Hedging with physical or cash settlement under transient multiplicative price impact

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 the 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 non-negativity of underlying asset prices while maintaining an arbitrage-free model. The basic (log-)linear example is a Black-Scholes model with relative price impact being proportional to the volume of shares traded, where the transience for impact on log-prices is being modelled like in Obizhaeva-Wang \cite{ObizhaevaWang13} for nominal prices. More generally, we allow for non-linear 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. Pricing equations under illiquidity extend no-arbitrage pricing a la Black-Scholes for complete markets in a non-paradoxical way (cf.\ {\c{C}}etin, Soner and Touzi \cite{CetinSonerTouzi10}) even without additional frictions, and can recover it in base cases.


Introduction
By using methods of stochastic target problems [29] and geometric dynamic programming in suitably choosen 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 because large volumes are to be traded, there is price impact and feedback effects from hedging can affect the minimal superhedging prices [19,28,20,3] and respective hedging strategies, which almost-surely super-replicate 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 physical units become relevant and this should show in pricing and hedging equations. As our results address hedging in terms of liquidation values, that means "real" instead of "paper" values [24], we recover such effects, whereas [19,20,13] study hedging in terms of book ("paper") values. The settlement constraints imposed for hedging (in Section 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 [3,16], [9,Rmk.3.3] and the comments after (2.11)) and overly excessive opportunities of manipulating derivative payoffs (as in [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 Θ of a large trader have additive linear impact (with parameter λ > 0) on the prevailing prices s of the underlying asset, via (log-)price: ds t = ds t + λdY t , with impact level: wheres is a given unaffected (fundamental) price evolution for the underlying, while Y is a market impact level process driven by Θ, whose mean-reverting dynamics is linear in the asset holdings Θ 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 (i.p. ifs is a martingale) and can simply serve to approximate multiplicative impact on short horizon. This is common in the literature on optimal trade execution, as explained by Busseti and Lillo in [15,Sect.6], who further describe [in Sect.5] how transient impact is calibrated additively to log-prices, that means multiplicatively in prices. See also the comparison in [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 this realm (see e.g. [4,1]) which are different to the superhedging and -pricing problem. An undesirable property of additive impact (1.1) in this context is that it can lead to negative asset prices s for the underlying. 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 = log S,s = logS to be log-prices instead of nominal prices S,S (affected, respective fundamental). Then impact on S =S exp(λY ) is multiplicate 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 motivating our transient multiplicative impact model, with unaffected price processS for the underlying asset being of Black-Scholes-Merton type. Our analysis moreover admits for non-linear and non-parametric resilience and price impact functions h and f (in (2.2), (2.3)). The model is a multiplicative variant of the (non-linear) additive impact model from [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 Y being their volume effect process (see [7,Sect.2.1]).
The contributions of the present paper are threefold: (1) We solve the superhedging problem under 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) value, such that physical units of the underlying risky asset and cash matter at maturity (and as well at initial) time, i.e. terminal (initial) price impact cannot be treated as null. Following terminology by [13] (cf. Remark 7.1), this means we solve the hedging problem for non-covered instead of covered options. (3) In this realm, the model we study is basically complete, with transient price impact being the only digression from the friction-less Black-Scholes model assumptions (forS), and it yields nontrivial extensions to the classical no-arbitrage pricing and hedging, while avoiding paradoxical effects from illiquidity modelling as mentioned in [16], without additional further frictions (like transaction costs, or constraints on trading strategies to be "small"). In particular, the large trader has neither the ability to "manipulate" [24,3] the market to achieve unreasonable profits (see Remark 2.3 and subsequent remarks) nor to sidestep liquidity costs entirely and trade in effect like a small trader by exploiting modelling artefacts that occur in lack of sensible continuity properties (cf. [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 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, what permits to characterize the superhedging price as the viscosity solution to a non-linear pricing PDE, which is a semi-linear extension of the Black-Scholes equation, whose non-linearity involves the (non-parametric) price impact and the resilience functions as well. If the PDE has a sufficiently regular solution, it yields an optimal strategy which is even replicating 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 the ones of the present paper as outlined in Section 7.
While there is a large literature on optimal execution and portfolio optimization problems under transient price impact, mostly for price impact being additive but also for multiplicative impact (see [26,2,15] or [21,8,9] and references therein), the literature on superhedging (or perfect hedging, i.e. replication) under price impact, as stated above, is mostly treating permanent and purely instantanuous price impact (transaction costs, possibly non-linear) or a combination of the two [19,28,3,16,20], with impact often being taken in multiplicative form. For the implications of option settlement specifications on hedging, only few papers admit for 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 cash units, as in [12]. Yet, most articles [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 price impact at maturity (and possibly initiation) in the analysis being effectively taken to be zero. Such relates to a different hedging problem for "covered" options (see Remark 7.1). A major difference to the work by Bouchard et al. [12], that 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 non-trivial effects from transience of price impact, that is multiplicative. Whereas 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 2.5) that is the log-linear variant of the model by [26]. More detailed comparisons are provided thoughout the paper.
The paper is organised as follows. Sections 2-3 introduce the model of transient multiplicative price impact and formulate the hedging problem. Effective coordinates for dynamic programming in "liquidation magnitudes" are explained in Section 4. Section 5 identifies hedging prices by viscosity solutions to semilinear PDEs (possibly degenerate, with delta constraints), with technical proofs deferred to Appendix A. Results are illustrated by numerical examples in Section 6. Finally, Section 7 extends 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.

