Hedging with physical or cash settlement under transient multiplicative price impact
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Becherer, Dirk; Bilarev, Todor Article — Published Version Hedging with physical or cash settlement under transient multiplicative price impact Finance and Stochastics Provided in Cooperation with: Springer Nature Suggested Citation: Becherer, Dirk; Bilarev, Todor (2024) : Hedging with physical or cash settlement under transient multiplicative price impact, Finance and Stochastics, ISSN 1432-1122, Springer, Berlin, Heidelberg, Vol. 28, Iss. 2, pp. 285-328, https://doi.org/10.1007/s00780-024-00531-7 This Version is available at: https://hdl.handle.net/10419/315064 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/
Finance and Stochastics (2024) 28:285–328 https://doi.org/10.1007/s00780-024-00531-7 Hedging with physical or cash settlement under transient multiplicative price impact Dirk Becherer1·Todor Bilarev2 Received: 12 July 2018 / Accepted: 7 March 2023 / Published online: 15 March 2024 © The Author(s) 2024 Abstract We solve the superhedging problem for European options in an illiquid extension of the Black–Scholes model, in which transactions have transient price impact and the costs and strategies for hedging are affected by physical or cash settlement requirements at maturity. Our analysis is based on a convenient choice of reduced effective coordinates of magnitudes at liquidation for geometric dynamic programming. The price impact is transient over time and multiplicative, ensuring nonnegativity of underlying asset prices while maintaining an arbitrage-free model. The basic (log-)linear example is a Black–Scholes model with a relative price impact proportional to the volume of shares traded, where the transience for impact on log-prices is modelled like in Obizhaeva and Wang (J. Financ. Mark. 16:1–32, 2013) for nominal prices. More generally, we allow nonlinear price impact and resilience functions. The viscosity solutions describing the minimal superhedging price are governed by the transient character of the price impact and by the physical or cash settlement specifications. The pricing equations under illiquidity extend no-arbitrage pricing à la Black–Scholes for complete markets in a non-paradoxical way (cf. Çetin et al. (Finance Stoch. 14:317–341, 2010)) even without additional frictions, and can recover it in base cases. Keywords Transient price impact ·Multiplicative impact ·Hedging ·Option settlement ·Resilience ·Viscosity solution ·Geometric dynamical programming · Effective coordinates Mathematics Subject Classification 49L20 ·49L25 ·60H30 ·91G20 ·93E20 Disclaimer: The opinions expressed in this publication are those of the authors. They do not purport to reflect views of their institutions. D. Becherer [email protected] T. Bilarev bilarev[email protected] 1Humboldt Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, D-10099 Berlin, Germany 2FactSet Research Systems Inc., 2 Srebama Str., 1407 Sofia, Bulgaria
286 D. Becherer, T. Bilarev JEL Classification C61 ·G12 ·G13 1Introduction By using methods of stochastic target problems (see Soner and Touzi [29]) and geometric dynamic programming in suitably chosen reduced effective coordinates of magnitudes at liquidation, we solve the superhedging problem for European derivatives in a market model with multiplicative transient price impact. If the market for the underlying is illiquid or if large volumes are to be traded, there is price impact and feedback effects from hedging can affect the minimal superhedging prices (see Frey [19], Schönbucher and Wilmott [28], Frey and Polte [20], Bank and Baum [3]) and the corresponding hedging strategies which almost surely superreplicate the option. Since trades at maturity can alter the price of the underlying and thereby the derivative payout, settlement specifications for the option (in cash or in physical units) become relevant and this shows in pricing and hedging equations. As our results address hedging in terms of liquidation values, i.e., “real” instead of “paper” values (see Jarrow [24]), we recover such effects, whereas Frey [19], Frey and Polte [20], Bouchard et al. [13] study hedging in terms of book (“paper”) values. The settlement constraints imposed for hedging (in Sect. 2) in combination with stability by a suitably chosen notion of value, which depends continuously on trading strategies, moreover help to avoid some known paradoxical effects in price impact modelling (see Bank and Baum [3], Çetin et al. [16], Becherer et al. [9, Remark 3.3], and cf. the comments about different notions of wealth after (2.9) and (2.10)) and overly excessive opportunities of manipulating derivative payoffs (as in Schönbucher and Wilmott [28, Sect. 4.1]). The best-known model for transient price impact is probably the one due to Obizhaeva and Wang [26]. It states that the dynamic holdings of a large trader have additive linear impact (with parameter λ>0) on the prevailing price sof the underlying asset via (log-)price: dst=d¯st+λdYt,with impact level: dYt=−βYtdt +dt=: −h(Yt)dt +dt,(1.1) where ¯sis a given unaffected (fundamental) price evolution for the underlying, while Yis a market impact level process, whose mean-reverting dynamics is driven by and is linear in the asset holdings of the large trader and transient over time, recovering at some resilience rate given by the parameter β>0 in the linear resilience function h. Assuming price impact to be additive helps for mathematical tractability (in particular if ¯sis a martingale) and can serve to approximate multiplicative impact on a short horizon. This is common in the literature on optimal trade execution, as explained in Busseti and Lillo [15, Sect. 6], who further describe [15, Sect. 5] how transient impact is calibrated additively to log-prices, hence multiplicatively in prices. See also the comparison in Becherer et al. [7, Example 5.5] for arguments in favour of impact to be multiplicative if combined with multiplicative price dynamics of Black–Scholes type. A large strand of literature investigates (linear–)quadratic control problems in
Hedging with transient price impact 287 this realm (see e.g. Bank et al. [4], Ackermann et al. [1]) which are different from superhedging and its respective pricing problem. An undesirable property of the additive impact (1.1) in this context is that it can lead to negative prices sfor the underlying asset. It is plausible that trading a quantity of stocks, that is, a fraction of company ownership, should have a relative (hence multiplicative) effect on the price. Indeed, already Bertsimas and Lo [10, Sect. 3] have argued that relative (percentage) price impact which is proportional to the traded number of stocks (i.e., additive impact with respect to log-price in first order approximation) is more plausible than absolute price impact, and they cite empirical evidence. A simple way to obtain a multiplicative impact variant is by a log-linear interpretation of the additive Obizhaeva–Wang model (1.1), simply by taking s=logS, ¯s=log ¯ Sto be log-prices instead of nominal prices S,¯ S(affected, respective fundamental). Then the impact on S=¯ Sexp(λY ) is multiplicative and log-linear, with the resilience and the (log-)price impact functions from (1.1) being linear. This is the basic log-linear example (see Example 2.1) which is covered by and motivates our transient multiplicative impact model, with unaffected price process ¯ Sfor the underlying asset of Black–Scholes–Merton type. Our analysis moreover allows nonlinear and nonparametric resilience and price impact functions hand fin (2.1), (2.2). The model is a multiplicative variant of the (nonlinear) additive impact model from Predoiu et al. [27], where price impact can be interpreted in terms of a limit order book shape that is static with respect to relative price perturbations with Ybeing their volume effect process (see Becherer et al. [7, Sect. 2.1]). The contributions of the present paper are threefold: (1) We solve the superhedging problem under a transient price impact which is multiplicative, instead of additive. (2) Our results account for settlement specifications imposed at maturity which require analysis in liquidation values instead of book (paper) values, so that physical units of the underlying risky asset and cash matter at maturity (and as well at the initial time), i.e., terminal (initial) price impact cannot be treated as null. Following the terminology by Bouchard et al. [13], this means that we solve the hedging problem for non-covered instead of covered options (however, see Remark 7.1 for extensions to covered options under transient multiplicative price impact). (3) In this realm, the model we study is basically complete, with the transient price impact being the only digression from the frictionless Black–Scholes model assumptions (for ¯ S), and it yields nontrivial extensions to the classical no-arbitrage pricing and hedging, while avoiding paradoxical effects from illiquidity modelling as mentioned in Çetin et al. [16], without adding further frictions (like transaction costs, or constraints on trading strategies to be “small”). In particular, the large trader neither has the ability to “manipulate” (see Jarrow [24], Bank and Baum [3]) the market to achieve unreasonable profits (see Remark 2.4 and subsequent remarks), nor can he sidestep liquidity costs entirely and trade in effect like a small trader by exploiting modelling artefacts that occur due to a lack of sensible continuity properties (cf. Becherer et al. [9, Sect. 3]). We formulate the superhedging problem as a stochastic target problem and prove a dynamic programming principle (DPP) along reduced coordinates for the effective price and impact processes, which represent the price and impact levels that would
288 D. Becherer, T. Bilarev prevail if the large trader were to unwind her (long or short) position in the underlying risky asset immediately. Along the reduced coordinates, the DPP provides a way to compare at stopping times the instantaneous liquidation wealth and the (minimal) superhedging price. This permits characterising the superhedging price as the viscosity solution to a nonlinear pricing PDE, which is a semilinear extension of the Black– Scholes equation, with the non-linearity involving the (nonparametric) price impact and resilience functions as well. If the PDE has a sufficiently regular solution, it yields an optimal strategy which even replicates the option payoff in the required settlement units. This strategy incorporates the transient nature of impact in that it depends on the effective level of impact. Our analysis is also motivated by analytical tractability. It shows how effects from transience of price impact arise in a basically complete model without other additional frictions from transaction costs or constraints, with scope for results beyond those of the present paper, as outlined in Sect. 7. While there is a large literature on optimal execution and portfolio optimisation problems under transient price impact, mostly for price impact being additive but also for multiplicative impact (see Obizhaeva and Wang [26], Alfonsi et al. [2], Busseti and Lillo [15] or Guo and Zervos [21], Becherer et al. [8,9] and references therein), the literature on superhedging (or perfect hedging, i.e., replication) under price impact, as stated above, mostly treats permanent and purely instantaneous price impact (transaction costs, possibly nonlinear) or a combination of the two (see Frey [19], Schönbucher and Wilmott [28], Bank and Baum [3], Çetin et al. [16], Frey and Polte [20]), with the impact often taken in multiplicative form. For the implications of option settlement specifications on hedging, only few papers allow a price impact also at maturity. Clearly, it requires some relevant non-zero price impact at maturity to obtain differences between settlement specifications for options in physical or in cash units, as in Bouchard et al. [12]. However, most articles (see [19,16,20]) treat another hedging problem which is not posed in terms of hedgable units of assets, but instead in terms of book (that is “paper”) value, with the price impact at maturity (and possibly initiation) in the analysis effectively taken to be zero. That relates to a different hedging problem for “covered” options (see Remark 7.1). A major difference to the work by Bouchard et al. [12], which offered a fresh view to the hedging problem and inspired ours, is that the analysis in [12] is for permanent and additive impact. In contrast, our hedging results show nontrivial effects from transience of price impact, given that the impact is multiplicative. While the basic example to [12] is the Bachelier model with additive impact, our basic example is a Black–Scholes-type model with transient multiplicative price impact (see Example 2.1 and Remark 2.6) which is the log-linear variant of the model by Obizhaeva and Wang [26]. More detailed comparisons are provided throughout the paper. The paper is organised as follows. Sections 2and 3introduce the model of transient multiplicative price impact and formulate the hedging problem. Effective coordinates for dynamic programming in “liquidation magnitudes” are explained in Sect. 4. Section 5identifies hedging prices by viscosity solutions to semilinear PDEs (possibly degenerate, with delta constraints), with technical proofs deferred to the Appendix. The results are illustrated by numerical examples in Sect. 6. Finally, Sect. 7 extends the results to combined transient and permanent impact, points out further possible extensions to cross-impact with multiple assets, and comments on related results to the different hedging problem for covered options.
