The case of exponential impact function

1−λf wS−f−wSY −λswSS

f(F⁻¹(f w_S+F))

+ +w_tS+sµ_tw_SS+ 1/2σ²s²w_SSS

f(F⁻¹(f wS+F)) , b(t, s, y) := σswSS

f(F⁻¹(f w_S+F)).

Hence, an optimal superhedging strategy will also account for the transient nature of price impact.

We close this section with a remark on Assumption 4.4.4 that implies the bijectivity ofF onR. In particular, this ensures that an optimal control θ^∗ can be defined. Similar conditions are also crucial for the results in [BB04] and [BLZ16], namely the surjectivity assumption A5 in [BB04] and the invertability assumption H2 in [BLZ16]. We will see in the next section how departing from this assumption leads naturally to singularity in the pricing pde with respect to the gradient. Indeed, the lack of invertability of F imposes a condition onw_S so thatθ^∗ could be defined. Thus, for the analysis there we will introduce constraints on the “delta”, i.e. the holdings in the risky asset, which in pde terms translates to constraints on the spacial gradientwS.

4.4.2 The case of exponential impact function

In this section, we consider the case of price impactf(x) = exp(λx) being exponential, meaning that the relative marginal price impact functionλ=f⁰/f >0 is constant. A peculiarity of this case is that at any time instantt, knowing the (marginal) priceS_t for the stock is sufficient to know the impact from an instant block trade, since after a block trade of size ∆ the price would beStf(Yt+ ∆) =Stexp(λ∆). Hence, the relative displacementf(Y^Θ) ofS from the fundamental priceS is immaterial to determine the price impact from a block trade, in difference to the situation of Section 4.4.1. Motivated by Remark 4.2.5, we consider trading with short-selling constraints, i.e. trading strategies are required to take values inK= [−K,∞) for someK >0.

4.4 The pricing PDEs and main results To derive (heuristically, at first) the pricing pde, let us apply formally Theorem 4.3.1 forv=w(t, s, y) att, s, y, τ =t+, provided thatwis smooth enough, to get the existence ofθ^∗∈ K such that, using Lemma 4.3.3, we have

L^θ∗w(t, s, y) dt−s(wS(t, s, y)−e^λθ∗/λ+ 1/λ)(σdWt+ηtdt)≥0, where ηt=µt−λh(y+θ^∗) and

L^θ∗w(t, s, y) :=−wt(t, s, y) +h(y+θ^∗)wY(t, s, y)−1

2σ²s²wSS(t, s, y).

As in Section 4.4.1, the diffusion part should vanish, giving the optimal control θ^∗= 1

λlog λwS(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_Kw(t, s, y)≥0, where for a smooth function ϕ

H_Kϕ(t, s, y) :=λϕ_S(t, s, y) + 1−e^−λK

Thus we obtain, just formally, that wshould be a solution to the variational inequality FK[w] := min{L^θ[w]w , HKw}= 0 on [0, T)×R+×R, (PDE^δ) where

θ[w](t, s, y) := 1/λ·log λwS(t, s, y) + 1

. (4.30)

It turns out that the gradient constraintsH_Kw≥0 on the value function, that hold on [0, T), propagate to the boundary, meaning that the correct boundary condition for (PDE^δ) is

min{w(T,·)−H,HKw}= 0. (BC^δ)

Next we state our main result for exponential price impact function.

Theorem 4.4.9. Suppose that the resilience function h is Lipschitz continuous and Assumption 4.4.3 is in force. Then the minimal superhedging pricew of an European option with maturityT and payoff profile(g0, g1)is the unique bounded viscosity solution of the variational inequality (PDE^δ) with boundary condition (BC^δ). In particular, w_∗=w^∗=w on[0, T]×R+×R.

Proof. The proofs are postponed for Section 4.8. The viscosity super-/sub-solution property are proved in Theorem 4.8.2 and Theorem 4.8.3 respectively, while uniqueness follows from the comparison result Theorem 4.8.6, see also Remark 4.8.7.

Corollary 4.4.10. In the setup from Theorem 4.4.9, suppose moreover that the payoff (g0, g1) does not depend on the level of impacty. Then the minimal superhedging price

is a function in(t, s)only and the pricing pde (4.30) simplifies to the Black-Scholes pde with gradient constraints.

In this case, if the face-lifted payoff F_K[H] is continuously differentiable with bounded derivative, where

FK[H](s) := sup

x≤0

H(s+x) +1−e^−λK

λ x

, s∈R+,

with the convention that H =H(0) on (−∞,0], then the minimal superhedging price coincides with the Black-Scholes price for the face-lifted payoffFK[H].

Proof. If (g₀, g₁) is a function of the price processsonly, then it is easy to see thatH is such as well and that the dimension of the state process can be reduces by ignoring the impact processY. In this case, the stochastic target problem in Section 4.3 could be formulated for the new state process and thus the value function would be a function on (t, s) only. The same analysis could be carried over to derive the pricing pde and to prove viscosity solution property of the value function. The pricing pde in this case would be the Black-Scholes pde with gradient constraints since the term h(Y)ϕ_Y in Lemma 4.3.3 would not be present. Hence, the minimal superhedging price in our large investor model would coincide with the minimal superhedging price under delta constraints in the small investor model for the payoffH (because it solves the same pde). In this one-dimensional setup, this price coincides with the Black-Scholes price for the face-lifted payoffF_K[H], cf. [CEK15, Proposition 3.1].

Example 4.4.11. Consider the contingent claim with payoffH(s) =s. In the friction-less Black-Scholes world, the present value of this claim is the price of the underlying, simply becausew^BS(t, s) =ssolves the Black-Scholes pde with the terminal condition H, and the replicating strategy in this case consists of holdingw_S^BS(t, s) = 1 asset, i.e. it is a buy-and-hold strategy. To see that a buy-and-hold strategy is also optimal for the large trader when f(y) = exp(λy), even in the case with transient price impact, note that for initial capital sin the riskless asset and impact levely, the large trader could buy at the beginning with an immediate block trade exactly θ(s, y) = 1/λlog(1 +λ) shares. The key property in the case of exponentialf is that pdoes not depend on s andy. Hence, after buyingθ(s, y) shares and holding them until maturityT, where the new price and impact would beS_T− andYT− respectively, the large trader performs a block trade to unwind his risky asset position and receives exactly

ST(F(YT−)−F(YT−−θ)) =STexp(λ(YT−−θ))exp(λθ)−1

λ =STf(YT−−θ) =ST

in cash, whereST will be the price after the final liquidation block trade. Hence, with this buy-and-hold strategy of 1/λlog(1 +λ) shares, that requires exactly capitalsat the beginning, the large trader will be able to replicate the claim with payoffH. Moreover, the arguments in Remark 4.3.4 show that the minimal initial capital for the large trader in this case cannot not be less that in the friction-less case, hence we just constructed an optimal hedging strategy for the large trader.

4.5 Combined transient and permanent price impact