A multiplicative transient price impact model
This section describes the model for this paper. An extension with additional permanent impact is described in Section 7. Let (Ω, F, P) be a complete probability space with countably generated F, a filtration F = (F t ) t≥0 satisfying the usual conditions and an F-Brownian motion W . We take semimartingales to have càdlàg paths, R + = (0, +∞) and inf ∅ = +∞.
The unaffected price processS of the underlying risky asset evolves, if the large trader (her) is inactive, according to the stochastic differential equation with constant σ > 0 and bounded progressive process µ. The càdlàg adapted process Θ 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 = Y Θ is defined pathwise on the Skorohod space of càdlàg paths, by for a resilience function h : R → R which is Lipschitz continuous function with sgn(x)h(x) ≥ 0, as in [7,9]. When the large trader trades dynamically according to strategy Θ, the risky asset price observed on the market, being the marginal price at which an additional infinitesimal quantity could be traded, is where the price impact function f : R → R + is increasing and in C 1 with f (0) = 1. In particular, λ := f /f is a non-negative and locally integrable C 0 function, satisfying Example 2.1. The basic example is transient proportional price impact with unaffected pricesS being geometric Brownian motion, as in the Black-Scholes model with µ ∈ R being constant, for resilience h(y) = βy and log-price impact log f (y) = λy being linear functions with constants β, λ ∈ R + . Then multiplicative price impact is proportional to the number of shares ∆Θ t = Θ t − Θ t− traded at time t, that is linear in log-prices with exponential decay log(S t+δ ) = log(S t ) exp(−βδ) over time, when there are no further trades within time period (t, t + δ]). For such linear choice of h and log f , log asset prices log S t under multiplicative impact evolve like nominal asset prices in the seminal model by [26] for additive transient price impact, as decribed in equations (1.1). Our setting also admits for resilience rate β := 0 (hence h) to be zero, what makes price impact permanent (cf. Section 7) and log-price impact log Next, we specify the large trader's proceeds (negative expenses) L, which are the variations of her cash account to fund the dynamic holdings Θ 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 Θ of finite variation, are the proceeds. And there is a unique continuous extension of the functional Θ → L(Θ) in (2.6) to general (bounded) semimartingale strategies Θ, that is given by To define L by (2.7) is thus natural as the continuous extension of L from (2.6) to all semimartingales.
For other possible applications, it is useful to note that, more generally, there is a unique continuous extension even beyond semimartingale strategies, see [9,Sect.3]; cf. [22,1] for use of related continuity arguments in different applications. But for our hedging problem, semimartingale strategies will suffice, see (4.3). In section Section 4, the superhedging problem of Definition 3.2 and the superhedging price (4.4) are defined with respect to a particular set of admissible strategies (see (4.3)). The form of this set (being as in [12]) plays a technical role for proofs for the geometric dynamical programming (of Theorem 4.1). One may ask, to which extent the particular choice of this set affects the superhedging price. We will see that in base cases, the superhedging prices w basically recovers impact-friction-less 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 w defined in (4.4) should be robust in that is does not depend on particularities of the said set. The almost-sure uniform approximation result by continuous finite variation strategies in [9,Proposition 3.12] explains, why such robustness indeed holds in the superhedging problem with the notion of liquidation wealth processes, which are given by the unique continuous extension from more elementary to more general trading strategies (than those in (4.3)).
The proceeds from a block trade of selling ∆Θ t shares at time t are (2.9) showing that the price per share that the large trader pays (resp. obtains) for a block buy (resp. sell) order is between the price before the trade f (Y Θ t− )S t and the price after the trade f (Y Θ t )S t . 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 Θ t− )S t and f (Y Θ t− + ∆Θ t )S t , see [7, Section 2.1], and Y can be understood as a volume effect process in spirit of [27].
For a self-financing strategy (B, Θ), in which the dynamic holdings in cash (the riskless asset, savings account) and in stock (the risky asset) evolve as B and Θ, the self-financing condition is 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 were forced to liquidate her position of Θ t stocks immediately by a single block trade at market prices, her liquidation wealth V liq This wealth process is mathematically conveniently tractable, evolving continuously with and V liq 0 = B 0 −, and it inherits from the proceeds (2.7) the stable continuous dependence properties (on Θ), mentioned above. The notion of liquidation wealth V liq (Θ) is relevant for the hedging application of Section 3 and it is different from the so-called book wealth process (2.12) 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 V liq In the terminology of Jarrow [24, cf. Sect.IV], V liq is real wealth whereas V book is paper wealth. Recently, Kolm and Webster [25] have given theoretical and practical reasons that accounting for value (respectively P&L, that means changes in value) of a risky asset position based on current market prices S () can be misleading and needs to be adjusted for price impact; in their terminology V liq corresponds to fundamental wealth whereas V book is accounting wealth, also referred to as mark-to-market wealth.
One obtains from (2.11) absence of arbitrage within the following set of admissible strategies

Proposition 2.2. The market is free of arbitrage up to any finite time horizon
In the terminology of [24, Sect.IV, eqn. (13)], the no-arbitrage result of Proposition 2.2 states that there exist no market manipulation trading strategies. Note that, in contrast, there is no reason to expect no-arbitrage in terms of book wealth V book ; there are simple counterexamples, see Example 2.4 for implications on (super-)hedging prices.

Remark 2.3.
In the seminal article by [23], a notion of no profitable round-trips (stronger than no-arbitrage) is defined, which (in our notation) requires that there exists no (selffinancing) strategy (B 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. They include permanent and additive impact. For comparison, under multiplicative permanent price impact, a linear log-price impact function log f is sufficient to conclude that V liq is a martingale under P (by (3.2)  2) The idea of proof is as in [9,Sect.4], where it was additionally required for admissible strategies that V liq is bounded from below. The latter condition however can be omitted in the present setup of bounded strategies. To see this, observe that for any Θ ∈ A NA there exists an equivalent measure Q Θ ∼ P (on F T ), constructed as in [9,proof of Thm.4.3], under which the wealth process V liq is a martingale.

Remark 2.5.
To highlight key differences to [12], let us explain in detail why the basic (log)-linear example for our setup is the Black-Scholes model forS (geometric Brownian motion) with multiplicative (proportional, relative) price impact (see Example 2.1), whereas for [12] the basic example is the Bachelier model (additive Brownian motion) with additive linear price impact. Note first that [12, see equation (2.1)] study a general model where price impact is permanent and additive, in the sense that (using notation as in our paper) resilience h = 0 is zero, thus Y = Θ for Y 0− = Θ 0− := 0, and the stock price after a small (infinitesimal) trade of size δ becomes s(θ + δ) ≈ s(θ) + δf(s(θ)) where f : R → (0, ∞) is a smooth function of the current stock price s(θ) which prevails if the large trader holds θ stocks just before the trade. That means, more precisely, d dθ s(θ) = f(s(θ)). For comparison, it is instructive to pretend, just formally, that one could choose a 'multiplicative' form f(x) := λ(x)x. With  [12]: Neither is x → λx (strictly) positive on R, nor is x → exp(λx) a surjective function from R → R. Observe that asset prices in [12] take values x in R (instead of (0, ∞)). The instructive basic example for their setting is the case of fixed (constant) impact with f(x) := λ > 0 where s(θ) =s + λθ, and with the unaffected asset pricē s evolving as in the Bachelier model (say), see [12,Section 3.4] with additive permanent price impact. In contrast, the basic example to our setup is transient proportional impact (being additive and linear in terms of log-prices) with respect to a Black-Scholes-type geometric Brownian motion forS, see Example 2.1.

Hedging under transient price impact
We solve in Sections 3-6 the common problem of dynamic hedging, where the issuer who wants to hedge the option receives at time t = 0 the option premium in cash. In an illiquid market setting with price impact, it is relevant to distinguish between cash settlement and physical settlement of an option payoff because, in contrast to frictionless models with unlimited liquidity, moving funds between the bank account and the risky asset account not only induces trading costs from price impact but also affects the price evolution of the underlying, what induces feedback effects [19,28]. Depending on the option's settlement specifications, a terminal block trade at maturity could affect an option's payoff in different ways (see Section 6). We consider contingent claims of the following type.