Hedging with transient price impact 289 2 A multiplicative transient price impact model This section describes the model for this paper. An extension with additional permanent impact is described in Sect. 7.Let(, F,P)be a complete probability space with countably generated F, a filtration F=(Ft)t≥0satisfying the usual conditions and an F-Brownian motion W. We take semimartingales to have càdlàg paths, R++ =(0,∞)and inf∅=+∞. The unaffected price process ¯ Sof the underlying risky asset evolves, if the large trader (she) is inactive, according to the stochastic differential equation d¯ St=¯ St(μtdt+σdWt), ¯ S0∈R++, with a constant σ>0 and a bounded progressive process μ. The càdlàg adapted process denotes the evolution of her holdings (in units of shares) in the risky asset, say a stock, which is the underlying for the derivative contingent claim in the hedging problem. The market impact process Y=Yis defined pathwise in the Skorohod space of càdlàg paths by dY t=−h(Y t)dt+dt,Y 0−=y∈R,(2.1) for a resilience function h:R→Rwhich is a Lipschitz-continuous function with sgn(x)h(x) ≥0, as in Becherer et al. [7,9]. When the large trader trades dynamically according to a strategy , the risky asset price observed on the market, which is the marginal price at which an additional infinitesimal quantity could be traded, is St:= S t:= f(Y t)¯ St,t≥0,(2.2) where the price impact function f:R→R++ is increasing and in C1with f(0)=1. In particular, λ:= f/f is a nonnegative and locally integrable C0-function satisfying f(x)=exp x 0λ(u) du,x∈R.(2.3) Example 2.1 The basic example is a transient proportional price impact with unaffected prices ¯ Sgiven by geometric Brownian motion, as in the Black–Scholes model with μ∈Rconstant, for resilience h(y) =βy and log-price impact logf(y) =λy linear functions with constants β,λ ∈R++. Then the multiplicative price impact is proportional to the number of shares t=t−t−traded at time t, that is, linear in log-prices with logSt+δt −log St−=λt with exponential decay logSt+δ=log(St)exp(−βδ) over time when there are no further trades within the time period (t, t +δ]. For such a linear choice of hand logf, the log-asset prices log Sunder multiplicative impact evolve like nominal asset prices in the seminal model by Obizhaeva and Wang [26] for additive transient price impact, as described in (1.1).
290 D. Becherer, T. Bilarev Our setting also allows the resilience rate β(hence h) to be zero, which makes the price impact permanent (cf. Sect. 7) and the log-price impact log(St/S0−)linear in Yt−Y0−=t−0−. Next, we specify the large trader’s proceeds (negative expenses) L, which are the variations of her cash account to fund the dynamic holdings in the risky asset. For simplicity, we assume zero interest and a riskless asset with constant price 1 as cash, i.e., prices are discounted in units of this numeraire asset. For continuous strategies of finite variation, L() =−· 0Sd(2.4) are the proceeds, and there is a unique continuous extension of the functional → L() in (2.4) to general (bounded) semimartingale strategies , given by L() := · 0F(Y t)d¯ St−· 0¯ St(f h)(Y t)dt−¯ SF(Y)−¯ S0F(Y 0−),(2.5) as shown in Becherer et al. [9, Theorem 3.8], with the function F(x):= x 0f(u)du, x ∈R.(2.6) More precisely, every (càdlàg) semimartingale can be approximated (in probability) in the Skorokhod space D([0,T])of càdlàg paths with the Skorokhod M1-topology (cf. [9, Sect. 3.1]) by a sequence of continuous processes of finite variation, and for semimartingales nP −→in (D([0,T]), M1)converging to a semimartingale ,we then have L(n)P −→L() in (D([0,T]), M1). To define Lby (2.5) is thus natural as the continuous extension of Lfrom (2.4) to all semimartingales. Remark2.2 In relation to the above continuous extension, we offer two general comments with regards to 1) literature and 2) subsequent results on hedging, which may be skipped at first reading. 1) For other potential applications, it seems helpful to note that more generally, there is a unique continuous extension even beyond semimartingale strategies; see [9, Sect. 3], and also Horst and Kivman [22] and Ackermann et al. [1] for similar continuity arguments in different applications. For our hedging problem in Sect. 4, however, semimartingale strategies will suffice; see e.g. in (4.3). 2) In Sect. 4, the superhedging problem of Definition 3.2 and the superhedging price (4.4) are going to be defined with respect to a particular set of admissible strategies (see (4.3)). The form of this set (which is as in Bouchard et al. [12]) plays a technical role in proofs for the geometric dynamical programming principle (cf. Theorem 4.1). It would be natural to ask to which extent the particular choice of this set affects the superhedging price. We can offer two (partial) answers to this questions, one of which is again related to suitable continuity properties. At first, we see that in base cases, the superhedging prices wbasically recover impactand frictionless
Hedging with transient price impact 291 Black–Scholes prices; see Corollary 5.12 and Remark 3.4 and likewise in [12] (with respect to the Bachelier model). This indicates that the superhedging price wdefined later in (4.4) is robust in the sense that it does not appear to depend on particularities of the said set. To explain, secondly, why such a robustness holds for the almost sure superhedging problem, an almost sure uniform approximation result (in terms of physical asset and cash holdings) of more general trading strategies by a suitable set of more elementary ones would be desirable in principle. Proposition 3.12 in [9] contributes such a result for the set of continuous finite-variation strategies; but this does not quite fit with the setup for the hedging problem in Sect. 4, as the respective set (4.3) there is different. The proceeds from a block trade of selling tshares at time tare −¯ Stt 0f(Y t−+x)dx, (2.7) showing that the price per share that the large trader pays (resp. obtains) for a block buy (resp. sell) order is between the price f(Y t−)¯ Stbefore the trade and the price f(Y t)¯ Stafter the trade. The form of proceeds and price impact from block trades can be interpreted from the perspective of a latent limit order book, where a block trade is executed against available orders in the order book for prices between f(Y t−)¯ St and f(Y t−+t)¯ St, see Becherer et al. [7, Sect. 2.1], and Ycan be understood as a volume effect process in the spirit of Predoiu et al. [27]. For a self-financing strategy (B, ) in which the dynamic holdings in cash (the riskless asset, savings account) and in the stock (the risky asset) evolve as Band , the self-financing condition is B=B0−+L(). In order to define a wealth dynamics for the large trader’s strategy, it remains to specify the value of the risky asset position in the portfolio in a suitable way. If the large trader is forced to liquidate her position of tstocks immediately by a hypothetical single block trade at market prices, her liquidation wealth Vliq t=Vliq t() at time t≥0 (before which the market impact is at Y t)is Vliq t() := Bt+¯ Stt 0f(Y t−x)dx =B0−+L()t+¯ Stt 0f(Y t−x)dx. (2.8) This wealth process is mathematically conveniently tractable, evolving continuously with dVliq t=F(Y t−)−F(Y t−−t−)d¯ St−¯ Stf(Y t−)−f(Y t−−t−)h(Yt)dt(2.9) and Vliq 0=B0−, and it inherits from the proceeds (2.5) the continuous dependence properties (on ) mentioned above. The notion of liquidation wealth Vliq() is relevant for the hedging application of Sect. 3, and is different from the so-called book
