A Repo Model of Fire Sales with VWAP and LOB Pricing Mechanisms
aa r X i v : . [ q -f i n . R M ] M a y A Repo Model of Fire Sales with VWAP and LOB PricingMechanisms
Maxim Bichuch ∗ Zachary Feinstein † Wednesday 13 th May, 2020
Abstract
We consider a network of banks that optimally choose a strategy of asset liquidations andborrowing in order to cover short term obligations. The borrowing is done in the form ofcollateralized repurchase agreements, the haircut level of which depends on the total liquidationsof all the banks. Similarly the fire-sale price of the asset obtained by each of the banks dependson the amount of assets liquidated by the bank itself and by other banks. By nature of thissetup, banks’ behavior is considered as a Nash equilibrium. This paper provides two forms formarket clearing to occur: through a common closing price and through an application of thelimit order book. The main results of this work are providing sufficient conditions for existenceand uniqueness of the clearing solutions (i.e., liquidations, borrowing, fire sale prices, and haircutlevels).
Keywords
Systemic Risk, Price-Mediated Contagion, Repurchase Agreements.
Historically, financial risk was typically measured for individual firms separately. After the financialcrisis of 2007-2009, a new understanding that risk can spread through the entire financial systemhas emerged. This is referred to as systemic risk – the risk that the distress of several banks canspread throughout the system to a degree that it may affect the viability of the entire system ora significant part of it. Such a propagation of risk is known as financial contagion. Two types ofcontagion are usually distinguished: those that happen due to local connections (e.g., obligationsbetween banks in the network), and those that happen due to their influence on the entire network(e.g., impact to asset prices). This study focuses on a form of global contagion through asset ∗ Department of Applied Mathematics and Statistics, Johns Hopkins University 3400 North Charles Street, Balti-more, MD 21218. [email protected] . Work is partially supported by NSF grant DMS-1736414. Research is partiallysupported by the Acheson J. Duncan Fund for the Advancement of Research in Statistics. † School of Business Stevens Institute of Technology, Hoboken NJ 07030, USA, [email protected] . Volume Weighted Average Pricing (VWAP) and a
Limit Order Book (LOB) based pricing scheme. Both of these schemes can beviewed as pricing limits as order sizes decrease to zero, but with different rates of liquidation. Thisallows us to incorporate notions of time dynamics into the static model proposed. The VWAPscheme determines prices if firms place orders at a rate proportional to their total desired liquida-tions; this, ultimately, results in the same average price for every bank. Such a pricing scheme wasintroduced in Banerjee and Feinstein [2019]. The LOB setting distinguishes prices by assuming allfirms place orders at the same speed; banks with smaller order volumes will receive a higher pricethan those with a larger order volume (as the latter will continue to eat through the book evenafter the former are done liquidating).As highlighted above, the innovation of this work is two-fold. First, we consider realistic pricingschemes that allow for banks to receive different prices based on the quantity of assets sold insteadof the, more standard, assumption that there is a unique price at which all transactions occur.Second, we consider collateralized borrowing of illiquid assets in a repo market in which the haircut2f this collateral also depends on the mark-to-market value of the asset. As opposed to the realizedliquidation prices, the haircut remains bank independent and only depends on the entire salevolume of the entire banking system since the deal depends on the value of the collateralizedasset rather than the riskiness of the individual banks. Under these constructions, we are able toinvestigate the sensitivity of the resulting market prices to the prevailing repo interest rate. Inparticular, regulators use interest rates as the primary control for financial stability. This wasseen in the emergency liquidity injection by the Federal Reserve in September 2019, in order tostabilize the repo market Ihrig et al. [2020], Afonso et al. [2020]. In fact, Gorton and Metrick[2012], Brunnermeier [2009] consider the 2007-2009 financial crisis as a run on the repo market.Therefore systematic consideration of repo markets and the impact of interest rates is of paramountimportance.The organization of this paper is as follows. Section 2 introduces the general model with generalinverse demand pricing functions. In that section we provide the existence of Nash equilibrium undera minimal set of assumptions. Section 3 introduces the VWAP and LOB based inverse demandpricing functions and discusses the conditions needed for maximal clearing solutions and uniquenessof Nash equilibrium. Numerical case studies and comparison of VWAP and LOB inverse demandpricing functions is in Section 4. The proofs for the main results are provided in the Appendix.Additionally in the Appendix, under the uniqueness conditions, we investigate the sensitivity ofthe clearing solutions to the prevailing repo rate.
We begin by assuming a system of n banks. In contrast to works that explicitly depend on thenetwork of interbank obligations, e.g., in Eisenberg and Noe [2001], Cifuentes et al. [2005], Aminiet al. [2016], Feinstein [2017], herein we will consider only fire sale effects and price mediatedcontagion as in, e.g., Greenwood et al. [2015], Braouezec and Wagalath [2018, 2019], Feinstein[2020], Banerjee and Feinstein [2019]. We will, for simplicity, assume that all the banks i = 1 , ..., n are facing a (cash) shortfall h i >
0, all while holding a i > h i cash in order to cover this shortfall. We assume thatthey can do so by either selling their illiquid asset, borrowing, or both. It will be assumed that theborrowing is going to be collateralized using the same illiquid asset. As is standard in the literature,due to the illiquidity, the price of the illiquid asset declines as assets are being sold; this is due tosupply-demand dynamics so that the equilibrium is maintained. The same effect is assumed for thecollateral value of the asset.Herein we introduce two “pricing” functions. Let ¯ f i : R n + → [0 ,
1] denote the average priceobtained by bank i = 1 , ..., n given the set of system liquidations ( s , ..., s n ) ∈ D := Q nj =1 [0 , a j ].Note that we implicitly impose a no short selling constraint throughout this work. Here, without3oss of generality, it was assumed that the current, highest price of the asset, before any saleshappened is 1, and it can only decrease thereafter. Notably, the construction of ¯ f i implies thatdifferent banks may obtain different prices in the market due to the market design or different ordersizes. Let g : R + → [0 ,
1] denote the price of the collateralized asset in the repurchase agreementsunder study, i.e., the function g (cid:16)P nj =1 s j (cid:17) encodes the haircut on the asset as a mapping of thetotal liquidations by all the banks. Note that while the price obtained by bank i may be unique dueto the different quantities different banks are selling, since the repo transaction is collateralized itis assumed that the repo market offers the same repo rate r to all banks and uses the same haircut g (cid:16)P nj =1 s j (cid:17) . Though we call g the “haircut”, it is more appropriate to denote 1 − g to be the truehaircut on the asset in the repo market. At various times in this work we will refer to g as thehaircut and others 1 − g will be given that name.Assuming banks sell s := ( s , ..., s n ) ∈ D , the realized loss to bank i from the sale is s i (1 − ¯ f i ( s )).The bank obtained s i ¯ f i ( s ) through this sale, therefore it needs to borrow an additional ( h i − s i ¯ f i ( s ))for the cost of r ( h i − s i ¯ f i ( s )). We will abuse notation and denote for convenience ¯ f i to be both¯ f i ( s i , s − i ) and ¯ f i ( s ). Therefore, bank i seeks to optimize: s ∗ i = s ∗ i ( s − i ) = arg min s i ∈ [0 ,a i ] s i (cid:0) − ¯ f i ( s i , s − i ) (cid:1) + r (cid:0) h i − s i ¯ f i ( s i , s − i ) (cid:1) s.t. s i ≤ h i ¯ f i ( s i , s − i ) , s i ≥ h i − a i g (cid:16)P nj =1 s j (cid:17) ¯ f i ( s i , s − i ) − g (cid:16)P nj =1 s j (cid:17) . (2.1)Here, the first inequality ensures that bank i does not obtain more than h i through the asset sale,and the second inequality constraint is used to ensure that h i − s i ¯ f i ( s i , s − i ) ≤ ( a i − s i ) g (cid:16)P nj =1 s j (cid:17) ,i.e., after the sale, bank i has enough collateral ( a i − s i ) g (cid:16)P nj =1 s j (cid:17) to cover its loan. The paperof Bichuch and Feinstein [2019] considers the case in which no haircut is taken, i.e., g ≡ i is solvent if and only if h i − a i g (cid:16)P nj =1 s j (cid:17) ¯ f i ( s i , s − i ) − g (cid:16)P nj =1 s j (cid:17) ≤ h i ¯ f i ( s i , s − i ) ≤ a i . By construction of the haircut for repurchase agreements 0 ≤ g (cid:16)P nj =1 s j (cid:17) < ¯ f i ( s i , s − i ). Undersuch a construction bank i is solvent if and only if h i ≤ a i ¯ f i ( s i , s − i ), i.e., if at the current pricerealized by bank i it is possible for said bank to cover its shortfall by liquidations alone. If bank i is insolvent then we will assume that it is forced to liquidate all of its asset holdings, i.e., s ∗ i = a i .For convenience, for the remainder of this work, denote ¯ q i = ¯ f i ( s i , s − i ) , i = 1 , ..., n, and q = g (cid:16)P nj =1 s j (cid:17) . With this notation, we modify (2.1) (similarly as in Bichuch and Feinstein42019]) such that we seek a Nash equilibrium of the game for each bank is ∗ i = s ∗ i ( s − i , q, ¯ q ) = arg min s i ∈ [0 ,a i ] s i (cid:0) − ¯ f i ( s i , s − i ) (cid:1) + r (cid:0) h i − s i ¯ f i ( s i , s − i ) (cid:1) (2.2)s.t. s i ≤ h i ¯ q i , s i ≥ h i − a i q ¯ q i − q . The goal is then to find a Nash equilibrium for (2.2), such that ( q, ¯ q ) are, additionally, fixed pointsof q = g n X j =1 s ∗ j , ¯ q i = ¯ f i ( s ∗ ) . As noted above, bank i is defaulting if h i ≥ a i ¯ q i and, in such a situation, s ∗ i = a i . Our goal isprimarily to find conditions for existence and uniqueness of this Nash game in the financial system.In order to do that we need assumptions on the inverse demand functions ¯ f i and g . Assumption 2.1.