Definition 3.1. A European option with maturity T ≥ 0 is specified by a measurable map
representing the payoff, with cash-settlement part g 0 and physical-delivery part g 1 at maturity. It entitles its holder to receive g 0 (S T , Y T ) in cash and g 1 (S T , Y T ) in units of the underlying risky asset, when (S T , Y T ) is the risky asset price and the level of market impact at maturity.
Henceforth, T is a fixed maturity time. The optimization task for the seller, that is the issuer, of the option with payoff (g 0 , g 1 ) is to do dynamic hedging at minimal cost to avoid potential losses from her obligation to deliver the payoff at maturity. Among her admissible trading strategies Γ (to be specified precisely in Section 4.1), she is going to look for the cheapest strategies to super-replicate the option's payout in the following sense.

Definition 3.2 (Hedging of an option). A superhedging strategy is a self-financing strategy
We emphasize that a hedging strategy has to deliver the physical component g 1 (S T , Y T ) at maturity exactly, and that any further (long or short) position in the underlying has to be unwound before options are settled at the resulting price S T and impact level Y T . In particular, a hedging strategy for a payoff with pure cash delivery part is a so-called round trip, i.e. it begins and ends with zero shares in the underlying, while the hedging strategy for a payoff with non-trivial physical delivery part should be such that the amount of risky assets held at maturity will meet exactly the physical delivery requirement. Thus, hedging strategies for European contingent claims with physical delivery can be different from those with pure cash delivery part, and we will see, that their respective prices can also differ.
The (minimal) superhedging price of an option with payoff (g 0 , g 1 ) is the minimal (infimum of) initial capital B 0− for which such a superhedging strategy (B, Θ) exists. Note that by the impact process Y , the hedging strategy Θ clearly affects the volatility of the price process S underlying the option payout, because price impact in (2.3) is multiplicative.
Options with pure cash settlement are described by g 1 = 0. Every (reasonable) option could be represented by a payoff with pure cash settlement. Indeed, if the Γ set is stable under adding additional jump at maturity time, meaning that Θ ∈ Γ implies that Θ + ∆1 {T } ∈ Γ for every F T -measurable ∆, then any European option can be represented by an option with pure cash settlement. To see this for an option with payoff (g 0 , g 1 ), let for (s, The value H(s, y) is the minimal amount of cash (riskless assets) needed to hedge the payoff (g 0 , g 1 ) with a single (instant) block trade at maturity, when just before that trade (at time T −) the level of impact is y and there are no holdings in the risky asset whose price is s. Indeed, a block trade of size θ will result in the new prices = sf (y + θ)/f (y) and impact y = y + θ, it will incur the cost s(F (y + θ) − F (y))/f (y). Thus, it will hedge the claim (g 0 , g 1 ) if θ = g 1 (s,ỹ) and we have enough capital to pay for the block trade and to cover the cash-delivery part that after the block trade equals g 0 s,ỹ), see Definition 3.2.
Example 3.3. 1. A cash-settled European call option with strike K is specified by the payoff (g 0 (s, y), g 1 (s, y)) = ((s − K) + , 0). 2. In comparison, a European call option with strike K and physical settlement has the payoff (−K1 {s≥K} , 1 {s≥K} ). Although the payoff profile (g 0 , g 1 ) does not directly depend on the level of impact y, the equivalent pure cash settlement profile H from (3.1) typically will depend on it, if the function λ is not constant. Indeed, the effect on the relative price change f (y + θ)/f (y) from a block trade θ can depend on the level y of impact before the trade in general, unless f (x) = exp(λx) for λ being constant (linear log-price impact).

Remark 3.4.
We discuss an example to show how the hedging problem for the large trader could be related to hedging in a market with perfect liquidity but with portfolio constraints, if F from (2.8) is not surjective onto R. In particular, in this case our market model will not be complete in the sense that not every contingent claim can be perfectly replicated. A prototypical example is the special case of purely permanent impact, i.e. h ≡ 0, with constant λ and log-linear price impact log f (x) = λx (as in Example 2.1), and an option whose payoff (H, 0) specifies settlement in cash only. Hence, we are in the setup of [3] with the smooth family of semimartingales P (x, t) := exp(λx)S t . If Y 0− = 0 and λ = 1, (2.11) takes the form By the conditions from Definition 3.2, any hedging strategy Θ satisfies Θ T = 0, and hence at maturity S T =S T and Y T = Y 0− = 0. Thus, the superreplication condition becomes . This means that, after a reparametrization Θ → exp(Θ)−1 of strategies, the superreplication problem in this large investor model becomes equivalent to the problem in the respective frictionless model (for instance, from Black-Scholes) with price processS for a small investor and with constraints on the delta (to be greater than -1 ), that is on the number of risky assets that a hedging strategy might hold. In particular, one should expect that in such situations (where F is not invertible) the pricing equation should contain gradient constraints. Note that this is different from [3] because for this particular f the crucial Assumption 5 there is violated, and also different from [12] because their assumption (H2) would not hold in this case.
In the presence of resilience for the market impact (h ≡ 0), the situation becomes more complex, however, since the evolution of the price and impact processes depend on the entire history of the trading strategy, and thus a simplification as above is not applicable. But we will see later in Section 5.2 that in the case f = exp(λ·) a lower bound on the delta will also emerge naturally in order to make sense of the pricing equation.

Superhedging by geometric dynamic programming
We formulate the superhedging problem as a stochastic target problem and prove a geometric Dynamic Programming Principle (DPP) for the control problem whose value function will be characterized. Notably, it will show that a DPP (Theorem 4.1) holds with respect to suitably chosen coordinates, which correspond to modified state processes describing the evolution of effective price and impact levels that would result from an immediate unwinding of the risky asset holdings by the large trader. With respect to these new effective coordinates, we will characterize the value function of the control problem as a viscosity solution to a partial differential equation, cf. (PDE) and (PDE δ ) in Section 5, that is the pricing PDE generalizing the (frictionless) Black-Scholes equation.