292 D. Becherer, T. Bilarev wealth process Vbook() := B+S =B0−+L() +S, (2.10) in which risky assets are evaluated at the current marginal market price S. Because of price impact (monotonicity of f, positivity of f,¯ S,S), clearly Vliq t≤Vbook t.In the terminology of Jarrow [24, Sect. IV], Vliq is real wealth whereas Vbook is paper wealth. Recently, Kolm and Webster [25] have given theoretical and practical reasons why accounting for the value (respectively the P&L, i.e., the changes in value) of a risky asset position based on current market prices Sas in (2.10) can be misleading and needs to be adjusted for price impact; in their terminology, Vliq corresponds to fundamental wealth whereas Vbook is accounting wealth, also referred to as mark-tomarket wealth. From (2.9), we obtain absence of arbitrage within the set of admissible strategies ANA := {(t)t≥0:is a bounded semimartingale with 0−=0 and t=0ont∈[T,∞)for some T∈(0,∞)}. Proposition2.3 The market is free of arbitrage up to any finite horizon T∈(0,∞)in the sense that there exists no ∈ANA with t=0on t∈[T,∞)such that for the self-financing strategy (B, ) with Vliq 0−:= B0−≤0, we have P[Vliq T≥0]=1and P[Vliq T>0]>0. Moreover,for any such (B, ),there exists a probability measure Qequivalent to P(on FT)such that Vliq is a Q-martingale. In the terminology of [24, Sect. IV, Eq. (13)], the no-arbitrage result of Proposition 2.3 states that there exist no market manipulation trading strategies.Note that in contrast, there is no reason to expect a no-arbitrage result in terms of book wealth Vbook; there are simple counterexamples, see Example 2.5 for implications on (super-)hedging prices. Remark 2.4 In the seminal article by Huberman and Stanzl [23], a notion of no profitable round-trips (stronger than no-arbitrage) is defined, which (in our notation) requires that there exists no (self-financing) strategy given by (B0−,)as in Proposition 2.3 with Vliq 0=0 and E[Vliq T]>0. This means that there is no such strategy from zero initial holdings (with Vliq 0=0) that achieves a terminal liquidation wealth which is positive in expectation, E[Vliq T]>0, within a compact time interval [0,T]. By definition, the liquidation wealth Vliq is the value of a cash-only position held after all stock holdings are liquidated. A much cited result from [23] states that price impact needs to be linear to exclude profitable round-trips. This is not in conflict with our modelling, as the proof in [23] relies of course on some assumptions. These include permanent and additive impact. For comparison, under multiplicative permanent price impact, a linear logprice impact function logfis sufficient to conclude that Vliq is a martingale under P(by (3.2) in Remark 3.4)if ¯ Sis a P-martingale (e.g. geometric Brownian motion, like in the Black–Scholes model under the risk-neutral measure). This implies E[Vliq T]=E[Vliq 0−], excluding profitable round-trips.
Hedging with transient price impact 299 studying the problem in suitably reduced coordinates which can be interpreted as quantities (for prices and impact y) at liquidation (of θ), and with respect to which a DPP and a viscosity characterisation are proved for the function w. While this idea is original and helps to make the analysis more transparent, in other aspects we can and do adapt techniques from Bouchard et al. [12]. To derive a dynamic programming principle for the function w, we want to compare it (evaluated at suitable coordinate processes) over time with the wealth process. Since by definition, wassumes zero initial risky asset holdings, it is natural to consider the (fictitious) state processes that would prevail if the trader were forced to liquidate her position in the risky asset immediately (with a block trade). To this end, let S(St,Y t, t):= ¯ Stf(Y t−t)=Stf(Y t−t)/f (Y t), Y(Y t, t):=Y t−t.(4.6) The process S(s,y,θ) is interpreted as the price of the asset that would prevail after θassets were liquidated, when sand yare the price of the risky asset and the market impact just before the trade, while Y(y, θ) would be the level of the market impact after this trade. In this sense, we refer to the processes S(S, Y ,) and Y(Y,) as the effective price and impact processes, respectively, for a self-financing trading strategy . Observe that both processes are continuous even though the trading strategy mayhavejumps. For the dynamic programming principle in Theorem 4.1, we compare the liquidation wealth Vliq defined in (2.8) with the value function walong evolutions of the effective price and effective impact processes (S(S, Y ,),Y(Y,)). Theorem 4.1 For the geometric DPP,we fix (t,s,y,v)∈[0,T]×R++ ×R×R. (i) If v>w(t,s,y),then there exist γ∈and θ∈Ksuch that Vliq,t,z,γ τ≥wτ,S(St,z,γ τ,Yt,z,γ τ, t,z,γ τ), Yt,z,γ τ−t,z,γ τ for all stopping times τ≥t,where z=(S(s, y, −θ),y +θ,θ,v). (ii) Let k≥1. If v<w 2k+2(t,s,y),then for every γ∈k,θ∈K∩[−k,k]and stopping time τ≥t,we have with z=(S(s, y, −θ),y +θ,θ,v) that PVliq,t,z,γ τ>w kτ,S(St,z,γ τ,Yt,z,γ τ, t,z,γ τ), Yt,z,γ τ−t,z,γ τ<1. Proof There are similarities and differences to Bouchard et al. [12, proof of Proposition 3.3] who treat the case for permanent additive impact; so we present the proof in full detail. As explained in Remark 3.4, the assumptions in [12] do not allow covering multiplicative price impact, and transience of impact naturally requires a further dimension in the DPP. The proof uses general ideas on dynamic programming for stochastic target problems and geometric flows; see Soner and Touzi [29]. We emphasise that for showing the DPP, our proof develops mathematical arguments in terms of effective coordinates and liquidation wealth Vliq, which simplifies the mathematical analysis and makes it more transparent. This also shows up in the possible extensions described in Sect. 7.
300 D. Becherer, T. Bilarev It is easy to see that for k≥2 and (t,s,y,θ)∈[0,T]×R++×R×(K∩[−k, k]), ¯wk(t,s,y,θ)≥wk+1t,S(s,y,θ),Y(y, θ),(4.7) wk−1t,S(s,y,θ),Y(y, θ)≥¯wk(t,s,y,θ). (4.8) Now suppose that v>w(t,s,y). Then by the definition of w, there exist θ∈Kand some γ∈G(t, z) for z=(S(s, y, −θ),y+θ,θ,v).Asin[29, proof of Theorem 3.1, Step 1], we have for all stopping times τ≥t(the first part of) the DPP for ¯w, namely Vliq,t,z,γ τ≥¯w(τ, St,z,γ τ,Yt,z,γ τ, t,z,γ τ). Then (i) follows from (4.7) by taking k→∞. To prove (ii), let v<w 2k+2(t,s,y) and suppose that there exist γ∈k,some ∈K∩[−k,k]and a stopping time τ≥tsuch that Vliq,t,z,γ τ>w kτ,S(St,z,γ τ,Yt,z,γ τ, t,z,γ τ), Yt,z,γ τ−t,z,γ τ for z=(S(s, y, −θ),y +θ,θ,v). Then by (4.8), we obtain Vliq,t,z,γ τ>¯wk+1(St,z,γ τ,Yt,z,γ τ, t,z,γ τ), and thus we get v≥¯w2k+1(t, S(s, y, −θ),y +θ,θ) by [29, proof of Theorem 3.1, Step 2]. In particular, we conclude from (4.7) that v≥w2k+2(t,s,y), which is a contradiction. Remark4.2 Part (ii) of the theorem is stated in terms of wkinstead of wbecause of a measurable selection argument employed in the proof; cf. [12, Remark 3.2]. To derive the pricing PDE from the dynamic programming principle in Theorem 4.1, we need the dynamics of the continuous processes t→ Vliq t−ϕt,S(St,Y t, t), Y(Y t, t)(4.9) for sufficiently smooth functions ϕ:[0,T]×R++ ×R,(t,s,y) → ϕ(t,s,y), that will later serve as test functions when characterising value functions by viscosity solutions. Lemma 4.3 For every γ=(a,b,ν) ∈and every ϕ∈C1,2,1([0,T]×R++ ×R), we have,for =γ, dVliq t−ϕ(t,St,Yt) =StF(Yt+t)−F(Yt) f(Yt)−ϕs(μt−λ(Yt)h(Yt+t)dt+σdWt +−ϕt−σ2S2 tϕss/2+h(Yt+t)ϕy+F(St,Yt, t)dt with F(s,y,θ)=sh(y+θ)λ(y)F(y +θ)−F(y) f(y) −f(y+θ)−f(y) f(y) ,
