ofprice impact. For a thorough discussion on the HJM framework in the classical interest ratetheory, we refer to Chapter 6 of the book by Filipovic [15].Given an integrable initial forward curve T → ˜ f (0 , T ) , we assume that the impactedforward rate process ˜ f ( · , T ) is given by ˜ f ( t, T ) = ˜ f (0 , T ) + (cid:90) t (cid:16) α ( s, T ) + J f ( s, T ) (cid:17) ds + (cid:90) t σ ( s, T ) dW P ( s ) , (3.37)for any ≤ t ≤ T and each maturity T > . Here W P is a Brownian motion under themeasure P and α ( · , T ) , J f ( · , T ) and σ ( · , T ) are assumed to be progressively measurableprocesses and satisfy for any T > (cid:90) T (cid:90) T ( | α ( s, t ) | + | J f ( s, t ) | ) dsdt < ∞ , sup s,t ≤ T | σ ( s, t ) | < ∞ . α ( · , T ) and σ ( · , T ) above are as in standard HJM model, the stochasticprocess J f represents the impact density relative to the forward rate and accounts for the factthat the forward curve is affected by the trading activity. From a modelling perspective, itplays a completely analogous role as the quantity J T defined in (3.13) for the impacted zero-coupon bond. In fact, in Proposition (3.14) we will determine the mathematical relationshiplinking these two quantities. Such relationship will allow us to understand how the forwardcurve is impacted by trading zero-coupon bonds.In this framework, the impacted short rate model is given by ˜ r ( t ) := ˜ f ( t, t ) = ˜ f (0 , t ) + (cid:90) t (cid:16) α ( s, t ) + J f ( s, t ) (cid:17) ds + (cid:90) t σ ( s, t ) dW P ( s ) , (3.38)and the impacted zero-coupon bond is defined as follows ˜ P ( t, T ) = e − (cid:82) Tt ˜ f ( t,u ) du . (3.39)Next we derive the explicit dynamics of { ˜ P ( t, T ) } ≤ t ≤ T . The following corollary is animpacted version of Lemma 6.1 in [15]. Corollary 3.13 (Impacted zero-coupon bond in HJM framework) . For every maturity T the impacted zero-coupon bond defined in (3.39) follows the dynamics ˜ P ( t, T ) = ˜ P (0 , T ) + (cid:90) t ˜ P ( s, T ) (cid:16) ˜ r ( s ) + ˜ b ( s, T ) (cid:17) ds + (cid:90) t ˜ P ( s, T ) ν ( s, T ) dW P ( s ) , t ≤ T, (3.40) where ˜ r is the impacted short rate defined in (3.38) and ν ( s, T ) := − (cid:90) Ts σ ( s, u ) du, ˜ b ( s, T ) := − (cid:90) Ts α ( s, u ) du − (cid:90) Ts J f ( s, u ) du + 12 ν ( s, T ) . (3.41)We now show that the impact J f can be expressed in terms of the impact relative tothe zero-coupon bond, and vice versa. In order to show this correspondence in terms ofagent’s trading speed, we need to make an additional assumption on the trading speeds onzero-coupon bonds. We assume that the price impact in the forward curve is a result oftrading by one or many agents over a continuum of zero-coupon bonds with maturities T and trading speeds { T ≥ v T ∈ A T } so that | ∂ T v T ( t ) | < ∞ , for all ≤ t ≤ T, P − a.s. (3.42)Note that this assumption in fact makes sense in bond trading, which has discrete matur-ities, as it claims that when there is a highly traded T i -bond, you would find that also theneighbouring T i − , T i +1 are liquid. Assumption (3.42) implies that ∂ T I T ( t ) is well definedas needed in the following Proposition. We recall that f represents the unimpacted forwardrate which is given by setting J f ≡ in (3.37).19 roposition 3.14 (Forward rate and zero-coupon bond price impact relation) . Let I T ( t ) be the overall impact defined in (3.11) and ˜ P ( · , T ) the impacted zero-coupon bond price in (3.39) . Assume ˜ f (0 , t ) = f (0 , t ) , meaning that the initial value of the forward curve is notaffected by trading. Then, the forward rate impact J f introduced in (3.37) is given by J f ( t, T ) = − ∂∂T log (cid:18) − I T ( t ) P ( t, T ) (cid:19) , for all ≤ t ≤ T such that ˜ P ( t, T ) > . (3.43)The proof of Proposition 3.14 is given in Section 7. Remark 3.15.
Note that the requirement that ˜ P ( t, T ) > ensures that the logarithm on theright-hand side of (3.43) is well defined, as (7.9) in the proof suggests. The proof also givesanother relation between J f ( · , T ) and I T which always holds but is perhaps not as direct. A well known feature of the classical HJM framework is that, under the risk neutralmeasure, the drift of the forward rate is completely specified by the volatility through theso called
HJM condition . In order to understand how this condition is affected by theintroduction of price impact, we will follow Theorem 6.1 of [15]. In particular, we have thefollowing key result.
Theorem 3.16 (HJM condition with price impact) . Let P be the real world measure underwhich the impacted forward rate as in (3.37) . Let ˜ Q ∼ P be an equivalent probability measureof the form d ˜ Q d P = exp (cid:26)(cid:90) t ˜ γ ( s ) dW P ( s ) − (cid:90) t ˜ γ ( s ) ds (cid:27) , (3.44) for some progressively measurable stochastic process ˜ γ = { ˜ γ ( t ) } t ≥ such that (cid:82) t ˜ γ ( s ) ds < ∞ ,for all t > , P -a.s. Then, ˜ Q is an equivalent (local) martingale measure if and only if ˜ b ( t, T ) = − ν ( t, T )˜ γ ( t ) , for all ≤ t ≤ T. (3.45) with ˜ b ( · , T ) and ν ( · , T ) defined as in (3.41) . In this case, the dynamics of the impactedforward rate under the measure ˜ Q is given by ˜ f ( t, T ) = ˜ f (0 , T ) + (cid:90) t (cid:18) σ ( s, T ) (cid:90) Ts σ ( s, u ) du (cid:19) ds + (cid:90) t σ ( s, T ) dW ˜ Q ( s ) . (3.46) Moreover, the prices of impacted zero-coupon bonds are ˜ P ( t, T ) = ˜ P (0 , T ) + (cid:90) t ˜ P ( s, T )˜ r ( s ) ds + (cid:90) t ˜ P ( s, T ) ν ( s, T ) dW ˜ Q ( s ) . (3.47)The proof of Theorem 3.16 is given in Section 7.In our context such a measure ˜ Q would be clearly interpreted as an impacted risk-neutralmeasure, completely analogous to the measure defined in (3.16). In fact, the stochasticprocess ˜ γ in the HJM condition (3.45) is the counterpart in the HJM framework, of the20mpacted market price of risk ˜ λ defined in Section 3. Indeed, combining equations (3.3) and(3.40) we obtain for ≤ t ≤ T , ˜ λ ( t ) = ˜ P ( t, T ) (cid:16) r ( t ) − ˜ b ( t, T ) (cid:17) − r ( t ) ˜ P ( t, T )˜ P ( t, T ) ν ( t, T ) = − − ˜ b ( t, T ) ν ( t, T ) = ˜ γ ( t ) . The HJM framework adjusted with price impact discussed in this section is thereforeperfectly consistent with the price impact model for zero-coupon bonds introduced in Section3.1.We remark once again that, in the classical theory of interest rates, the meaning of theHJM condition lies in the fact that the drift of the forward rate is constrained under the riskneutral measure. Similarly, looking at the market price of risk, we notice that a constraint ispresent for the drift of the zero-coupon bond. In particular, its drift, under the risk neutralmeasure, has to be precisely the risk free interest rate. The interesting point we would liketo make here is that, once we incorporate price impact, the same kind of constraints holds,only under the newly defined impacted measure ˜ Q .We conclude this section by making two remarks. We first address the question of whenthe measure defined in Theorem 3.16 is an equivalent martingale measure, instead of justlocal martingale measure. The second remark concerns the Markov property of the impactedshort rate. In both cases, we see that the classical results carry over to the price impactframework, thanks to the key fact that the impact component affects only the drift of theforward rate. Remark 3.17 (Impacted risk neutral measure is an EMM) . Let ν ( t, T ) be defined as in (3.41) . From Corollary 6.2 of [15] it follows that the measure ˜ Q defined in Theorem 3.16 isan equivalent martingale measure if either E ˜ Q (cid:104) e (cid:82) T ν ( t,T ) dt (cid:105) < ∞ , for all T ≥ , or f ( t, T ) ≥ , for all ≤ t ≤ T. Remark 3.18 (Markov property of the short rate) . As pointed out in Chapter 5 of [11],one of the main drawbacks of HJM theory is that the implied short rate dynamics is usuallynot Markovian. Here we simply remark that, since the volatility of the forward rate σ ( t, T ) is not affected by price impact, if the Markov property of the short rate r ( t ) is ensured underthe measure Q when there is no trading, hence no price impact, then it is also preserved inthe presence of price impact under the ˜ Q . In this section we illustrate the argument outlined in Section 3.3 by computing the explicitprice of a Eurodollar-futures contract when the underlying short rate follows an Ornstein-21hlenbeck process [31]. The dynamics under the risk neutral measure Q is given by dr ( t ) = k ( θ − r ( t )) dt + σdW Q ( t ) , (4.1)with k, θ, σ positive parameters. The dynamics of the short rate under the real world measure P can be expressed as dr ( t ) = k ( θ − r ( t )) dt + σ ( dW P ( t ) − λ ( t ) dt ) , (4.2)where we highlight the classical market price of risk process λ defined in (3.3). Anotherrepresentation for r ( t ) under P is dr ( t ) = ˜ k (˜ θ − r ( t )) dt + σ ( dW P ( t ) − ˜ λ ( t ) dt ) , (4.3)where ˜ λ is the impacted market price of risk defined in (3.15) and ˜ k, ˜ θ are positive constants.Combining the two equivalent representations (4.2) and (4.3), we see that