Pricing and hedging contingent claims with liquidity costs and market impact

We study the influence of taking liquidity costs and market impact into account when hedging a contingent claim, first in the discrete time setting, then in continuous time. In the latter case and in a complete market, we derive a fully non-linear pricing partial differential equation, and characterizes its parabolic nature according to the value of a numerical parameter naturally interpreted as a relaxation coefficient for market impact. We then investigate the more challenging case of stochastic volatility models, and prove the parabolicity of the pricing equation in a particular case. Introduction There is a long history of studying the effect of transaction costs and liquidity costs in the context of derivative pricing and hedging. Transaction costs due to the presence of a Bid-Ask spread are well understood in discrete time, see [8]. In continuous time, they lead to quasi-variational inequalities, see e.g. [17], and to imperfect claim replication due to the infinite cost of hedging continuously over time. In this work, the emphasis is put rather on liquidity costs, that is, the extra price one has to pay over the theoretical price of a tradable asset, due to the finiteness of available liquidity at the best possible price. A reference work for the modelling and mathematical study of liquidity in the context of a dynamic hedging strategy is [3], see also [14], and our results can be seen as partially building on the same approach. It is however unfortunate that a major drawback occurs when adding liquidity costs: as can easily be seen in [3] [12] [14], the pricing and hedging equation are not unconditionally parabolic anymore and, therefore, only a local existence and uniqueness of smooth solutions may be available. Note that this drawback can easily be inferred from the very early heuristics in [9]: the formula suggested by Leland makes perfectly good sense for small perturbation of the initial volatility, but is meaningless when the modified volatility becomes negative. An answer to this problem is proposed in [4], where the authors introduce super-replicating strategies and show that the minimal cost of a super-replicating strategy solves a well-posed parabolic equation. Still, a partial conclusion is that incorporating liquidity cost leads to ill-posed pricing equation for large option positions, a situation which cannot be considered satisfactory and hints at the fact that some ingredient may be missing in the physical modelling of the market. It turns out that this missing ingredient is precisely the market impact of the delta-hedger, as will become clear from our results. This fact is already observed by the second author in [11], where a well posed, fully non-linear parabolic equation is obtained using a simple market impact model. Motivated by the need for quantitative approaches to algorithmic trading, the study of market impact in order-driven markets has become a very active research subject in the past decade. In a very elementary way, there always is an instantaneous market impact termed virtual impact in [16] whenever a transaction 1 ha l-0 08 02 40 2, v er si on 3 13 A pr 2 01 3 takes place, in the sense that the best available price immediately following a transaction may be modified if the size of the transaction is larger than the quantity available at the best limit in the order book. As many empirical works show, see e.g. [2] [16], a relaxation phenomenon then takes place: after a trade, the instantaneous impact decreases to a smaller value, the permanent impact. This phenomenon is named resilience in [16], it can be interpreted as a rapid, negatively correlated response of the market to large price changes due to liquidity effects. In the context of derivative hedging, it is clear that there are realistic situations e.g., a large option on an illiquid stock where the market impact of an option hedging strategy is significant. This situation has already been addressed by several authors, see in particular [15] [6] [5] [13], where various hypothesis on the dynamics, the market impact and the hedging strategy are proposed and studied. One may also refer to [7] [10] [14] for more recent related works. It is however noteworthy that in these references, liquidity costs and market impact are not considered jointly, whereas in fact, the latter is a rather direct consequence of the former. As we shall demonstrate, the level of permanent impact plays a fundamental role in the well-posedness of the pricing and hedging equation, a fact that was overlooked in previous works on liquidity costs and impact. Also, from a practical point of view, it seems relevant to us to relate the well-posedness of the modified Black-Scholes equation to a parameter that can be measured empirically using high frequency data. This paper aims at contributing to the field by laying the grounds for a reasonable model of liquidity costs and market impact for derivative hedging. We start in a discrete time setting, where notions are best introduced and properly defined, and then move on to the continuous time case. Liquidity