Hedging with transient price impact 301 where St=S(St,Y t, t),Yt=Y(Y t, t)and the derivatives of ϕare evaluated at (t, St,Yt). Proof Since St=S(St,Y t, t)equals ¯ Stf(Y t−t), the product rule and f=λf imply dSt=Stμt−λ(Y t−t)h(Y t)dt+σdWt.(4.10) By Itô’s formula, we obtain dϕ(t,St,Y t−t) =ϕtdt+ϕsdSt+ϕyd(Y t−t)+ϕss/2d[S]t =ϕt−λ(Y t−t)h(Y t)Stϕs−h(Y t)ϕy+σ2S2 tϕss/2dt +μtStϕsdt+σStϕsdWt.(4.11) With reference to (2.9), we have dVliq t=−h(Y t)St f(Y t)−f(Y t−t) f(Y t−t)dt+μtSt F(Y t)−F(Y t−t) f(Y t−t)dt +σSt F(Y t)−F(Y t−t) f(Y t−t)dWt.(4.12) Combining (4.11) and (4.12) and rearranging terms completes the proof. Remark 4.4 Consider the case when λis constant, i.e., f(x) =exp(λx). Then we have F≡0 and the dynamics of Vliq can be stated in a surprisingly simple form, namely dVliq t=F(t)dSt, where St=S(St,Y t, t)has the dynamics (4.10). As a consequence, the superhedging price (of the large investor) of an option with maturity Tand pure cash settlement H(ST)is at least the small investor’s price of H, in the absence of the large trader, when the price process is ¯ Sinstead. Indeed, for each (bounded) superhedging strategy (of the large investor) with initial capital v, there exists P≈P (on FT) such that S=S0−E(σ W)under Pfor a P-Brownian motion W. Hence Vliq() is a P-martingale and thus v≥EP[H(ST)]=EP[H(ST)](recall that T=0, implying ST=ST). On the other hand, a Feynman–Kac argument shows that EP[H(ST)]is just the classical Black–Scholes price for a small investor in a frictionless market with risky asset process ¯ S.Aswas an arbitrary superhedging strategy with initial capital v, taking the infimum yields the claim. The above observation shows a notable difference to the model in Bank and Baum [3, Theorem 5.3], where the price for the large investor is typically smaller. This is mainly due to a different specification of superhedging strategies with less
302 D. Becherer, T. Bilarev stringent settlement constraints, according to which a large trader may be able to reduce at maturity the payoff of the option to a larger extent by exploiting her price impact on the underlying at maturity. In other words, she can vary at maturity her risky asset position in order to minimise the payoff with fewer constraints, and immediately afterwards can unwind any residual risky asset position at no additional cost (by the absence of a bid–ask spread). In contrast, our setup is more restrictive by imposing as settlement constraint on the strategies that they have to replicate the physical delivery part exactly, i.e., after settlement, the hedging strategy has to hold a nonnegative cash position without residual holdings in the risky asset. We note that an argument as above does not apply in the general case with nonconstant λfor our price impact model. In fact, the examples in Sect. 6also reveal situations where superhedging is cheaper for the large trader; cf. Example 6.1. 5 The pricing PDEs and main results Next, we determine the terminal value for the function wat maturity date Tthat will serve as a boundary condition for the pricing PDE. Recall that Kis the (constraint) set in which trading strategies take values and set Kn=K∩[−n, n]for n∈N. Lemma 5.1 For the PDE boundary conditions,for n∈N,let Hn(s, y) := infg0sf(y+θ) f(y) ,y+θ+sF(y +θ)−F(y) f(y) : θ∈Kn,θ =g1sf(y+θ) f(y) ,y+θ. Then we have wn(T, ·)=Hn(·)and w(T, ·)=H(·),where the function His given by H:= inf n≥0Hn.(5.1) Proof At the maturity time T, the hedger of the option must do a block trade of size θ in order to meet the physical-delivery part specified by g1, thereby moving the price of the underlying from sto sf(y+θ) f(y) and the impact level from yto y+θ. Such a block trade incurs costs of size sF(y+θ)−F(y) f(y) , and hence it superreplicates the payoff (g0,g 1)if the hedger can cover these costs and the required cash-delivery part, which after the block trade is g0(s f(y+θ) f(y) ,y+θ). Remark 5.2 Note that H(s,y) =+∞if the equation θ=g1(s f(y+θ) f(y) ,y +θ) does not have a solution θin K.
Hedging with transient price impact 303 As we do not know at this point whether the value function wis continuous, we need to work with discontinuous viscosity solutions and hence consider the relaxed semilimits w∗(t,s,y):= lim inf (t,s,y,k)→(t,s,y,∞)wk(t,s,y), (5.2) w∗(t,s,y):= lim sup (t,s,y,k)→(t,s,y,∞) wk(t,s,y), (5.3) where the limits are taken over t<T. Recall that wis a (discontinuous) viscosity solution (of our pricing equations, see Sects. 5.1 and 5.2)ifw∗(resp. w∗)isa supersolution (resp. subsolution). For proving the viscosity property, we make the following assumption. Assumption 5.3 The functions w∗and w∗are bounded on [0,T]×R++×R, and the payoff function Hfrom (5.1) is regular in the sense that it is continuous, bounded, and the monotone convergence Hn↓Hholds uniformly on compacts. In particular, Assumption 5.3 implies that w(T, ·)is finite. This means that the payoff is well behaved in terms of the physical-delivery part, i.e., if the trader was supposed to fulfil her obligation from selling the option immediately, she would be able to do so in any situation (in any state (s, y)) with an admissible trade, provided that she has enough capital. 5.1 Case study for a general bounded price impact function f In this section, the following assumption is supposed to hold. Assumption 5.4 The resilience function his Lipschitz and bounded; the price impact function fis bounded away from 0 and ∞, i.e., infRf>0 and supRf<∞;λ is bounded and continuously differentiable with bounded derivative; and K=R(no delta constraints). Under Assumption 5.4, the antiderivative Ffrom (2.6) and its inverse F−1are bijections R→Rand Lipschitz-continuous with Lipschitz constants supRf<∞ and 1/infRf, respectively. To derive the pricing PDE just formally (at first, to be justified later) in this case, let (t,s,y)∈[0,T)×R++×Rand formally apply part (i) of the DPP in Theorem 4.1 to v=w(t,s, y) (assuming that the infimum in the definition of wis attained) and τ=t+, together with Lemma 4.3 for ϕ=w, assuming that wis smooth enough. Thus we get the existence of θ∗such that 0≤sF(y +θ∗)−F(y) f(y) −ws(t,s,y) μt−λ(y)h(y +θ∗)dt+σdWt +−wt(t,s,y)−σ2s2wss(t,s,y)/2 +h(y +θ∗)wy(t,s,y)+F(s,y,θ∗)dt.
304 D. Becherer, T. Bilarev Still arguing at a formal level, this cannot hold unless F(y +θ∗)=f(y)w s(t,s,y)+F(y), −wt(t,s,y)−σ2s2wss(t,s,y)/2+h(y +θ∗)wy(t,s,y)+F(s,y,θ∗)≥0.(5.4) In particular, θ∗=θ∗(t,y,s)=F−1(f (y)ws(t,s,y)+F(y))−y. The second part of the DPP in Theorem 4.1 will actually give that the drift term must be 0, i.e., we should have equality in (5.4). This formally motivates that the form of the pricing PDE for wshould be 0=−wt−1 2σ2s2wss +˜ h(t, s, y)wy+sλ(y)ws +s˜ h(t, s, y)1−˜ f(t,s,y) f(y) ,(5.5) where for (t,s,y)∈[0,T)×R++ ×R,weset ˜ h(t, s, y) := h◦F−1f(y)w s(t,s,y)+F(y) , ˜ f(t,s,y):= f◦F−1f(y)w s(t,s,y)+F(y) . Observe that the PDE is semilinear and degenerate (since it does not contain secondorder derivatives involving the y-variable). Our main result is as follows. Theorem 5.5 Under Assumptions 5.3 and 5.4,the value function wof the superhedging problem is continuous and is the unique bounded viscosity solution to (5.5)with the boundary condition w(T, ·)=H(·),where His defined in (5.1). Proof The viscosity property, i.e., that w∗(respectively w∗) is a viscosity supersolution (resp. subsolution), follows by the dynamic programming principle in Theorem 4.1 together with Lemma 4.3. The key arguments are presented in the Appendix in detail for the case where λis constant, which actually leads to a slightly more involved pricing PDE (5.11) (including gradient constraints) requiring additional justifications. The comparison result of Theorem A.5 proves uniqueness and continuity; cf. Remark A.7. Let us conclude this section by commenting on some consequences from Theorem 5.5 for the superhedging price and the existence of a corresponding hedging strategy. A numerical example is presented in Sect. 6. Remark 5.6 Like in the classical case of liquid markets (without price impact), the superhedging price does not depend on the drift in the unperturbed price process. This may be seen more directly by working under the equivalent martingale measure for ¯ S from the beginning. On the other hand, the superhedging price depends nontrivially on the initial level of impact yand the resilience function h, and can do so even for option payoffs of the form (g0(s), 0), i.e., payoffs not depending on the level of impact. So it turns out that for pricing and hedging (cf. Remark 5.8), the deviation of the market price from the ‘unaffected’ value, determined by the impact level y,isa relevant state variable.