Stochastic target formulation
We consider strategies that take values in the constraint set K ⊆ R, for one of the two cases K = [−K, +∞) for some K > 0, or (4.1) The short-selling constraints (4.1) will be needed when F is not surjective onto R, see Remark 3.4, in which case we will consider in Section 5.2 f (x) = exp(λx) for some λ > 0, while K = R will be in force when f is bounded away from 0 and +∞, meaning that the (relative) change of the price from a block trade cannot be arbitrarily big.
For the analysis, we need to allow for jumps in the admissible trading strategies in order to obtain a DPP, following [12]. [0, t]) is adapted to the underlying filtration. Note that the elements of U k have the representation where 0 ≤ τ 1 < · · · < τ k ≤ T are stopping times and δ i is a [−k, k]-valued F τi -random variable (might take values 0 as well). Consider also U := k≥1 U k .
The admissible trading strategies Θ that we consider are bounded, take values in K and have the representation In this sense, we identify the trading strategies by triplets (a, b, ν) ∈ A × U. For k ∈ N set and let Γ := k≥1 Γ k . To reformulate the superhedging problem in our price impact model as a stochastic target problem, consider for (t, z) where the processes S t,z,γ , Y t,z,γ , Θ t,z,γ and V liq,t,z,γ correspond to the price, impact, risky asset position and instantaneous liquidation wealth processes on [t, T ] for the control Θ t,z,γ associated with γ (from the decomposition like (4.3) but on [t, T ]), when started at time t− at s, y, θ and v, respectively. Following the discussion in Section 3, for an European option whose payoff in cash-and physical units at maturity T is described by a measurable map (s, y) ∈ R + ×R → (g 0 (s, y), g 1 (s, y)), γ ∈ Γ is a dynamic superhedging strategy if its state process is a.s. at maturity T within the set which we call the target set. The superhedging strategies are for θ denoting the initial position in the risky asset, and with Our aim is to derive the (minimal) superhedging price in the case where the hedger holds no assets of the underlying initially. Let us note that the value function depends on the constraint set K (via the target set G). Note also that the set of admissible superhedging strategies (identified with G(t, s, y, 0, v)) is a subset of A NA , meaning that the superhedging price of a non-negative payoff H (considered as pure cash delivery equivalent of (g 0 , g 1 ) in (3.1) that is positive with non-zero probability, is strictly positive.

Effective coordinates and dynamic programming principle
For stochastic target problems usually a form of the dynamic programming principle holds and plays a crucial role in deriving a PDE that characterizes the value function (in a viscosity sense). The aim of this section is to provide a suitable DPP.
Let us first note that the formulation for the superhedging problem above looks not timeconsistent, because in the definition (4.4) of the superhedging price w it is assumed that the initial position in risky assets is zero, whereas at later times it typically will not be. To obtain a time-consistent formulation, the first naive idea could be to make the risky asset position a new variable, that means to work with the functionw defined on [ But the functionw(t, ·, ·, ·) would have to respect a functional relation along suitable orbits of the coordinates (s, y, θ) at any time t, because of the equations (2.3) and (2.9), namelȳ This suggests that one coordinate dimension is redundant and a 'PDE on curves' may be required to describew. Indeed, for our transient price impact problem we show how the state space can be reduced to make the analysis more transparent, by studying the problem in suitably reduced coordinates which can be interpreted as quantities (for price and impact s, y) at liquidation (of θ)', and with respect to which a DPP and a viscosity characterization is proven for the function w. Otherwise, we can follow arguments by [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 w assumes zero initial risky asset holding, it is natural to consider the (fictitious) state processes that would prevail if the trader would be forced to liquidate her position in the risky asset immediately (with a block trade). To this end, let The process S(s, y, θ) is interpreted as the price of the asset that would prevail after θ assets were liquidated, when s and y are 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 t , Y Θ t , Θ t ) and Y(Y Θ t , Θ t ) as the effective price and impact processes, respectively, for a self-financing trading strategy Θ. Observe that both processes are continuous, even though the trading strategy Θ may have jumps.
For the dynamic programming principle in Theorem 4.1, we are going to compare the liquidation wealth V liq , defined in (2.10), with the value function w along evolutions of the effective price and effective impact processes (S(S, Y Θ , Θ), Y(Y Θ , Θ)).
Proof. There are similarities and differences to [12,proof of Prop.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 admit to cover multiplicative price impact. And transience of impact naturally requires a further dimension in the DDP. The proof uses general ideas on dynamic programming for stochastic target problems and geometric flows [29]. We emphasize that for showing the DPP, our proof develops mathematical arguments in terms of effective coordinates and liquidation wealth V liq , what simplifies mathematical analysis and makes it more transparent. Such shows also in the ability for extensions, described in Section 7.
To prove (ii), let v < w 2k+2 (t, s, y) and suppose that there exists γ ∈ Γ k , Θ ∈ K ∩ [−k, k] and a stopping time τ ≥ t such that V liq,t,z,γ ) and thus, by [29, proof of Thm.3.1, Step 2], we get that v ≥w 2k+1 (t, S(s, y, −θ), y + θ, θ). In particular, by (4.7) we conclude that v ≥ w 2k+2 (t, s, y), hence a contradiction. To derive the pricing PDE from the dynamic programming principle in Theorem 4.1, we need the dynamics of the continuous processes for sufficiently smooth functions ϕ : s, y), that will later serve as test functions when characterizing value functions by viscosity solutions.

Lemma 4.3. For every
By Itô's formula, we obtain (4.11) With reference to (2.11), we have Combining (4.11) and (4.12) and rearranging the terms completes the proof.

Remark 4.4.
Consider the case when λ is constant, i.e. f = exp(λ·). Then we simply have F ≡ 0 and the dynamics of V liq can be stated in a surprisingly simplified form, namely where S t = S(S t , Y Θ t , Θ t ) has the dynamics (4.10). As a consequence, the superhedging price (for the large investor) of an option with maturity T and pure cash settlement H(S T ) is at least the small investor's price of H, in absence of the large trader, when the price process isS instead. Indeed, for each (bounded) superhedging strategy Θ (by the large investor) with initial capital v there exists . On the other hand, a Feynman-Kac argument shows that E P Θ [H(S T )] is just the classical Black-Scholes price for a small investor in a frictionless market with risky asset processS. As Θ was 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 [3,Thm. 5.3], where the price for the large investor would be typically smaller. This is mainly due to a different specification of superhedging strategies with less stringent settlement constraints, according to which a large trader may be able to reduce at maturity the payoff of the option to a larger extend, by exploiting her price impact on the underlying at maturity. That means, she can vary at maturity her risky asset position in order to minimize the payoff with less constraints, and immediately afterwards could unwind any residual risky asset position at no additional cost (by absence of 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 non-negative cash position without residual holdings in the risky asset.
We note that an argument as above does not apply in the general case with non-constant λ for our price impact model. In fact, examples in Section 6 also reveal situations where superhedging could be cheaper for the large trader, cf. Example 6.1.

The pricing PDEs and main results
Next, we determine the terminal value for the function w at maturity date T , that will serve as a boundary condition for the pricing PDE. Recall that K is the (constraint) set in which trading strategies take values and set K n = K ∩ [−n, n] for n ∈ N.
Then we have w n (T, ·) = H n (·) and w(T, ·) = H(·), where the function H is given by Proof. At maturity time T , the hedger of the option has to do a block trade of size θ in order to meet the physical delivery part specified by g 1 , thereby moving the price of the underlying from s to s f (y+θ) f (y) and the impact level from y to y + θ. Such a block trade incurs costs of size s F (y+θ)−F (y) f (y) and hence it superreplicates the payoff (g 0 , g 1 ) if the hedger can cover this costs and the required cash delivery part, which after the block trade is g 0 s f (y+θ) f (y) , y + θ . w k (t , s , y ), (5.2) where the limits are taken over t < T . Recall that w is a (discontinuous) viscosity solution (of our pricing equations, see Sections 5.1 and 5.2) if w * (resp. w * ) is a supersolution (resp. subsolution). For proving the viscosity property we make the following assumption. 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 fulfill his obligation from selling the option immediately, he would be able to do so in any situation (in any state of (s, y)) with an admissible trade, provided that he has enough capital.

