The case of covered options

the self-financing portfolio (β,Θ) with initial cash holdingsβ0−=v^BS(0, S0−) we have by (4.21), (4.23) and (4.33) (recall that S₀₋=S0)

V_T^liq=v^BS(0,S0) + Z T

0

∂Sv^BS(t,St) dSt

− Z T

0

Sth(Y_t^Θ)

f(Y_t^Θ)−f(Y_t^Θ−Θt)

F(Y_t^Θ)−F(Y_t^Θ−Θt)−λ(Y_t^Θ−Θ_t)

dt

=H(ST)− Z T

0

Sth(Y_t^Θ)

f(Y_t^Θ)−f(Y_t^Θ−Θ_t)

F(Y_t^Θ)−F(Y_t^Θ−Θt)−λ(Y_t^Θ−Θ_t)

dt. (4.34) In particular, if the integrand in (4.34) is negative on [0, T], then (β,Θ) would be a superhedging strategy for the large trader with initial capital β0−=v^BS(0, S0−) and hence

v(0, S0−, Y0−)≤v^BS(0, S0−). (4.35)

One could show that the integrand will be negative for instance when Y^Θ≥0 on [0, T] andλis strictly decreasing (at least on a compact set containing the range of Y^Θ and Y^Θ−Θ). Such a situation could arise if for example Y₀₋ is large enough. However, this should be intuitively favorable for the large trader due to the additional negative drift in the price that would suppress the underlying and hence the payout at maturity, i.e. in this case it could be expected that superhedging for the large trader might be cheaper. Let us point out that equality in (4.35) cannot hold for all values ofS₀₋, Y₀₋

since the two functions are solutions of different pdes; for non-constant λ, (PDE) does not simplify to the Black-Scholes pde like in Section 4.4.2.

4.7 The case of covered options

A key conclusion from [BLZ16, BLZ17] is that the way the hedger forms the hedging strategy and delivers the payoff is crucial for the pricing equation. In our setup so far the initial and terminal actions of the hedger have impact on the price and the minimal superhedging price is characterized by a semi-linear pde. We consider now the case ofcovered options, that is when the buyer of the option could be asked to provide the required initial hedging position and to accept a mix of cash and stocks (at their current market price) as a final payment, thus allowing the hedger of the option to escape initial and terminal impact of forming and unwinding the hedging position respective. The pricing equation turns out to be fully non-linear and degenerate in the second-order term. Since this is not our main contribution of this chapter, we restrict ourselves to a sketch of the derivation of the pricing pde and how one could adapt directly the analysis from [BLZ16].

Let us consider continuous hedging strategies that are Itô processes

dΘt=a(t) dt+b(t) dWt, Θ0=θ0∈R, (4.36) where aand b are continuous processes with some integrability conditions. For such controls Θ, the market impact process and the perturbed price process take the form:

dYt= (−h(Yt) +a(t)) dt+b(t) dWt, Y₀=y

with S₀=f(y)¯s(initial impact of acquiringθ₀shares in the beginning is omitted). Note thatθ0 needs to be determined for a replicating strategy.

Performing integration by parts in (4.6), we can rewrite the gains from trading for continuous strategy Θ as

For a self-financing strategy (β,Θ) the book wealth at timeT is (recall (4.10)) V_T^book(Θ) =β0+LT(Θ) + ΘTST.

Consider a contingent claim of the formH =g(ST) written on the risky asset price.

For a superhedging strategy Θ with initial capital p, the hedger needs to set up the initial position in the risky asset Θ₀ that incurs the cost ΘS₀. Hence, at maturity we have

p+LT(Θ) + ΘTST −Θ0S0≥g(ST). (4.39)

Using (4.38), the change in the book wealth satisfies dL_t+ d(S_tΘ_t) = Θ_tdS_t+1

2d[S,Θ]_t−1

2σS_td[Θ, W]_t

= ΘtdSt+1

2Stλ(Yt)b²(t) dt (4.40)

Therefore, a replicating strategy Θ with initial capitalpshould satisfy p+

To construct a replicating strategy we look for a pair of processes (a, b) (or equivalently a strategy Θ) such that the processGt:=G0+Rt

0ΘudSu+Rt 0 1

2Suλ(Yu)b²(u) dusatisfies

4.7 The case of covered options GT =g(ST). To find such a process, we try the following Ansatz: Gt=v(t, St) for a smooth enough functionv: (0,∞)×R→R. Applying Itô’s formula we get

dv(t, St) =vt(t, St) dt+vs(t, St) dSt+¹₂vss(t, St) d[S]t

= [vt+¹₂S_t²(σ+λ(Yt)b(t))²vss+Stξ(t)vs] dt +S_t(σ+λ(Y_t)b(t))v_s(t, S_t) dW_t. Comparing the drift and the diffusion terms we need