the following holdsfor any t ≥ kθ − kr ( t ) − σλ ( t ) = ˜ k ˜ θ − ˜ kr ( t ) − σ ˜ λ ( t ) . (4.4)Similarly to what is done in the standard theory (see Brigo and Mercurio [11], section 3.2.1),we assume the short rate r ( t ) has the same kind of dynamics under the measures P , Q and ˜ Q , that is λ ( t ) = λr ( t ) , ˜ λ ( t ) = ˜ λr ( t ) , (4.5)with λ, ˜ λ constants. The whole impact is then encapsulated in the constant ˜ λ . By plugging(4.5) into (4.4), we deduce ˜ k = k − σ (˜ λ − λ ) , ˜ θ = kθk − σ (˜ λ − λ ) . (4.6)Clearly, in order to ensure all parameters are positive, we must require k > σ (˜ λ − λ ) . In this way, the short rate r ( t ) is normally distributed under all three measures. Inparticular, plugging the Girsanov transformation from the measure P to the measure ˜ Q ,defined in (3.16), into equation (4.3), the short rate dynamics under ˜ Q can be convenientlyrewritten as dr ( t ) = ˜ k (˜ θ − r ( t )) dt + σdW ˜ Q ( t ) . (4.7)Since the short rate under ˜ Q is Gaussian, { (cid:82) Tt r ( s ) ds } t ≥ is also a Gaussian process. Atthe same time, we recall the well known fact that if X is a normal random variable withmean µ X and variance σ X , then E (exp( X )) = exp( µ X + σ X ) . Following the same argumentas in (Brigo and Mercurio [11], Chapters 3.2.1, 3.3.2 and Chapter 4), we can use (4.7) inorder to express the impacted zero-coupon bond price as follows ˜ P ( t, T ) = A ( t, T ) e − B ( t,T ) r ( t ) A ( t, T ) = exp (cid:26)(cid:18) ˜ θ − σ k (cid:19) [ B ( t, T ) − T + t ] − σ k B ( t, T ) (cid:27) ,B ( t, T ) = 1˜ k (cid:16) − e − ˜ k ( T − t ) (cid:17) . (4.8)Hence, the key expectation needed to compute the impacted Eurodollar future fair price inequation (3.24) is equal to E ˜ Q (cid:20) P ( t, T ) (cid:21) = 1 A ( t, T ) E ˜ Q (cid:104) e B ( t,T ) r ( t ) (cid:105) . (4.9)Since r ( t ) is normally distributed, B ( t, T ) r ( t ) will be normally distributed as well with meanand variance respectively equal to (see Brigo and Mercurio [11], Eq. (3.7)) E ˜ Q [ B ( t, T ) r ( t )] = B ( t, T ) (cid:104) r (0) e − ˜ kt + θ (1 − e − ˜ kt ) (cid:105) , Var ˜ Q [ B ( t, T ) r ( t )] = B ( t, T ) (cid:20) σ k (1 − e − kt ) (cid:21) . Therefore in order to get the impacted price of a Eurodollar-future contract we need tocompute the expectation in the right hand side of (4.9) which can be written explicitly as E ˜ Q (cid:20) P ( t, T ) (cid:21) = 1 A ( t, T ) ×× exp (cid:26) B ( t, T )[ r (0) e − ˜ kt + θ (1 − e − ˜ kt )] + 12 B ( t, T ) (cid:20) σ k (1 − e − kt ) (cid:21)(cid:27) . The main conclusion here is that defining the short rate under the impacted risk neutralmeasure preserves analytical tractability of interest rate derivatives precises.
In this section we compute the explicit price of a Eurodollar-futures contract when theunderlying short rate follows a Hull White model [24]. We start with the classical frameworkwhere there is not price impact. In this case the short rate is given by dr ( t ) = [ θ ( t ) − ar ( t )] dt + σdW Q ( t ) , where a and σ are positive constants and the function θ is chosen in order to fit exactlythe term structure of interest rates being currently observed in the market. Denoting by P M (0 , T ) the unimpacted market discount factor for the maturity T and defining the (un-impacted) market instantaneous forward rate at time for the maturity Tf M (0 , T ) := − ∂∂T ln P M (0 , T ) , the function θ is given by (see e.g. Brigo and Mercurio [11], Chapter 3, Eq. (3.34)) θ ( t ) = ∂f M (0 , t ) ∂T + af M (0 , t ) + σ a (cid:0) − e − at (cid:1) , ∂f M (0 ,t ) ∂T denotes the partial derivative of f M with respect to its second variable. Westart by computing the price under the classical risk neutral measure Q . According to eq.(3.36)–(3.37) in Chapter 3 of [11], the short rate is normally distributed with mean andvariance respectively equal to E Q [ r ( t ) | F s ] = r ( s ) e − a ( t − s ) + α ( t ) − α ( s ) e − a ( t − s ) Var Q [ r ( t ) | F s ] = σ a (cid:104) − e − a ( t − s ) (cid:105) , where α ( t ) := f M (0 , t ) + σ a (1 − e − at ) . As before, we notice that the integral of the short rate will be normally distributed as well,hence the price of a zero-coupon bond under the classical risk neutral measure is given by(see eq. (3.39) in Chapter 3 of [11]), P ( t, T ) = A ( t, T ) e − B ( t,T ) r ( t ) , where A ( t, T ) = P M (0 , T ) P M (0 , t ) exp (cid:26) B ( t, T ) f M (0 , t ) − σ a (1 − e − at ) B ( t, T ) (cid:27) ,B ( t, T ) = 1 a (cid:104) − e − a ( T − t ) (cid:105) . Moreover, the term B ( t, T ) r ( t ) is still normally distributed and we immediately have E Q [ B ( t, T ) r ( t ) | F s ] = B ( t, T ) (cid:16) r ( s ) e − a ( t − s ) + α ( t ) − α ( s ) e − a ( t − s ) (cid:17) , Var Q [ B ( t, T ) r ( t ) | F s ] = B ( t, T ) σ a (cid:104) − e − a ( t − s ) (cid:105) . This implies that the expectation we are interested in, under the classical risk neutral meas-ure Q , can be written explicitly as (see Section 13.12.1 in [11]) E Q (cid:20) P ( t, T ) (cid:21) = 1 A ( t, T ) exp (cid:26) B ( t, T ) E Q [ r ( t )] + 12 B ( t, T ) Var Q [ r ( t )] (cid:27) . Next we derive the corresponding expression under the impacted risk neutral measure ˜ Q in(3.16). We assume as in Section 4.1 that the market price of risk and impacted market priceof risk are given by λ ( t ) = λr ( t ) , ˜ λ ( t ) = ˜ λr ( t ) , for some constants λ, ˜ λ . Using the Girsanov change of measure from Q to ˜ Q defined inSection 3.2, it follows that the short rate under ˜ Q is given by dr ( t ) = [ θ ( t ) − ar ( t )] dt + σdW Q ( t )= [ θ ( t ) − ar ( t )] dt + σdW ˜ Q ( t ) + σ (˜ λ − λ ) r ( t ) dt = (cid:104) θ ( t ) − ( a − σ (˜ λ − λ )) r ( t ) (cid:105) dt + σdW ˜ Q ( t ) . ˜ a := a − σ (˜ λ − λ ) . The pricing formula for E ˜ Q (cid:104) P ( t,T ) (cid:105) is then derived by following the same steps as in theclassical case. Similarly to the Vasicek model, analytical tractability is preserved. In this section we give a few numerical examples for the behaviour of the yield curve underprice impact in the framework of short-rate affine models, which was described in Section3.4. In order to compute the cross price impact, we need the drift and the volatility of thezero-coupon bond. For the sake of simplicity, we assume the short rate is described by aVasicek model (4.1) dr ( t ) = k ( θ − r ( t )) dt + σdW Q ( t ) , with k, θ, σ positive parameters. Then, the drift and the diffusion coefficients of the unim-pacted zero-coupon bond are given by (3.27): µ T ( t, r ( t )) = e − B ( t,T ) r ( t ) (cid:20) ∂A∂t − A ( t, T ) ∂B∂t r ( t ) + 12 A ( t, T ) B ( t, T ) σ − A ( t, T ) B ( t, T ) k ( θ − r ( t )) (cid:21) ,σ T ( t, r ( t )) = − σB ( t, T ) A ( t, T ) e − B ( t,T ) r ( t ) , where the functions A, B are given as in (4.8) and their derivatives are given by ∂B∂t = − e − k ( T − t ) , ∂A∂t = A ( t, T ) (cid:20)(cid:18) θ − σ k (cid:19) (cid:18) ∂B∂t + 1 (cid:19) − σ k B ( t, T ) ∂B∂t (cid:21) . We can then plug all these quantities in equation (3.29) to determine the dynamics of thecross-impacted zero-coupon bond and therefore the corresponding impacted yield. We setthe following values for the parameters in (4.1): k = 0 . , θ = 0 . , σ = 0 . , r = 0 . . We consider zero-coupon bonds with maturities T := { , , , , } years and assume thatan agent is trading on the bond with maturity T = 5 years. All the other zero-coupon bondsexperience cross price impact during the trading period. We fix the execution time horizonto be τ = 10 days. All bonds are simulated over the time interval [0 , months ] , discretizedin N = 365 subintervals with time step ∆ t = 1 / . The short rate r defined in (4.1) issimulated via Euler-Maruyama scheme. Since we are going to describe the average behaviourof the yield curve under market impact, we also set the number of Monte Carlo simulationsto M = 10 . . As we shall see below in the detailed algorithm, for each realization of theshort rate, we will have a corresponding impacted yield curve. The idea is then to plot theaverage of such curves. 25or the sake of simplicity, we discuss the benchmark trading strategy v T ( s ) := (cid:40) c, if s ≤ τ , otherwise (5.1)with c some positive constant if we buy, negative if we sell. In our simulations we choose c = 2 . The transient impact defined in (3.8) reads as Υ vT ( t ) = ye − ρt + γe − ρt (cid:90) t e ρs c s ≤ τ ds, (5.2)where the parameters are set to ρ = 2 , γ = 1 , y = 0 . . The functions l, K introduced in (3.7) are assumed to be of the form l ( t, T ) = κ (cid:18) − tT (cid:19) α , K ( t, T ) = (cid:18) − tT (cid:19) β with κ ≥ , α, β ≥ . In particular, we choose α = 1 , β = 1 , κ = 0 . . Following (3.12), the price of the impacted bond in T = 5 y is ˜ P ( t, T ) = P ( t, T ) + (cid:90) t J T ( s ) ds, where J T , which was defined in (3.13), is specified to be J T ( t ) = − κT v T ( t ) + (cid:18) − tT (cid:19) [ − ρ Υ vT + v T ( t )] − Υ vT ( t ) The algorithm we implemented to simulate the impacted yield curve consists of the followingsteps.
