Lions and Lasry (2007) introduced a utility-maximization theory for a “large” investor whose trading moves an asset price through a linear permanent impact law: they computed the fair (liquidation) value of a large position, the exponential-utility indifference price of options, and the induced effective volatility, proving that the optimal hedge feeds back into the asset’s realized volatility through the option’s gamma. Several items the authors sketched only formally or stated without proof within this model’s scope remain untreated in the literature: a rigorous theory for nonlinear impact laws k(u)k(u) such as threshold impact, for state-dependent impact k(S,α)k(S,\alpha), and for influence laws with memory or relaxation effects; a proof of the stated λ→+∞\lambda\to+\infty limit of the indifference price in the multiplicative-impact model, which requires the envelope of the payoff; and the higher-order terms of the small-impact asymptotic expansion, which the authors state can be obtained but do not carry out. Extending the utility-based fair-value, indifference-pricing, and effective-dynamics results of the Lions–Lasry model from constant linear impact to the influence laws its authors enumerated, and completing the two asymptotic programs, remains open.

Background

When a trader holds a position that is large relative to market depth, their own buying and selling moves the transaction price against them — the phenomenon of market impact or price impact. The simplest continuous-time model, going back to Kyle’s framework of a linear “elastic” price law, takes the impact of the trading rate utu_t on the price to be linear and permanent (the displacement persists after the trade). In the additive form the price follows dSt=σ(St) dBt−k ut dtdS_t=\sigma(S_t)\,dB_t-k\,u_t\,dt, where BtB_t is a standard Brownian motion, σ>0\sigma>0 is an exogenous volatility function (assumed bounded above and below away from zero and Lipschitz), and k>0k>0 measures the (inverse) market depth; in the multiplicative form the drift term acts on the relative price, dSt/St=σ(St) dBt−k ut dtdS_t/S_t=\sigma(S_t)\,dB_t-k\,u_t\,dt. The large investor’s position αt\alpha_t evolves as dαt=−ut dtd\alpha_t=-u_t\,dt and the cash account as dCt=utSt dtdC_t=u_t S_t\,dt. Because liquidating a sizable position depresses the price, the mark-to-market value αS\alpha S overstates the cash the position can actually generate.

Lions and Lasry (2007) — the first paper in a series devoted to reconciling option pricing theory with trading feedback — solved two problems in this model by utility maximization over liquidation strategies subject to the terminal constraint αT=0\alpha_T=0. First, the fair (liquidation) value of a position is P=αS−kα2/2P=\alpha S-k\alpha^2/2 in the additive model and P=1−e−kαkSP=\frac{1-e^{-k\alpha}}{k}S in the multiplicative model; these formulas are independent of the utility function and of σ\sigma. The proofs rest on a new class of singular stochastic control problems (control problems in which the control may be a process of unbounded variation, so that classical Hamilton–Jacobi–Bellman (HJB) arguments fail), introduced in Lasry and Lions (2000): the value function is invariant under explicit one-parameter flows of the state (C,S,α)(C,S,\alpha), which reduces the dimension and yields the closed forms. Second, for exponential utility Uλ(z)=1−e−λzU_\lambda(z)=1-e^{-\lambda z} (absolute risk aversion λ\lambda) and a European payoff written on the impact-adjusted underlying XT=ST−kαTX_T=S_T-k\alpha_T (resp. XT=e−kαTSTX_T=e^{-k\alpha_T}S_T; a choice that prevents the trader from manipulating the payoff), they derived the nonlinear partial differential equation for the utility indifference price π\pi — the cash amount making the investor indifferent between writing and not writing the option — characterized the optimal hedge, and computed the induced effective dynamics: for constant σ\sigma the price is the usual Black–Scholes price (the classical no-impact value πBS(S,t)=E[Φ(ST)∣St=S]\pi_{BS}(S,t)=E[\Phi(S_T)\mid S_t=S], with the associated hedging delta and gamma ∂2πBS/∂S2\partial^2\pi_{BS}/\partial S^2) while the hedged asset follows dSt=σ(1+k ∂2πBS/∂S2) dBt+…dS_t=\sigma(1+k\,\partial^2\pi_{BS}/\partial S^2)\,dB_t+\dots, i.e. the hedging feedback modifies realized volatility through the option’s gamma. The PDEs are understood in the viscosity-solution sense (weak solutions appropriate for fully nonlinear degenerate equations). The authors also proved the limits of π\pi as λ→0+\lambda\to0_+ and λ→+∞\lambda\to+\infty in the additive model, derived the first-order small-kk expansion σeff(S)=σ(S){1+k ∂2πBS/∂S2+O(k2)}\sigma_{\mathrm{eff}}(S)=\sigma(S)\{1+k\,\partial^2\pi_{BS}/\partial S^2+O(k^2)\} and its small-payoff analogue, and extended the fair-value and effective-volatility formulas to multi-asset models with a non-diagonal impact matrix KK, where hedging creates cross-correlations between initially independent assets.