Let M ≥ P ni =1 a i be the total initial market capitalization of the illiquid asset.For i = 1 , ..., n we assume that ¯ f i : R n + → [0 , are each continuous and strictly decreasing inevery argument with ¯ f i (0 , ...,
0) = 1 and ¯ f i ( a , ..., a n ) > . Additionally, for i = 1 , ..., n and s − i ∈ Q nj =1 ,j =1 [0 , a j ] we assume that s i s i ¯ f i ( s, s − i ) is concave.The haircut function g : R + → [0 , is continuous and strictly decreasing, with min ≤ i ≤ n ¯ f i ( s ) >g (cid:16)P nj =1 s j (cid:17) for every s ∈ D . Existence of a Nash equilibrium easily follows as a consequence of Brouwer’s fixed-point theorem:
Theorem 2.2 (Existence of Nash Equilibrium) . Assume the inverse demand functions ¯ f i , i =1 , ..., n and haircut function g satisfy Assumption 2.1. Then there exists a Nash equilibrium liquidat-ing strategy s ∗∗ ∈ D with equilibrium prices ( q ∗∗ , ¯ q ∗∗ , ..., ¯ q ∗∗ n ) = (cid:0) g ( P ni =1 s ∗∗ i ) , ¯ f ( s ∗∗ ) , ..., ¯ f n ( s ∗∗ ) (cid:1) .Proof of Theorem 2.2. Fix bank i and consider (2.2) as a function of ( s − i , q, ¯ q i ) such that 0 ≤ q < ¯ q i ,with ¯ f i ( a , ..., a n ) ≤ ¯ q i , and s ∗− i ∈ Q nj =1 , = i [0 , a j ]. Since the objective function of (2.2) is convex in s i and the constraint set is a convex interval, the set of minimizers for a fixed set of parameters( s − i , q, ¯ q i ) is convex. An application of Berge maximum theorem (on ¯ q i ≥ h i a i due to the continuityof the objective and constraint functions) yields upper continuity and convex-valuedness of the setof maximizers. This is extended for the region of insolvency by the assumption that s ∗ i = a i on h i > a i ¯ q i . Thus a joint equilibrium ( s ∗∗ , q ∗∗ , ¯ q ∗∗ , ..., ¯ q ∗∗ n ) can be found via Kakutani’s fixed pointtheorem.It turns out that the conditions for existence of equilibrium are very mild, compared to theuniqueness conditions. This is not surprising considering the following example. Example 2.3.
Consider an n = 1 bank setting with r = 0 repo rate. This bank has assets andshortfall so that a ¯ f ( a ) < h < a g (0) . Therefore, two possible solutions exist: . If the bank liquidates no assets then ( q ∗∗ , ¯ q ∗∗ , s ∗∗ ) = ( g (0) , , is an equilibrium solution;2. If the bank defaults and liquidates all its assets then ( q ∗∗ , ¯ q ∗∗ , s ∗∗ ) = ( g ( a ) , ¯ f ( a ) , a ) is anequilibrium solution. We now concentrate our efforts into understanding properties of these equilibria and find con-ditions to guarantee their uniqueness. In what follows we will concentrate on two examples for theinverse demand functions ¯ f i , i = 1 , ..., n . We now concentrate our efforts into understanding when the above equilibrium is unique. In whatfollows we will concentrate on two sample functions. However, instead of specifying the inversedemand functions ¯ f i directly, we derive them from a density function of limit order book togetherwith some trading rules. Let this density be given by f : R + → [0 , f can beviewed as a the price of the next infinitely small trade. We concentrate on two realistic examplesof price constructions given the liquidations, i.e., market rules, to construct the price of the tradewith functional forms ¯ f i : R n + → [0 ,
1] which provides the average price obtained by firm i giventhe set of system liquidations.1. Volume Weighted Average Price (VWAP):
For i = 1 , ..., n set ¯ f i ( s ) := R P nj =1 sj f ( σ ) dσ P nj =1 s j .Note that in this case ¯ f i ( s ) = ¯ f j ( s ) for i, j ∈ { , , ..., n } .2. Limit Order Book Based Price (LOB):
For i = 1 , ..., n set¯ f i ( s ) := 1 s i k X j =1 n − ( j − Z P jl =1 ( n − ( l − s [ l ] − s [ l − ) P j − l =1 ( n − ( l − s [ l ] − s [ l − ) f ( σ ) dσ, where 0 =: s [0] ≤ s [1] ≤ s [2] ≤ ... ≤ s [ n ] are the order statistics and s i = s [ k ] .Note that the VWAP example corresponds to how some exchanges calculate the closing price(e.g., in Mexico, India and Saudi Arabia ). Therefore, given our assumption that this is an illiquidasset, this is a good representation of price paid by banks given the amounts of trades they (collec-tively) want to make. Whereas the LOB example is an example of how to price market trades allcoming at the same time using an existing limit order trades already in the book. This is a veryinteresting and novel example, as in this case, different banks pay different prices. As far as theauthors are aware, this LOB construction has never previously been formulated.Alternatively, these specific pricing functionals can be viewed as a limit as order sizes decreaseto zero at different rates. VWAP can be viewed as the limit when all banks submit their orders ata rate proportional to the total desired liquidation; as such, every bank finishes trading at the same“time” and thus all banks obtain the same average price. In contrast, the LOB is the limit when research.ftserussell.com/products/downloads/Closing_Prices_Used_For_Index_Calculation.pdf f : Assumption 3.1.
Let M ≥ P ni =1 a i be the total initial market capitalization of the illiquid asset.The order book density function f : R + → [0 , is strictly decreasing and twice continuously differ-entiable, with f (0) = 1 and f ( s ) > for any s ∈ [0 , M ] . Additionally it will be assumed that thefirst derivative f ′ : R + → − R + is nondecreasing. Throughout the remainder of this work we often wish to consider a comparison of vectors of( q, ¯ q ); this is accomplished in the usual way, i.e., ( q , ¯ q ) ≥ ( q , ¯ q ) if and only if q ≥ q and¯ q i ≥ ¯ q i for every i = 1 , ..., n .Our next goal is to ultimately establish uniqueness-type properties of the Nash equilibrium.In order to do so, similarly to Bichuch and Feinstein [2019], we consider the problem with fixedliquidation price(s) and the haircut value as described in (2.2). As opposed to Theorem 2.2 above,we first show that there exist unique Nash equilibrium liquidations for these fixed prices as shownin Proposition 3.2 below, the proof of which is delayed until Appendix A.1. Proposition 3.2.
Let b Q := { ( q, ¯ q ) ∈ (0 , × (0 , n | q < ¯ q i ∀ i = 1 , , ..., n } . Under VWAP or LOBstructure and Assumption 3.1, given ( q, ¯ q , ..., ¯ q n ) ∈ b Q there exists a unique set of equilibriumliquidations ¯ s ( q, ¯ q , ..., ¯ q n ) = s ∗ (¯ s ( q, ¯ q , ..., ¯ q n ) , q, ¯ q , ..., ¯ q n ) to (2.2) . From Example 2.3 it is clear the that uniqueness of the equilibrium does not hold without furtherassumptions, but we can show, as done in Theorem 3.3, that the set of all fixed points prices ( q ∗ , ¯ q ∗ )in the Nash equilibrium of (2.2) is a lattice under a VWAP pricing scheme; notably, as demonstratedbelow in Example 3.4, the LOB pricing scheme does not satisfy the typical conditions for this result.The proof of the theorem is presented in the Appendix A.2. Theorem 3.3.
Under the VWAP structure and Assumption 3.1, the set of clearing haircuts andprices is a lattice; in particular, there exists a greatest and least clearing haircut and set of clearingprices: ( q ↑ , ¯ q ↑ , ..., ¯ q ↑ n ) ≥ ( q ↓ , ¯ q ↓ , ..., ¯ q ↓ n ) .Sketch of proof. Taking advantage of Proposition 3.2, we find that the sum P ni =1 ¯ s i is monotonicin ( q, ¯ q , ..., ¯ q n ). Therefore we apply Tarski’s fixed point theorem. The details are provided inAppendix A.2.Importantly, Theorem 3.3 is not applied to the LOB structure. To demonstrate the lack of sucha result for the LOB, we present here a counterexample to the monotonicity property utilized inthe proof of Theorem 3.3. Example 3.4.
Consider an n = 2 bank system with r = 5% repo rate. Let bank 1 have shortfall h = 1 . and a = 10 assets. Let bank 2 have shortfall h = 5 and a = 10 assets. Consider linear order density function f ( s ) = 1 − . s and haircut function g ( s ) = 0 . − . s .We then consider two possible inputs to the clearing system ( q , ¯ q ) := (0 . , (1 , and ( q , ¯ q ) :=(0 . , (0 . , . .1. Under initial prices of ( q , ¯ q ) , the banks sell ¯ s ( q , ¯ q ) = (1 . , . assets each. This resultsin LOB based prices of ¯ q , † := 0 . and ¯ q , † := 0 . for banks 1 and 2 respectively. Theresulting haircut is q , † := 0 . .2. Under initial prices of ( q , ¯ q ) , the banks sell ¯ s ( q , ¯ q ) = (2 , assets each. This results inLOB based prices of ¯ q , † := 0 . and ¯ q , † := 0 . for banks 1 and 2 respectively. Theresulting haircut is q , † := 0 . .Notably, though ( q , ¯ q ) ≥ ( q , ¯ q ) , this monotonicity does not hold for the resulting prices as ( q , † , ¯ q , † ) ( q , † , ¯ q , † ) . Finally, we introduce additional assumptions and establish uniqueness of the equilibrium inTheorem 3.6 below, the proof of which is delayed until the Appendix A.3. For such a result weintroduce a simplified notation, let ∂ x := ∂∂x denote the partial derivative operator with respect tosome variable x . Definition 3.5.
We will say that bank i ∈ { , ..., n } is fundamentally solvent if it is able to coverits shortfall in any case, that is if h i ≤ a i ¯ f i ( a ) , where a = ( a , ..., a n ) ⊤ . Remark 1.
If bank i is fundamentally solvent then there is a feasible solution to the maximizationproblem (2.2) , provided ( q, ¯ q ) = ( g ( P ni =1 s i ) , ¯ f ( s )) for some s ∈ D , since the feasible region isnon-empty. Indeed,1. h i ¯ q i ≤ a i if and only if h i ≤ a i ¯ q i .2. h i − a i q ¯ q i − q ≤ a i if and only if h i ≤ a i ¯ q i .3. h i − a i q ¯ q i − q ≤ h i ¯ q i if and only if h i ≤ a i ¯ q i . Theorem 3.6.
Assume all banks are fundamentally solvent. Under VWAP or LOB structure andAssumption 3.1, if additionally, − cM (cid:0) min j,k ∂ s k ¯ f j ( n ) ∧ g ′ (0) (cid:1) < min j min s ∈D (cid:0) ¯ f j ( s ) − g ( P ni =1 s i ) (cid:1) with c = 3 and c = n in case of VWAP and LOB, respectively, then there exists a unique clearinghaircut and set of actualized prices ( q ∗ , ¯ q ∗ ) .Sketch of proof. The proof follows from the Banach fixed point theorem and is presented in Ap-pendix A.3.
Remark 2.
At this point we wish to recall Example 2.3 which highlights a case of non-uniquenessof the clearing solution. In that single bank setting, the bank is not fundamentally solvent since, byconstruction, a ¯ f ( a ) < h . This highlights the importance of the assumption that all banks arefundamentally solvent in Theorem 3.6 for the uniqueness of the clearing prices. emark 3. With the consideration of existence and uniqueness of the clearing solution, the sensi-tivity of the equilibrium liquidations and prices to the repo rate r is of great interest. This is studiedmathematically in Appendix B. Intuitively, we expect that as the repo rate rises, and borrowingbecomes more expensive the liquidation of the illiquid asset increase. It also follows from here that,the higher the interest rate, the lower the terminal asset price. Alternatively, from a regulator’sperspective, if the goal is to limit the extent of the fire sales, it can be achieved by controlling theinterest rates, as was done recently in September 2019, and was also used extensively during thecrisis (see Quinn et al. [2020] and Cecchetti [2009] respectively). We refer to the case studies inSection 4 for visualizations of this notion. Before considering specific examples, we will first introduce a consideration for the computationof the clearing prices ( q, ¯ q ) = ( g ( P ni =1 ¯ s i ( q, ¯ q )) , ¯ f (¯ s ( q, ¯ q ))) . This approach will always converge forthe VWAP setting due to Theorem 3.3; though the LOB setting does not satisfy monotonicity, thisalgorithm converged for every choice of parameters attempted by the authors indicating a strongerresult than found thus far. Specifically, these are computed via Picard iterations beginning from( q , ¯ q ) := n +1 . However, ¯ s ( q, ¯ q ) will require consideration for computation itself due to its gametheoretic construction. As provided in Proposition 3.2 these liquidations exist and are unique. Infact, due to the construction of the problem as discussed in the proof of that proposition, we are ableto apply the algorithm provided in Rosen [1965]. This is summarized in Algorithm 2 of Bichuch andFeinstein [2019] for the VWAP setting. We wish to note that in the LOB setting, the computationcan be improved significantly via an iterative approach of determining the banks liquidating thefewest number of assets.In this section we will consider two primary case studies. The first is a consideration of theVWAP and LOB structures to determine their relative ordering, i.e., is one better than the other.This is important from a mechanism design perspective as different markets consider the closingprice using different rule sets. The second case study we will consider is an implementation ofEuropean banking data to determine the impacts of interest rates and haircut functions on theclearing prices. In this first case study, we will investigate two networks in detail in order to show that some systemconstructions find that VWAP has more total liquidations with less system-wide use of the repomarkets than LOB, while other constructions have the reverse ordering. In particular, we will firstconsider a system of n identical banks and second a specific system of n = 2 banks only.9 .1.1 Symmetric case study Consider a system of n identical banks. Each of these banks has shortfall h > a > r ∈ (0 , ). For the purposes of this example, consider theorder book density f ( s ) = 1 − αs and haircut function g ( s ) = − αs for α ∈ (cid:16) r (1+ r )( n +1) a , na (cid:17) ;notably these constructions satisfy Assumption 3.1 and taken so as to construct an example inwhich firms have a choice of behavior. Consider now our two market mechanisms: VWAP andLOB.1. VWAP:
By construction ¯ f i ( s ) := 1 − α P nj =1 s j for every bank i in the VWAP construction.Additionally, we take advantage of the symmetric setup to conclude that all banks shouldfollow the same strategy, i.e., s V W AP = s V W AP n for some singleton s V W AP ∈ [0 , a ] and¯ q V W AP = ¯ q V W AP n for some singleton ¯ q V W AP ∈ [0 , q, ¯ q ) with q < ¯ q : s ∗ i ( q, ¯ q n ) = arg min s i ∈ [0 ,a ] α r ) X j = i s ∗ j ( q, ¯ q n ) + s i s i + r ( h − s i ) (cid:12)(cid:12)(cid:12)(cid:12) s i ∈ (cid:20) h − aq ¯ q − q , h ¯ q (cid:21) = arg min s i ∈ [0 ,a ] (cid:26) α r ) s i + h α r )( n − s V W AP ( q, ¯ q ) − r i s i + rh (cid:12)(cid:12)(cid:12)(cid:12) s i ∈ (cid:20) h − aq ¯ q − q , h ¯ q (cid:21) (cid:27) = h − aq ¯ q − q ∨ (cid:20) r (1 + r ) α − n − s V W AP ( q, ¯ q ) (cid:21) ∧ h ¯ q , if h < a ¯ q (and s V W AP ( q, ¯ q ) = a if h ≥ a ¯ q ). In particular, this provides a single fixed pointproblem to find s V W AP ( q, ¯ q ), i.e., s V W AP ( q, ¯ q ) = h − aq ¯ q − q ∨ (cid:20) r (1 + r ) α − n − s V W AP ( q, ¯ q ) (cid:21) ∧ h ¯ q , ⇒ s V W AP ( q, ¯ q ) = h − aq ¯ q − q ∨ (cid:20) rα (1 + r )( n + 1) (cid:21) ∧ h ¯ q if h < a ¯ q . We wish to note that the existence of s V W AP ( q, ¯ q ) justifies our choice of s V W AP = s V W AP n as, due to Proposition 3.2, s V W AP is unique and thus must equal s V W AP n . Finally,it remains to find the equilibrium prices ( q V W AP , ¯ q V W AP ): q V W AP = − + √ − αnh if h ∈ H V W AP , − rn (1+ r )( n +1) if h ∈ H V W AP , − αna − p αn ( h − a ) + 4( αna ) if h ∈ H V W AP , − αna if h ∈ H V W AP , q V W AP = √ − αnh if h ∈ H V W AP , − rn (1+ r )( n +1) if h ∈ H V W AP − αna − p αn ( h − a ) + 4( αna ) if h ∈ H V W AP , − α na if h ∈ H V W AP , with borrowing/liquidation regions H V W AP = (cid:20) , rα (1 + r )( n + 1) (cid:18) − rn (1 + r )(1 + n ) (cid:19)(cid:19) , H V W AP = (cid:20) rα (1 + r )( n + 1) (cid:18) − rn (1 + r )(1 + n ) (cid:19) , rα (1 + r )( n + 1) (cid:18)
12 + rn (1 + r )( n + 1) (cid:19) + a (cid:18) − rn (1 + r )( n + 1) (cid:19)(cid:19) , H V W AP = (cid:20) rα (1 + r )( n + 1) (cid:18)
12 + rn (1 + r )( n + 1) (cid:19) + a (cid:18) − rn (1 + r )( n + 1) (cid:19) , a (cid:16) − α na (cid:17)(cid:19) , H V W AP = h a (cid:16) − α na (cid:17) , ∞ (cid:17) . We wish to note that all square roots are well defined on the intervals on which they areconsidered. Additionally, q V W AP and ¯ q V W AP are continuous in h ; as such the closures ofthe bounding intervals can be chosen arbitrarily. Though this setting does not satisfy theuniqueness conditions of Theorem 3.6, the simplicity of the symmetric system still admits aunique clearing solution.2. LOB:
By construction ¯ f [ i ] ( s ) := 1 − α s [ i ] hP i − k =1 s [ k ] (2 s [ i ] − s [ k ] ) + ( n − ( i − s i ] i for everybank [ i ] (i.e., the bank liquidating the i th most assets) in the LOB construction. Additionally,we take advantage of the symmetric setup to conclude that all banks should follow the samestrategy, i.e., s LOB = s LOB n for some singleton s LOB ∈ [0 , a ] and ¯ q LOB = ¯ q LOB n for somesingleton ¯ q LOB ∈ [0 , q, ¯ q ) with q < ¯ q : s ∗ i ( q, ¯ q n ) = arg min s i ∈ [0 ,a ] α (1 + r ) h ns i I { s i ≤ s LOB ( q, ¯ q ) } +(( n − s LOB ( q, ¯ q ) + 2( n − s LOB ( q, ¯ q ) s i + s i ) I { s i >s LOB ( q, ¯ q ) } i + r ( h − s i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s i ∈ (cid:20) h − aq ¯ q − q , h ¯ q (cid:21) = h − aq ¯ q − q ∨ h rα (1+ r ) n i ∧ h ¯ q if rα (1+ r ) n ≤ s LOB ( q, ¯ q ) , h − aq ¯ q − q ∨ h rα (1+ r ) − ( n − s LOB ( q, ¯ q ) i ∧ h ¯ q if rα (1+ r ) n > s LOB ( q, ¯ q ) , if h < a ¯ q (and s LOB ( q, ¯ q ) = a if h ≥ a ¯ q ). In particular, this provides a single fixed point11roblem to find s LOB ( q, ¯ q ), i.e., if h < a ¯ qs LOB ( q, ¯ q ) = h − aq ¯ q − q ∨ h rα (1+ r ) n i ∧ h ¯ q if rα (1+ r ) n ≤ s LOB ( q, ¯ q ) , h − aq ¯ q − q ∨ h rα (1+ r ) − ( n − s LOB ( q, ¯ q ) i ∧ h ¯ q if rα (1+ r ) n > s LOB ( q, ¯ q ) , ⇒ s LOB ( q, ¯ q ) = h − aq ¯ q − q ∨ (cid:20) rα (1 + r ) n (cid:21) ∧ h ¯ q as both provided cases result in the same fixed point. We wish to note that the existence of s LOB ( q, ¯ q ) justifies our choice of s LOB = s LOB n as, due to Proposition 3.2, s LOB is uniqueand thus must equal s LOB n . Finally, it remains to find the equilibrium prices ( q LOB , ¯ q LOB ): q LOB = − + √ − αnh if h ∈ H LOB − r r if h ∈ H LOB − αna − p αn ( h − a ) + 4( αna ) if h ∈ H LOB − αna if h ∈ H LOB ¯ q LOB = √ − αnh if h ∈ H LOB , − r r ) if h ∈ H LOB , − αna − p αn ( h − a ) + 4( αna ) if h ∈ H LOB , − α na if h ∈ H LOB , with borrowing/liquidation regions H LOB = (cid:20) , rα (1 + r ) n (cid:18) − r r ) (cid:19)(cid:19) , H LOB = (cid:20) rα (1 + r ) n (cid:18) − r r ) (cid:19) , r α (1 + r ) n (cid:18) r r (cid:19) + a (cid:18) − r r (cid:19)(cid:19) , H LOB = (cid:20) r α (1 + r ) n (cid:18) r r (cid:19) + a (cid:18) − r r (cid:19) , a (cid:16) − α na (cid:17)(cid:19) , H LOB = h a (cid:16) − α na (cid:17) , ∞ (cid:17) . We wish to note that all square roots are well defined on the intervals on which they areconsidered. Additionally, q V W AP and ¯ q V W AP are continuous in h ; as such the closures ofthe bounding intervals can be chosen arbitrarily. Though this setting does not satisfy theuniqueness conditions of Theorem 3.6, the simplicity of the symmetric system still admits aunique clearing solution.Notably, s V W AP ( q, ¯ q ) ≥ s LOB ( q, ¯ q ) for any choice of ( q, ¯ q ) by construction. In fact, if thereexists n ≥ H V W AP ∩ H LOB , i.e., h ∈ (cid:16) rα (1+ r ) n (cid:16) − r r ) (cid:17) , rα (1+ r )( n +1) (cid:16) + rn (1+ r )( n +1) (cid:17) + a (cid:16) − rn (1+ r )( n +1) (cid:17)(cid:17) .
