Trading with small nonlinear price impact

Cayé, Thomas; Herdegen, Martin; Muhle‐Karbe, Johannes

doi:10.1214/19-aap1513

Cited by 14 publications

(49 citation statements)

References 56 publications

(154 reference statements)

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“…It generalizes the ODE derived by Guasoni and Weber (2020), which is also a crucial element of the solution in Cayé et al. (2020) in the setting of one‐dimensional markets with nonlinear price impact.…”

Section: Introductionmentioning

confidence: 56%

“…This equation is linear and its solution is the first‐order utility loss in (37), the function

a

governs the first‐order utility loss induced by the presence of friction, and we have the Feynman–Kac representation

u false(t, w, s false) = E [\int_{t}^{T} a (() r, W_{r}^{0}, S_{r}) d r | W_{t}^{0} = w, S_{t} = s] .

(iii)In the one‐dimensional case, studied by Guasoni and Weber (2020) and Cayé et al. (2020), the first corrector equation simplifies to an ordinary differential equation. Its solution, a couple consisting of a function and a constant, gives similarly both the speed of trading as a function of the displacement from the frictionless optimizer and the leading order utility loss induced by the presence of frictions. (iv)As the small parameter

ε

in our model appears inside the function

f

in Equation (19), it corresponds to

λ^{\frac{1}{α}}

in Guasoni and Weber (2020) and Cayé et al.…”

Section: Resultsmentioning

confidence: 99%

“…We fix

α \in (0, 1)

(and therefore

m > 2

) and define

m^{*} = \frac{α}{α + 3} = \frac{1}{3 m - 2}

. (ii)Note that the function

f

is such that the instantaneous transaction costs function

θ \to θ \cdot f (s, θ)

is nonnegative, convex, and positive homogeneous of order

α + 1 = \frac{m}{m - 1} \in false(1, 2 false)

: this is the

d

‐dimensional version of the subquadratic trading cost as it appears in Guasoni and Weber (2020) and Cayé et al. (2020) where the market is composed of a single risky asset and a bank account. The importance of the dual friction

normalΦ

was already identified by Dolinsky and Soner (2013) in discrete time frictional markets and used by Guasoni and Rásonyi (2015) to provide a characterization of the optimizer of the solution of the nonlinear frictions in a one‐dimensional set‐up using convex duality.…”

Section: The Modelmentioning

confidence: 99%

“…These are expressed in terms of the solution of an ordinary differential equation that depends on

α

. More recently, the problem was solved in Cayé, Herdegen, and Muhle‐Karbe (2020) for constant absolute risk averse (CARA) investors and general one‐dimensional markets where the price follows a not necessarily Markovian Itô diffusion. The Ordinary Differential Equation (ODE) found in Guasoni and Weber (2020) is still a crucial building block of the solution.…”

Section: Introductionmentioning

confidence: 99%

“…See Cayé et al. (2020), for example, where the case of a smoothed and convexified put option with exponential utility of final wealth is considered.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotics for small nonlinear price impact: A PDE approach to the multidimensional case

Bayraktar

Cayé

Ekren

2020

Mathematical Finance

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We provide an asymptotic expansion of the value function of a multidimensional utility maximization problem from consumption with small nonlinear price impact. In our model, cross‐impacts between assets are allowed. In the limit for small price impact, we determine the asymptotic expansion of the value function around its frictionless version. The leading order correction is characterized by a nonlinear second‐order PDE related to an ergodic control problem and a linear parabolic PDE. We illustrate our result on a multivariate geometric Brownian motion price model.

show abstract

Section: Introductionmentioning

confidence: 56%

“…This equation is linear and its solution is the first‐order utility loss in (37), the function

a

governs the first‐order utility loss induced by the presence of friction, and we have the Feynman–Kac representation

u false(t, w, s false) = E [\int_{t}^{T} a (() r, W_{r}^{0}, S_{r}) d r | W_{t}^{0} = w, S_{t} = s] .

ε

in our model appears inside the function

f

in Equation (19), it corresponds to

λ^{\frac{1}{α}}

in Guasoni and Weber (2020) and Cayé et al.…”

Section: Resultsmentioning

confidence: 99%

“…We fix

α \in (0, 1)

(and therefore

m > 2

) and define

m^{*} = \frac{α}{α + 3} = \frac{1}{3 m - 2}

. (ii)Note that the function

f

is such that the instantaneous transaction costs function

θ \to θ \cdot f (s, θ)

is nonnegative, convex, and positive homogeneous of order

α + 1 = \frac{m}{m - 1} \in false(1, 2 false)

: this is the

d

normalΦ

Section: The Modelmentioning

confidence: 99%

“…These are expressed in terms of the solution of an ordinary differential equation that depends on

α

Section: Introductionmentioning

confidence: 99%

“…See Cayé et al. (2020), for example, where the case of a smoothed and convexified put option with exponential utility of final wealth is considered.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations