Solution for Risk Neutral Investors

estimated from below as

− Z T

0

u¯^j(s)gu¯^j(s) +u^−j(s)ds+H^j(P^u¯j(T)) +

Z T 0

u^j(s)gu^j(s) +u^−j(s)ds−H^j(P^uj(T))

≥ Z T

0

u¯^j(s)gu^j(s) +u^−j(s)−gu¯^j(s) +u^−j(s)ds +

Z T 0

u^j(s)−u¯^j(s)gu^j(s) +u^−j(s)ds−λ(X^j(T)−Y^j(T))kH_pk_∞. Note that u^j(s) +u^−j(s) ≥0 on A and thus g u^j(s) +u^−j(s)≥0 due to Assumption 3.2.2. Furthermore, g u^j(s) +u^−j(s)−g u¯^j(s) +u^−j(s)≥ u^j(s)−u¯^j(s), again by Assumption 3.2.2. The difference in the payoffs is therefore larger than

Z T 0

u¯^j(s)u^j(s)−u¯^j(s)ds−λh Z T

0

u^j(s)−u¯^j(s)ds

= Z

u^j(s)>¯u^j(s)

¯u^j(s)−λh u^j(s)−u¯^j(s)ds

On the set u^j(s)>u¯^j(s) we have ¯u^j(s) =K = ^λ_ε(h+ 1) and the above expression is strictly positive, a contradiction. This shows that u^j(s) is bounded above by K on the set A for eachj ≤N. Still on the set A, we get the following lower bound:

u^j(s) =

N

X

i=1

uⁱ(s) +^X

i6=j

−uⁱ(s)≥0−(N −1)K. (3.4)

A symmetric argument on the set B , ⁿ(s, ω)∈[0, T]×Ω :^PN_i=1uⁱ(s, ω)≤0^o com-pletes the proof.

3.3. Solution for Risk Neutral Investors

In this section we use dynamic programming to show that Problem 3.2.5 admits a unique solution (in a certain class) for risk neutral agents. Here the preference functional is Ψ^j_t(Z) =E[Z|F_t] for eachj≤N. We also show that the solution can be given in closed form for the special case of a linear cost function.

The idea is to consider the value function associated to Problem 3.2.5 for player j, where his competitors’ strategies are fixed, and to characterise it as the solution of the HJB PDE. Solving the resulting coupled system of PDEs for all players simultaneously then provides an equilibrium point of the stochastic differential game, cf. Friedman [1972]. To begin with, we fix the strategies (uⁱ)_i6=j and define the value function for

playerj≤N as subject to the state dynamics

dP(s) =σdW(s) +λ p∈Rthe value function represents the conditional expected portfolio value at maturity that playerj can achieve by trading optimally, given the other players’ strategies. The associated HJB equation is, cf. Fleming and Soner [1993] Theorem IV.3.1,

0 =v_t^j+1

2σ²v_pp^j + sup

c^j∈R

hλc^j+u^−jv_p^j−c^jgc^j+u^−ji, (3.5) with terminal condition v^j(T, p) = H^j(p), where v_t and v_p denote time and spatial derivatives, respectively. The HJB equation is formulated in terms of the candidate value functionsv¹, ..., v^N instead of the actual value functions V¹, ..., V^N. We first need to show existence and uniqueness of a smooth solution to (3.5) before we can identifyvⁱ withVⁱ. Given the aggregate trading strategyu^−j of all the other agents, a candidate for the maximiserc^j =u^j in (3.5) should satisfy

0 =λv_p^j−gc^j+u^−j−c^jg⁰c^j+u^−j. (3.6) We have one equation of this type for each playerj≤N. Summing them up and defining theaggregate trading speed as

u^ag, In view of Assumption 3.2.2 the map z7→ N g(z) +zg⁰(z) is strictly increasing. Hence condition (3.7) admits a unique solution u^ag which depends on ^PN_i=1vⁱ_p. Plugging u^ag

3.3. Solution for Risk Neutral Investors

back into (3.6) allows to compute the candidate optimal control for playerj ≤N as c^j =u^j = λv_p^j−g(u^ag)

g⁰(u^ag) . (3.8)

This expression is well defined since g⁰ > 0 again by Assumption 3.2.2. Plugging this candidate optimal control into the HJB equation, we see that the system of HJB PDEs now takes the form

0 =v^j_t+ 1

2σ²v^j_pp+λ

u^ag− g(u^ag) g⁰(u^ag)

v^j_p+ g(u^ag)²

g⁰(u^ag) (3.9) with terminal conditionv^j(T, p) =H^j(p) for j≤N. Note that the coupling stems from the aggregate trading speed u^ag via condition (3.7).

Remark 3.3.1. Looking back, we have turned the individual HJB equations (3.5) into the system of coupled PDEs (3.9). Systems of this form appear naturally in the theory of differential games, but we did not find a reference which covers this particular case.

Theorem 1 of Friedman [1972] for instance is valid only on a bounded state space. We shall use our a priori estimates of Proposition 3.2.9 in order to prove existence of a unique solution to (3.9).

The following theorem, whose proof is given in Appendix A.3.1, shows that the system of PDEs (3.9) has a unique classical solution ifH^j ∈ C_b², i.e. H^j is twice continuously dif-ferentiable and its derivatives up to order 2 are bounded, for eachj. Similarly,C^1,2is the space of functions which are continuously differentiable in time and twice continuously differentiable in space.

Theorem 3.3.2. Let H∈ C²_b. Then the Cauchy problem (3.9) admits a unique classical solution in C^1,2, which is the vector of value functions.

Remark 3.3.3. An alternative way of solving the system (3.9) is the following: If we sum up the N equations, we get a Cauchy problem for the aggregate value function v,^PNi=1vⁱ, namely

0 =v_t+ 1

2σ²v_pp+u^ag[λv_p−g(u^ag)] (3.10) with terminal conditionv(T, p) =^PN_i=1Hⁱ(p). Existence and uniqueness of a solution to this one-dimensional problem can be shown using Theorem V.8.1 in Ladyzenskaja et al.

[1968]. Once the solution is known, we can plug it back into (3.9) and get N decoupled equations. This technique is applied in the following section where we construct an explicit solution for linear cost functions.

It is hard to find a closed form solution for the coupled PDE (3.9). However, for the particular choice g(z) = κz with a liquidity parameter κ >0 the solution to (3.9) can

be given explicitly. Here and throughout, we denote by f_µ,σ²(z), 1

√

2πσ²exp −(z−µ)² 2σ²

!

the normal density with meanµand varianceσ².

Proposition 3.3.4. Let g(z) = κz. Then the solution of (3.9) can be given in closed form as the solution to a nonhomogeneous heat equation.

Proof. The optimal trading speed from (3.8) and the aggregate trading speed from (3.7) are

u^j = λ

κ v_p^j− 1 N + 1

N

X

i=1

v_pⁱ

!

(3.11) u^ag=

N

X

i=1

uⁱ = λ κ(N+ 1)

N

X

i=1

v_pⁱ = λ

κ(N+ 1)v_p. (3.12) Equation (3.9) for playerj’s value function now becomes

0 =v^j_t +1

2σ²v^j_pp+κ(u^ag)².

Combining this with (3.12) and summing up forj = 1, ..., N yields the following PDE for the aggregate value functionv=^PN_i=1vⁱ:

0 =vt+ 1

2σ²vpp+ λ²N

κ(N+ 1)²v_p² (3.13)

with terminal conditionv(T, p) =^PN_i=1Hⁱ(p). This PDE is a variant of Burgers’ equa-tion, cf. Rosencrans [1972]. It allows for an explicit soluequa-tion, which we cite in Lemma A.3.3 in the appendix. With this solution at hand, we can solve for each single investor’s value function. We plug the solutionvback into the equations (3.11) and (3.12) for the trading speeds, and those into the PDE (3.9). This yields

0 =v^j_t+ 1

2σ²v_pp^j + λ² κ(N+ 1)²v²_p

with terminal condition v^j(T, p) =H^j(p). This is now a PDE in the unknown function v^j with known function vp. We see that it is a nonhomogeneous heat equation with solution given by

v^j(T−t, p) = Z

R

H^j(z)f_p,σ²_t(z)dz+ λ² κ(N+ 1)²

Z t 0

Z

R

v_p²(s, z)f_p,σ²_(t−s)(z)dzds wherev is known from Lemma A.3.3 (in particular it is bounded and integrable).

3.3. Solution for Risk Neutral Investors

Figure 3.1.:Trading speed and surplus for one risk neutral investor holding a European Call option.

Let us conclude this section with some numerical illustrations. For risk neutral players and a linear cost structure, we reduced the system of PDEs to the one-dimensional PDE (3.13) for the aggregate value function. This can be interpreted as the value function of the representative agent. Such reduction to a representative agent is not always possible for more general utility functions. In the sequel we illustrate the optimal trading speed u(s, p) and surplus of a representative agent as functions of time and spot prices for a Eu-ropean call option H(P(T)) = (P(T)−K)⁺ and digital option H(P(T)) =1{P(T)≥K}, respectively.³ By surplus, we mean the difference between the representative agent’s op-timal expected portfolio valuev(t, p) and the conditional expected payoffEt,p[H(P(T))]

in the absence of any market impact. It represents the expected net benefit due to price manipulation.

We choose a linear cost function, strike K = 100, maturity T = 1, volatility σ = 1 and liquidity parameters λ =κ = 0.01. We see from Figure 3.1 that for the case of a call option both the optimal trading speed and the surplus increases with the spot; the latter also increases with the time to maturity. Furthermore, the increase in the trading speed is maximal when the option is at the money. For digital options (figure 2) the trading speed is highest for at the money options close to maturity as the trader tries to push the spot above the strike. If the spot is far away from the strike, the trading speed is very small as it is unlikely that the trader can push the spot above the strike before expiry. For both option types a high spread renders manipulation unattractive. Figures 3 and 4 show the optimal trading speed and the surplus at time t= 0 for the Call and Digital option for a representative agent. We used the cost function

g(z) =κz+c·sign(z) for different spreadsc∈ {0,0.001,0.002,0.003,0.004} (3.14) with the remaining parameters as above. We see that the higher the spread, the smaller

3Note that the cost function in (3.14) is not smooth, and the Call and Digital options are not smooth and bounded, so Theorem 3.3.2 does not apply directly. There are two ways to overcome this difficulty:

We could either approximategand H by smooth and bounded functions. Or we could interpretv not as a classical, but only as a viscosity solution of (3.5), cf. Fleming and Soner [1993] chapterV.

Figure 3.2.:Trading speed and surplus for one risk neutral investor holding a Digital option.

Figure 3.3.:Trading speed and surplus for a risk neutral investor holding a European Call option for different spread sizess= 0 (black), 0.001 (blue), 0.002 (red), 0.003 (green), 0.004 (brown). The higher the spread, the smaller the trading speed and the surplus.

the trading speed and the surplus. This is intuitive as frequent trading, in particular when the option is at the money, incurs high spread crossing costs. The same is true for fixed transaction costs which also discourage frequent trading.

Im Dokument Stochastic control in limit order markets (Seite 125-130)