12n contrast, by construction of the order book density f , the borrowing by each firm at equi-librium (and therefore total system wide borrowing) is greater under the VWAP framework thanthe LOB framework, i.e., h − s V W AP ( q V W AP , ¯ q V W AP )¯ q V W AP ≤ h − s LOB ( q LOB , ¯ q LOB )¯ q LOB , withstrict ordering on the same interval as given above.
In contrast to the symmetric system above, we now wish to consider a system in which the VWAPsetting results in fewer liquidations and more borrowing than the LOB framework. To do this, let’sconsider a simple heterogeneous n = 2 bank setting with r = 0 . a = (1 , h = (0 . , . f ( s ) = 1 − αs and haircut function g ( s ) = − αs , but with the specific price impact parameter α = 0 .
05. With this construction, theclearing liquidations and prices can be determined numerically to be • s V W AP = (0 , . q V W AP = 0 . q V W AP = (0 . , . . • s LOB = (0 . , . q LOB = 0 . q LOB = (0 . , . . As desired at the beginning of this example, total liquidations are less for both banks (i.e., s V W AP < s LOB ), but borrowing by both banks has the opposite order (i.e., h i − s V W APi ¯ q V W APi > h i − s LOBi ¯ q LOBi , i = 1 , . ). This is the opposite order from the symmetric case study considered above;as such, there is no consistent order between the VWAP and LOB settings that can be determined. As shown in the prior two examples, there is no consistent ordering between the VWAP and LOBsettings. For symmetric systems and, more generally, systems close to symmetric, if the marketregulators wish to promote borrowing over liquidations, then the LOB framework is preferable;however, for certain heterogeneous systems, the VWAP framework may be preferable to that sameregulator. As such, the use of stress testing of different market mechanisms is of the paramountimportance in order to determine the optimal market mechanism.We wish to make one final consideration on the comparison of the VWAP and LOB frame-works. We conjecture that the distinction between the two setting occurs only if some bank is bothliquidating and borrowing. Most prior works, e.g., Amini et al. [2016], consider only the situationin which firms can only liquidate in order to cover their liabilities. Without borrowing allowed, theVWAP and LOB frameworks will always coincide at the aggregate level. As such the mechanismchoice of ¯ f is irrelevant when considered in the standard literature (which is written in a VWAPstyle manner). We conclude this work with a consideration of financial system calibrated to 2011 European bankingdata. This stress test data has been utilized in numerous prior studies for studying interbank13
Interest rate r -3 T o t a l li qu i da t i on s Impact of Interest Rates on Liquidations
VWAPLOB (a) Total liquidations
Interest rate r -3 T o t a l bo rr o w i ng Impact of Interest Rates on Borrowing
VWAPLOB (b) Total borrowing
Figure 1: Summary statistics of the European banking sector’s response to a changing interest rateenvironment.liability networks (e.g., Gandy and Veraart [2016], Chen et al. [2016], Feinstein [2019]). We willcalibrate and utilize this EBA dataset in much the same way as in Bichuch and Feinstein [2019],i.e., to have a more realistic system but one that still requires heuristics and, as such, is fordemonstration purposes only.As a stylized bank balance sheet, we will consider two categories of assets: cash assets c i and illiquid assets a i . We will additionally consider two categories of liabilities: external liabilities ¯ p i and capital C i . In order to determine these values, we calibrate the system as in Bichuch andFeinstein [2019] but ignoring all interbank obligations considered as cash so as to discount defaultcontagion and focus solely on price-mediate contagion as discussed in the remainder of this work.The total assets T i and capital C i are provided by this dataset directly for each bank i . The externalliabilities ¯ p i = T i − C i are computed by balance sheet construction. It remains to split the totalassets into cash and illiquid assets; we make this split according to the tier 1 capital ratio R i , i.e., c i = R i T i and a i = (1 − R i ) T i .In order to complete our model, we need to consider the remaining parameters of the system.We set the market capitalization M = P ni =1 a i to be the total number of shares of the illiquidassets held by the banks. For this example we consider the linear order book density function f ( s ) = 1 − αs and haircut function g ( s ) = − αs for α = M (i.e., a 0 .
30 euro haircut ischarged on top of the “market price” f ( s )). By construction, this setting satisfies all conditionsof Theorem 3.6. We will focus on the impacts of altering the interest rate environment in orderto compare the VWAP and LOB settings. This is undertaken in the prevailing low interest rateenvironment during the period from which this data is collected. For this study, no external shockis applied to the financial system.For our consideration, we compare the VWAP and LOB settings while varying the interestrate environment. The results of varying the interest rate is displayed in Figure 1. As expected,14otal liquidations (Figure 1a) increase as the interest rate increases, whereas the total borrowing(Figure 1b) is exactly the reverse of the total liquidations and, as such, is decreasing as the interestrate increases. Notably, as discussed in the case studies of Section 4.1, under some interest rateenvironments VWAP encourages more borrowing than LOB and vice versa under other interestrate environments. We find that, system-wide, there is less selling and more borrowing in the LOBsetting for higher interest rates. Notably, the LOB setting results in a non-smooth response as afunction of the interest rate r . This results from the heterogeneous prices actualized by all banks;due to these varying prices, each bank switches strategies at varying interest rates. This is incontrast to the VWAP setting in which, though the banks are heterogeneous, the strategies of thebanks mostly overlap. With this notion, it becomes clear that LOB provides greater flexibility foran intervention to control fire sales through the manipulation of interest rates. In this work, we have considered a model of a system of banks that need to raise funds to covertheir liquidity shortfalls. These firms decide on an optimal combination to raise the money throughborrowing in a repo market and selling an illiquid asset in a fire-sale, with both the haircut and thefire-sale prices dependent on actions of other banks. We focused on two frameworks to determinethe fire-sale prices: the volume weighted average price and a notion of the limit order book inorder to capture notions of pricing dynamics in this, otherwise, static model. We found sufficientconditions for existence and uniqueness of the Nash equilibrium in this game. Finally, we havecompared the VWAP and the LOB settings analytically when the banks are identical and performa numerical study using the 2011 EBA data.
References
Gara Afonso, Kyungmin Kim, Antoine Martin, Ed Nosal, Simon Potter, and Sam Schulhofer-Wohl. Monetarypolicy implementation with an ample supply of reserves.
FRB of New York Staff Report , (910), 2020.Hamed Amini, Damir Filipovi´c, and Andreea Minca. Systemic risk with central counterparty clearing. SwissFinance Institute Research Paper No. 13-34, Swiss Finance Institute, 2013.Hamed Amini, Damir Filipovi´c, and Andreea Minca. Uniqueness of equilibrium in a payment system withliquidation costs.
Operations Research Letters , 44(1):1–5, 2016.Kartik Anand, Ben Craig, and Goetz Von Peter. Filling in the blanks: Network structure and interbankcontagion.
Quantitative Finance , 15(4):625–636, 2015.Tathagata Banerjee and Zachary Feinstein. Price mediated contagion through capital ratio requirements.2019. Working paper.Marco Bardoscia, Paolo Barucca, Adam Brinley Codd, and John Hill. The decline of solvency contagionrisk.
Bank of England Staff Working Paper , 662, 2017. axim Bichuch and Zachary Feinstein. Optimization of fire sales and borrowing in systemic risk. SIAMJournal on Financial Mathematics , 10:68–88, 2019.Michael Boss, Helmut Elsinger, Martin Summer, and Stefan Thurner. Network topology of the interbankmarket.
Quantitative Finance , 4(6):677–684, 2004.Yann Braouezec and Lakshithe Wagalath. Risk-based capital requirements and optimal liquidation in astress scenario.
Review of Finance , 22(2):747–782, 2018.Yann Braouezec and Lakshithe Wagalath. Strategic fire-sales and price-mediated contagion in the bankingsystem.
European Journal of Operational Research , 274(3):1180–1197, 2019.Markus K. Brunnermeier. Deciphering the liquidity and credit crunch 2007-2008.
The Journal of EconomicPerspectives , 23(1):77–100, 2009.Agostino Capponi, Peng-Chu Chen, and David D. Yao. Liability concentration and systemic losses in financialnetworks.
Operations Research , 64(5):1121–1134, 2016.Stephen G Cecchetti. Crisis and responses: the federal reserve in the early stages of the financial crisis.
Journal of Economic Perspectives , 23(1):51–75, 2009.Nan Chen, Xin Liu, and David D. Yao. An optimization view of financial systemic risk modeling: Thenetwork effect and the market liquidity effect.
Operations Research , 64(5), 2016.Rodrigo Cifuentes, Hyun Song Shin, and Gianluigi Ferrucci. Liquidity risk and contagion.
Journal of theEuropean Economic Association , 3(2-3):556–566, 2005.Larry Eisenberg and Thomas H. Noe. Systemic risk in financial systems.
Management Science , 47(2):236–249, 2001.Matthew Elliott, Benjamin Golub, and Matthew O. Jackson. Financial networks and contagion.
AmericanEconomic Review , 104(10):3115–3153, 2014.Helmut Elsinger. Financial networks, cross holdings, and limited liability. ¨Osterreichische Nationalbank(Austrian Central Bank) , 156, 2009.Helmut Elsinger, Alfred Lehar, and Martin Summer. Network models and systemic risk assessment. In
Handbook on Systemic Risk , pages 287–305. Cambridge University Press, 2013.Zachary Feinstein. Financial contagion and asset liquidation strategies.
Operations Research Letters , 45(2):109–114, 2017.Zachary Feinstein. Obligations with physical delivery in a multi-layered financial network.
SIAM Journalon Financial Mathematics , 10(4):877–906, 2019.Zachary Feinstein. Capital regulation under price impacts and dynamic financial contagion.
EuropeanJournal of Operational Research , 281(2):449–463, 2020.Zachary Feinstein and Fatena El-Masri. The effects of leverage requirements and fire sales on financialcontagion via asset liquidation strategies in financial networks.
Statistics & Risk Modeling , 34(3-4):109–114, 2017. rasanna Gai and Sujit Kapadia. Contagion in financial networks. Bank of England Working Papers 383,Bank of England, 2010.Prasanna Gai, Andrew Haldane, and Sujit Kapadia. Complexity, concentration and contagion. Journal ofMonetary Economics , 58(5):453–470, 2011.Axel Gandy and Luitgard A.M. Veraart. A Bayesian methodology for systemic risk assessment in financialnetworks.
Management Science , 2016. DOI: 10.1287/mnsc.2016.2546.Paul Glasserman and H. Peyton Young. How likely is contagion in financial networks?
Journal of Bankingand Finance , 50:383–399, 2015.Gary B. Gorton and Andrew Metrick.
Journal of Financial Economics , 104(3):425–451, 2012.Christian Gouri´eroux, J.-C. H´eam, and Alain Monfort. Bilateral exposures and systemic solvency risk.
Canadian Journal of Economics , 45(4):1273–1309, 2012.Robin Greenwood, Augustin Landier, and David Thesmar. Vulnerable banks.
Journal of Financial Eco-nomics , 115(3):3–28, 2015.Grzegorz Halaj and Christoffer Kok. Modelling the emergence of the interbank networks.
QuantitativeFinance , 15(4):653–671, 2015.Anne-Caroline H¨user. Too interconnected to fail: A survey of the interbank networks literature.
Journal ofNetwork Theory in Finance , 1(3):1–50, 2015.Jane Ihrig, Zeynep Senyuz, and Gretchen C Weinbach. The feds ample-reserves approach to implementingmonetary policy. 2020.Erland Nier, Jing Yang, Tanju Yorulmazer, and Amadeo Alentorn. Network models and financial stability.
Journal of Economic Dynamics and Control , 31(6):2033–2060, 2007.Stephen Quinn, William Roberds, and Charles M Kahn. Standing repo facilities, then and now.
FederalReserve Bank of Atlanta’s Policy Hub , (1), 2020.Leonard C.G. Rogers and Luitgard A.M. Veraart. Failure and rescue in an interbank network.
ManagementScience , 59(4):882–898, 2013.J. Ben Rosen. Existence and uniqueness of equilibrium points for concave n-person games.
Econometrica ,33(3):520–534, 1965.Jeremy Staum. Counterparty contagion in context: Contributions to systemic risk. In
Handbook on SystemicRisk , pages 512–548. Cambridge University Press, 2013.Christian Upper. Simulation methods to assess the danger of contagion in interbank markets.
Journal ofFinancial Stability , 7(3):111–125, 2011.Stefan Weber and Kerstin Weske. The joint impact of bankruptcy costs, fire sales and cross-holdings onsystemic risk in financial networks.
Probability, Uncertainty and Quantitative Risk , 2(1):9, June 2017. Proofs of Section 3
A.1 Proof of Proposition 3.2
In both the VWAP and LOB settings, for a fixed ( q, ¯ q , ..., ¯ q n ) ∈ b Q the existence of an equilibrium ¯ s ( q, ¯ q , ..., ¯ q n )follows along the same steps as the proof of Theorem 2.2. We next show the uniqueness of ¯ s ( q, ¯ q , ..., ¯ q n ) byutilizing the results of Rosen [1965] on convex games. A.1.1 Volume weighted averaged price
Proof.
In this case, the uniqueness of ¯ s ( q, ¯ q , ..., ¯ q n ) follows from Bichuch and Feinstein [2019][Theorem 3.2],as soon as we verify that the assumptions of that theorem hold.Recall that ¯ f i is independent of the index i . Also, note that that we can write ¯ f i as a function of thetotal liquidation s − i , where s − i = n X j = i,j =1 s j , (A.1)¯ f i ( s i , s − i ) =: ˆ f ( s i + s − i ) = 1 s i + s − i Z s i + s − i f ( s ) ds. (A.2)We assume that f satisfies Assumption 3.1, and proceed to verify that Bichuch and Feinstein [2019] [As-sumption 2.1] is also satisfied by ˆ f . Indeed, ˆ f ′ ( s ) = − s R s f ( u ) du + f ( s ) s ≤ − sf ( s ) s + f ( s ) s = 0 . It is alsoeasily seen that d ds ( s ˆ f ( s )) = f ′ ( s ) < . Lastly we need to show that ˆ f ′′ ≥ . Calculate that s ˆ f ′′ ( s ) = s R s f ( u ) du − f ( s ) + sf ′ ( s ) . Since f isconvex, we have that f ( s ) − f (0) ≤ sf ′ ( s ) , and thus it is sufficient to show that s R s f ( u ) du − f (0) − f ( s ) ≥ . Using the fact that f is convex, we have that λf ( s )+(1 − λ ) f ( s ) ≤ f ( λs +(1 − λ ) s ) , λ ∈ [0 , λ ∈ [0 ,
1] gives f ( s )+ f ( s )2 ≤ s − s R s s f ( u ) du, which gives the desired result. A.1.2 Limit order book
Proof.
Recall from Rosen [1965] that for s ∈ R n , the function s H ( s ; ρ ) is diagonally strictly convex, if forsome (fixed) ρ ∈ R n + and for every s , s ∈ R n , s = s , we have ( s − s ) ⊤ γ ( s ; ρ ) − ( s − s ) ⊤ γ ( s ; ρ ) < γ ( s ; ρ ) = ∂ s H ( s ; ρ )... ∂ s n H n ( s ; ρ n ) = ρ (cid:16) − (1 + r ) f (cid:16)P k ℓ =1 ( n − ( ℓ − s [ ℓ ] − s [ ℓ − ) (cid:17)(cid:17) ... ρ n (cid:16) − (1 + r ) f (cid:16)P k n ℓ =1 ( n − ( ℓ − s [ ℓ ] − s [ ℓ − ) (cid:17)(cid:17) , where k i is such that s [ k i ] = s i . Additionally, [Rosen, 1965, Theorem 6] shows that a sufficient condition for H to be diagonally strictly convex is if Γ( s ; ρ ) + Γ( s ; ρ ) ⊤ is a symmetric positive definite matrix for every s ∈ R n and some ρ ∈ R n + , where Γ is the Jacobian matrix of γ with respect to s . Without loss of generality,for fixed value s , assume k i = i for every bank.Set ρ i = r then (cid:2) Γ( s ; ρ ) + (Γ( s ; ρ )) ⊤ (cid:3) ij = − (1 + I { i = j } [2( n − i ) + 1]) f ′ i ∨ j − X ℓ =1 s ℓ + ( n − ( i ∨ j − s i ∨ j ! . hus, in full matrix notation, we find Γ( s ; ρ ) + Γ( s ; ρ ) ⊤ = A ( s ) + n X j =1 B j ( s ) A ( s ) = − diag [2( n − i ) + 1] f ′ i − X ℓ =1 s ℓ + ( n − ( i − s i !! B j ( s ) = (cid:20) f ′ (cid:18) j − P ℓ =1 s ℓ + ( n − ( j − s j (cid:19) − f ′ (cid:18) j P ℓ =1 s ℓ + ( n − j ) s j +1 (cid:19)(cid:21) j × j j × ( n − j ) ( n − j ) × j ( n − j ) × ( n − j ) if j < n − f ′ ( P nℓ =1 s ℓ ) n × n if j = n. For any liquidations s , by construction, the matrix A ( s ) is positive definite and B j ( s ) is positive semidefinite(by nondecreasing property of f ′ ). The uniqueness of ¯ s ( q, ¯ q , ..., ¯ q n ) follows from Rosen [1965][Theorem2]. A.2 Proof of Theorem 3.3
Our goal is to apply Tarski’s fixed point theorem, to do which, we need to prove that n X i =1 ¯ s i ( q ↓ , ¯ q ↓ ) ≥ n X i =1 ¯ s i ( q ↑ , ¯ q ↑ ) , (A.3)for ( q ↓ , ¯ q ↓ ) ≤ ( q ↑ , ¯ q ↑ ). Proof.
Recall the definitions of ˆ f and s − i given in (A.2) and (A.1) respectively. Therefore, we can assumethat ¯ q := ¯ q = ... = ¯ q n . First we note that for i = 1 , ..., n ¯ s i ( q, ¯ q n ) = a i if h i ≥ a i ¯ q, (cid:16) h i − a i q ¯ q − q (cid:17) + ∨ h s i (¯ s − i ( q, ¯ q n )) ∧ h i ¯ q i if h i < a i ¯ q, (A.4)where s i is the solution to1 − (1 + r )( ˆ f ( s i + ¯ s − i ( q, ¯ q n )) + s i ˆ f ′ ( s i + ¯ s − i ( q, ¯ q n ))) = 0 . (A.5)With this construction we wish to note that bank i is defaulting and has no other option but to liquidateall its assets if and only if h i ≥ a i ¯ q . Indeed, as noted previously in the body of this work, in the oppositecase h i < a i ¯ q , we have that:1. h i ¯ q < a i if and only if h i < a i ¯ q .2. h i − a i q ¯ q − q < a i if and only if h i < a i ¯ q .3. h i − a i q ¯ q − q < h i ¯ q if and only if h i < a i ¯ q .Therefore, ¯ s i ( q, ¯ q n ) , i = 1 , ..., n is well defined in (A.4), and ¯ s i ( q, ¯ q ) < a i in all those cases.First consider the case when all banks keep at the same liquidation strategy, in other words the definitionof ¯ s i in (A.4) is equal to the same term (i.e., among a i , h i ¯ q , (cid:16) h i − a i q ¯ q − q (cid:17) + , and s i ). Then for i = 1 , ..., n : • If ¯ s i = a i then ∂ q ¯ s i , ∂ ¯ q ¯ s i = 0. If ¯ s i = h i ¯ q , then ∂ q ¯ s i = 0 , ∂ ¯ q ¯ s i < • If ¯ s i = (cid:16) h i − a i q ¯ q − q (cid:17) + , first assume that h i ≥ a i q , in addition to h i < a i ¯ q . The former results in ∂ ¯ q ¯ s i ≤ ∂ q ¯ s i <
0. If, instead, h i < a i q then ∂ ¯ q ¯ s i = ∂ q ¯ s i = 0. Note that wehave also used our assumption that ¯ q > q . • The last case to consider is when ¯ s i = s i . This is the most interesting case because ( s i ) ′ ∈ ( − , ∂ ¯ q s i = ( s i ) ′ ∂ ¯ q s − i ≥
0. This case, requires a more careful analysis as follows.Let I be the set of banks j = 1 , ..., n , such that ¯ s j = s j , then differentiating (A.5) w.r.t. ¯ q , and usingthe fact that ( s i ) ′ ∈ ( − , ∂ ¯ q ¯s I = − (diag( | I | − c ) + c1 ⊤| I | ) − c X j I ∂ ¯ q ¯ s j , for some c ∈ [0 , | I | . First, we wish to show that diag( | I | − c ) + c1 ⊤| I | is invertible:det (cid:16) diag( | I | − c ) + c1 ⊤| I | (cid:17) = det c i · · · c i c i · · · c i ... ... . . . ... c i | I | c i | I | · · · = det − (1 − c i ) · · · − (1 − c i ) c i − c i · · · c i | I | · · · − c i | I | = − c i ) X i ∈ I \{ i } c i − c i Y i ∈ I \{ i } (1 − c i )= − c i ) X i ∈ I c i − c i − (1 − c i ) c i − c i ! Y i ∈ I \{ i } (1 − c i )= X i ∈ I c i − c i ! Y i ∈ I (1 − c i ) , where the 2nd line follows from subtracting the first column from every subsequent column and the3rd line by using the Schur complement to determine the determinant. Thus we find thatdet (cid:16) diag( | I | − c ) + c1 ⊤| I | (cid:17) = X i ∈ I c i − c i ! Y i ∈ I (1 − c i ) > . Taking this all together: n X i =1 ∂ ¯ q ¯ s i = X j I ∂ ¯ q ∂ ¯ s j − ⊤| I | (diag( | I | − c ) + c1 ⊤| I | ) − c X j I ∂ ¯ q ¯ s j = (cid:16) − ⊤| I | (diag( | I | − c ) + c1 ⊤| I | ) − c (cid:17) X j I ∂ ¯ q ¯ s j . (A.6) oreover 1 − ⊤| I | (diag( | I | − c ) + c1 ⊤| I | ) − c ≥
0, since: ⊤| I | (diag( | I | − c ) + c1 ⊤| I | ) − c = ⊤| I | diag( | I | − c ) − − diag( | I | − c ) − c1 ⊤| I | diag( | I | − c ) − ⊤| I | diag( | I | − c ) − c ! c = X i ∈ I c i − c i −
11 + P i ∈ I c i − c i X i ∈ I X j ∈ I c i c j (1 − c i )(1 − c j )= −
11 + P i ∈ I c i − c i X j ∈ I c j − c j X i ∈ I c i − c i = X i ∈ I c i − c i ! − X i ∈ I c i − c i ≤ , where the first equality follows from the Sherman-Morrison matrix identity.It now follows from (A.6) that P ni =1 ∂ ¯ q ¯ s i ≤
0, as desired. The same calculation also shows that P ni =1 ∂ q ¯ s i ≤
0, and therefore (A.3) holds.Finally, in the case, that some banks may switch liquidation strategies, we use the fact that the map-pings ¯ s i ( · , · ) , s i ( · ) i = 1 , ..., n are continuous. If there is a switch in strategies for bank i at some fixedpoint q , ¯ q , then by continuity, of all the mappings in (A.4) it follows that both one sided derivatives P ni =1 ∂ +¯ q ¯ s i , P ni =1 ∂ − ¯ q ¯ s i ≤
0. Therefore P ni =1 ¯ s i is decreasing in ¯ q . Similar result also holds for q . Weconclude that (A.3) holds. A.3 Proof of Theorem 3.6
To show uniqueness, we consider the equilibrium prices, as a mapping of ( q ∗ , ¯ q ∗ ) to liquidating positions ofbanks ¯ s ( q ∗ , ¯ q ∗ ), and then to the resulting prices, and show the uniqueness of a fixed point to this mapping.To simplify notation throughout this proof, let Q i , i ∈ { , n } , denote the set of attainable prices. The caseof VWAP corresponds to i = 1 and Q := n ( g ( s ) , ˆ f ( s )) | s ∈ [0 , M ] o and the case of LOB corresponds to i = 1, and Q n := (cid:8) ( g ( P ni =1 s i ) , ¯ f ( s ) , ..., f n ( s )) | s ∈ D (cid:9) . Moreover, for convenience define I := (cid:8) i ∈ { , ..., n } | ¯ s i = s i (cid:9) , I U := (cid:26) i ∈ { , ..., n } | ¯ s i = h i ¯ q i (cid:27) , (A.7) I a := { i ∈ { , ..., n } | ¯ s i = a i } , I L := ( i ∈ { , ..., n } | ¯ s i = (cid:18) h i − a i q ¯ q i − q (cid:19) + ) . (A.8)As before, we divide the proof into the VWAP and LOB cases: A.3.1 Volume weighted average price
Proof.