Several parts of this program are left without rigorous treatment in the source. Interest rates are declared a straightforward omission. Section 4 states, without proofs, “variants and extensions”: influence laws with memory or relaxation effects (impact responses depending on the recent history of trading rather than the instantaneous rate) and an impact coefficient kk depending on the price level StS_t and possibly on the position αt\alpha_t — adaptations the authors declare “straightforward” but do not carry out — and, “slightly more delicate”, a nonlinear impact law in which k utk\,u_t is replaced by k(ut)k(u_t) with kk Lipschitz and k(x)−kxk(x)-kx bounded on R\mathbb{R} (the motivating example is threshold impact that switches on only when the trading rate exceeds a level), for which only a formal HJB equation and one special-case value formula are given. In the multiplicative model the admissible hedge is constrained by β≥−1/k\beta\ge-1/k; the λ→+∞\lambda\to+\infty limit there is described as “much more delicate”, requires replacing the payoff Φ\Phi by its envelope Φ^\hat\Phi (the least dominating function whose derivative is bounded below by −1/k-1/k), and the stated convergence is given without proof — in contrast to the additive case, where the corresponding limits are proved. The authors also state that higher-order terms of the small-kk asymptotic expansion “can be obtained” but do not carry them out (“we shall not do so here”). Finally, the announced self-consistent multi-agent theory of volatility formation was pursued by the authors in the companion paper Lions and Lasry (2006) and grew into mean field game theory (Lasry and Lions, 2007) — the theory of differential games with a continuum of asymptotically negligible agents — so that direction is no longer open in its original form.

Problem Statement

Within the Lions–Lasry large-investor framework — additive impact dSt=σ(St) dBt−k ut dtdS_t=\sigma(S_t)\,dB_t-k\,u_t\,dt or multiplicative impact dSt/St=σ(St) dBt−k ut dtdS_t/S_t=\sigma(S_t)\,dB_t-k\,u_t\,dt, with position dαt=−ut dtd\alpha_t=-u_t\,dt, cash dCt=utSt dtdC_t=u_t S_t\,dt, σ\sigma bounded above and below away from zero and Lipschitz, utility-based valuation with terminal liquidation (or terminal inventory penalties), and European payoffs written on the impact-adjusted underlying XT=ST−kαTX_T=S_T-k\alpha_T (resp. XT=e−kαTSTX_T=e^{-k\alpha_T}S_T) — extend the rigorous theory of the fair (liquidation) value, the utility indifference price, the optimal hedge, and the induced effective price dynamics from the constant linear impact law to the influence laws the authors identified but did not treat, and complete the two asymptotic programs they left unfinished. The four items below are jointly required by the scientific target. Concretely:

(1) Nonlinear impact laws. Replace the term k ut dtk\,u_t\,dt by k(ut) dtk(u_t)\,dt where kk is Lipschitz on R\mathbb{R} and k(x)−kxk(x)-kx is bounded (source eq. (43)); the motivating case is threshold impact, e.g. k(x)=0k(x)=0 on [−a,b][-a,b], k(x)=k(x−b)k(x)=k(x-b) for x≥bx\ge b, k(x)=k(x+a)k(x)=k(x+a) for x≤−ax\le-a (as in the source’s own notation, the same letter kk denotes both the nonlinear impact law and the constant slope beyond the thresholds). The source gives only a formal HJB equation and one special-case value formula; provide the rigorous theory: well-posedness of the singular control problem, verification, fair-value formulas where they survive, the indifference-pricing equation, and the effective dynamics and effective-volatility analogues.

(2) State-dependent and transient impact. Section 4 lists impact coefficients of the form k(St,αt)k(S_t,\alpha_t) and influence laws with memory or relaxation effects among its “variants and extensions”, declaring the adaptations straightforward, but supplies no proof or sketch; treat these laws rigorously, preserving or delimiting the utility-based fair-value and indifference-pricing conclusions. The memory/relaxation class consists of influence laws whose instantaneous impact is a prescribed decaying linear functional of the past trading rate, given by a memory kernel GG: the drift term k ut dtk\,u_t\,dt is replaced by k ηt dtk\,\eta_t\,dt with ηt=∫0tG(t−s) us ds\eta_t=\int_0^t G(t-s)\,u_s\,ds, the exponentially relaxing kernel G(t)=ρ e−ρtG(t)=\rho\,e^{-\rho t} (equivalently dηt=ρ (ut−ηt) dtd\eta_t=\rho\,(u_t-\eta_t)\,dt) being the model case. Influence laws whose impact is not representable by such a memory kernel are outside the item.

(3) Infinite risk aversion in the multiplicative model. Prove the convergence stated without proof in the source: as λ→+∞\lambda\to+\infty, the exponential-utility indifference price converges to the viscosity solution of the limiting equation (source eq. (29′)) with terminal condition the envelope Φ^\hat\Phi of the payoff (the infimum of all Φ~≥Φ\tilde\Phi\ge\Phi with Φ~′≥−1/k\tilde\Phi'\ge-1/k), including uniqueness for the limiting problem — or refute it with an explicit instance.

(4) Higher-order small-impact asymptotics. Compute and rigorously justify the expansion in powers of kk of the indifference price, the optimal hedge, and the effective volatility to all orders — the complete asymptotic series together with a proved remainder bound — in both impact laws and in the multi-asset extension with non-diagonal impact matrix; the source restricts itself to π=πBS+O(k)\pi=\pi_{BS}+O(k), α=∂πBS/∂S+O(k)\alpha=\partial\pi_{BS}/\partial S+O(k), σeff=σ{1+k ∂2πBS/∂S2+O(k2)}\sigma_{\mathrm{eff}}=\sigma\{1+k\,\partial^2\pi_{BS}/\partial S^2+O(k^2)\} and explicitly declines the more precise expansion (“it is in fact possible to make a more precise expansion but we shall not do so here”) in the additive single-asset discussion, gives only first-order analogues in the multiplicative and multi-asset settings, and states that in the multi-asset setting all preceding arguments can be followed and trivially adapted; the higher-order program is accordingly posed here for all of these settings.

Answers must retain the source’s model conventions (impact-adjusted payoff, both impact laws, exponential or general utilities as in the source) rather than substituting a different impact mechanism such as temporary impact, quadratic transaction costs, or limit-order-book models.

References

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