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 h is Lipschitz and bounded, the price impact function f is bounded away from 0 and ∞, i.e. inf R f > 0 and sup R f < +∞, λ is bounded and continuously differentiable with bounded derivative, and K = R (no delta constraints).
Under Assumption 5.4, the antiderivative F from (2.8) and its inverse F −1 are bijections R → R and Lipschitz continuous with Lipschitz constants sup R f < +∞ and 1/ inf R f , respectively.
To derive the pricing PDE just formally (at first, to be justified later) in this case, let (t, s, y) ∈ [0, T ) × R + × R and apply formally part (i) of DPP in Theorem 4.1 to v = w(t, s, y) (assuming that the infimum in the definition of w is attained) and τ = t+, together with Lemma 4.3 for ϕ = w, assuming that w is smooth enough. Thus we get the existence of θ * such that − w s (t, s, y) (µ t − λ(y)h(y + θ * )) dt + σ dW t + − w t (t, s, y) − 1 2 σ 2 s 2 w ss (t, s, y) + h(y + θ * )w y (t, s, y) + F(s, y, θ * ) dt. Still arguing just at a formal level, this cannot hold unless F (y + θ * ) = f (y)w s (t, s, y) + F (y) and In particular, θ * = θ * (t, y, s) = F −1 f (y)w s (t, s, y) + F (y) − y. The second part of DPP in Theorem 4.1 will actually give that the drift term must be 0, i.e. we should have equality in (5.3). This formally motivates that the form of the pricing PDE for w should be where for (t, s, y) Observe that the PDE is semilinear and degenerate (since not containing second order derivatives involving the y-variable). Our main result is as follows.

Theorem 5.5. Under Assumption 5.3 and Assumption 5.4, the value function w of the superhedging problem is continuous and is the unique bounded viscosity solution to (PDE) with the boundary condition w(T, ·) = H(·), where H is defined in (BC).
Proof. The viscosity property, i.e. that w * (respectively w * ) is a viscosity supersolution (respectively subsolution), follows by the dynamic programming principle in Theorem 4.1 together with Lemma 4.3. The key arguments are presented in Appendix A in detail for the case where λ is constant, which actually leads to a slightly more involved pricing PDE (PDE δ ) (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 respective hedging strategy. A numerical example is presented in Section 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 forS from the beginning. On the other hand, the superhedging price depends non-trivially on the initial level of impact y and the resilience function h, and can do so even for option payoffs of the form (g 0 (s), 0), i.e. not depending on the level of impact. So it turns out that for the pricing and hedging (cf. Remark 5.8) the deviation of the market price from the 'unaffected' value, determined by the impact level y, is a relevant state variable.

Remark 5.7.
Observe that for only permanent impact, that means for h ≡ 0, (PDE) simplifies to the classical (frictionless) Black-Scholes pricing equation and hence the superhedging price for the large trader 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 is perfectly replicating the option payout from the (minimal) superhedging price. This means, 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 ∈ C 1,3,1 b ([0, T ] × R + × R) solves the pricing PDE (PDE) with the boundary condition w(T, ·) = H(·). Then for any ε > 0 a superhedging strategy with an initial cost of w(0, s, y) + ε can be constructed as follows. Consider the self-financing strategy (B, Θ) with B 0− = w(0, s, y) + ε, Θ 0 = F −1 (f (y)w s (0, s, y) + F (y)) − y, meaning that a block trade of size ∆Θ 0 = Θ 0 is performed at time 0, and where Y Θ = Y Θ − Θ. Then by Lemma 4.3, together with (5.4) and (PDE) we conclude that where the last line follows from (5.5). By definition of H, having H + ε in cash at time T will be enough to superreplicate the European claim with payoff (g 0 , g 1 ) by doing a possible additional final block trade of size ∆ ε . Note that such a block trade would not affect V liq T . Hence, the strategy Θ + 1 {T } ∆ ε will be superreplicating for the European claim. Note that one could take ε = 0 if the constructed strategy is bounded and the infimum in the definition of H n is attained (cf. Lemma 5.1), i.e. we have a replicating strategy in this case.
An application of Itô's formula gives that a strategy Θ satisfying the fixed-point problem (5.4) can be obtained, under suitable regularity, by solving the following system of SDEs 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 superhedging strategy accounts for the transient nature of price impact.

Remark 5.9.
To describe how replicating hedging strategies in our model are decribed by coupled Forward-Backward SDEs, suppose that Θ is a replicating strategy for an option with cash-equivalent payoff H and let (Y, S) be the effective impact and price processes. By a change of measure argument, we can assume w.l.o.g. that µ = 0.
and using (4.12) leads to the following coupled FBSDE: where the driver of the FBSDE g : R × R + × R → R is given by Example 5.10. 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 v BS (s) = s and a (minimal) 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 (PDE) with the boundary condition H being given by the function where c : [0, T ] × R → R is a solution to the following backwards transport equation In particular, by the dynamics of c it then holds for any strategy Θ than c(t, where Y Θ is the effective impact process corresponding to Θ. In particular, by (5.4) a minimal replicating strategy satisfies on [0, T ) the equation Hence, a buy-and-hold strategy is also optimal for the large trader. We can observe: 1.) Purely permanent impact (that means h = 0) would yield the Black-Scholes price w(t, s, y) = s and the buy-and-hold strategy of c(0, y) = c(T, y) = F −1 (f (y) + F (y)) − y shares, that does not depend on the maturity T .
2.) In comparison, if price impact is not permanent but transient (h = 0), the price (5.7) depends non-trivially on the maturity T , in addition to the price impact and resilience functions f and h respectively.  c(t, y) > c(T, y). Moreover, there are situations where this condition holds and situations where it is violated. The reason is that there are two counterbalancing effects: At initiation where the large trader buys shares to set up the initial delta hedge, moving prices in an unfavourable direction, and at maturity when she liquidates the delta and could move prices in a direction favourable to her. Which of these two effects dominates depends non-trivially on the level of liquidity initially and at maturity, and the settlement specificaions of the option; See discussion in Example 6.1.
Let us comment here on Assumption 5.4 that implies bijectivity of F on R. Observe that its inverse F −1 is used to describe the optimal control θ * . Similar conditions are also crucial for the results in [3] and [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 singularity in the pricing PDE with respect to the gradient. Indeed, the lack of invertibility of F requires conditions on w s so that θ * could be derived. Therefore, the analysis there will involve constraints on the 'delta', that means on the holdings in the risky asset, what in PDE terms translates to constraints on the spacial gradient w s .