We first fix ¯ q = ˆ f ( s ) , q = g ( s ) for some s ∈ [0 , M ] (recall the definition of ˆ f from (A.2)) and lookfor an equilibrium ¯ s i ( q, ¯ q n ) = s ∗ i ( P j = i ¯ s j ( q, ¯ q n ) , q, ¯ q n ) for all i = 1 , ..., n . That is for the modified Nashequilibrium given by (2.2) and formulated explicitly in (A.4).The next goal is to show ( q, ¯ q ) (Φ( q, ¯ q ) , ¯Φ( q, ¯ q )) = ( g ( P nj =1 ¯ s j ( q, ¯ q n )) , ˆ f ( P nj =1 ¯ s j ( q, ¯ q n ))) , is acontraction mapping. That is, our goal is to show that (cid:12)(cid:12) ¯Φ( q , ¯ q ) − ¯Φ( q , ¯ q ) (cid:12)(cid:12) ≤ ¯ L (cid:13)(cid:13) ( q , ¯ q ) − ( q , ¯ q ) (cid:13)(cid:13) ∞ ,and (cid:12)(cid:12) Φ( q , ¯ q ) − Φ( q , ¯ q ) (cid:12)(cid:12) ≤ L (cid:13)(cid:13) ( q , ¯ q ) − ( q , ¯ q ) (cid:13)(cid:13) ∞ with L, ¯ L < q , ¯ q ) , q , ¯ q ) ∈ Q . Without loss of generality, for this proof we will assume q ≤ q ; therefore ( q , ¯ q ) ∈ b Q aswell.Indeed, with the convention that 0 / (cid:12)(cid:12) ¯Φ( q , ¯ q ) − ¯Φ( q , ¯ q ) (cid:12)(cid:12) k ( q , ¯ q ) − ( q , ¯ q ) k ∞ ≤ (cid:12)(cid:12) ¯Φ( q , ¯ q ) − ¯Φ( q , ¯ q ) (cid:12)(cid:12) k ( q , ¯ q ) − ( q , ¯ q ) k ∞ + (cid:12)(cid:12) ¯Φ( q , ¯ q ) − ¯Φ( q , ¯ q ) (cid:12)(cid:12) k ( q , ¯ q ) − ( q , ¯ q ) k ∞ ≤ (cid:12)(cid:12) ¯Φ( q , ¯ q ) − ¯Φ( q , ¯ q ) (cid:12)(cid:12) | ¯ q − ¯ q | + (cid:12)(cid:12) ¯Φ( q , ¯ q ) − ¯Φ( q , ¯ q ) (cid:12)(cid:12) | q − q | = 1 | ¯ q − ¯ q | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ f n X j =1 ¯ s j ( q , ¯ q n ) − ˆ f n X j =1 ¯ s j ( q , ¯ q n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 1 | q − q | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ f n X j =1 ¯ s j ( q , ¯ q n ) − ˆ f n X j =1 ¯ s j ( q , ¯ q n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ − ˆ f ′ (0) max ( q, ¯ q ) ∈Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X j =1 ∂ ¯ q ¯ s j ( q, ¯ q n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + max ( q, ¯ q ) ∈Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X j =1 ∂ q ¯ s j ( q, ¯ q n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (A.9)Similarly for Φ( q, ¯ q ). Thus to be a contraction mapping, it is sufficient to show that − ˆ f ′ (0) max ( q, ¯ q ) ∈Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X j =1 ∂ ¯ q ¯ s j ( q, ¯ q n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + max ( q, ¯ q ) ∈Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X j =1 ∂ q ¯ s j ( q, ¯ q n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < , − g ′ (0) max ( q, ¯ q ) ∈Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X j =1 ∂ ¯ q ¯ s j ( q, ¯ q n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + max ( q, ¯ q ) ∈Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X j =1 ∂ q ¯ s j ( q, ¯ q n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < . In order to show this, consider the sensitivity of ¯ s ( q, ¯ q n ) with respect to q, ¯ q . Recall the construc-tion of ¯ s given by (A.4). Recall the definitions of I U , I L , I from (A.7) and (A.8). Assume that a i , h i ¯ q , s i ( P j = i ¯ s j ( q, ¯ q n )), h i − a i q ¯ q − q are all different for all i = 1 , ..., n , so that together with the continuity of s itfollows that ¯ s is differentiable with respect to q, ¯ q and its derivatives for a given bank i are given by ∂ ¯ q ¯ s i ( q, ¯ q n ) = − I { i ∈ I U } h i ¯ q − I { i ∈ I L } h i − a i q (¯ q − q ) + ( s i ) ′ ( X j = i ¯ s j ( q, ¯ q n ))( X j = i ∂ ¯ q ¯ s j ( q, ¯ q n )) I { i ∈ I } ! ,∂ q ¯ s i ( q, ¯ q n ) = I { i ∈ I L } h i − a i ¯ q (¯ q − q ) + ( s i ) ′ ( X j = i ¯ s j ( q, ¯ q n ))( X j = i ∂ q ¯ s j ( q, ¯ q n )) I { i ∈ I } ! . (A.10)Here, the derivative of the optimal liquidations ( s i ( s − i )) can be found via implicit differentiation: ( s i ) ′ ( s − i ) = − ˆ f ′ ( s − i + s i ( s − i ))+ s i ( s − i ) ˆ f ′′ ( s − i + s i ( s − i ))2 ˆ f ′ ( s − i + s i ( s − i ))+ s i ( s − i ) ˆ f ′′ ( s − i + s i ( s − i )) . Therefore ( s i ) ′ ( s − i ) ∈ ( − ,
0] for all banks i such that ¯ s i = s i ifˆ f ′ ( s ) + s ˆ f ′′ ( s ) ≤ s ∈ [0 , M ]. olving the system (A.10), it follows that ∂ ¯ q ¯ s ( q, ¯ q )= − I − diag ( s i ) ′ ( X j = i ¯ s j ( q, ¯ q n ))( X j = i ∂ ¯ q ¯ s j ( q, ¯ q n )) I { i ∈ I } i =1 ,...,n ( n × n − I ) − × (cid:18) diag (cid:16)(cid:2) I { i ∈ I U } (cid:3) i =1 ,...,n (cid:17) h ¯ q + diag (cid:16)(cid:2) I { i ∈ I L } (cid:3) i =1 ,...,n (cid:17) h − q a (¯ q − q ) (cid:19) ,∂ q ¯ s ( q, ¯ q )= I − diag ( s i ) ′ ( X j = i ¯ s j ( q, ¯ q n ))( X j = i ∂ q ¯ s j ( q, ¯ q n )) I i ∈ I } i =1 ,...,n ( n × n − I ) − × diag (cid:16)(cid:2) I i ∈ I L } (cid:3) i =1 ,...,n (cid:17) h − ¯ q a (¯ q − q ) . Using the fact that ( s i ) ′ ( s − i ) ∈ ( − ,
0] for i = 1 , ..., n as follows from the sufficient assumption of thetheorem, it thus follows that (cid:12)(cid:12) ⊤ n ∂ ¯ q ¯ s ( q, ¯ q ) (cid:12)(cid:12) ≤ max d ∈ [0 , n (cid:12)(cid:12)(cid:12)(cid:12) ⊤ n ( I + diag( d )( n × n − I )) − (cid:18) diag (cid:16)(cid:2) I { d i =0 ,i ∈ I U } (cid:3) i =1 ,...,n (cid:17) h ¯ q + diag (cid:16)(cid:2) I { d i =0 ,i ∈ I L } (cid:3) i =1 ,...,n (cid:17) h − q a (¯ q − q ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . To compute this maximum, let B ( d ) := I + diag( d )( n × n − I ) = diag ( n − d ) + d1 ⊤ n . By the Sherman-Morrison formula B ( d ) − = diag ( n − d ) − − ⊤ n diag( n − d ) − d diag ( n − d ) − d1 ⊤ n diag ( n − d ) − . Itnow follows that for any j = 1 , ..., n n X i =1 (cid:0) B ( d ) − (cid:1) ij I { d j =0 } = 11 + P nk =1 d k − d k n X k =1 d k − d k − X k = j d k − d k I { d j =0 } = I { d j =0 } P nk =1 d k − d k . Together with Remark 1 we conclude thatmax ( q, ¯ q ) ∈Q (cid:12)(cid:12) ⊤ n ∂ ¯ q ¯ s ( q, ¯ q ) (cid:12)(cid:12) ≤ max ( q, ¯ q ) ∈Q , d ∈ [0 , n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⊤ n B ( d ) − (A.11) × (cid:18) diag (cid:16)(cid:2) I { d i =0 ,i ∈ I U } (cid:3) i =1 ,...,n (cid:17) h ¯ q + diag (cid:16)(cid:2) I { d i =0 ,i ∈ I L } (cid:3) i =1 ,...,n (cid:17) h − q a (¯ q − q ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max ¯ q ∈ [ ˆ f ( M ) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⊤ n h ¯ q ∧ a ¯ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + max ( q, ¯ q ) ∈Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⊤ n h − q a ¯ q − q ∧ a ¯ q − q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max ¯ q ∈ [ ˆ f ( M ) , P ni =1 a i ¯ q + max ( q, ¯ q ) ∈Q P ni =1 a i ¯ q − q ≤ M ˆ f ( M ) + M min s ∈ [0 ,M ] (cid:16) ˆ f ( s ) − g ( s ) (cid:17) . imilarly, max ( q, ¯ q ) ∈Q (cid:12)(cid:12) ⊤ n ∂ q ¯ s ( q, ¯ q ) (cid:12)(cid:12) ≤ max ( q, ¯ q ) ∈Q , d ∈ [0 , n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⊤ n B ( d ) − diag (cid:16)(cid:2) I { d i =0 ,i ∈ I L } (cid:3) i =1 ,...,n (cid:17) h − ¯ q a (¯ q − q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max ( q, ¯ q ) ∈Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⊤ n h − ¯ q a ¯ q − q ¯ q − q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max ( q, ¯ q ) ∈Q P ni =1 a i ¯ q − q ≤ M min s ∈ [0 ,M ] (cid:16) ˆ f ( s ) − g ( s ) (cid:17) , where in the last inequality we have used that fact that a i ≥ h i − a i q ¯ q − q ≥ h i − a i ¯ q ¯ q − q = − a i + h i − a i q ¯ q − q ≥ − a i . Recalling(A.9), we conclude that (Φ , ¯Φ) is a contraction mapping if − M ( ˆ f ′ (0) ∧ ¯ g ′ (0)) < min s ∈ [0 ,M ] (cid:16) ˆ f ( s ) − g ( s ) (cid:17) .Finally, it can be seen that ˆ f ′ ( s ) = f ( s ) − ˆ f ( s ) s . Therefore, ˆ f ′ (0) = f ′ (0) . Recall that it was assumed that a i , h i ¯ q , h i − a i q ¯ q − q , s i ( P j = i ¯ s j ( q, ¯ q n )) are all different. If this assumption isviolated, say s i ( P j = i ¯ s j ( q, ¯ q n )) < h i − a i q ¯ q − q = h i ¯ q , then we need to consider one-sided derivatives. In that case,the derivative from the right ∂ ¯ q + ¯ s i ( q, ¯ q n ) = − h ¯ q , while the derivative from the left ∂ ¯ q − ¯ s i ( q, ¯ q n ) = − h i − a i q (¯ q − q ) . In this case, both one-sided derivatives would satisfy (A.11). The other cases, can be treated similarly.
A.3.2 Limit order book
Proof.