costs are modelled by a simple, stationary order book, characterized by its shape around the best price, and the permanent market impact is measured by a numerical parameter γ, 0 6 γ 6 1: γ = 0 means no permanent impact, so the order book goes back to its previous state after the transaction is performed, whereas γ = 1 means no relaxation, the liquidity consumed by the transaction is not replaced. This simplified representation of market impact rests on the hypothesis that the characteristic time of the derivative hedger may be different from the relaxation time of the order book, a realistic hypothesis since delta-hedge generally occurs at a lower frequency than does liquidity providing. What we consider as our main result is Theorem 4.1, which states that, in the complete market case, the range of parameter for which the pricing equation is unconditionally parabolic is 23 6 γ 6 1. This result, which we find quite nice in that it is explicit in terms of the parameter γ, obviously explains the ill-posedness of the pricing equations in the references [3] [12] that correspond to the case γ = 0, or [7] [10] that correspond to the case to the case γ = 1 2 within our formulation. In particular, Theorem 4.1 implies that when re-hedging occurs at the same frequency as that at which liquidity is provided to the order book that is, when γ = 1 the pricing equation is well-posed. This result was already obtained by the second author in [11]. The paper is organized as follows: after recalling some classical notations and concepts, Section 1 presents the order book model that will be used to describe liquidity costs. Then, in Section 2, we write down the model for the observed price dynamics and study the associated risk-minimizing strategy taking into account liquidity costs and market impact. Section 3 is devoted to the continuous time version of these results. The pricing and hedging equations are then worked out and characterized in the case of a complete market, in the single asset case in Section 4, and in the multi-asset case in Section 5. Finally, Section 6 touches upon the case of stochastic volatility models, for which partial results are presented. 1 Basic notations and definitions To ease notations, we will assume throughout the paper that the risk-free interest rate is always 0, and that the assets pay no dividend. Discrete time setting The tradable asset price is modelled by a stochastic process Sk, (k = 0, · · · , T ) on a probability space (Ω,F , P ) . Fk denotes the σ−field of events observable up to and including time k. Sk is assumed to be 2 ha l-0 08 02 40 2, v er si on 3 13 A pr 2 01 3 adapted and square-integrable. A contingent claim is a square-integrable random variable H ∈ L(P ) of the following form H = δST +β with δ and β , FT -measurable random variables. A trading strategy Φ is given by two stochastic processes δk, (k = 0, · · · , T ) and βk, (k = 0, · · · , T ). δk (resp. βk) is the amount of stock (resp. cash) held during period k, (= [tk, tk+1)) and is fixed at the beginning of that period, i.e. we assume that δk (resp. βk) is Fk−measurable (k = 0, · · · , T ). Moreover, δ and β are in L(P ). The theoretical value of the portfolio at time k is given by Vk = δkSk + βk, (k = 1, · · · , T ). A strategy is H−admissible iff each Vk is square-integrable and VT = H. Continuous time setting In the continuous case, (Ω,F , P ) is a probability space with a filtration (Ft)0≤t≤T satisfying the usual conditions of right-continuity and completeness. T ∈ R∗+ denotes a fixed and finite time horizon. Moreover, F0 is trivial and FT = F . The risky asset S = (St)0≤t≤T is a strictly positive, continuous Ft-semimartingale, and a trading strategy Φ is a pair of càdlàg and adapted processes δ = (δt)0≤t≤T , β = (βt)0≤t≤T , while a contingent claim is described by a random variable H ∈ L(P ), with H = δST +β , δ and β being FT−measurable random variables. H−admissible strategies are defined as follows: Definition 1.0.1 A trading strategy will be called H-admissible iff  δT = δ H P − a.s. βT = β H P − a.s. δ has finite and integrable quadratic variation β has finite and integrable quadratic variation δ and β have finite and integrable quadratic covariation. Order book, transaction cost and impact A constant, symmetric order-book profile is considered around the price Ŝt of the asset S at a given time t before the option position is delta-hedged think of Ŝt as a theoretical price in the absence of the option hedger. The relative density μ(x) > 0 of the order book is the derivative of the function M(x) ≡ ∫ x 0 μ(t)dt ≡ number of shares one can buy (resp. sell) between the prices Ŝt and Ŝt(1 +x) for positive (resp. negative) x. The instantaneous virtual in the terminology of [16] market impact of a transaction of size is then Ivirtual( ) = ŜtM −1( ), (1.1) it is precisely the difference between the price before and immediately after the transaction is completed. The level of permanent impact is then measured via a parameter γ: Ipermanent( ) = γŜtM −1( ). (1.2) It seems reasonable to assume that 0 6 γ 6 1, but we do not impose a priori this constraint. The actual cost of the same transaction is