Hedging with transient price impact 305 Remark 5.7 Observe that for only permanent impact, i.e., h≡0, (5.5) simplifies to the classical (frictionless) Black–Scholes pricing equation. Hence the superhedging price for the large trader then equals the Black–Scholes price for the option with payoff H. Remark 5.8 Under sufficient regularity, it turns out that a strategy can be constructed that perfectly replicates the option payout from the (minimal) superhedging price. This means that we have dynamic hedging in the sense of replication, like in the frictionless complete Black–Scholes model. To this end, suppose that a function w∈C1,3,1 b([0,T]×R++×R)solves the pricing PDE (5.5) with the boundary condition w(T, ·)=H(·). Then for any ε>0, a superhedging strategy with an initial cost of w(0,s,y)+εcan be constructed as follows. Consider the self-financing strategy (B, ) with B0−=w(0,s,y) +ε, 0=F−1(f (y)ws(0,s,y) +F(y)) −y, meaning that a block trade of size 0=0is performed at time 0, and t=F−1f(Y t)wst,S(St,Y t, t), Y t+F(Y t)−Y t for t∈[0,T), (5.6) T=0,i.e., T=−T−,(5.7) where Y=Y−. Then by Lemma 4.3 together with (5.6) and (5.5), we conclude that ε=Vliq 0() −w(0,s,y) =Vliq T() −wT,S(ST,Y T, T), Y T =Vliq T() −HS(ST,Y T, T), Y T =Vliq T() −H(ST,Y T), T=0, where the last line follows from (5.7). By the definition of H,havingH+εin cash at time Tis enough to superreplicate the European claim with payoff (g0,g 1)by doing a possible additional final block trade of size ε. Note that such a block trade does not affect Vliq T. Hence the strategy +1{T}εis superreplicating for the European claim. Note that one can take ε=0 if the constructed strategy is bounded and the infimum in the definition of Hnis attained (cf. Lemma 5.1), i.e., we get a replicating strategy in this case. An application of Itô’s formula gives that a strategy satisfying the fixed-point problem (5.6) can be obtained, under suitable regularity, by solving the system
306 D. Becherer, T. Bilarev of SDEs dSt=Stμt−λ(Y t)h(Y t+t)dt+σdWt, dt=a(t,St,Y t, t)dt+b(t, St,Y t)dWt, dY t=−h(Y t+t)dt, (5.8) with initial conditions S0=s,Y 0=yand 0=F−1(f (y)ws(0,s,y)+F(y))−y, where a(t,s,y,θ) := h(y +θ)1−λf ws−f−wsy −λswss f(F−1(f ws+F)) +wts +sμtwss +1 2σ2s2wsss f(F−1(f ws+F)) , b(t, s, y) := σswss f(F−1(f ws+F)), and where we write f=f(y),λ=λ(y), etc., when arguments of functions have not been specified, to ease the notation. Thus an optimal (i.e., cheapest) superhedging strategy accounts for the transient nature of price impact, which shows up by the presence of the resilience function hof the impact in the formulas above. Remark5.9 To describe how replicating hedging strategies in our model are described by coupled forward–backward SDEs, suppose that is a replicating strategy for an option with cash-equivalent payoff Hand let (Y,S)be the effective impact and price processes. By a change of measure argument, we can assume without loss of generality that μ=0. Setting Zt:= σStF( Yt+t)−F(Yt) f(Yt),giving t=F−1(σ−1S−1 tf(Yt)Zt+F(Yt)) −Yt, and using (4.12) leads to the coupled FBSDE dYt=−(h ◦F−1)σ−1S−1 tf(Yt)Zt+F(Yt)dt, dSt=St−λ(Yt)(h ◦F−1)σ−1S−1 tf(Yt)Zt+F(Yt)dt+σdWt, dVliq t=g(Yt,St,Z t)dt+ZtdWt,V liq T=H(ST,YT), where the driver g:R×R++ ×R→Rof the FBSDE is given by g(y,s,z)=−s(h ◦F−1)σ−1s−1f(y)z+F(y) ×(f ◦F−1)(σ−1s−1f(y)z+F(y))−f(y) f(y) . Example5.10 As instructive example, consider an option with maturity T>0 whose payout at maturity is the spot price of the asset, i.e., H(s,y) =s. In the frictionless Black–Scholes model, its arbitrage-free price is vBS(s) =sand a (minimal, i.e.,
Hedging with transient price impact 307 cheapest) replicating strategy is to buy one share at initiation and hold it until maturity, where it is liquidated at the spot price. For the solution in our price impact model, let us consider the classical solution to (5.5) with the boundary condition H given by the function w(t,s, y) =F(y +c(t, y)) −F(y) f(y) s, (5.9) where c:[0,T]×R→Ris a solution to the backward transport equation −ct+h(y +c)cy=0on[0,T)×R, c(T, y) =F−1f(y)+F(y) −yon R. In particular, by the dynamics of c, it holds for any strategy that c(t, Y t)=c(0,Y 0)for t∈[0,T], where Yis the effective impact process corresponding to . In particular, by (5.6), a minimal replicating strategy satisfies on [0,T)the equation ∗ t=c(t, Y∗ t)=c(0,Y∗ 0)=c(0,Y 0−). Hence a buy-and-hold strategy is also optimal for the large trader. We can observe the following: 1) Purely permanent impact (h≡0) yields the Black–Scholes price w(t, s,y) =s and the buy-and-hold strategy with c(0,y)=c(T, y) =F−1f(y)+F(y) −y shares, which does not depend on the maturity T. 2) In comparison, if the price impact is not permanent but transient (h≡ 0), the price (5.9) depends nontrivially on the maturity T, in addition to the price impact and resilience functions fand h, respectively. 3) The large trader’s price w(t, s, y) dominates the Black–Scholes price vBS(t, s) (which is equal to sin this example) if and only if c(t, y) > c(T, y). Moreover, there are situations where this condition holds and situations where it is violated. The intuitive reason is that there are two counterbalancing effects: at initiation, where the large trader buys shares to set up the initial delta hedge, thereby moving prices in an unfavourable direction, and at maturity, when she liquidates the delta and moves prices in a direction favourable to her. Which of these two effects dominates overall depends in a nontrivial way on the level of liquidity at initiation and at maturity, and on the settlement specifications of the option; see the discussion in Example 6.1. Let us comment here on Assumption 5.4 which implies bijectivity of Fon R. Observe that the inverse F−1is used to describe the optimal control θ∗. Similar conditions are also crucial for the results in Bank and Baum [3] and Bouchard et al. [12]; see the surjectivity assumption (A5) in [3] and the invertibility assumption (H2) in [12]. The next section shows how departing from this assumption leads naturally to singularities in the pricing PDE with respect to the gradient. Indeed, the lack of invertibility of Frequires conditions on wsso that θ∗can be derived. Therefore, the analysis there will involve constraints on the ‘delta’, i.e., on the holdings in the risky asset, which in PDE terms translates to constraints on the spatial gradient ws.
308 D. Becherer, T. Bilarev 5.2 Case study for price impact of exponential form We extend the analysis to a natural case where the antiderivative of the price impact function is not assumed to be surjective. To this end, the price impact function is taken to be of exponential form f(x) =exp(λx) with λa constant (i.e., log fis linear), meaning that the relative marginal price impact function λ=f/f > 0 is constant. A distinctive feature of this case is that at any time t, knowing the (marginal) stock price Stis sufficient to determine the impact from an instant block trade, since after a block trade of size , the price is ¯ Stf(Y t+) =Stexp(λ). Hence the relative displacement f(Y)of Sfrom the fundamental price ¯ Sis immaterial to determine the price impact from a block trade, in contrast to the situation of Sect. 5.1. Motivated by Remark 3.4, we impose short-selling constraints by requiring trading strategies to evolve in K=[−K,∞)for some K>0. To derive (only heuristically at first, we justify it rigorously later) the pricing PDE, let us apply formally Theorem 4.1 for v=w(t, s, y) at t,s,y,τ=t+, provided that wis smooth enough, to get the existence of θ∗∈Ksuch that using Lemma 4.3,we have Lθ∗w(t,s, y) dt−sws(t,s,y)−eλθ∗/λ +1/λ(σ dWt+ηtdt) ≥0,(5.10) where ηt=μt−λh(y +θ∗)and Lθ∗w(t,s, y) := −wt(t,s,y)+h(y +θ∗)wy(t,s,y)−1 2σ2s2wss(t,s,y). As in Sect. 5.1, the diffusion part in (5.10) should vanish, giving the optimal control θ∗=1 λlogλws(t,s,y)+1, and from the drift part, we identify the pricing PDE Lθ∗w(t, s, y) =0. The constraint θ∗∈Kis now equivalent to HKw(t, s, y) ≥0, where for a smooth function ϕ,weset HKϕ(t,s,y) := λϕs(t,s,y)+1−e−λK . Thus we conclude formally that wshould be a solution to the variational inequality FK[w]:=min{Lθ[w]w,HKw}=0on[0,T)×R++ ×R,(5.11) where θ[w](t,s,y):= 1 λlogλws(t,s,y)+1.(5.12) As usual, the gradient constraints propagate to the boundary, meaning that the boundary condition for (5.11) should be min{w(T, ·)−H,HKw}=0.(5.13) After this motivation, we state the main result for the exponential price impact function f(x)=exp(λx).
