Integral representation of martingales motivated by the problem of endogenous completeness in financial economics

Let $\mathbb{Q}$ and $\mathbb{P}$ be equivalent probability measures and let $\psi$ be a $J$-dimensional vector of random variables such that $\frac{d\mathbb{Q}}{d\mathbb{P}}$ and $\psi$ are defined in terms of a weak solution $X$ to a $d$-dimensional stochastic differential equation. Motivated by the problem of \emph{endogenous completeness} in financial economics we present conditions which guarantee that every local martingale under $\mathbb{Q}$ is a stochastic integral with respect to the $J$-dimensional martingale S_t \set \mathbb{E}^{\mathbb{Q}}[\psi|\mathcal{F}_t]. While the drift $b=b(t,x)$ and the volatility $\sigma = \sigma(t,x)$ coefficients for $X$ need to have only minimal regularity properties with respect to $x$, they are assumed to be analytic functions with respect to $t$. We provide a counter-example showing that this $t$-analyticity assumption for $\sigma$ cannot be removed.Comment: A stronger version of the main theorem is obtained. The "financial" part of the previous version is remove

Optimal Execution in a General One-Sided Limit-Order Book and Endogenous Dynamic Completeness of Financial Models

This thesis consists of two parts. The first one is a result obtained under the supervision of Steven Shreve and with the collaboration of Gennady Shaikhet. Our work yielded a detailed description of the optimal strategies for a large investor, when she needed to buy a large amount of shares of a stock over a finite time horizon. The dynamics of the limit order book of the underlying stock is a generalization of known results to continuous time and to arbitrary distributions of the said limit order book. See the introduction section in chapter 2 for a more detailed discussion. The second part is a result obtained under the supervision of Dmitry Kramkov. Our work yielded a sufficient condition on the structure of the economic factors, dividends of traded assets and total endowment in a single-agent economy, such that in an Arrow - Debreu - Radner equilibrium the market is complete. The main result is formulated as an integral representation theorem. Our work generalizes and complements fairly recent results in this direction (at the time of this thesis) by requiring less smoothness of the driving diffusion process at the expense of seemingly stronger conditions on the terminal dividends of the assets. See the introduction section in chapter 3 for a more detailed discussion.</p

Integral representation of martingales motivated by the problem of endogenous completeness in financial economics

Let $\mathbb{Q}$ and $\mathbb{P}$ be equivalent probability measures and let $\psi$ be a $J$-dimensional vector of random variables such that $\frac{d\mathbb{Q}}{d\mathbb{P}}$ and $\psi$ are defined in terms of a weak solution $X$ to a $d$-dimensional stochastic differential equation. Motivated by the problem of \emph{endogenous completeness} in financial economics we present conditions which guarantee that every local martingale under $\mathbb{Q}$ is a stochastic integral with respect to the $J$-dimensional martingale S_t \set \mathbb{E}^{\mathbb{Q}}[\psi|\mathcal{F}_t]. While the drift $b=b(t,x)$ and the volatility $\sigma = \sigma(t,x)$ coefficients for $X$ need to have only minimal regularity properties with respect to $x$, they are assumed to be analytic functions with respect to $t$. We provide a counter-example showing that this $t$-analyticity assumption for $\sigma$ cannot be removed.

Integral representation of martingales motivated by the problem of endogenous completeness in financial economics

<p>Let Q and P be equivalent probability measures and let ψ be a Jdimensional vector of random variables such that dQ dP and ψ are defined in terms of a weak solution X to a d-dimensional stochastic differential equation. Motivated by the problem of endogenous completeness in financial economics we present conditions which guarantee that every local martingale under Q is a stochastic integral with respect to the J-dimensional martingale St , E Q[ψ|Ft]. While the drift b = b(t, x) and the volatility σ = σ(t, x) coefficients for X need to have only minimal regularity properties with respect to x, they are assumed to be analytic functions with respect to t. We provide a counter-example showing that this t-analyticity assumption for σ cannot be removed</p

Optimal Execution in a General One-Sided Limit-Order Book

We construct an optimal execution strategy for the purchase of a large number of shares of a financial asset over a fixed interval of time. Purchases of the asset have a nonlinear impact on price, and this is moderated over time by resilience in the limit-order book that determines the price. The limit-order book is permitted to have arbitrary shape. The form of the optimal execution strategy is to make an initial lump purchase and then purchase continuously for some period of time during which the rate of purchase is set to match the order book resiliency. At the end of this period, another lump purchase is made, and following that there is again a period of purchasing continuously at a rate set to match the order book resiliency. At the end of this second period, there is a final lump purchase. Any of the lump purchases could be of size zero. A simple condition is provided that guarantees that the intermediate lump purchase is of size zero.</p

Optimal execution in a general one-sided limit-order book

We construct an optimal execution strategy for the purchase of a large number of shares of a financial asset over a fixed interval of time. Purchases of the asset have a nonlinear impact on price, and this is moderated over time by resilience in the limit-order book that determines the price. The limit-order book is permitted to have arbitrary shape. The form of the optimal execution strategy is to make an initial lump purchase and then purchase continuously for some period of time during which the rate of purchase is set to match the order book resiliency. At the end of this period, another lump purchase is made, and following that there is again a period of purchasing continuously at a rate set to match the order book resiliency. At the end of this second period, there is a final lump purchase. Any of the lump purchases could be of size zero. A simple condition is provided that guarantees that the intermediate lump purchase is of size zero

Hedging with Temporary Price Impact

© 2016, Springer-Verlag Berlin Heidelberg. We consider the problem of hedging a European contingent claim in a Bachelier model with temporary price impact as proposed by Almgren and Chriss (J Risk 3:5–39, 2001). Following the approach of Rogers and Singh (Math Financ 20:597–615, 2010) and Naujokat and Westray (Math Financ Econ 4(4):299–335, 2011), the hedging problem can be regarded as a cost optimal tracking problem of the frictionless hedging strategy. We solve this problem explicitly for general predictable target hedging strategies. It turns out that, rather than towards the current target position, the optimal policy trades towards a weighted average of expected future target positions. This generalizes an observation of Gârleanu and Pedersen (Dynamic portfolio choice with frictions. Preprint, 2013b) from their homogenous Markovian optimal investment problem to a general hedging problem. Our findings complement a number of previous studies in the literature on optimal strategies in illiquid markets as, e.g., Gârleanu and Pedersen (Dynamic portfolio choice with frictions. Preprint, 2013b), Naujokat and Westray (Math Financ Econ 4(4):299–335, 2011), Rogers and Singh (Math Financ 20:597–615, 2010), Almgren and Li (Option hedging with smooth market impact. Preprint, 2015), Moreau et al. (Math Financ. doi:10.1111/mafi.12098, 2015), Kallsen and Muhle-Karbe (High-resilience limits of block-shaped order books. Preprint, 2014), Guasoni and Weber (Mathematical Financ. doi:10.1111/mafi.12099, 2015a; Nonlinear price impact and portfolio choice. Preprint, 2015b), where the frictionless hedging strategy is confined to diffusions. The consideration of general predictable reference strategies is made possible by the use of a convex analysis approach instead of the more common dynamic programming methods