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 λ being a constant (i.e. log f is 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 S t is sufficient to determine the impact from an instant block trade, since after a block trade of size ∆ the price would beS t f (Y t + ∆) = S t exp(λ∆). Hence, the relative displacement f (Y Θ ) of S from the fundamental priceS is immaterial to determine the price impact from a block trade, in difference to the situation of Section 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 will 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 w is smooth enough, to get the existence of θ * ∈ K such that, using Lemma 4.3, we have where η t = µ t − λh(y + θ * ) and As in Section 5.1, the diffusion part in (5.8) should vanish, giving the optimal control θ * = 1 λ log λw s (t, s, y) + 1 , and from the drift part we identify the pricing PDE L θ * w(t, s, y) = 0. The constraint θ * ∈ K is now equivalent to H K w(t, s, y) ≥ 0, where for a smooth function ϕ we set Thus we conclude, just formally, that w should be a solution to the variational inequality where θ[w](t, s, y) := 1/λ · log λw s (t, s, y) + 1 . (5.9) As usual, the gradient constraints propagate to the boundary, meaning that the boundary condition for (PDE δ ) should be After this motivation, we state the main result for exponential price impact f = exp(λ ·). Proof. If (g 0 , g 1 ) is a function of the price s of the underlying only (but not of y), then it is easy to see that H is such as well and that the dimension of the state process can be reduced by omitting the impact process Y . In this case, the stochastic target problem in Section 4 could be formulated for the new state process and thus the value function would be a function on (t, s) only. The same analysis can be carried over to derive the pricing PDE and to prove viscosity solution property of the value function.

Numerical examples
We discuss numerical calculations of the superhedging price w characterized by (PDE), cf. Theorem 5.5, to illustrate results. For the computations we consider an impact function f (x) = 1 + arctan(x)/10, x ∈ R, (6.1) satisfing Assumption 5.4. Note that λ(x) = 1/(10(1+x 2 )f (x)) varies most within the range of about (−4, 4); Here, the change in impact is significant, see Figure 1a. Apart from satisfying our assumptions and having F (x) = x + (x arctan(x) − 1/2 log(1 + x 2 ))/10 in explicit form, being useful for the implementation, it turns out that similar shape of impact has been observed in the calibration of a related propagator model to real data, see [15,Appendix].
For h(y) = βy with β = 1, we compare the large trader's price of a European call option with physical delivery at maturity T = 0.5 and strike K = 50, and the option's frictionless price, i.e. the classical Black-Scholes price of a European call option for the same model parameters. Let us recall that the case f = 1 in our price impact model coincides with the Black-Scholes model. The volatility σ is set to 0.3. The payoff for the large trader is − K 1 {s≥K} that we "smooth out" by approximating the indicator function from above by linearly interpolating 0 and 1 between K − 0.5 and K.
To approximate both prices, we solve the corresponding PDEs using a (semi-implicit) finite difference scheme in the bounded region Indeed, for initial impact y close to -20 or +20 the impact function is approximately constant and until maturity T resilience would be unlikely to bring back the level of impact to the region where the changes in f are significant, see Figure 1a; Thus we might expect that the price would not depend that much on the level of impact. On the other hand, for larger values of s one may expect the price to depend approximately linearly on s (like the payoff profile). The difference between the Black-Scholes price and the large trader's price (as a function of the risky asset price s and the level of impact y) is shown in Figure 1b. Let us point out that the Black-Scholes price does not depend on level of impact y.
Numerical results of Figure 1c illustrate that the superreplication price for the large trader dominates the frictionless Black-Scholes price for the call option with physical delivery. But we note that such property does not need to hold in general. For instance, it does not appear to do so in case of a European call with pure cash delivery, where numerical computations show that for the large investor the price could also be smaller, typically if the impact level at inception is away from zero, see also Example 6.1. The intuition for this more complex behaviour is that for pure cash delivery, the net turnover until maturity of traded assets for a (super-)hedging strategy must be zero (as Θ 0− = Θ T = 0 then), while non-zero resilience (h = 0) induces additional drift, that also turn out to be less costly to the large trader, if moving the underlying price paths into regions with lower (or zero) option payout.
On the other hand, superhedging becomes more expensive for the large trader when she has to deliver the underlying asset physically at maturity, since, if the call option settles in-the-money, she needs to do a final block trade to buy what is lacking (in the pre-terminal delta position) for the one physical unit required. But this last price impact at maturity is costly in that it further increases the issuer's call option payout for physical delivery, in comparison to cash settlement, where selling the long delta position decreases the payout.
In addition, observe that the presence of resilience renders the level of impact (or the displacement from the fundamental price) to be a relevant state variable for the problem. For the setup of our numerical example for instance, the price of a European call option with physical delivery, when hedging is initiated at neutral impact level (y = 0), is cheaper in the presence of resilience than in the case of no resilience, i.e. only permanent impact, see Figure 1d. This is however not always the case, for example if impact at initiation is negative (y < 0). To conclude, the dependence in y of the option's price is complex: apart from the drift on the prices that the level of impact induces, it also determines the price impact from intermediate tradings and the final trade (enforced by settlement rules). And we have mentioned examples where superhedging could be less or more expensive for the large investor in the presence or absence of resilience.
Example 6.1. The price of an European option in the Black-Scholes model (for s small investor) could indeed be greater than the superhedging price for the large trader of this option with pure cash delivery. To see this, consider for maturity T > 0 the solution v BS of the Black-Scholes PDE with bounded and smooth terminal condition H that has bounded derivatives, where we moreover assume that ∂ S H ≥ 0, for instance a smooth approximation In particular, if the integrand in (6.3) is negative on [0, T ], then (B, Θ) would be a superhedging strategy for the large trader with initial capital One can show that the integrand will be negative for instance when Y Θ ≥ 0 on [0, T ] and λ is strictly decreasing (at least on a compact set containing the range of Y Θ and Y Θ − Θ); Such a situation could arise if for example Y 0− is large enough. Alternatively, a negative integrand could also occur if for instance Y Θ is negative on [0, T ], for instance if Y 0− is small enough, and λ is strictly increasing. Let us mention that equality in (6.4) cannot hold in general, for all values of S 0− , Y 0− , as this would imply that v would not depend on the initial level of impact Y 0− , that is not the case for general payoff functions H, see e.g. Figure 1b.