vt+ 1/2S_t²(σ+λ(Yt)b(t))²vss+vsStξ(t) =1

2Stλ(Yt)b²(t) + ΘtStξ(t), (4.41) (σ+λ(Yt)b(t))Stvs(t, St) = ΘtSt(σ+λ(Yt)b(t)). (4.42) Now (4.42) is satisfied for Θt=vs(t, St) and for this choice of Θ, (4.41) reduces to

vt(t, St) +¹₂S_t²(σ+λ(Yt)b(t))²vss(t, St)−¹₂Stλ(Yt)b²(t) = 0. (4.43) To get the form ofb, we have by Itô’s formula

a(t) dt+b(t) dW_t= dΘ_t= dv_s(t, S_t) =

=vstdt+vssdSt+ 1/2vsssd[S]t

and comparing the diffusion coefficients we get thatb(t) =vssSt(σ+λ(Yt)b(t)), i.e.

b(t) = σStvss(t, St)

1−λ(Yt)Stvss(t, St). (4.44)

Similarly, we get

a(t) =v_st+v_ssS_tξ(t) +¹₂v_sssS_t²(σ+λ(Y_t)b(t))².

Using the definition ofξin (4.37), we get (with λandλ⁰ evaluated atY_t)

a(t) = v_st+v_ssS_t[µ−λh(Y_t) + 0.5(λ²+λ⁰)b²(t) +λσb(t)] + 1/2v_sssS²_t(σ+λb(t))² 1−λStvss

. Note that σ+λ(Yt)b(t) =σ/(1−λ(Yt)Stvss(t, St)). Thus, (4.43) yields the PDE

vt(t, s) +1 2

σ²s²vss(t, s)

1−λ(y)svss(t, s)= 0. (4.45)

Note that this pricing pde is (structurally) very similar to the equations derived in [LY05, FP11, BLZ17]. Due to the singularity atλ(y)svss= 1 in (4.45), constraints on sv_ss (upper boundγ:R+→R) need to be imposed in order to have a well-posed pde.

Following the analysis in [BLZ17], it turns out that (4.45) characterizes the minimal superhedging price, after appropriate gamma constraints are imposed. Indeed, let’s write

(4.36) as

dΘt=σ_Θ^a,b(St) dSt+µ^a,b_Θ (St) dt, with S_tσ_Θ^a,b(S_t) = b_t

σ+λ(Yt)bt

, µ^a,b_Θ (S_t) =a_t−ξ_tS_tσ^a,b_Θ (S_t).

Like in [BLZ17], we restrict the admissible trading strategies Θ = (a, b) to these with Lipschitz continuous and bounded a, b, for whichb is an Itô diffusion with Lipschitz continuous and bounded drift and diffusion processes, and such thatStσ^a,b_Θ (St) is bounded from above by γ(St) and also bounded from below. Then the arguments in [BLZ17]

would carry over to our setup, under conditions on γandλ(cf. Remark 4.7.1 below), and would give that the minimal superhedging price

v_γ(t, y, s) = inf{p| ∃admissible Θ so that (4.39) holds}

satisfiesvγ(t, y, s) =v^y_γ(t, s), wherev_γ^y(t, s) is the unique viscosity solution of the pricing pde

F^y[ϕ](t, s) := min

−ϕt(t, s)−1 2

σ²s²ϕss(t, s)

1−λ(y)sϕss(t, s), γ(s)−sϕss

= 0 on [0, T)×R+, (4.46) with terminal condition given by the face-lifted payoff ˆg, where ˆgis the smallest function aboveg that is a viscosity supersolution of the equationγ−sϕss≥0.

We conclude this section by stressing some features of the minimal superhedging price in this case and pointing important differences to the case of non-covered options.

Remark 4.7.1. The arguments from [BLZ17] could be adapted to the present setup for bounded continuousγthat satisfies

sup

y∈R,s∈R+

σsγ(s)

1−λ(y)γ(s) ∈R+,

and continuous bounded payoffsg. The main reason is that for everyy∈Rands∈R+, the map M ∈ (−∞, γ(s)] 7→ _1−λ(y)M^σ2sM is non-decreasing and convex, like in [BLZ17, Remark 3.1], ensuring that the smoothing techniques from [BLZ17, Section 3.1] go through here as well.

Remark 4.7.2. 1. The resilience function h does not appear in the pricing pde (4.46). Note that this is different from the results in Section 4.4.1, where the resilience function enters the pricing equation in a non-trivial way. However, the price of a covered option will depend on the initial level of impact y throughλ.

2. The minimal superheding price is decreasing in the impact λin the sense that if λ≥λ, then˜ v^λ_γ ≥v_γ^˜λ, andv^λ_γ ≥v^BS, where v^BS is the Black-Scholes price for the option with payoffg, i.e.v^BS solves−∂tv^BS−1/2σ²s²∂_ss²v^BS= 0 on [0, T)×R+

with terminal conditionv^BS(T,·) =g(·), see [BLZ17, Remark 2.9].

Im Dokument Feedback Effects in Stochastic Control Problems with Liquidity Frictions (Seite 105-109)