We first fix ¯ q = ¯ f ( s ) , q = g ( P ni =1 s i ) for some s ∈ D and look for an equilibrium ¯ s i ( q, ¯ q ) = s ∗ i (¯ s − i ( q, ¯ q ) , q, ¯ q ) which is explicitly provided by¯ s i ( q, ¯ q ) = a i if h i ≥ a i ¯ q i (cid:16) h i − a i q ¯ q i − q (cid:17) + ∨ h s i (¯ s − i ( q, ¯ q )) ∧ h i ¯ q i i if h i < a i ¯ q i (A.12)where s i (¯ s − i ) solves the first order condition1 − (1 + r ) f n X j =1 I { ¯ s j − I { ¯ s j
Initially, as in the prior proofs, assume that for each i = 1 , ..., n , the possible solutions to the optimizationfor ¯ s i from (A.4), namely a i , h i ¯ q , h i − a i q ¯ q − q , s i ( P j = i ¯ s j ), are all different. We want to study ∂ r ¯ s i for i = 1 , ..., n .From the previous assumption it follows that ∂ r ¯ s i = i ∈ I a − h i ¯ q ∂ r ¯ q if i ∈ I U (cid:16) h i − a i ¯ q (¯ q − q ) (cid:17) ∂ r q − (cid:16) h i − a i q (¯ q − q ) (cid:17) ∂ r ¯ q if i ∈ I L ∂ r s i if i ∈ I where I a , I U , I L , I were defined in (A.7) and (A.8).Before continuing, we will consider ∂ r ¯ s i for i ∈ I . By construction, we have − ( ˆ f ( s i + X j = i ¯ s j ) + s i ˆ f ′ ( s i + X j = i ¯ s j )) − (1 + r )(2 ˆ f ′ ( s i + X j = i ¯ s j ) + s i ˆ f ′′ ( s i + X j = i ¯ s j )) ∂ r ¯ s i − (1 + r ) X j = i ( ˆ f ′ ( s i + X j = i ¯ s j ) + s i ˆ f ′′ ( s i + X j = i ¯ s j )) ∂ r ¯ s j = 0 . ecall that every bank i ∈ I will satisfy the same condition, i.e., ∂ r ¯ s i = ∂ r ¯ s j for every i, j ∈ I . For nota-tional simplicity let s = s i , ∂ r s = ∂ r s i for arbitrary i ∈ I . Let c = ˆ f ′ ( | I | s + P j I ¯ s j )+ s ˆ f ′′ ( | I | s + P j I ¯ s j )2 ˆ f ′ ( | I | s + P j I ¯ s j )+ s ˆ f ′′ ( | I | s + P j I ¯ s j ) and d = − ˆ f ( | I | s + P j I ¯ s j )+ s ˆ f ′ ( | I | s + P j I ¯ s j )(1+ r )(2 ˆ f ′ ( | I | s + P j I ¯ s j )+ s ˆ f ′′ ( | I | s + P j I ¯ s j )) . Recall that by our Assumption 2.1, 0 ≤ c < d >
0. Therefore, it can be shown that ∂ r s = d c ( | I | − − c c ( | I | − X j I ∂ r ¯ s j . We can now consider the joint sensitivity of the haircut q and price ¯ q to interest rates: ∂ r q = X i ∈ I ∂ r s + X i I ∂ r ¯ s i g ′ ( n X i =1 ¯ s i )= | I | d c ( | I | −
1) + 1 − c c ( | I | − X j I ∂ r ¯ s i g ′ ( n X i =1 ¯ s i ) ∂ r ¯ q = X i ∈ I ∂ r s + X i I ∂ r ¯ s i ˆ f ′ ( n X i =1 ¯ s i )= | I | d c ( | I | −
1) + 1 − c c ( | I | − X j I ∂ r ¯ s i ˆ f ′ ( n X i =1 ¯ s i ) . To simplify notation, let ˜ c = − c c ( | I |− and ˜ d = | I | d c ( | I |− . Therefore ∂ r q = (cid:20) ˜ d + ˜ c (cid:18) h − a ¯ q (¯ q − q ) [ I { i ∈ I L } ] i − h ¯ q (cid:19) ∂ r q − ˜ c (cid:18) h − aq (¯ q − q ) [ I { i ∈ I L } ] i [ I { i ∈ I U } ] i (cid:19) ∂ r ¯ q (cid:21) g ′ ( n X i =1 ¯ s i ) ,∂ r ¯ q = (cid:20) ˜ d + ˜ c (cid:18) h − a ¯ q (¯ q − q ) [ I { i ∈ I L } ] i − h ¯ q (cid:19) ∂ r q − ˜ c (cid:18) h − aq (¯ q − q ) [ I { i ∈ I L } ] i [ I { i ∈ I U } ] i (cid:19) ∂ r ¯ q (cid:21) ˆ f ′ ( n X i =1 ¯ s i ) . That is, the sensitivity of the haircut and prices ( q, ¯ q ) w.r.t. the interest rate r is the solution of a linearsystem ∂ r q∂ r ¯ q ! = [ I − W ] − g ′ ( P ni =1 ¯ s i ) ˜ d ˆ f ′ ( P ni =1 ¯ s i ) ! = I + g ′ ( P ni =1 ¯ s i )ˆ f ′ ( P ni =1 ¯ s i ) ! (cid:16) h − a ¯ q (¯ q − q ) [ I { i ∈ I L } ] i − h h − aq (¯ q − q ) [ I { i ∈ I L } ] i + h ¯ q [ I { i ∈ I U } ] i i (cid:17) ˜ c − ˜ c h(cid:16) h − a ¯ q (¯ q − q ) [ I { i ∈ I L } ] i (cid:17) g ′ ( P ni =1 ¯ s i ) − (cid:16) h − aq (¯ q − q ) [ I { i ∈ I L } ] i + h ¯ q [ I { i ∈ I U } ] i (cid:17) ˆ f ′ ( P ni =1 ¯ s i ) i × g ′ ( P ni =1 ¯ s i ) ˜ d ˆ f ′ ( P ni =1 ¯ s i ) ! ,W = g ′ ( P ni =1 ¯ s i )ˆ f ′ ( P ni =1 ¯ s i ) ! (cid:16) h − a ¯ q (¯ q − q ) [ I { i ∈ I L } ] i − h h − aq (¯ q − q ) [ I { i ∈ I L } ] i + h ¯ q [ I { i ∈ I U } ] i i (cid:17) ˜ c. oreover, it also follows that ∂ r n X i =1 ¯ s i = ∂ r qg ′ ( P ni =1 ¯ s i )= 1 + ˜ c (cid:16)(cid:16) h − a ¯ q (¯ q − q ) [ I { i ∈ I L } ] i (cid:17) g ′ ( P ni =1 ¯ s i ) − (cid:16) h − aq (¯ q − q ) [ I { i ∈ I L } ] i + h ¯ q [ I { i ∈ I U } ] i (cid:17) ˆ f ′ ( P ni =1 ¯ s i ) (cid:17) − ˜ c h(cid:16) h − a ¯ q (¯ q − q ) [ I { i ∈ I L } ] i (cid:17) g ′ ( P ni =1 ¯ s i ) − (cid:16) h − aq (¯ q − q ) [ I { i ∈ I L } ] i + h ¯ q [ I { i ∈ I U } ] i (cid:17) ˆ f ′ ( P ni =1 ¯ s i ) i = 11 − ˜ c h(cid:16) h − a ¯ q (¯ q − q ) [ I { i ∈ I L } ] i (cid:17) g ′ ( P ni =1 ¯ s i ) − (cid:16) h − aq (¯ q − q ) [ I { i ∈ I L } ] i + h ¯ q [ I { i ∈ I U } ] i (cid:17) ˆ f ′ ( P ni =1 ¯ s i ) i . It follows that ∂ r P ni =1 ¯ s i > (cid:16) h − a ¯ q (¯ q − q ) [ I { i ∈ I L } ] i (cid:17) g ′ ( P ni =1 ¯ s i ) − (cid:16) h − aq (¯ q − q ) [ I { i ∈ I L } ] i + h ¯ q [ I { i ∈ I U } ] i (cid:17) ˆ f ′ ( P ni =1 ¯ s i ) < c , which happens if, for example, ˆ f ′ is small enough. B.2 Limit order book
Initially, again assume that for each i = 1 , ..., n , the possible solutions ( a i , h i ¯ q i , h i − a i q ¯ q i − q , s i ( P j = i ¯ s j )) to theoptimization (A.12) are all different. As in the VWAP case, we want to study ∂ r ¯ s i for i ∈ { , ..., n } . Fromthe previous assumption it follows that ∂ r ¯ s i = i ∈ I a , − h i ¯ q i ∂ r ¯ q i if i ∈ I U , (cid:16) h i − a i ¯ q i (¯ q i − q ) (cid:17) ∂ r q − (cid:16) h i − a i q (¯ q i − q ) (cid:17) ∂ r ¯ q i if i ∈ I L ,∂ r s i if i ∈ I , where I a , I U , I L , I were defined in (A.7) and (A.8).Recall s i (¯ s − i ) solves the first order condition1 − (1 + r ) f n X j =1 I { ¯ s j ¯ s j } h j − a j q j (¯ q j − q ) ∂ s j ¯ f i (¯ s ) ! j,i ≤ i,j ≤ n +1 . Note that N is such that N n +1 = 0 . Therefore, we have that W − = ( I + N ) − D − = (cid:0) I − N + N + ... + ( − n N n (cid:1) D − . Finally, W − = W − − W − (0 , , ..., , ⊤ h I { j ∈ I L } h j − a j q (¯ q j − q ) g ′ − I { j = n +1 } g ′ P i ∈ I L h i − a i q (¯ q i − q ) i j =1 ,...,n +1 W − h I { j ∈ I L } h j − a j q (¯ q j − q ) g ′ − I { j = n +1 } g ′ P i ∈ I L h i − a i q (¯ q i − q ) i j =1 ,...,n +1 W − (0 , , ..., , ⊤ ⊤ . To calculate ∂ r P ni =1 ¯ s i , recall that ∂ r ¯ q i = 0 for i ∈ I U , and from (B.2) it follows that ∂ r n X i =1 ¯ s i = X i ∈ I ∪ I L ∂ r ¯ s i = ∂ r qg ′ ( P ni =1 ¯ s i ) = (0 , , ..., , ⊤ W − b g ′ ( P ni =1 ¯ s i ) . It follows that ∂ r P ni =1 ¯ s i ≥ , , ..., , ⊤ W − b ≤ ..