Trading With Small Price Impact

Moreau, Ludovic; Muhle‐Karbe, Johannes; Soner, H. Meté

doi:10.1111/mafi.12098

Cited by 59 publications

(148 citation statements)

References 72 publications

(316 reference statements)

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“…As a consequence, the asymptotically optimal (relative) trading speeds in this “high‐resilience regime” are

{normalΛ}^{- 1} Q_{d}^{⊤} = {normalΛ}^{- 1} + o false(R^{- 1} false), and {normalΛ}^{- 1} Q_{h}^{⊤} = {normalΛ}^{- 1 / 2} {({normalΛ}^{- 1 / 2} γ Σ {normalΛ}^{- 1 / 2})}^{1 / 2} {normalΛ}^{1 / 2} + o false(1 false),

R \to \infty

. The second formula shows that as the resilience grows, we recover the asymptotically optimal trading rate for the model with purely temporary trading costs (Moreau et al., , Theorem 4.7). In particular, this tracking speed only depends on the market, preference, and cost parameters, but not the optimal trading strategy at hand.…”

Section: Resultsmentioning

confidence: 82%

“…The quadratic variation of the Merton portfolio is multiplied by the positive‐definite matrix A 2 determined from the Riccati equation . If the resilience R becomes large compared to the price impact parameters Λ and C , one readily verifies that A 2 converges to the solution of the matrix equation

γ Σ = A_{2} {normalΛ}^{- 1} A_{2}

, that is,

A_{2} = {normalΛ}^{1 / 2} {({normalΛ}^{- 1 / 2} γ Σ {normalΛ}^{- 1 / 2})}^{1 / 2} {normalΛ}^{1 / 2} + o false(1 false), as R \to \infty .

Whence, as resilience grows, temporary trading costs become the dominant friction and A 2 recovers the factor for purely temporary quadratic costs (Moreau et al., , Remarks 4.5 and 4.6) . The other comparative statics of this term are discussed in more detail in Section 3.5 for the one‐dimensional case, where the Riccati equation can be solved explicitly.…”

Section: Resultsmentioning

confidence: 84%

“…Moreau et al. () and the references therein. Whence, this “portfolio gamma” or “activity rate” is the crucial sensitivity of trading strategies with respect to small frictions also in the present setting where part of their effect only wears off gradually.…”

Section: Resultsmentioning

confidence: 99%

“…In addition to the dynamic component discussed so far, the value expansion also includes several terms that depend on the initial conditions. The quadratic form ϖ is similar to its counterpart for purely temporary costs (Moreau et al., , Theorem 4.3), in that, it penalizes squared deviations of the initial portfolio from the frictionless optimum. Here, however, the initial distortion of the prices relative to their unaffected values also comes into play.…”

Section: Resultsmentioning

confidence: 99%

“…Indeed, the representation of the first‐order loss from Moreau et al. () remains valid in the present context after updating the scaling weight to account for the additional model parameters. In contrast, the optimal trading rate is more complex.…”

Section: Introductionmentioning

confidence: 99%

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Portfolio choice with small temporary and transient price impact

Ekren

Muhle‐Karbe

2019

Mathematical Finance

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We study portfolio selection in a model with both temporary and transient price impact introduced by Garleanu and Pedersen. In the large‐liquidity limit where both frictions are small, we derive explicit formulas for the asymptotically optimal trading rate and the corresponding minimal leading‐order performance loss. We find that the losses are governed by the volatility of the frictionless target strategy, like in models with only temporary price impact. In contrast, the corresponding optimal portfolio not only tracks the frictionless optimizer, but also exploits the displacement of the market price from its unaffected level.

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“…As a consequence, the asymptotically optimal (relative) trading speeds in this “high‐resilience regime” are

{normalΛ}^{- 1} Q_{d}^{⊤} = {normalΛ}^{- 1} + o false(R^{- 1} false), and {normalΛ}^{- 1} Q_{h}^{⊤} = {normalΛ}^{- 1 / 2} {({normalΛ}^{- 1 / 2} γ Σ {normalΛ}^{- 1 / 2})}^{1 / 2} {normalΛ}^{1 / 2} + o false(1 false),

R \to \infty

Section: Resultsmentioning

confidence: 82%

γ Σ = A_{2} {normalΛ}^{- 1} A_{2}

, that is,

A_{2} = {normalΛ}^{1 / 2} {({normalΛ}^{- 1 / 2} γ Σ {normalΛ}^{- 1 / 2})}^{1 / 2} {normalΛ}^{1 / 2} + o false(1 false), as R \to \infty .

Section: Resultsmentioning

confidence: 84%

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Portfolio choice with small temporary and transient price impact

Ekren

Muhle‐Karbe

2019

Mathematical Finance

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Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations

2019

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High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (i) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (ii) a merged formulation of the PDE and the 2BSDE problem, (iii) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (iv) a stochastic gradient descent-type optimization procedure. Numerical results obtained using TensorFlow in Python illustrate the efficiency and the accuracy of the method in the cases of a 100-dimensional Black-Scholes-Barenblatt equation, 1 arXiv:1709.05963v1 [math.NA] 18 Sep 2017 a 100-dimensional Hamilton-Jacobi-Bellman equation, and a nonlinear expectation of a 100-dimensional G-Brownian motion.

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