Step
1: Simulate a path of the short rate r ( t ) given in (4.1) for t ∈ [0 , months ] . Step
2: Compute the unimpacted zero-coupon bond price P ( t, T ) for the trading maturity T = 5 years using equation (3.25) for t ∈ [0 , months ] . Step
3: Compute the unimpacted yield Y ( t, T ) by plugging P ( t, T ) in (3.26) for t ∈ [0 , months ] . Step
4: Compute the (directly) impacted zero-coupon bond ˜ P ( t, T ) using (3.12) for t ∈ [0 , months ] . Step
5: Compute the (directly) impacted yield ˜ Y ( t, T ) by plugging ˜ P ( t, T ) into (3.30) for t ∈ [0 , months ] . Step
6: For all other maturities S = 1 , , , years, compute the cross impacted zero-couponbond price ˜ P ( t, S ) using equation (3.29) for t ∈ [0 , months ] .26 tep
7: Compute the cross impacted yield ˜ Y ( t, S ) by plugging ˜ P ( t, S ) into (3.30) for t ∈ [0 , months ] . Step
8: Repeat these steps M = 10 . times and compare the average of Y ( t, T ) with theaverage of ˜ Y ( t, T ) .In Figure 1 we visualize for all maturities the average classical yield E [ Y ( t, T )] versusthe average impacted yield E [ ˜ Y ( t, T )] at times t = 5 days (middle of trading), t = 11 days(right after trading is ended) and t = 270 days (after months).Figure 1: Trading zero-coupon bond with maturity T = 5 years. Average unimpacted yieldcurve and average impacted yield curve in the middle of trading (top left panel), right aftertrading is concluded (top right panel) and after nine months (bottom panel).In the top panels we see that the yield has decreased over all maturities as result oftrading. This is consistent with the fact that a buy strategy of bonds pushes their prices updue to price impact, hence the yield decreases. Clearly, the almost parallel shift of the yieldcurve is a just a consequence of the very simple (constant) trading strategy we defined inequation (5.1). We expect to observe much more complicated behaviours when implementingmore sophisticated strategies. In the bottom panel, instead, we observe that, roughly ninemonths after performing the trades, the two yield curves pretty much coincide. This isdue to the transient component in the price impact model, which induces impacted yield27urve to converge to its classical counterpart as time goes by. When analysing price impactdue to zero-coupon bond trading, one aspect that certainly can’t be ignored is the specialnature of the assets we are trading. Unlike what happens with stocks, the time evolution ofzero-coupon bonds is constrained, specifically by the fact that they must reach value atmaturity. It therefore appears that two fundamental forces are in play: the intrinsic pull topar effect, which makes both the impacted and unimpacted bond price go to , hence thecorresponding yields to , and the price impact effect, which induces the bond price to firstincrease (if we buy) or decrease (if we sell), and then revert back to its unimpacted value.Interestingly, when trading stocks, it will take the impacted asset forever to converge to theunimpacted counterpart, as the transient impact converges to as time t goes to infinity.When trading bonds, though, this convergence occurs in finite time. In order to betterunderstand the role played by price impact, in Figure 2 we compare directly the behaviourof the impacted bond ˜ P ( t, T ) and of the classical bond P ( t, T ) for different maturities T .Figure 2: Trading bond with maturity T = 5 years. Averaged impacted zero-coupon bondvs averaged classic zero-coupon bond for maturities years (top left panel), year (top rightpanel), years (middle left panel), years (middle right panel), (bottom panel). Allcurves are observed over the interval [0 , year ] .We observe that, over one year, the pull-to-par effect is somehow stronger than thetransient impact effect in bonds with short maturity ( T = 1 , . By this, we mean that28he unimpacted and impacted bonds meet at, or very close to, maturity. For bonds withlong maturity ( T = 5 , , ), instead, the transient effect is prominent. This causes theimpacted bond curve and the unimpacted bond curve to cross each other significantly beforetheir maturity. In fact, we can numerically compute the first instant the two curves meetand we observe that the longer the maturity, the sooner this happens. This is illustrated inFigure 3.Figure 3: Trading bond with maturity T = 5 years. First instant (in days) impacted bondcurve and unimpacted bond curve cross for maturities , , years. All curves are observedover the interval [0 , year ] .The interplay between the cross price impact effect, averaged over . realizations,and the pull to par effect is demonstrated in Figure 4 for the price of a zero-coupon bondwith maturity S = 1 year when trading a bond with maturity T = 5 years. Trading takesplace on the first 10 days of the year, while the time scale in the graph is of one year. Weillustrate this effect for various values of the transient impact parameter ρ in equation (5.2). It can be observed that the price of the cross-impacted zero-coupon bond with maturity S = 1 year isnot at expiration, as it should be, but slightly higher (top left panel). This is not a numerical error, butrather a consequence of our model not being able to ensure the cross-impacted bonds reach value precisely at their respective maturities. See Remark 3.12. . realizations isdemonstrated for the price of a bond with maturity S = 1 year when trading a bond withmaturity T = 5 years, for various values of ρ . Trading takes place on the first 10 days of theyear, while the time scale in the graph is of one year.It can be observed that the higher ρ , the more aggressively the price is "pulled down"close to the original price before the trades. At the beginning, far from maturity, thetransient impact component dominates and the price decreases. After some time, though,the bond intrinsic nature takes over and the price starts to increase.Another phenomenon which is revealed in our framework is the interplay between themean reversion of the short rate model and the price impact. Recall that in Section 4.1 wefound that the mean reversion speed k under the measure Q and the mean reversion ˜ k underthe price-impacted measure ˜ Q are linked by (4.4) as follows ˜ k = k − σ (˜ λ − λ ) , with ˜ λ, λ representing the impacted market price of risk and the classical market price ofrisk respectively. We stress that the higher k , the faster the short rate r under Q andits counterpart under ˜ Q converge to their respective stationary distributions. At the sametime, since the variance of the stationary distribution is σ / (2 k ) , large values of k reducethe overall variance of the model, thereby making the two types of rates that we consider30loser to each other. This, in turn, implies that after a long time ( T = 10 , years) thezero-coupon bond P and impacted zero-coupon bond ˜ P , hence their yields, will be closer toeach other. Conversely, if k is small, the two short rates are quite far from each other andthe overall variance of the model is large. Furthermore, looking again at (4.4), we noticethat the larger k , the less significant the impact component − σ (˜ λ − λ ) , and vice versa. Ina way, the speed of mean reversion works in an opposite direction to the price impact. Wedemonstrate this in Figure 5 for k = 0 . (top panel) and for k = 0 . (bottom panel).As above, we trade the zero-coupon bond with maturity T = 5 years, trading occurs forthe first days and the yields are observed after months. The difference in behaviouris evident for long maturities ( T = 5 , , . While in the bottom panel unimpacted yieldand impacted yield are really close to each other (as in Figure (1), right panel), in the toppanel the distance between the two yields is rather significant.Figure 5: Impacted and unimpacted yield curves for k = 0 . (top panel) and for k = 0 . (bottom panel) when trading zero-coupon bond with maturity T = 5 years. Trading occursduring the first days. Yield curves are observed after nine months.31 Optimal execution of bonds in presence of price impact
