Numerical example - Feedback Effects in Stochastic Control Problems with Liquidity Frictions

Let us first discuss the setup of Section 4.4.1 which essentially requiredF to be invertible on R. In this case, the pricing pde will have the same structure as (PDE) with the following modifications: replace ˜hand ˜f by ˜h^η and ˜f^η respectively with

˜h^η(t, s, y) =h

1+ηF⁻¹((1 +η)f(y)ϕ_S(t, s, y) +F(y)) +_1+η^η y , f˜^η(t, s, y) =f◦F⁻¹((1 +η)f(y)ϕ_S(t, s, y) +F(y)).

An optimal hedging strategy Θ^∗, if it exists, would satisfy (cf. Remark 4.4.8 forη= 0) (1 +η)Θ^∗_t =F⁻¹((1 +η)f(Y^∗t)ϕ_S(t,S^∗t,Y^∗t) +F(Y^∗t))−Y^∗t,

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 “unaffected” price process tracked byY^Θ.

In the setup of Section 4.4.2, we again need to consider portfolio constraintsθ∈ K forK= [−K,+∞) in order to derive the pricing pde. Moreover,F^η = 0 and thus the pricing pde takes the following form: ∀(t, s, y)∈[0, T)×R+×R

min{−w_t−¹₂σ²s²w_SS+h(y+θ^∗)w_Y, λ(1 +η)ϕ_S+ 1−e^−λ(1+η)K}= 0, where θ^∗= _λ(1+η)¹ log λ(1 +η)w_S+ 1

, with boundary condition min{w(T,·)−H, λ(1 +η)ϕS+ 1−e^−λ(1+η)K}= 0,

where H is the modified boundary condition from Lemma 4.4.1 as explained above. In particular, the pricing pde with permanent as well as transient impact coincides with the pricing pde with pure transient impact but with modifiedλ, in this caseλ(1 +η).

4.6 Numerical example

In this section, we discuss numerical results on the minimal superhedging price w characterized by (PDE), cf. Theorem 4.4.5. For our numerical simulations we consider impact function

f(x) = 1 + arctan(x)/10, x∈R, (4.32)

4.6 Numerical example that satisfies Assumption 4.4.4. In this case the changes ofλ(x) = 1/(10(1 +x²)f(x)) are most significant in (−4,4) where the change in impact is significant, see Fig-ure 4.1a. Apart from satisfying our assumptions and having explicit antiderivative F(x) =x+ (xarctan(x)−1/2 log(1 +x²))/10, being useful in the numerical implemen-tation, it turns out that similar shape of the impact function was observed when the related Propagator model was calibrated to real data, see [BL12, Appendix] for details.

For h(y) =βy withβ = 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 its Black-Scholes price, i.e. the Black-Black-Scholes price of a European call option for the same model parameters; let us recall that the case f = 1 in our market impact model gives the Black-Scholes model. The volatility σ is set to 0.3. The payoff for the large trader isH(s, y) =

sF(y+1)−F(y)

f(y) −K

1{s≥K} that we “smooth out” by approximating the indicator function by linearly interpolating 0 and 1 betweenK−0.5 andK.

To approximate both prices, we solve the corresponding pdes using (semi-implicit) finite difference scheme in the bounded region (y, s)∈[−20,20]×[0,200]. For our simulation we set the following boundary condition for t < T: ^∂w_∂s = (F(y+ 1)−F(y))/f(y) on [−20,20]× {200}, ^∂w_∂y = 0 on{−20,20} ×[0,200]∪[−20,20]× {0}. Indeed, for initial impacty close to -20 or 20 the impact function is approximately constant and until maturityT resilience will not be able bring back the level of impact to the region where the changes inf are significant, see Figure 4.1a, thus we should expect that the price would not depend that much on the level of impact. On the other hand, for larger values ofsone expects the price to depend linearly ins(like the payoff profile). The difference between the Black-Scholes price and the large trader’s price (as a function of the risky asset pricesand the level of impacty) is shown in Figure 4.1b. Let us point out that the Black-Scholes price does not depend on level of impacty.

Although our numerical results suggest that the value of the option with physical delivery in our large trader model dominates its Black-Scholes price, this does not seem to be the case for the European call with pure cash delivery. In this case numerical simulations show that the price for the large investor can be smaller, typically when the initial impact is away from zero, i.e. in regions where the level of impact affects significantly the price when trading. The intuition is that for pure cash delivery, the net number of traded assets for a (super-)hedging strategy is zero (recall that Θ₀₋= ΘT = 0 for a superhedging Θ), while the presence of resilience incurs additional drift that could be favorable for the large trader, typically pushing the prices down if the hedging strategy consists of holding positive number of risky assets (and initial displacement is not small);