The case of exponential impact function

In this section, we consider the case of price impact f (x) = exp(λx) being exponential, meaning that the relative marginal price impact function λ = f0/f > 0 is constant. A

peculiarity of this case is that at any time instant t, knowing the (marginal) price St

for the stock is sufficient to know the impact from an instant block trade, since after a block trade of size ∆ the price would be Stf (Yt+ ∆) = Stexp(λ∆). Hence, the relative

displacement f (YΘ) of S from the fundamental price S 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 in K = [−K, ∞) for some K > 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 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 4.3.3, we have

Lθ∗w(t, s, y) dt − s(wS(t, s, y) − eλθ

∗

/λ + 1/λ)(σ dWt+ ηtdt) ≥ 0,

where ηt= µt− λh(y + θ∗) and

Lθ∗_{w(t, s, y) := −w} t(t, s, y) + h(y + θ∗)wY(t, s, y) − 1 2σ 2_s2_w SS(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 Lθ∗_{w(t, s, y) = 0. The constraint}

θ∗∈ K is now equivalent to HKw(t, s, y) ≥ 0, where for a smooth function ϕ

HKϕ(t, s, y) := λϕS(t, s, y) + 1 − e−λK

Thus we obtain, just formally, that w should be a solution to the variational inequality FK[w] := min{Lθ[w]w , HKw} = 0 on [0, T ) × R+× R, (PDEδ)

where

θ[w](t, s, y) := 1/λ · log λwS(t, s, y) + 1
. (4.30)

It turns out that the gradient constraints HKw ≥ 0 on the value function, that hold

on [0, T ), propagate to the boundary, meaning that the correct boundary condition for (PDEδ) is

min{w(T, ·) − H, HKw} = 0. (BCδ)

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 price w of an European option with maturity T and payoff profile (g0, g1) is the unique bounded viscosity solution

of the variational inequality (PDEδ) with boundary condition (BCδ). 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

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 FK[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 payoff FK[H].

Proof. If (g0, g1) is a function of the price process s only, then it is easy to see that H is

such as well and that the dimension of the state process can be reduces by ignoring the impact process Y . 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 )ϕY 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 payoff H (because it solves the same pde). In this one-dimensional setup, this price coincides with the Black-Scholes price for the face-lifted payoff FK[H],

cf. [CEK15, Proposition 3.1].

Example 4.4.11. Consider the contingent claim with payoff H(s) = s. In the friction-

less Black-Scholes world, the present value of this claim is the price of the underlying, simply because wBS_{(t, s) = s solves the Black-Scholes pde with the terminal condition}

H, and the replicating strategy in this case consists of holding w_SBS(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 s in the riskless asset and impact level y, the large trader could buy at the beginning with an immediate block trade exactly θ(s, y) = 1/λ log(1 + λ) shares. The key property in the case of exponential f is that p does not depend on s and y. Hence, after buying θ(s, y) shares and holding them until maturity T , where the new price and impact would be ST −and YT − 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, where ST will be the price after the final liquidation block trade. Hence, with

this buy-and-hold strategy of 1/λ log(1 + λ) shares, that requires exactly capital s at the beginning, the large trader will be able to replicate the claim with payoff H. 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