In this section we consider a problem of an agent who tries to liquidate a large inventoryof T -bonds within a finite time horizon [0 , τ ] where τ < T . We assume that the agent’stransactions create both temporary and transient price impact and that the performance ofthe agent is measured by a revenue-cost functional that captures the transaction costs whichresult by price impact, and the risk of holding inventory for long time periods. Our optimalexecution framework is closely related to the framework which was proposed for executionof equities in Section 3.1 of [29]. The main difference between the two frameworks is thatin our framework the price impact has to vanish at the bond’s maturity in order to satisfythe boundary condition ˜ P ( T, T ) = 1 .Let
T > denote the bond’s maturity. We assume that the unimpacted bond price P ( · , T ) is given by (3.2) and we consider the canonical decomposition P ( · , T ) = A ( · , T ) +¯ M ( · , T ) , where A ( t, T ) := (cid:90) t µ T ( s, r ( s )) ds, ≤ t ≤ T, is a predictable finite-variation process and ¯ M ( t, T ) := (cid:90) t σ T ( s, r ( s )) dW P ( s ) , ≤ t ≤ T, local martingale. We assume that the coefficients σ T , µ T in (3.2) are such that we have E [ (cid:104) ¯ M ( · , T ) (cid:105) τ ] + E (cid:34)(cid:18)(cid:90) τ | dA ( · , T ) | (cid:19) (cid:35) < ∞ . (6.1)In this case we say that a bond price { P ( t, T ) } t ∈ [0 ,T ] is in H .The initial position of the agent’s inventory is denoted by x > and the number ofshares the agent holds at time t ∈ [0 , τ ] is given by X v T ( t ) (cid:44) x − (cid:90) t v T ( s ) ds (6.2)where { v T ( t ) } t ∈ [0 ,τ ] denotes the trading speed. We say that the trading speed is admissibleif it belongs to the following class of admissible strategies A (cid:44) (cid:26) v T : v T progressively measurable s.t. E (cid:20)(cid:90) τ v T ( s ) ds (cid:21) < ∞ (cid:27) . (6.3)We assume that the trader’s trading activity causes price impact on the bond’s price asdescribed by { ˜ P ( t, T ) } t ∈ [0 ,T ] in (3.7).As in Section 2 of [29], we now suppose that the trader’s optimal trading objective is tounwind her initial position x > in the presence of temporary and transient price impact32hrough maximizing the following performance functional J ( v ) := E (cid:20) (cid:90) τ (cid:18) P ( t, T ) − K ( t, T )Υ vT ( t ) (cid:19) v T ( t ) dt − (cid:90) τ l ( t, T ) v T ( t ) dt + X vT ( τ ) P ( τ, T ) − φ (cid:90) τ ( X vT ( t )) dt − (cid:37) ( X vT ( τ )) (cid:19)(cid:21) . (6.4)The first, second and third terms in J represent the trader’s terminal wealth, meaning thefinal cash position including the accrued trading costs induced by temporary and transientprice impact, as well as the remaining final risky asset position’s book value. The fourthand fifth terms, instead, account for the penalties φ, (cid:37) > on the trader’s running penalty(i.e. the risk aversion term) and the penalty of holding any terminal inventory, respectively.Since T is fixed, for the sake of readability we will omit the subscripts T for the rest ofthis section. The main result of this section is the derivation of the unique optimal admissiblestrategy, namely J ( v ) → max v ∈ A . (6.5)and exhibiting an explicit expression for the optimal trading strategy. We define A ( t ) := − − ρ γ − φ Λ( t ) ρK ( t, T )Λ( t ) − Λ (cid:48) ( t ) K ( t, T ) − Λ( t ) ∂ t K ( t, T ) 0 Λ (cid:48) ( t ) + ρ Λ( t )0 0 K ( t, T ) γ ρ , (6.6)where Λ( t ) := 12 l ( t, T ) . (6.7)Note that Λ( t ) is well defined for ≤ t ≤ τ since l ( t, T ) > on this interval by (3.9). Let Φ be the fundamental solution of the matrix-valued ordinary differential equation ddt Φ( t ) = A ( t )Φ( t ) , Φ(0) = Id . (6.8)Let us define the matrix Ψ( t, τ ) := Φ − ( τ )Φ( t ) . (6.9)33e also define the vector G : G ( t, τ ) := (cid:37)l ( τ, T ) Ψ ( t, τ ) − K ( τ, T )2 l ( τ, T ) Ψ ( t, τ ) − Ψ ( t, τ ) ,G ( t, τ ) := (cid:37)l ( τ, T ) Ψ ( t, τ ) − K ( τ, T )2 l ( τ, T ) Ψ ( t, τ ) − Ψ ( t, τ ) ,G ( t, τ ) := (cid:37)l ( τ, T ) Ψ ( t, τ ) − K ( τ, T )2 l ( τ, T ) Ψ ( t, τ ) − Ψ ( t, τ ) ,G ( t, τ ) := (cid:37)l ( τ, T ) Ψ ( t, τ ) − K ( τ, T )2 l ( τ, T ) Ψ ( t, τ ) − Ψ ( t, τ ) . (6.10)Next, we define the process Γ ˆ v ( t ) := Λ (cid:48) ( t )Λ( t ) (cid:18) P ( t, T ) + ˜ M ( t ) − φ (cid:90) t X ˆ v ( u ) du (cid:19) , (6.11)where ˜ M is the square integrable martingale ˜ M ( s ) := E s (cid:20) φ (cid:90) τ X ˆ v ( u ) du + 2 (cid:37)X ˆ v ( τ ) − P ( τ, T ) (cid:21) , (6.12)and E t denotes the expectation conditioned on the filtration F t for all t ∈ [0 , τ ] . Finally wedefine the following functions on ≤ t ≤ τ , v ( t, τ ) := (cid:18) − G ( t, τ )Ψ ( t, τ ) G ( t, τ )Ψ ( t, τ ) (cid:19) − ,v ( t, τ ) := (cid:18) G ( t, τ )Ψ ( t, τ ) G ( t, τ )Ψ ( t, τ ) − G ( t, τ ) G ( t, τ ) (cid:19) ,v ( t, τ ) := (cid:18) G ( t, τ )Ψ ( t, τ ) G ( t, τ )Ψ ( t, τ ) − G ( t, τ ) G ( t, τ ) (cid:19) ,v ( t, τ ) := G ( t, τ ) G ( t, τ ) . (6.13)In order for the optimal strategy to be well defined, we will need additional assumptions.Note that if l, K are positive constants these assumptions translate to Assumption 3.1 andLemma 5.5 in [29]. Assumption 6.1.
We assume that the following hold:(A.1) inf ≤ t ≤ τ | G ( t, τ )Ψ ( t, τ ) − G ( t, τ )Ψ ( t, τ ) | > , (A.2) sup ≤ t ≤ τ | Ψ j ( t, τ ) | < ∞ , sup ≤ t ≤ τ | G j ( t, τ ) | < ∞ , j ∈ { , , , } (A.3) inf ≤ t ≤ τ | Ψ ( t, τ ) | > , inf ≤ t ≤ τ | G ( t, τ ) | > . emark 6.2. At this point we stress the fact that the conditions in Assumption 6.1 arenot very general, however the purpose of this section is to show how to incorporate optimalexecution into the impacted bonds framework. Future work may improve the theoreticalresults on this topic.
Next we present the main result of this section, which derives the unique optimal tradingspeed.
Theorem 6.3 (Optimal trading strategy) . Under Assumption 6.1, there exists a uniqueoptimal strategy ˆ v ∈ A which maximises (6.5) and it is given by the following feedback form v ( t ) = v ( t, τ ) (cid:32) v ( t, τ ) X v ( t ) + v ( t, τ )Υ v ( t )+ v ( t, τ ) E t (cid:20)(cid:90) τt Λ( s )Ψ ( s, τ )Ψ ( t, τ ) ( µ ( s ) + Γ ˆ v ( s )) ds (cid:21) − E t (cid:20)(cid:90) τt Λ( s ) G ( s, τ ) G ( t, τ ) ( µ ( s ) + Γ ˆ v ( s )) ds (cid:21) (cid:33) , (6.14) for all t ∈ (0 , τ ) . The proof Theorem 6.3 is given in Section 8.
Proof of Theorem 3.4.