Extensions: Permanent impact, covered options, multiple illiquid assets with cross-impact
This section explains possible extensions and variations of the previous results on hedging under multiplicative transient price impact. We show first, how the results generalize to combined transient and permanent price impact, and explain, how working in suitable effective coordinates further enables extensions to multiple illiquid assets. We also comment and give references for the solution to the different but related hedging problem for covered options.
For η ≥ 0, the marginal price of the risky asset (for an extra infinitesimal quantity) is in a generalization of equation (2.3), with Y Θ being given by (2.2). Following the arguments in [9,Section 5.4], the (stable) proceeds from a general semimartingale strategy Θ arẽ In particular, a block trade ∆Θ t yields the proceeds −S t 1 1+η Thus, following the discussion in Section 2, the volume effect process (in the spirit of [27]) in this case is ηΘ + Y Θ and thereby has a permanent and a transient component. The dynamics of the instantaneous liquidation value processṼ liq now satisfies It is worth noting that the generalization by an additional permanent impact is not changing the effective price and impact processes S(S, Y Θ , Θ) and Y(Y Θ , Θ) because the permanent component vanishes for asset holdings with zero shares in the risky asset. Therefore, the previous analysis carries over to additional permanent impact, with adjustments as follows: • The boundary condition in Lemma 5.1 needs to be modified by adding the prefactor 1 + η to θ, when θ appears as an argument of a function; • In Lemma 4.3, F (Y Θ ) should be substituted by F (ηΘ + Y Θ ), all the fractions should be divided by 1 + η and F will now become F η with .
An optimal hedging strategy Θ * , if it exists, satisfies (as in Remark 5.8 for η = 0) where S * = S(S, Y Θ * , Θ * ) and Y * = Y(Y Θ * , Θ * ). Hence, the large trader's optimal strategy also reflects the permanent component in addition to the displacement from the fundamental price process tracked by Y Θ . In the setup from Section 5.2, we again consider portfolio constraints θ ∈ K = [−K, +∞) in order to derive the pricing PDE. Thanks to F η = 0, the pricing PDE here simplifies to where H is the modified boundary condition from Lemma 5.1, as explained above. In particular, the pricing PDE with permanent impact coincides with the pricing PDE with purely transient impact but with suitably modified λ, that in this case becomes λ(1 + η).
Remark 7.1. We explain how to obtain further results and note key differences in the related hedging problem for so-called covered options, as in [13] but under multiplicative transient price impact, where our analysis carries over similarly, by adopting arguments of [13] from the case of additive permanent price impact, as shown in [6,Sect.8].
In contrast to the problem studied in the main body of the present paper and in [12] for non-covered options there is no price impact at inception and at maturity in the hedging problem for covered options. Such makes the stochastic target problem very different. The financial interpretation is, that the buyer of a covered option provides (at discretion of the hedger) the required initial (delta) hedging position and accepts any mix of cash and stocks (at 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 when forming and unwinding the hedging position for covered options. We mention that similar assumptions are made in the literature [19,20,16] where analysis is in terms of book value, instead of liquidation value [3,12].
In the previous sections, the superhedging price for (non-covered) options under transient multiplicative price impact is characterized by a degenerate semilinear PDE, whose nonlinearity involves the resilience function h and the price impact function f . It can involve gradient constraints (that means 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 non-linear and singular in the second-order term. This induces gamma constraints, whereas for non-covered options singularity arises in the first order derivative and induces delta constraints, see Section 5.2. For covered options, it can be shown [6,Sect.8] that the resilience of price impact is immaterial for the hedging price, irrespectively of a particular form for the resilience function, what has been observed likewise in [13,Section 4] for additive impact. We emphasize that this is very different to Section 5.1, where the resilience function enters the pricing equation in a non-trivial 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 [6, see Remark 8.2,2)] that the superheding price is decreasing in the impact function λ, in the sense that if λ ≥λ, then the price with respect to λ dominates the one with respect toλ. For a dual formulation for the hedging of covered options we refer to [14].

Remark 7.2.
Working in effective coordinates, as explained in Section 4, further permits to extend results about transient price impact, in additive or multiplicative form, to multiple risky assets with cross-impact from transactions across different assets (being described in [11, ch.5, cf. 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 (cf. [11,Thm.5.1.4]). Thereby, results like from previous sections (or [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 characterize superhedging prices, which does involve the resilience function (h) of transient impact (see [11,Sect.5.3.2]). And under certain conditions one recovers as instructive reference case again results as in a multi-dimensional Bachelier model with its natural pricing formula [11][in Rem.5.3.8]), that is not involving the price impact. This extends (to multiple dimensions) the instructive 1-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, though closely related to the usual Bachelier-delta-hedging strategy, by being computed at liquidation magnitudes of the stock price. This is entirely analogous to Black-Scholes quantities occuring under (permanent) multiplicative impact the basic log-linear example of our model (see Example 2.1 and Remark 2.5).

A. Proofs
This section provides the proofs delegated from Section 4, 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 (5.9) for the form of the optimal strategy to be finitely defined.
First, we verify in Appendix A.1 that if the pricing PDE (PDE δ ) admits a sufficiently smooth classical solution, then a replicating strategy in feedback form can be constructed. Such a construction will be needed also for the contradiction argument in the proof of the subsolution property in Appendix 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. Suppose further that w is sufficiently regular (see the subsequent remark) so that there exists an admissible strategy Θ ∈ Γ of the form

A.1. Verification argument for exponential impact function
In particular, Θ 0 = 1/λ log(λw s (0, s, y) + 1) and ∆Θ T ∈ K. Consider the self-financing portfolio (β, Θ) with β 0− = w (0, s, y). Then as in Remark 5.8 we get By definition of H, this shows that V liq T (Θ) at maturity T is enough capital to (super-)replicate the European claim with payoff (g 0 , g 1 ) with a possible additional block trade (provided that the infima in the definition of H, cf. Lemma 5.1, are attained). Hence, (β, Θ) will be a (super-)replicating strategy for the European claim (g 0 , g 1 ) with initial capital w(0, s, y), meaning that its price is exactly w (0, s, y).
Remark A.1 (On the form of a replicating strategy). To construct a replicating strategy (A.1), suppose moreover that w ∈ C 1,3,1 ([0, T ] × R + × R) and apply Itô's formula, similarly as in Remark 5.8, to get for t < T the equation where for S t := S(S t , Θ t ) and with all the derivatives of w above being evaluated at (t, S(S t , Θ t ), Y t −Θ t ). Thus, a replicating strategy, which is superhedging the payout at a minimal cost, can be constructed as the (Θ t ) t∈[0,T ) -part (plus a terminal block trade) from a solution, if it exists, to the SDE system for t ∈ [0, T ], with initial condition S 0 = s, Y Θ 0 = y and Θ 0 = 1/λ log(λw s (0, s, y) + 1).