Hedging with transient price impact 315 of additive permanent price impact, as shown in detail in Becherer and Bilarev [6, Sect. 8]. In contrast to the problem studied in the main body of the present paper and in Bouchard et al. [12]fornon-covered options, the stochastic target problem for covered options is very different in that there is no price impact at inception and at maturity in the hedging problem for covered options. The reason (see [13]) is that the buyer of a covered option has to provide (upon request and at the discretion of the hedger) the required initial (delta) hedging position as a part of the option premium, and accepts any mix of cash and stocks (at a suitable book value if evaluated at current marginal market prices S) as an option settlement. In this way, the hedger is not exposed to initial and terminal impact for meeting settlement specifications when forming and unwinding the hedging position for covered options. We mention that similar assumptions are made in the literature by Frey [19], Frey and Polte [20], Çetin et al. [16], where the analysis is in terms of book value instead of liquidation value; see also Bank and Baum [3] and Bouchard et al. [12]. In the previous sections, the superhedging price for (non-covered) options under transient multiplicative price impact was characterised by a degenerate semilinear PDE, whose non-linearity involves the resilience function hand the price impact function f. It can involve gradient constraints (i.e., delta constraints), reducing to the Black–Scholes equation with gradient constraints in the situation of Corollary 5.12. In contrast, for covered options, the corresponding pricing equation turns out to be fully nonlinear and singular in the second-order term. This induces gamma constraints, whereas for non-covered options, a singularity arises in the first-order derivative and induces delta constraints; see Sect. 5.2. For covered options, it can be shown (see Becherer and Bilarev [6, Sect. 8]) that the resilience of the price impact is immaterial for the hedging price, irrespectively of a particular form for the resilience function, which has been observed likewise in [13, Sect. 4] for additive impact. We emphasise that this is very different to Sect. 5.1 where the resilience function enters the pricing equation in a nontrivial way. It turns out that the current deviation of the asset price from the unaffected price becomes a relevant state variable for describing the solution. Moreover, one can show (see Becherer and Bilarev [6, Remark 8.2, 2)]) that the superhedging price is decreasing in the impact function λin the sense that if λ≥˜ λ, then the price with respect to λdominates the one with respect to ˜ λ. For a dual formulation for the hedging of covered options, we refer to Bouchard and Tan [14]. Remark 7.2 As explained in Sect. 4, working in effective coordinates further permits extending results about transient price impact, in additive or multiplicative form, to multiple risky assets with cross-impact from transactions across different assets (described in Bilarev [11, Chap. 5, see Example 5.1.6]). To this end, a key idea is that the impact function needs to be the gradient field of a suitable potential in order to avoid a form of instantaneously profitable round-trips (see [11, Theorem 5.1.4]). Thereby, results like from previous sections (or Bouchard et al. [12] for permanent impact) can be extended to multiple assets in an additive transient cross-impact model. One obtains a geometric DPP and a viscosity PDE to characterise superhedging prices, which involves the resilience function hof the transient impact (see [11, Sect. 5.3.2]). Moreover, under certain conditions, one recovers as instructive reference case again
316 D. Becherer, T. Bilarev results as in a multidimensional Bachelier model with its natural pricing formula ([11, Remark 5.3.8]) that does not involve the price impact. This extends to multiple dimensions the instructive one-dimensional linear permanent impact example from [12, Sect. 2.4], which also yields the familiar Bachelier pricing formula. Notice that the hedging strategy is affected by the price impact, though closely related to the usual Bachelier delta-hedging strategy formula, by being computed at liquidation magnitudes for the stock price (i.e., in effective coordinates, analogously to those in (4.6) with Sinstead of S). This is entirely analogous to Black–Scholes formula related quantities (for pricing and hedging) occurring under (permanent) multiplicative impact in the basic log-linear example of our model (see Example 2.1 and Remark 2.6). Appendix: Proofs This section provides the proofs relegated from Sect. 5, in particular the proof of Theorem 5.11. Recall that in this case, f(x) =exp(λx) for λ>0, and thus the effective price simplifies to S(s,y,θ)=se−λθ =: S(s, θ), i.e., the level of impact is not needed in order to determine the price change of a block trade, given the price before the trade. We consider strategies taking values in K=[−K,∞)for K>0. This yields a gradient constraint for the PDE that is needed because of a singularity in the PDE, for the expression (5.12) for the form of the optimal strategy to be finitely defined. First, we verify in Appendix A.1 that if the pricing PDE (5.11) admits a sufficiently smooth classical solution, a replicating strategy in feedback form can be constructed. Such a construction is also needed for the contradiction argument in the proof of the subsolution property in Sect. A.2 where, using smooth test functions, one constructs locally strategies which, roughly speaking, behave like replicating strategies. The viscosity property proofs are collected in Appendix A.2, and in Appendix A.3, we prove comparison results that imply uniqueness of the viscosity solutions of the pricing PDEs and continuity of the value function for the superhedging problem. A.1 Verification argument for exponential impact function Suppose that the function w∈C1,2,1([0,T]×R++ ×R)has the property that for any (t,s,y)∈[0,T]×R++ ×R,wehave 1) θ[w](t,s,y)∈K, recalling the definition in (5.12); 2) Lθ[w](t,s,y)w(t,s, y) =0 when t<T; 3) w(T,s,y) =H(s,y). Suppose further that wis sufficiently regular (see Remark A.1 below) so that there exists an admissible strategy ∈of the form t=1 λlogλwst,S(St, t), Yt−t+1for t∈[0,T), T=0,i.e., T=−T−.(A.1)
Hedging with transient price impact 317 In particular, 0=log(λws(0,s,y)+1)/λ and T∈K. Consider the self-financing portfolio (β, ) with β0−=w(0,s,y). Then as in Remark 5.8, we get Vliq T() =H(ST,Y T), T=0. By the definition of H, this shows that Vliq T() at maturity Tis enough capital to (super-)replicate the European claim with payoff (g0,g 1)with a possible additional block trade (provided that the infima in the definition of H, see Lemma 5.1,are attained). Hence (β, ) will be a (super-)replicating strategy for the European claim (g0,g 1)with initial capital w(0,s,y), which is equal to the superreplication price w(0,s,y), meaning that a replicating hedging strategy as described exists and is an optimal (i.e., cheapest) superreplication strategy under the given assumptions. RemarkA.1 To construct a replicating strategy as in (A.1), we suppose moreover that w∈C1,3,1([0,T]×R++ ×R)and apply Itô’s formula, similarly as in Remark 5.8, to get for t<Tthe equation dt=1 λ1 λws+1d(λws+1)−1 2(λws+1)2d[λws+1]t =a(t,St,Y t, t)dt+b(t, St,Y t)dWt, where for St:=S(St, t)and Y t=Y t−t,weset a(t,St,Y t, t):= 1 λws+1wts +wssStμt−λh(Y t)−wsyh(Y t) +1 2wsssσ2S2 t−λ2σ2S2 twss 2(λws+1), b(t, St,Y t):= σStwss λws+1, with all derivatives of wevaluated at (t, S(St, t), Yt−t). Thus a replicating strategy, which is superhedging the payout at a minimal cost (see the arguments preceding the current remark), can be constructed as the (t)t∈[0,T )-part (plus a terminal block trade) from a solution, if it exists, to the SDE system, for t∈[0,T], dSt=Stμt−λh(Y t+t)dt+σdWt, dt=a(t,St,Y t, t)dt+b(t, St,Y t)dWt, dY t=−h(Y t+t)dt, (A.2) with initial condition S0=s,Y 0=yand 0=log(λws(0,s,y)+1)/λ. A.2 Viscosity solution property of wfor exponential impact function For the results from Sect. 5.2, we now prove the viscosity property.