We adapt the argument by Bjork in Section 3.2 of [7] to our case.We fix two maturities T and S , and we consider a portfolio V consisting of S -bonds and T -bonds. We further assume that both bonds are traded with admissible trading speeds v T and v S which correspond by (3.13) to impact densities J T and J S .From (3.2) and (3.12) we can write the dynamics of the impacted bonds as follows: d ˜ P ( t, T ) = µ T ( t, r ( t )) dt + J T ( t ) dt + σ T ( t, r ( t )) dW P ( t ) ,d ˜ P ( t, S ) = µ S ( t, r ( t )) dt + J S ( t ) dt + σ S ( t, r ( t )) dW P ( t ) . (7.1)Let ˜ h T , ˜ h S by locally bounded predictable processes representing the weights of the T and S bonds, respectively. We denote by ˜ V ( t ) the portfolio value process, i.e. ˜ V ( t ) ≡ ˜ V ( t ; ˜ h ) := ˜ h T ( t ) ˜ P ( t, T ) + ˜ h S ( t ) ˜ P ( t, S ) . Since, by assumption, the impacted-portfolio is self-financing, it holds at any time t (seeDefinition 3.2) d ˜ V ( t ; ˜ h ) = ˜ h T ( t ) d ˜ P ( t, T ) + ˜ h S ( t ) d ˜ P ( t, S ) . It is convenient to define the relative (impacted) weights α T i ( t ) := ˜ h T i ( t ) ˜ P ( t, T i )˜ V ( t ; ˜ h ) , T i ∈ { T, S } ,
35e conclude that if the impacted portfolio is self financing, then d ˜ V ( t )˜ V ( t ) = α T ( t ) d ˜ P ( t, T )˜ P ( t, T ) + α S ( t ) d ˜ P ( t, S )˜ P ( t, S ) . (7.2)In order to ease the notation, we suppress the dependence on r ( t ) in the drift and volatility.Substituting the dynamics (7.1) into (7.2), we have d ˜ V ( t )˜ V ( t ) = α T ( t )˜ P ( t, T ) ( µ T ( t ) − J T ( t )) dt + α S ( t )˜ P ( t, S ) ( µ S ( t ) − J S ( t )) dt ++ (cid:18) α S ( t ) σ S ( t )˜ P ( t, S ) + α T ( t ) σ T ( t )˜ P ( t, T ) (cid:19) dW P ( t ) . (7.3)At this point, we choose the relative weights so that the diffusive part of the equation abovevanishes, that is, α T ( t ) + α S ( t ) = 1 ,α T ( t ) σ T ( t )˜ P ( t, T ) + α S ( t ) σ S ( t )˜ P ( t, S ) = 0 . (7.4)Solving this system gives α S ( t ) = σ T ( t ) / ˜ P ( t, T ) σ T ( t ) / ˜ P ( t, T ) − σ S ( t ) / ˜ P ( t, S ) ,α T ( t ) = − σ S ( t ) / ˜ P ( t, S ) σ T ( t ) / ˜ P ( t, T ) − σ S ( t ) / ˜ P ( t, S ) . (7.5)Notice that the above expressions are well defined. Indeed, if the denominator was approach-ing zero, then the sum of the two weights would be zero and this would contradict (7.4).Next, we substitute (7.5) into (7.3). Following again Bjork’s argument, we use the fact thatour impacted portfolio is locally risk-free (as in Definition 3.3) by assumption and deducethe following relationship must hold: µ T ( t ) − J T ( t )˜ P ( t, T ) (cid:32) − σ S ( t ) / ˜ P ( t, S ) σ T ( t ) / ˜ P ( t, T ) − σ S ( t ) / ˜ P ( t, S ) (cid:33) ++ µ S ( t ) − J S ( t )˜ P ( t, S ) (cid:32) σ T ( t ) / ˜ P ( t, T ) σ T ( t ) / ˜ P ( t, T ) − σ S ( t ) / ˜ P ( t, S ) (cid:33) = r ( t ) . Multiplying both sides by the term σ T ( t )˜ P ( t, T ) − σ S ( t )˜ P ( t, S ) , we obtain (cid:18) µ S ( t ) − J S ( t )˜ P ( t, S ) − r ( t ) (cid:19) (cid:18) σ T ( t )˜ P ( t, T ) (cid:19) = (cid:18) µ T ( t ) − J T ( t )˜ P ( t, T ) − r ( t ) (cid:19) (cid:18) σ S ( t )˜ P ( t, S ) (cid:19) .
36t follows that, (cid:18) µ S ( t ) − J S ( t )˜ P ( t, S ) − r ( t ) (cid:19) (cid:32) ˜ P ( t, S ) σ S ( t ) (cid:33) = (cid:18) µ T ( t ) − J T ( t )˜ P ( t, T ) − r ( t ) (cid:19) (cid:32) ˜ P ( t, T ) σ T ( t ) (cid:33) , and rearranging we deduce µ S ( t ) − J S ( t ) − r ( t ) ˜ P ( t, S ) σ S ( t ) = µ T ( t ) − J T ( t ) − r ( t ) ˜ P ( t, T ) σ T ( t ) . (7.6)Notice that the left hand side of (7.6) depends on S but not on T , while the right hand sideof (7.6) depends on T , but not on S . Since S and T are arbitrary, we conclude that bothsides of (7.6) depend only on t and r ( t ) . Proof of Theorem 3.9.
The proof is similar to the proof of Proposition 1.1 in Chapter 1.2of [7] (see also Harrison and Kreps [20] Theorem 2 and relative Corollary in Section 3 andHarrison and Pliska [21], Theorem 2.7, Section 2). For the sake of completeness, we givethe proof here, translated in our price impact environment. Let
T < + ∞ be some finitematurity. Let ˜ h be an arbitrage portfolio and ˜ V the corresponding value process. Then, giventhe positivity of the discount factor (bank account) defined in (3.19) and the equivalencebetween the real world measure P and the impacted risk neutral measure ˜ Q , we immediatelydeduce ˜ Q (cid:32) ˜ V ( T ) B ( T ) ≥ (cid:33) = 1 , ˜ Q (cid:32) ˜ V ( T ) B ( T ) > (cid:33) > . (7.7)Moreover we have V (0) = ˜ V (0) B (0) = E ˜ Q (cid:34) ˜ V ( T ) B ( T ) (cid:35) > , where the first equality comes from the definition of arbitrage, the second from the factthat B (0) = 1 and the third from the fact that ˜ V ( t ) /B ( t ) is a martingale under ˜ Q . Finally,the positivity of the expectation is a consequence of (7.7). We get a contradiction so weconclude that absence of arbitrage must hold. Proof of Proposition 3.14.
We start by writing the impacted forward rate defined in (3.37)as ˜ f ( t, T ) = f ( t, T ) + (cid:90) t J f ( s, T ) ds, where f represents the unimpacted forward rate (see e.g. Chapter 6, of [15]) and we usedthe assumption ˜ f (0 , t ) = f (0 , t ) . Then, using (3.39), we deduce ˜ P ( t, T ) = exp (cid:26) − (cid:90) Tt ˜ f ( t, u ) du (cid:27) = exp (cid:26) − (cid:90) Tt f ( t, u ) du − (cid:90) Tt J f ( s, u ) du (cid:27) = P ( t, T ) exp (cid:26) − (cid:90) Tt J f ( s, u ) du (cid:27) , (7.8)37here P denotes the unimpacted zero-coupon bond and we used the well known relationbetween P ( t, T ) and f ( t, T ) . From (3.7) and (3.11) we have ˜ P ( t, T ) = P ( t, T ) − I T ( t ) . (7.9)Substituting this last expression into (7.8) and rearranging, we obtain exp (cid:26) − (cid:90) Tt J f ( s, u ) du (cid:27) = ˜ P ( t, T )˜ P ( t, T ) + I T ( t ) . By taking logarithms on both sides yields and using (7.9) we get (cid:90) Tt J f ( s, u ) du = − log (cid:32) ˜ P ( t, T )˜ P ( t, T ) + I T ( t ) (cid:33) = − log (cid:18) − I T ( t ) P ( t, T ) (cid:19) . Differentiating with respect to maturity, we get (3.43).
Proof of Theorem 3.16 .
Let B ( t ) be the bank account defined in (3.19) and let the impactedzero-coupon bond ˜ P follow the dynamics (3.40). By applying Ito’s formula to the discountedimpacted zero-coupon bond price, we immediately find d ˜ P ( t, T ) B ( t ) = ˜ P ( t, T ) B ( t ) ˜ b ( t, T ) dt + ˜ P ( t, T ) B ( t ) ν ( t, T ) dW P ( t ) , with ˜ b and ν defined as in (3.41). Changing measure form the real world P to the impactedrisk neutral ˜ Q as in (3.44) implies d ˜ P ( t, T ) B ( t ) = ˜ P ( t, T ) B ( t ) (cid:16) ˜ b ( s, T ) + ν ( t, T )˜ γ ( t ) (cid:17) dt + ˜ P ( t, T ) B ( t ) ν ( t, T ) dW ˜ Q ( t ) . Therefore, we clearly see that ˜ P ( t, T ) B ( t ) local martingale under ˜ Q ⇐⇒ ˜ b ( s, T ) = − ν ( t, T )˜ γ ( t ) . This is our new HJM condition. Notice also that since both functions ν and ˜ b are continuouswith respect to T , this condition is equivalent to saying that the impacted measure ˜ Q is anequivalent local martingale measure. Following Theorem 6.1 in [15], Chapter 6, we can plugin the explicit expression for ˜ b in (3.41) and write the HJM condition (3.45) as − (cid:90) Ts α ( s, u ) du − (cid:90) Ts J f ( s, u ) du + 12 ν ( s, T ) = − ν ( t, T )˜ γ ( t ) . (7.10)Differentiating both sides with respect to the maturity T yields the equation − α ( t, T ) + σ ( t, T ) (cid:90) Tt σ ( t, u ) du − J f ( t, T ) = σ ( t, T )˜ γ ( t ) , α ( t, T ) + J f ( t, T ) = σ ( t, T ) (cid:90) Tt σ ( t, u ) du − σ ( t, T )˜ γ ( t ) . (7.11)Substituting (7.11) in the dynamics of the forward rate (3.37) and using Girsanov yields(3.46). Using (3.45) along with (3.40) and Girsanov gives (3.47). The uniqueness of the optimal strategy follows by a standard convexity argument for theperformance functional (6.4). Hence we only need to derive the optimal strategy.We start by deriving a system of coupled forward-backward stochastic differential equa-tions (FBSDEs) which is satisfied by the solution to the stochastic control problem.