A.2. Viscosity solution property of w for exponential impact function
For the result from Section 5.2 we now prove the viscosity property.
Theorem A.2. The function w * from (5.1) is a viscosity supersolution of (PDE δ ) on Case 1: Suppose that H K ϕ(t 0 , s 0 , y 0 ) < 0. By continuity of the operator H K there exists an open neighborhood O of (t 0 , s 0 , y 0 ) whose closure is contained in [0, T ) × R + × R, such that H K ϕ(t, s, y) < −ε in O for some ε > 0. Therefore, after possibly decreasing the neighbourhood O, there exists a constant k ε > 0 such that Let (t n , s n , y n ) n ⊂ O be a sequence converging to (t 0 , s 0 , y 0 ) with w(t n , s n , y n ) → w * (t 0 , s 0 , y 0 ) (note that w * is the lower-semicontinuous envelope of w). Set v n := w(t n , s n , y n ) + 1/n.
Since v n > w(t n , s n , y n ), Theorem 4.1 implies the existence of θ n ∈ K and strategies γ n ∈ Γ such that for stopping times τ n ≥ t n (to be suitably chosen later) we have P-a.s. where z n = (s n e λθn , y n +θ n , θ n , v n ). To abbreviate notation, in the sequel we write n as superscript instead of (t n , z n , γ n ), with S n := S(S tn,zn,γn , Θ tn,zn,γn ), Y n := Y tn,zn,γn − Θ tn,zn,γn . Take τ n = inf{t ≥ t n (t, S n t , Y n t ) ∈ O}, which is the first entrance time of the parabolic boundary of the open region O. In particular, τ n < T . Since w ≥ w * ≥ ϕ and w * − ϕ has a strict local minimum at (t 0 , s 0 , y 0 ), there exists ι > 0 such that Hence, V liq,n τn − ϕ(τ n , S n τn , Y n τn ) ≥ ι. Now, Lemma 4.3 together with the fact that S n tn = s n , Y n tn = y n , gives that P-a.s.
with η n t := µ t − λh(Y n t ). Note that ζ n t is well-defined on [t n , τ n ] and uniformly bounded, noting (A.3) and the fact that Y n is bounded since Θ n is so. Hence, by Girsanov's theorem, there exists a measure P n that is equivalent to P such that is a square-integrable martingale under P n as the integrand of the stochastic integral is uniformly bounded, because of the definition of τ n , the continuity of ϕ S and the boundedness of the range of Θ, noting τ n ≤ T . Taking expectation under P n of the right-hand side of (A.5) leads to v n − ϕ(t n , s n , y n ) ≥ ι > 0, what yields a contradiction as by our choice of v n and the sequence (t n , s n , y n ) n v n − ϕ(t n , s n , y n ) −→ w * (t 0 , s 0 , y 0 ) − ϕ(t 0 , s 0 , y 0 ) = 0. In particular, by continuity of the functions involved we have (after possibly decreasing the open set O) that for every (t, s, y) ∈ O and for some r > 0 L θ ϕ(t, s, y) < −ε whenever |ϕ S (t, s, y) + 1/λ − e λθ /λ| ≤ r .
As in Case 1, consider a sequence (t n , s n , y n ) in O which converges to (t 0 , s 0 , y 0 ) and such that w(t n , s n , y n ) → w * (t 0 , s 0 , y 0 ). Set v n := w(t n , s n , y n )+1/n and let θ n ∈ K and strategies γ n ∈ Γ be such that the dynamic programming principle (A.4) holds for the stopping times τ n that are the first exit times of (·, S n , Y n ) from the set O. Now, a contradiction follows similarly as in Case 1 with the following adjustment: We have where we set with the functions ϕ and ϕ S in the expressions above being evaluated at (·, S n · , Y n · ). The contradiction now follows by taking expectation under P n ≈ P and letting n → ∞.
(strict) max First assume that t 0 < T . To ease the notations, we will use the variable x to denote the pair (s, y). Because of the special form of the second part of DPP, cf. Theorem 4.1 (ii), we need to employ w k (instead of w as we did in the proof of the supersolution property). By [5, Lemma 6.1] we can take a sequence (k n , t n , x n ) n≥1 such that k n → ∞, any (t n , x n ) is a local maximum of w * kn − ϕ, and (t n , x n , w kn (t n , x n )) → (t 0 , x 0 , w * (t 0 , x 0 )). Assume that F K [ϕ](t 0 , x 0 ) > 0 and let ϕ n (t, x) = ϕ(t, x) + |t − t n | 2 + |y − y n | 2 + |s − s n | 4 . Then F K [ϕ n ] > 0 holds in a neighborhood B of (t 0 , x 0 ) that contains (t n , x n ), for all n large enough. Since we will be working on the local neighborhood B where also H K ϕ n > 0, we can modify (in a smooth way) the functions h and ϕ n outside of B to be supported on a slightly bigger compact set where H K ϕ n > 0 holds. Thus, after possibly passing to a suitable subsequence, there exists γ n ∈ Γ kn such that  Proof. To prove super-(resp. sub-) solution property, let (t 0 , s 0 , y 0 ) ∈ [0, T ) × R + × R be any point andφ ∈ C ∞ b ([0, T ] × R + × R) be a test function forũ at (t 0 , s 0 , y 0 ), i.e., min We haveφ s (t, s, y) = e κt f (y)ϕ s (t, sf (y), y) ϕ ss (t, s, y) = e κt f 2 (y)ϕ ss (t, sf (y), y) ϕ y (t, s, y) = e κt λ(y)f (y)ϕ s (t, sf (y), y) + e κt ϕ y (t, sf (y), y) = λ(y)φ s (t, s, y) + e κt ϕ y (t, sf (y), y) ϕ t (t, s, y) = e κt ϕ t (t, sf (y), y) + κe κt ϕ(t, sf (y), y).

A.3. Comparison results for viscosity solutions
By direct application of these identities we derive from the right-hand side of (A.7) forφ evaluated at (t 0 , s 0 , y 0 ) exactly the right-hand side of (A.6) for ϕ at (t 0 , s 0 f (y 0 ), y 0 ). By the viscosity property of u and (A.9) we thus conclude that (A.7) holds forφ at (t 0 , s 0 , y 0 ) with greater or equal (resp. less or equal). This proves the claim.
By Lemma A.4 it now suffices to prove comparison for equation (A.7) since this would imply a comparison result for (A.6). This is done in the following result. and where S 2 denotes the set of 2 × 2 symmetric non-negative matrices and I 2 ∈ S 2 is the identity matrix. Using the viscosity property of u and v at (t n , s n 1 , y n 1 ) and (t n , s n 2 , y n 2 ) respectively, we have κu(t n , s n 1 , y n 1 ) − b n − 1 2 σ 2 (s n 1 ) 2 X n 11 + L(s n 1 , y n 1 , p n , q n ) ≤ 0 κv(t n , s n 2 , y n 2 ) − b n − 1 2 σ 2 (s n 2 ) 2 Y n 11 + L(s n 2 , y n 2 , p n , q n ) ≥ 0, where L(t, s, y, p, q) := −B 1 (y, e −κt p)q+λ(y)B 1 (y, e −κt p)p−sB 2 (y, e −κt p)p−e κt sf (y)B 3 (y, e −κt p).