318 D. Becherer, T. Bilarev Theorem A.2 The function w∗from (5.2)is a viscosity supersolution of (5.11)on [0,T)×R++ ×Rwith the boundary condition (5.13)on {T}×R++ ×R. Proof First, let (t0,s 0,y 0)∈[0,T)×R++ ×Rand ϕ∈C∞ b([0,T]×R++ ×R)be a smooth function such that we have a strict (meaning uniquely attained) minimum, (strict) min [0,T ]×R++×R(w∗−ϕ) =(w∗−ϕ)(t0,s 0,y 0)=0. Case 1: Suppose that HKϕ(t0,s 0,y 0)<0. By the continuity of the operator HK, there exists an open neighbourhood Oof (t0,s 0,y 0)whose closure is contained in [0,T)×R++ ×Rsuch that HKϕ(t,s,y) < −εin Ofor some ε>0. Therefore, after possibly shrinking the neighbourhood O, there exists a constant kε>0 such that s|ϕS(t,s,y)+1/λ −eλθ /λ|≥kεfor all θ∈K,(t,s,y)∈O.(A.3) Let (tn,s n,y n)n∈N⊆Obe a sequence converging to (t0,s 0,y 0)with w(tn,s n,y n)−→ w∗(t0,s 0,y 0), where w∗is the lower-semicontinuous envelope of w. Set vn:= w(tn,s n,y n)+1/n. Since vn>w(t n,s n,y n), Theorem 4.1 implies the existence of θn∈Kand strategies γn∈such that for stopping times τn≥tn(to be suitably chosen later), we have P-a.s. for t∈[tn,T]that Vliq,tn,zn,γn t∧τn≥w·,S(Stn,zn,γn, tn,zn,γn), Y tn,zn,γn−tn,zn,γnt∧τn,(A.4) where zn=(sneλθn,y n+θn,θ n,v n). To abbreviate notation, we write in the sequel nas superscript instead of the full argument (tn,z n,γ n), so that Sn:= S(Stn,zn,γn, tn,zn,γn), Yn:= Ytn,zn,γn−tn,zn,γn. Take τn=inf{t≥tn:(t, Sn t,Yn t)∈ O}, which is the first entrance time of the parabolic boundary of the open region O. In particular, τn<T. Since w≥w∗≥ϕ and w∗−ϕhas a strict local minimum at (t0,s 0,y 0), there exists ι>0 such that (w −ϕ)(τn,Sn τn,Yn τn)≥ι. Hence P-a.s., we have Vliq,n τn−ϕ(τn,Sn τn,Yn τn)≥ι. Now Lemma 4.3 together with the fact that Sn tn=sn,Yn tn=yngives that P-a.s., ι≤vn−ϕ(tn,s n,y n) −τn tn Sn uϕS(u, Sn u,Yn u)+1/λ −eλn u/λ(σ dWu+ζn udu), (A.5) where ζn t:=ηn t−Ln tϕ Sn t(ϕS(u, Sn t,Yn t)+1/λ −eλn t/λ) for t∈[tn,τ n]
Hedging with transient price impact 319 with ηn t:= μt−λh(Yn t). Note that ζn tis well defined on [tn,τ n]and uniformly bounded, noting (A.3) and the fact that Ynis bounded since nis. Hence by Girsanov’s theorem, there exists a measure Pnequivalent to Psuch that t∧τn tn SuϕS(u, Sn u,Yn u)+1/λ −eλu/λ(σ dWu+ζn udu), t ≥tn, is a square-integrable martingale under Pn, as the integrand of the stochastic integral is uniformly bounded because of the definition of τn, the continuity of ϕSand the boundedness of the range of , noting τn≤T. Taking expectations under Pnof the right-hand side of (A.5) leads to vn−ϕ(tn,s n,y n)≥ι>0, which yields a contradiction as by our choice of vnand of the sequence (tn,s n,y n)n∈N,wehave vn−ϕ(tn,s n,y n)−→ w∗(t0,s 0,y 0)−ϕ(t0,s 0,y 0)=0. Case 2: From Case 1, we know that HKϕ(t0,s 0,y 0)≥0. Hence θ[ϕ](t0,s 0,y 0)=1 λlogλϕS(t0,s 0,y 0)+1 is well defined (also in a neighbourhood of (t0,s 0,y 0)). Let us suppose that Lθ[ϕ]ϕ(t0,s 0,y 0)<0. By the continuity of the operator L, there exist then an open neighbourhood O⊆[0,T]×R++ ×Rof (t0,s 0,y 0)and some r>0 and ε>0 such that Lθϕ(t,s,y) < −εfor (t,s,y)∈O,θ ∈θ[ϕ](t,s,y)−r, θ[ϕ](t,s,y)+r. In particular, by the continuity of the involved functions, we have (after possibly shrinking the open set O)thatforevery(t,s,y)∈Oand for some r>0, Lθϕ(t,s,y) < −εwhenever |ϕS(t,s,y)+1/λ −eλθ /λ|≤r. As in Case 1, consider a sequence (tn,s n,y n)in Owhich converges to (t0,s 0,y 0)and such that w(tn,s n,y n)→w∗(t0,s 0,y 0). Set vn:=w(tn,s n,y n)+1/n and let θn∈K and strategies γn∈be such that the dynamic programming principle (A.4) holds for the stopping times τnthat are the first exit times of (·,Sn,Yn)from the set O. Now a contradiction follows similarly as in Case 1 with the following adjustment: We have Vliq,n t∧τn−ϕ(·,Sn,Yn)t∧τn =vn−ϕ(tn,s n,y n) −t∧τn tn Sn u(ϕS+1/λ −eλn u/λ)(σ dWu+ζn udu) +t∧τn tnLn uϕ(u,Sn u,Yn u)1{|ϕS+1/λ−eλn u/λ|≤r}du ≤vn−ϕ(tn,s n,y n)−t∧τn tn Sn u(ϕS+1/λ −eλn u/λ)(σ dWu+ζn udu),
320 D. Becherer, T. Bilarev where we set ζn t:=ηn t−Ln tϕ Sn t(ϕS+1/λ −eλn t/λ)1{|ϕS+1/λ−eλn t/λ|≥r}for t∈[tn,τ n], with the functions ϕand ϕSabove evaluated at (·,Sn ·,Yn ·). The contradiction now follows by taking expectations under Pn≈Pand letting n→∞. Boundary condition. Let (s0,y 0)∈R++ ×Rand ϕbe a smooth function with (strict) min [0,T ]×R++×R(w∗−ϕ) =(w∗−ϕ)(T , s0,y 0)=0. Suppose that min{w∗(T, s0,y 0)−H(s0,y 0), HKϕ(T,s0,y 0)}<0. The case HKϕ(T,s0,y 0)<0 leads to a contradiction by the same arguments as in Case 1 above, using that HKϕ<0 in a small neighbourhood of (T, s0,y 0). Hence we obtain HKϕ(T,s0,y 0)≥0. Now if w∗(T , s0,y 0)<H(s 0,y 0), then also ϕ(T,s0,y 0)−H(s0,y 0)<0. After possibly modifying the test function ϕby (t,s,y) → ϕ(t,s,y) −√T−t, we can assume that ∂tϕ(t,s,y) →∞when t→T, uniformly on compacts. Hence in an ε-neighbourhood [T−ε, T ) ×Bε(s0,y 0)around (T, s0,y 0),wehave Lθ[ϕ]ϕ<0. Moreover, after possibly decreasing ε,wehaveϕ(T, ·)≤H(·)−ι1 on Bε(s0,y 0)for some ι1>0. We can argue as in Cases 1 and 2 above, by starting from (tn,s n,y n)in [T−ε, T ) ×Bε(s0,y 0), with (tn,s n,y n)→(T, s0,y 0)and w(tn,s n,y n)→w∗(T, s0,y 0), stopping at the (parabolic) boundary at time τnand using w(T, ·)=H(·), to get Vliq,n τn−ϕ·,S(Sn, n), Yn−nτn≥ι1∧ι2, where ι2:= inf[T−ε,T )×∂Bε(s0,y0)(w∗−ϕ) > 0. A contradiction follows as in Case 2 above. Now we prove the subsolution property. Theorem A.3 The function w∗from (5.3)is a viscosity subsolution of (5.11)on [0,T)×R++ ×Rwith the boundary condition (5.13)on {T}×R++ ×R. Proof The proof is similar to and inspired by the one for the subsolution property in [12, Theorem 3.7]. The reason is that in this case, the gradient constraints ensure that a test function ϕthat would possibly contradict the subsolution property must satisfy HKϕ>0 locally and hence is sufficiently “nice” to define (locally) control processes (employing the verification argument in Remark A.1) that lead to a contradiction like in [12]. For completeness, we outline differences in the line of proof and sketch the main steps. Let ϕ∈C∞ b([0,T],R++ ×R)be a test function with the property that the point (t0,s 0,y 0)∈[0,T]×R++ ×Ris a strict (local) maximum of w∗−ϕ, i.e., (strict) max [0,T ]×R++×R(w∗−ϕ) =(w∗−ϕ)(t0,s 0,y 0)=0.