Lemma 8.1 (FBSDE system) . A control ˆ v ∈ A solves the optimization problem (6.5) if and only if the processes ( X ˆ v , Υ ˆ v , ˆ v, Z ˆ v ) satisfy the coupled forward-backward stochasticdifferential equations dX ˆ v ( t ) = − ˆ v ( t ) dt, X ˆ v (0) = x,d Υ ˆ v ( t ) = − ρ Υ ˆ v ( t ) dt + γ ˆ v ( t ) dt, Υ ˆ v (0) = y,d ˆ v ( t ) = Λ( t ) dP ( t, T ) − t ) φX ˆ v ( t ) dt +Υ ˆ v ( t ) [ − Λ (cid:48) ( t ) K ( t, T ) − Λ( t ) ∂ t K ( t, T ) + ρK ( t, T )Λ( t )] dt + Z ˆ v ( t ) [Λ (cid:48) ( t ) + ρ Λ( t )] dt + Λ( t )Γ ˆ v ( t ) dt + dM ( t ) , ˆ v ( τ ) = (cid:37)l ( τ,T ) X ˆ v ( τ ) − K ( τ,T )2 l ( τ,T ) Υ ˆ v ( τ ) ,dZ ˆ v ( t ) = (cid:0) ρZ ˆ v ( t ) + K ( t, T ) γ ˆ v ( t ) (cid:1) dt + dN ( t ) , Z ˆ v ( τ ) = 0 , (8.1) for two suitable square integrable martingales M = ( M ( · , T )) ≤ t ≤ τ and N = ( N ( · , T )) ≤ t ≤ τ ,where the Λ , Γ ˆ v and ˜ M are defined in (6.7) , (6.11) and (6.12) respectively.Proof. The proof follows the same lines as Lemmas 5.1 and 5.2 in [29]. Since for all v ∈ A the map v → J ( v ) is strictly concave, we can study the unique critical point at which theGateaux derivative of J , which is defined as (cid:104) J (cid:48) ( v ) , α (cid:105) := lim (cid:15) → J ( v + (cid:15)α ) − J ( v ) (cid:15) , is equal to for any α ∈ A . This derivative can be computed analytically as follows. Let (cid:15) > and v, α ∈ A . Since for all t ∈ [0 , τ ] , X v + (cid:15)α ( t ) = x − (cid:90) t ( v ( s ) + (cid:15)α ( s )) ds = X v ( t ) − (cid:15) (cid:90) t α ( s ) ds Υ v + (cid:15)α ( t ) = Υ v ( t ) + (cid:15)γ (cid:90) t e − ρ ( t − s ) α ( s ) ds, (8.2)39rom (6.4) and (8.2) we have J ( v + (cid:15)α ) == E (cid:34) (cid:90) τ (cid:18) P ( t, T ) − K ( t, T )Υ v ( t ) − K ( t, T ) (cid:15)γ (cid:90) t e − ρ ( t − s ) α ( s ) ds (cid:19) ( v ( t ) + (cid:15)α ( t )) dt − (cid:90) τ l ( t, T ) v ( t ) + (cid:15) l ( t, T ) α t + 2 l ( t, T ) v ( t ) (cid:15)α ( t ) dt + X v ( τ ) P ( τ, T ) − (cid:15)P ( τ, T ) (cid:90) τ α ( s ) ds − φ (cid:90) τ ( X v ( t )) + (cid:15) (cid:18)(cid:90) t α ( s ) ds (cid:19) − X v ( t ) (cid:15) (cid:90) t α ( s ) dsdt − (cid:37) (cid:32) ( X v ( τ )) + (cid:15) (cid:18)(cid:90) τ α ( s ) ds (cid:19) − X v ( τ ) (cid:15) (cid:90) τ α ( s ) ds (cid:33) (cid:35) . It follows that J ( v + (cid:15)α ) − J ( v ) = (cid:15) E (cid:34) (cid:90) τ ( P ( τ, T ) − K ( t, T )Υ v ( t )) α ( t ) dt − (cid:90) τ K ( t, T ) v ( t ) (cid:90) t γe − ρ ( t − s ) α ( s ) dsdt − (cid:90) τ l ( t, T ) v ( t ) α ( t ) dt + 2 φ (cid:90) τ X v ( t ) (cid:90) t α ( s ) dsdt + 2 (cid:37)X v ( τ ) (cid:90) τ α ( s ) ds − P ( τ, T ) (cid:90) τ α ( s ) ds (cid:35) + (cid:15) E (cid:34) γ (cid:90) τ K ( t, T ) α ( t ) (cid:90) t e − ρ ( t − s ) α ( s ) dsdt − (cid:90) τ l ( t, T ) α ( t ) dt − φ (cid:90) τ (cid:18)(cid:90) t α ( s ) ds (cid:19) dt − (cid:37) (cid:18)(cid:90) τ α ( s ) ds (cid:19) (cid:35) . Note that all the terms above are finite since (cid:96) and K are bounded functions and since α, v ∈ A . Applying Fubini’s theorem twice, we obtain (cid:104) J (cid:48) ( v ) , α (cid:105) = E (cid:34) (cid:90) τ α ( s ) (cid:32) P ( s, T ) − K ( s, T )Υ v ( s ) − (cid:90) τs K ( t, T ) e − ρ ( t − s ) γv ( t ) dt + − l ( s, T ) v ( s ) + 2 φ (cid:90) τs X v ( t ) dt + 2 (cid:37)X v ( τ ) − P ( τ, T ) (cid:33) ds (cid:35) , for any α ∈ A . We get the following condition on the optimal strategy E (cid:34) (cid:90) τ α ( s ) (cid:32) P ( s, T ) − K ( s, T )Υ v ( s ) − (cid:90) τs K ( t, T ) e − ρ ( t − s ) γv ( t ) dt − l ( s, T ) v ( s ) + 2 φ (cid:90) τs X v ( t ) dt + 2 (cid:37)X v ( τ ) − P ( τ, T ) (cid:33) ds (cid:35) = 0 . (8.3)Next we show that given the optimal strategy ˆ v ∈ A , the vector ( X ˆ v , Υ ˆ v ) satisfies the firstorder condition (8.3) if and only if the vector ( X ˆ v , Υ ˆ v , ˆ v, Z ˆ v ) solves a FBSDE system, forsome auxiliary process Z . 40or any s > we denote by E s the conditional expectation with respect to the filtration F s . Assume ˆ v ∈ A maximizes the functional J . Applying the optional projection theoremwe obtain E (cid:34) (cid:90) τ α ( s ) (cid:18) P ( s, T ) − K ( s, T )Υ v ( s ) − E s (cid:20) (cid:90) τs K ( t, T ) e − ρ ( t − s ) γ ˆ v ( t ) dt (cid:21) − l ( s, T )ˆ v ( s )+ E s (cid:20) φ (cid:90) τs X ˆ v ( t ) dt + 2 (cid:37)X ˆ v ( τ ) − P ( τ, T ) (cid:21)(cid:19) ds (cid:35) = 0 , for all α ∈ A . This implies P ( s, T ) − K ( s, T )Υ ˆ v ( s ) − e ρs E s (cid:20) (cid:90) τs K ( t, T ) e − ρt γ ˆ v ( t ) dt (cid:21) − l ( s, T )ˆ v ( s )+ E s (cid:20) φ (cid:90) τs X ˆ v ( t ) dt + 2 (cid:37)X ˆ v ( τ ) − P ( τ, T ) (cid:21) = P ( s, T ) − K ( s, T )Υ ˆ v ( s ) − e ρs (cid:18) E s (cid:20) (cid:90) τ K ( t, T ) e − ρt γ ˆ v ( t ) dt (cid:21) − (cid:90) s K ( t, T ) e − ρt γ ˆ v ( t ) dt (cid:19) − l ( s, T )ˆ v ( s ) + E s (cid:20) φ (cid:90) τ X ˆ v ( t ) dt + 2 (cid:37)X ˆ v ( τ ) − P ( τ, T ) (cid:21) − φ (cid:90) s X ˆ v ( t ) dt = 0 , d P ⊗ ds a.e. on Ω × [0 , τ ] . (8.4)Next, we define the square-integrable martingale ˜ N ( s ) := E s (cid:20)(cid:90) τ K ( t, T ) e − ρt γ ˆ v ( t ) dt (cid:21) (8.5)and the auxiliary square-integrable process Z ˆ v ( s ) := e ρs (cid:18) (cid:90) s K ( t, T ) e − ρt γ ˆ v ( t ) dt − ˜ N ( s ) (cid:19) , for all s ∈ [0 , τ ] . Note that since both l and K are assumed to be uniformly bounded and v ∈ A , we have that P ( τ, T ) ∈ L (Ω , F τ , P ) . Therefore, we obtain P ( s, T ) − K ( s, T )Υ ˆ v ( s ) + Z ˆ v ( s ) − l ( s, T )ˆ v ( s ) + ˜ M ( s ) − φ (cid:90) s X ˆ v ( t ) dt = 0 (8.6)almost everywhere on Ω × [0 , τ ] , where ˜ M is the square-integrable martingale defined in(6.12), and we immediately see that the process Z ˆ v satisfies the BSDE dZ ˆ v ( t ) = (cid:16) ρZ ˆ v ( t ) + K ( t, T ) γ ˆ v ( t ) (cid:17) dt − e ρt d ˜ N ( t ) , Z ˆ v ( τ ) = 0 . From (3.8) we get that Υ ˆ v satisfies d Υ ˆ v ( t ) = − ρ Υ ˆ v ( t ) dt + γ ˆ v ( t ) dt, Υ ˆ v (0) = y. Λ was defined in (6.7). From (8.6) it follows that ˆ v satisfies the backwardstochastic differential equation d ˆ v ( s ) = Λ (cid:48) ( s ) (cid:18) P ( s, T ) − K ( s, T )Υ ˆ v ( s ) + Z ˆ v ( s ) + ˜ M ( s ) − φ (cid:90) s X ˆ v ( u ) du (cid:19) ds + Λ( s ) (cid:18) dP ( s, T ) − ∂ s K ( s, T )Υ ˆ v ( s ) ds − K ( s, T ) d Υ ˆ v ( s ) + dZ ˆ v ( s )+ d ˜ M ( s ) − φX ˆ v ( s ) ds (cid:19) = Λ (cid:48) ( s ) (cid:18) P ( s, T ) − K ( s, T )Υ ˆ v ( s ) + Z ˆ v ( s ) + ˜ M ( s ) − φ (cid:90) s X ˆ v ( u ) du (cid:19) ds + Λ( s ) dP ( s, T ) − Λ( s ) ∂ s K ( s, T )Υ ˆ v ( s ) ds + ρK ( s, T )Υ ˆ v ( s )Λ( s ) ds + Λ( s ) ρZ ˆ v ( s ) ds − s ) φX ˆ v ( s ) ds + Λ( s ) d ˜ M ( s ) − Λ( s ) e ρs d ˜ N ( s )ˆ v ( τ ) = (cid:37)l ( τ, T ) X ˆ v ( τ ) − K ( τ, T )2 l ( τ, T ) Υ ˆ v ( τ ) , Putting these equations together with (3.5), we obtain the FBSDE system (8.1) with
M, N square-integrable martingales defined as M ( t ) := (cid:90) t Λ( s ) d ˜ M ( s ) − (cid:90) t Λ( s ) e ρs d ˜ N ( s ) N ( t ) := − (cid:90) t e ρs d ˜ N ( s ) . In order to check the integrability of M , recall that Λ was defined in (6.7). Since l is boundedaway from on [0 , τ ] (see (3.9)) we have sup ≤ t ≤ τ | Λ( t ) | < ∞ . Then, it holds E [ M ( t )] ≤ E (cid:20)(cid:90) t Λ ( s ) d [ ˜ M ] s (cid:21) + E (cid:20)(cid:90) t Λ ( s ) e ρs d [ ˜ N ] s (cid:21) ≤ C E [ ˜ M ] T + C E [ ˜ N ] T < ∞ for some constants C , C , where in the last inequality we used the fact that both ˜ M and ˜ N are square integrable martingales.Next, assume that (ˆ v, X ˆ v , Υ ˆ v , Z ˆ v ) is a solution to the FBSDE system (8.1) and ˆ v ∈ A .We will show that ˆ v satisfies the first order condition (8.3), hence it maximizes the costfunctional (6.4). First, note that the BSDE for ˆ v can be solved explicitly and the solution42s indeed given in (8.4) ˆ v ( s ) = 12 l ( s, T ) (cid:32) P ( s, T ) − K ( s, T )Υ ˆ v ( s ) + Z ˆ v ( t ) + ˜ M ( s ) − φ (cid:90) s X ˆ v ( t ) dt (cid:33) = 12 l ( s, T ) (cid:32) P ( s, T ) − K ( s, T )Υ ˆ v ( s ) − e ρs (cid:18) ˜ N ( s ) − (cid:90) s K ( t, T ) e − ρt γ ˆ v ( t ) dt (cid:19) + ˜ M ( s ) − φ (cid:90) s X ˆ v ( u ) du (cid:33) , with ˜ N , ˜ M defined in (8.5) and (6.12), respectively. Plugging this into the first order condi-tion (8.3) yields E (cid:34) (cid:90) τ (cid:18) e ρs (cid:18) ˜ N ( s ) − (cid:90) τ K ( t, T ) e − ρt γ ˆ v ( t ) dt (cid:19) − ˜ M ( s )+ 2 φ (cid:90) τ X ( t ) dt + 2 (cid:37)X ˆ v ( τ ) − P ( τ, T ) (cid:19) ds (cid:35) = E (cid:34) (cid:90) τ α ( s ) (cid:18) e ρs ( ˜ N ( s ) − ˜ N ( τ )) − ˜ M ( s ) + ˜ M ( τ ) (cid:19) ds (cid:35) = E (cid:34) (cid:90) τ α ( s ) (cid:18) e ρs ( ˜ N ( s ) − E s [ ˜ N ( τ )]) − ˜ M ( s ) + E s [ ˜ M ( τ )] (cid:19) ds (cid:35) = 0 , for all α ∈ A . Since ˜ N , ˜ M are martingales, hence the first order condition (8.3) is satisfiedand ˆ v ∈ A is the optimal strategy.Before giving the proof of our main theorem, we will need the following Lemma, whichwill help us to show the optimal strategy in (6.3) is indeed admissible. Lemma 8.2.
Let Γ ˆ v be defined as in (6.11) . Then, there exist constants C , C > suchthat E (cid:20)(cid:90) τ (cid:0) Γ ˆ v ( s ) (cid:1) ds (cid:21) ≤ C + C E (cid:20)(cid:90) τ v ( s ) ds (cid:21) . Proof.
Firstly, by the assumptions on l (see (3.9) and (3.10)) it follows that sup ≤ t ≤ τ (cid:12)(cid:12)(cid:12)(cid:12) Λ (cid:48) ( t )Λ( t ) (cid:12)(cid:12)(cid:12)(cid:12) = sup ≤ t ≤ τ (cid:12)(cid:12)(cid:12)(cid:12) ∂ t l ( t, T ) l ( t, T ) (cid:12)(cid:12)(cid:12)(cid:12) < ∞ , (8.7)where Λ is given in (6.7). Therefore, from (8.7), (6.11) and Jensen’s inequality we get that43here exist constants C , C > such that E (cid:20)(cid:90) τ Γ ˆ v ( s ) ds (cid:21) ≤ E (cid:20)(cid:90) τ (cid:16) Λ (cid:48) ( s )Λ( s ) (cid:17) (cid:18) P ( s, T ) + ˜ M ( s ) + 4 φ (cid:16) (cid:90) s X ˆ v ( u ) du (cid:17) (cid:19) ds (cid:21) ≤ C E (cid:20)(cid:90) τ (cid:18) P ( s, T ) + ˜ M ( s ) + 4 φ (cid:16) (cid:90) s X ˆ v ( u ) du (cid:17) (cid:19) ds (cid:21) ≤ C + 4 φ E (cid:20)(cid:90) τ (cid:16) (cid:90) s X ˆ v ( u ) du (cid:17) ds (cid:21) , where we used (6.1) and the fact that the martingale ˜ M defined in (6.12) is square-integrable.Next, using the definition of X ˆ v in (6.2) and Jensen’s inequality twice, we deduce E (cid:20)(cid:90) τ (cid:16) (cid:90) s X ˆ v ( u ) du (cid:17) ds (cid:21) = E (cid:20)(cid:90) τ (cid:16) (cid:90) s (cid:0) x − (cid:90) u ˆ v ( y ) dy (cid:1) du (cid:17) ds (cid:21) ≤ C + C E (cid:20)(cid:90) τ (cid:90) s (cid:90) u ˆ v ( y ) dyduds (cid:21) ≤ C + C E (cid:20)(cid:90) τ (cid:90) τ (cid:90) τ ˆ v ( y ) dyduds (cid:21) ≤ C + C E (cid:20)(cid:90) τ ˆ v ( y ) ds (cid:21) , for some constants C , C , and we are done.We are now ready to prove Theorem 6.3. Proof of Theorem 6.3.