Hedging with transient price impact 321 First assume that t0<T. To ease notation, we use the variable xto denote the pair (s, y). Because of the special form of the second part of the DPP in Theorem 4.1(ii), we need to employ wk(instead of was in the proof of the supersolution property). By Barles [5, Lemma 6.1], we can take a sequence (kn,t n,x n)n∈Nsuch that kn→∞,any(tn,x n)is a local maximum of w∗ kn−ϕand (tn,x n,w kn(tn,x n)) →(t0,x 0,w∗(t0,x 0)). Assume that FK[ϕ](t0,x 0)>0 and let ϕn(t, x) =ϕ(t,x) +|t−tn|2+|y−yn|2+|s−sn|4. Then FK[ϕn]>0 holds in a neighbourhood Bof (t0,x 0)that contains (tn,x n)for all nlarge enough. Since we work on the local neighbourhood Bwhere also HKϕn>0, we can modify (in a smooth way) the functions hand ϕnoutside of Bto be supported on a slightly bigger compact set where HKϕn>0 holds. Thus after possibly passing to a suitable subsequence, there exist γn∈knsuch that tn,zn,γn t=1 λlogλ∂ϕn ∂s (t, Stn,zn,γn t,Ytn,zn,γn t)+1,t≥tn, where we set Stn,zn,γn t=S(Stn,zn,γn t, tn,zn,γn t)and Ytn,zn,γn t=(Y −)tn,zn,γn tfor zn=(sn,y n,0,w kn(tn,x n)−n−1);seeRemarkA.1.Letτnbe the first time after tn at which the process (Stn,zn,γn t,Ytn,zn,γn t)t≥tnleaves B.Likein[12, proof of Theorem 3.7], we conclude by applying Itô’s formula, using Lemma 4.3 and FK[ϕn]>0 on Bthat P-a.s., Vliq,tn,zn,γn τn≥ϕnτn,Stn,zn,γn τn,(Y −)tn,zn,γn τn+vn−ϕn(tn,x n). Now a contradiction follows as in [12, proof of Theorem 3.7, subsolution property, (a)]. For the boundary condition, i.e., the case t0=T, the arguments are exactly the same as in [12, proof of Theorem 3.7, subsolution property, (b)]. A.3 Comparison results for viscosity solutions First we provide a comparison result for the pricing PDE (5.5), needed for the proof of Theorem 5.5. Note that (5.5) has the structure 0=−ϕt−σ2s2 2ϕss −B1y,f(y)ϕsϕy−sB2y,f(y)ϕsϕs−sB3y,f(y)ϕs,(A.6) where Bi:R2→R,i=1,2,3, are bounded and Lipschitz-continuous functions. By a change of coordinates, one can transform the PDE as follows. Lemma A.4 Let ube a viscosity subsolution (resp.supersolution)of the PDE (A.6). Fix κ>0. Then the function ˜udefined by ˜u(t, s, y) =eκtut, sf (y), yfor (t,s,y)∈[0,T]×R++ ×R
322 D. Becherer, T. Bilarev is a viscosity subsolution (resp.supersolution)of the PDE 0=κϕ −ϕt−σ2s2 2ϕss −B1(y, e−κtϕs)ϕy+λ(y)B1(y, e−κtϕs)ϕs −sB2(y, e−κtϕs)ϕs−eκtsf (y)B3(y, e−κtϕs). (A.7) Proof To prove the super-(resp. sub-)solution property, take any point (t0,s 0,y 0) in [0,T)×R++ ×Rand a test function ˜ϕ∈C∞ b([0,T]×R++ ×R)for ˜uat (t0,s 0,y 0), i.e., min [0,T ]×R++×R(resp. max)( ˜u−˜ϕ) =˜u(t0,s 0,y 0)−˜ϕ(t0,s 0,y 0)=0.(A.8) Consider ϕ(t,s,y) := e−κt ˜ϕ(t,s/f(y),y) for (t,s,y) ∈[0,T]×R++ ××R. We have by definition that eκtϕ(t, sf (y), y) =˜ϕ(t,s,y) for (t,s,y) ∈[0,T]×R++ ××R. In particular, ϕis a test function for uat (t0,s 0f(y 0), y0)since by (A.8), we get min [0,T ]×R++×R(resp. max)(u −ϕ) =ut0,s 0f(y 0), y0−˜ϕt0,s 0f(y 0), y0 =0.(A.9) We also have ˜ϕs(t,s,y)=eκtf(y)ϕ st, sf (y), y, ˜ϕss(t,s,y)=eκtf2(y)ϕsst, sf (y), y, ˜ϕy(t,s,y)=eκtλ(y)f(y)ϕst, sf (y), y+eκtϕyt, sf (y), y =λ(y) ˜ϕs(t,s,y)+eκtϕyt, sf (y), y, ˜ϕt(t,s,y)=eκtϕtt, sf (y), y+κeκtϕt, sf (y), y. By direct application of these identities, we derive from the right-hand side of (A.7)for ˜ϕevaluated at (t0,s 0,y 0)exactly the right-hand side of (A.6)forϕat (t0,s 0f(y 0), y0). By the viscosity property of uand (A.9), we thus conclude that (A.7) holds for ˜ϕat (t0,s 0,y 0)with “≥” (resp. “≤”). This proves the claim. By Lemma A.4, it now suffices to prove comparison for (A.7) since this implies a comparison result for (A.6). This is done in the following result. Theorem A.5 Let u(respectively v)be a bounded upper-semicontinuous subsolution (resp.lower-semicontinuous supersolution)on [0,T)×R++ ×Rof (A.7). Suppose that u≤von {T}×R++ ×R.Then u≤von [0,T]×R++ ×R. Proof To prove the claim by contradiction, let us suppose that sup (t,s,y)∈[0,T ]×R++×R(u −v)(t,s,y)>0.
Hedging with transient price impact 323 Then we can find R>1 such that with OR:= (1/R, R),wehave sup (t,s,y)∈[0,T ]×OR×[−R,R] (u −v)(t,s,y)>0. In particular, there exist δ>0 and (t0,s 0,y 0)∈OR×[−R,R]with the property that (u −v)(t0,s 0,y 0)=δ>0. Now consider for n∈Nthe bounded upper-semicontinuous function n(t, s1,s 2,y 1,y 2):= u(t, s1,y 1)−v(t,s2,y 2)−n 2(s1−s2)2−n 2(y1−y2)2. It attains its maximum at some (tn,sn 1,sn 2,yn 1,yn 2)∈[0,T]×O2 R×[−R,R]2by compactness of that set, and we clearly have n(tn,sn 1,sn 2,yn 1,yn 2)≥δfor all n∈N.(A.10) By arguments as in [12, proof of Lemma 3.11], one obtains (after possibly passing to a subsequence) that n(sn 1−sn 2)2+n(yn 1−yn 2)2−→ 0asn→∞.(A.11) Note that (A.11) also implies n(sn 1−sn 2)(yn 1−yn 2)→0asn→∞. Now by Ishii’s lemma as stated in Crandall et al. [18, Theorem 8.3], there exist (bn,Xn,Yn)in R×S2×S2such that with pn=n(sn 1−sn 2)and qn=n(yn 1−yn 2), we have bn,(pn,qn), Xn∈¯ P2,+ Oau(tn,sn 1,yn 1), bn,(pn,qn), Y n∈¯ P2,− Oav(tn,sn 2,yn 2), where Xnand Ynsatisfy Xn0 0−Yn≤3nI2−I2 −I2I2.(A.12) Here, S2denotes the set of 2 ×2 symmetric nonnegative matrices and I2∈S2is the identity matrix. Using the viscosity property of uand vat (tn,sn 1,yn 1)and (tn,sn 2,yn 2), respectively, we have κu(tn,sn 1,yn 1)−bn−1 2σ2(sn 1)2Xn 11 +L(sn 1,yn 1,pn,qn)≤0, κv(tn,sn 2,yn 2)−bn−1 2σ2(sn 2)2Yn 11 +L(sn 2,yn 2,pn,qn)≥0, where L(t, s, y, p, q) := −B1(y, e−κtp)q +λ(y)B1(y, e−κtp)p −sB2(y, e−κtp)p −eκtsf (y)B3(y, e−κtp).
324 D. Becherer, T. Bilarev As a consequence, 0<κδ<κ u(tn,sn 1,yn 1)−v(tn,sn 2,yn 2) ≤−1 2σ2(sn 2)2Yn 11 +1 2σ2(sn 1)2Xn 11 +L(tn,sn 2,yn 2,pn,qn)−L(tn,sn 1,yn 1,pn,qn). (A.13) On the other hand, by (A.12), we get that 1 2σ2(sn 1)2Xn 11 −1 2σ2(sn 2)2Yn 11 ≤3 2σ2n(sn 1−sn 2)2, which converges to 0 for n→∞due to (A.11). Let us now analyse the difference L(tn,sn 2,yn 2,pn,qn)−L(tn,sn 1,yn 1,pn,qn). With C(resp. CR) denoting a Lipschitz constant (depending on R) that may change from line to line, we get estimates for the corresponding terms via |B1(yn 1,e −κtnpn)qn−B1(yn 2,e −κtpn)qn|≤C|yn 1−yn 2||qn|, |λ(yn 1)B1(yn 1,e −κtnpn)pn −λ(yn 2)B1(yn 2,e −κtnpn)pn|≤C|yn 1−yn 2||pn|, |sn 1B2(yn 1,e −κtnpn)pn−sn 2B2(yn 2,e −κtnpn)pn|≤C|(sn 1−sn 2)pn| +CR|(yn 1−yn 2)pn|, |eκtnsn 1f(yn 1)B3(yn 1,e −κtnpn) −eκtnsn 2f(yn 2)B3(yn 2,e −κtnpn)|≤CR(|sn 1−sn 2|+|yn 1−yn 2|). As all estimates from above vanish for n→∞, the right-hand side in (A.13)is bounded by something that converges to 0 as n→∞. But this yields a contradiction for large n. Because of lack of a precise reference, we provide a comparison result also in the case of delta constraints leading to the variational inequality (5.11). Theorem A.6 Suppose that the resilience function his Lipschitz-continuous and Assumption 5.3 holds.Let u(resp.v)be a bounded upper-(resp.lower-)semicontinuous viscosity subsolution (resp.supersolution)of the variational inequality (5.11)with the terminal condition (5.13). Then u≤von [0,T]×R++ ×R. Proof We argue by contradiction. For any a>0, set Oa:= [a,∞)×[−1/a, 1/a]. If sup[0,T ]×R++×R(u −v) > 0, there exists a>0 with sup[0,T ]×Oa(u −v) > 0. For κ>0, consider ˜u:= eκtuand ˜v:= eκtv. Then ˜u(resp. ˜v) is a viscosity sub-(resp. super-)solution of min{κϕ +˜ L[ϕ],HK,t ϕ}=0