Define X v ( t ) := X ˆ v ( t )Υ ˆ v ( t )ˆ v ( t ) Z ˆ v ( t ) , M ( t ) := P ( t, T ) + (cid:82) t Γ ˆ v ( s ) ds + (cid:82) t Λ − ( s ) dM ( s ) (cid:82) t Λ − ( s ) dN ( s ) , where Λ and Γ ˆ v are defined in (6.7) and (6.11) respectively. The FBSDE system (8.1) canbe written as d X ˆ vt = A ( t ) X ˆ vt dt + Λ( t ) d M ( t ) , ≤ t ≤ τ, where the matrix A ( t ) is defined in (6.6), with initial conditions X ˆ v, (0) = x, X ˆ v, (0) = y, and terminal conditions (cid:18) (cid:37)l ( τ, T ) , − K ( τ, T )2 l ( τ, T ) , − , (cid:19) X ˆ v ( τ ) = 0 , (0 , , , X ˆ v ( τ ) = 0 . (8.8)Exploiting linearity, the unique solution can be expressed as X ˆ v ( τ ) = Φ( τ )Φ − ( t ) X ˆ v ( t ) + (cid:90) τt Φ( τ )Φ − ( s )Λ( s ) d M ( s ) , Φ solves the ODE (6.8). Moreover, it can be immediately seen that the first terminalcondition in (8.8) yields G ( t, τ ) X ˆ v ( t ) + G ( t, τ )Υ ˆ v ( t ) + G ( t, τ )ˆ v ( t ) + G ( t, τ ) Z ˆ v ( t )+ (cid:90) τt Λ( s ) (cid:16) G ( s, τ ) (cid:16) dP ( s, T ) + Γ ˆ v ( s ) ds + Λ − ( s ) dM ( s ) (cid:17) + G ( s, τ )Λ − ( s ) dN ( s ) (cid:17) with G = ( G , G , G , G ) defined in (6.10). Solving for the trading speed v , taking ex-pectations and using that P ∈ H , together with the fact that both M and N are squareintegrable martingales, implies ˆ v ( t ) = − G ( t, τ ) G ( t, τ ) X ˆ v ( t ) − G ( t, τ ) G ( t, τ ) Υ ˆ v ( t ) − G ( t, τ ) G ( t, τ ) Z ˆ v ( t ) − E t (cid:20)(cid:90) τt Λ( s ) G ( s, τ ) G ( t, τ ) ( µ ( s ) + Γ ˆ v ( s )) ds (cid:21) . (8.9)Recall that Ψ was defined in (6.9). Then the second terminal condition in (8.8) implies , , , t, τ ) X ˆ v ( t ) + (0 , , , (cid:90) τt Ψ( s, τ )Λ( s ) d M ( s )= Ψ ( t, τ ) X ˆ v ( t ) + Ψ ( t, τ )Υ ˆ v ( t ) + Ψ ( t, τ )ˆ v ( t ) + Ψ ( t, τ ) Z ˆ v ( t )+ (cid:90) τt Λ( s ) (cid:16) Ψ ( s, τ ) (cid:16) dP ( s, T ) + Γ ˆ v ( s ) ds + Λ − ( s ) dM ( s ) (cid:17) + Ψ ( s, τ )Λ − ( s ) dN ( s ) (cid:17) . Hence, taking expectation and solving for Z u yields Z ˆ v ( t ) = − Ψ ( t, τ )Ψ ( t, τ ) X ˆ v ( t ) − Ψ ( t, τ )Ψ ( t, τ ) Υ ˆ v ( t ) − Ψ ( t, τ )Ψ ( t, τ ) ˆ v ( t ) − E t (cid:20)(cid:90) τt Λ( s )Ψ ( s, τ )Ψ ( t, τ ) ( µ ( s ) + Γ ˆ v ( s )) ds (cid:21) . (8.10)Therefore, plugging (8.10) into (8.9) gives ˆ v ( t ) = − G ( t, τ ) G ( t, τ ) X ˆ v ( t ) − G ( t, τ ) G ( t, τ ) Υ ˆ v ( t ) + G ( t, τ )Ψ ( t, τ ) G ( t, τ )Ψ ( t, τ ) X ˆ v ( t )+ G ( t, τ )Ψ ( t, τ ) G ( t, τ )Ψ ( t, τ ) Υ ˆ v ( t ) + G ( t, τ )Ψ ( t, τ ) G ( t, τ )Ψ ( t, τ ) ˆ v ( t )+ G ( t, τ ) G ( t, τ ) E t (cid:20)(cid:90) τt Λ( s )Ψ ( s, τ )Ψ ( t, τ ) ( µ ( s ) + Γ ˆ v ( s )) ds (cid:21) − E t (cid:20)(cid:90) τt Λ( s ) G ( s, τ ) G ( t, τ ) ( µ ( s ) + Γ ˆ v ( s )) ds (cid:21) . Rearranging and using the Definitions 6.13, we obtain the linear feedback form (6.14). Fi-nally, we prove that the optimal trading strategy is admissible, that is, ˆ v ∈ A , as defined in(6.3). Thanks to assumptions (A.1) and (A.2), we immediately see that sup ≤ t ≤ τ | v ( t, τ ) | < ∞ . v and v are both bounded on [0 , τ ] . Exploiting again assumptions (A.1)-(A.3), together with (6.1) we get that sup ≤ t ≤ τ (cid:12)(cid:12)(cid:12)(cid:12) E t (cid:20) (cid:90) τt Λ( s )Ψ ( s, τ )Ψ ( t, τ ) ( µ ( s ) + Γ ˆ v ( s )) ds − E t (cid:90) τt Λ( s ) G ( s, τ ) G ( t, τ ) ( µ ( s ) + Γ ˆ v ( s )) ds (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C E (cid:20)(cid:90) τ ( | µ ( s ) | + | Γ ˆ v ( s ) | ) ds (cid:21) ≤ ˜ C + ˜ C (cid:18) E (cid:20)(cid:90) τ Γ ˆ v ( s ) ds (cid:21)(cid:19) / ≤ ˜ C + ˜ C (cid:18) E (cid:20)(cid:90) τ ˆ v ( s ) ds (cid:21)(cid:19) / , where we have used Jensen’s inequality and Lemma 8.2 in the last two inequalities. Usingthe above bound, together with equations (6.2) and (3.8) we get from (6.14) that E [ˆ v ( t )] ≤ C + C (cid:90) τ E [ˆ v ( s )] ds, ≤ t ≤ τ, for some positive constants C , C , where we used again Jensen’s inequality. Thanks toGronwall’s lemma, we get that sup ≤ t ≤ τ E [ˆ v ( t )] < ∞ , which implies (cid:90) τ E [ˆ v ( s )] ds < ∞ . Hence Fubini’s theorem, we conclude that ˆ v ∈ A .46 eferences [1] F. Abergel and G. Loeper , Pricing and hedging contingent claims with liquiditycosts and market impact , Available at SSRN 2239498, (2013).[2]
A. Alfonsi, A. Fruth, and A. Schied , Optimal execution strategies in limit orderbooks with general shape functions , Quant. Finance, 10 (2010), pp. 143–157.[3]
R. Almgren and N. Chriss , Optimal execution of portfolio transactions , Journal ofRisk, 3 (2000), pp. 5–39.[4]
E. Bacry, A. Luga, M. Lasnier, and C. A. Lehalle , Market Impacts and theLife Cycle of Investors Orders , Market Microstructure and Liquidity, 1 (2015).[5]
C. Bellani, D. Brigo, A. Done, and E. Neuman , Static vs adaptive strategies foroptimal execution with signals , arXiv:1811.11265, (2018).[6]
D. Bertsimas and A. W. Lo , Optimal control of execution costs , Journal of FinancialMarkets, 1 (1998), pp. 1–50.[7]
T. Björk , Interest rate theory , in Financial Mathematics, Springer, 1997, pp. 53–122.[8]
G. Bormetti, D. Brigo, M. Francischello, and A. Pallavicini , Impact ofmultiple curve dynamics in credit valuation adjustments under collateralization , Quant-itative Finance, 18 (2018), pp. 31–44.[9]
B. Bouchard, G. Loeper, and Y. Zou , Almost-sure hedging with permanent priceimpact , Finance and Stochastics, 20 (2016), pp. 741–771.[10]
B. Bouchard, G. Loeper, and Y. Zou , Hedging of covered options with linearmarket impact and gamma constraint , SIAM Journal on Control and Optimization, 55(2017), pp. 3319–3348.[11]
D. Brigo and F. Mercurio , Interest rate models-theory and practice: with smile,inflation and credit , Springer, Berlin, Heidelberg, 2006.[12]
D. Brigo, A. Pallavicini, and R. Torresetti , Credit models and the crisis : ajourney into CDOs, copulas, correlations and dynamic models. , Wiley, 2010.[13]
R. Carmona and K. Webster , The self-financing equation in high frequency mar-kets , arXiv preprint arXiv:1312.2302, (2013).[14]
Á. Cartea, S. Jaimungal, and J. Penalva , Algorithmic and High-Frequency Trad-ing (Mathematics, Finance and Risk) , Cambridge University Press, 1 ed., Oct. 2015.[15]
D. Filipovic , Term-Structure Models. A Graduate Course. , Springer, 2009.[16]
J. Gatheral, A. Schied, and A. Slynko , Exponential resilience and decay of mar-ket impact , in Econophysics of Order-driven Markets, F. Abergel, B. Chakrabarti,A. Chakraborti, and M. Mitra, eds., Springer-Verlag, 2011, pp. 225–236.4717]
J. Gatheral, A. Schied, and A. Slynko , Transient linear price impact and Fred-holm integral equations , Math. Finance, 22 (2012), pp. 445–474.[18]
O. Guéant , The Financial Mathematics of Market Liquidity: From Optimal Executionto Market Making , Chapman and Hall/CRC, Apr. 2016.[19]
O. Guéant, C. A. Lehalle, and J. Fernandez-Tapia , Optimal Portfolio Liquid-ation with Limit Orders , SIAM Journal on Financial Mathematics, 13 (2012), pp. 740–764.[20]
J. M. Harrison and D. M. Kreps , Martingales and arbitrage in multiperiod secur-ities markets , Journal of Economic theory, 20 (1979), pp. 381–408.[21]
J. M. Harrison and S. R. Pliska , Martingales and stochastic integrals in the theoryof continuous trading , Stochastic processes and their applications, 11 (1981), pp. 215–260.[22]
D. Heath, R. Jarrow, and A. Morton , Bond pricing and the term structure of in-terest rates: A new methodology for contingent claims valuation , Econometrica: Journalof the Econometric Society, (1992), pp. 77–105.[23]
M. Henrard , Interest Rate Modelling in the Multi-Curve Framework: Foundations,Evolution and Implementation , Applied Quantitative Finance, Palgrave Macmillan UK,2014.[24]
J. Hull and A. White , Pricing interest-rate-derivative securities , The review offinancial studies, 3 (1990), pp. 573–592.[25]
I. Kharroubi and H. Pham , Optimal portfolio liquidation with execution cost andrisk , SIAM Journal on Financial Mathematics, 1 (2010), pp. 897–931.[26]
C. A. Lehalle, S. Laruelle, R. Burgot, S. Pelin, and M. Lasnier , MarketMicrostructure in Practice , World Scientific publishing, 2013.[27]
G. Loeper et al. , Option pricing with linear market impact and nonlinear black–scholes equations , The Annals of Applied Probability, 28 (2018), pp. 2664–2726.[28]
E. Neuman and A. Schied , Optimal portfolio liquidation in target zone models andcatalytic superprocesses , 2015.[29]
E. Neuman and M. Voß , Optimal signal-adaptive trading with temporary and tran-sient price impact , arXiv preprint arXiv:2002.09549, (2020).[30]
A. A. Obizhaeva and J. Wang , Optimal trading strategy and supply/demand dy-namics , Journal of Financial Markets, 16 (2013), pp. 1 – 32.[31]
O. Vasicek , An equilibrium characterization of the term structure , Journal of financialeconomics, 5 (1977), pp. 177–188.[32]