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Register a new userEquilibrium price and optimal insider trading strategy under stochastic liquidity with long memory
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In this paper, the Kyle model of insider trading is extended by characterizing the trading volume with long memory and allowing the noise trading volatility to follow a general stochastic process. Under this newly revised model, the equilibrium conditions are determined, with which the optimal insider trading strategy, price impact and price volatility are obtained explicitly. The volatility of the price volatility appears excessive, which is a result of the fact that a more aggressive trading strategy is chosen by the insider when uninformed volume is higher. The optimal trading strategy turns out to possess the property of long memory, and the price impact is also affected by the fractional noise.
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Trading frictions are stochastic. They are, moreover, in many instances fast-mean reverting. Here, we study how to optimally trade in a market with stochastic price impact and study approximations to the resulting optimal control problem using singular perturbation methods. We prove, by constructing sub- and super-solutions, that the approximations are accurate to the specified order. Finally, we perform some numerical experiments to illustrate the effect that stochastic trading frictions have on optimal trading.
An unconventional approach for optimal stopping under model ambiguity is introduced. Besides ambiguity itself, we take into account how ambiguity-averse an agent is. This inclusion of ambiguity attitude, via an $alpha$-maxmin nonlinear expectation, renders the stopping problem time-inconsistent. We look for subgame perfect equilibrium stopping policies, formulated as fixed points of an operator. For a one-dimensional diffusion with drift and volatility uncertainty, we show that every equilibrium can be obtained through a fixed-point iteration. This allows us to capture much more diverse behavior, depending on an agents ambiguity attitude, beyond the standard worst-case (or best-case) analysis. In a concrete example of real options valuation under volatility uncertainty, all equilibrium stopping policies, as well as the best one among them, are fully characterized. It demonstrates explicitly the effect of ambiguity attitude on decision making: the more ambiguity-averse, the more eager to stop -- so as to withdraw from the uncertain environment. The main result hinges on a delicate analysis of continuous sample paths in the canonical space and the capacity theory. To resolve measurability issues, a generalized measurable projection theorem, new to the literature, is also established.
An investor trades a safe and several risky assets with linear price impact to maximize expected utility from terminal wealth. In the limit for small impact costs, we explicitly determine the optimal policy and welfare, in a general Markovian setting allowing for stochastic market, cost, and preference parameters. These results shed light on the general structure of the problem at hand, and also unveil close connections to optimal execution problems and to other market frictions such as proportional and fixed transaction costs.
We consider the problem of optimal inside portfolio $pi(t)$ in a financial market with a corresponding wealth process $X(t)=X^{pi}(t)$ modelled by begin{align}label{eq0.1} begin{cases} dX(t)&=pi(t)X(t)[alpha(t)dt+beta(t)dB(t)]; quad tin[0, T] X(0)&=x_0>0, end{cases} end{align} where $B(cdot)$ is a Brownian motion. We assume that the insider at time $t$ has access to market information $varepsilon_t>0$ units ahead of time, in addition to the history of the market up to time $t$. The problem is to find an insider portfolio $pi^{*}$ which maximizes the expected logarithmic utility $J(pi)$ of the terminal wealth, i.e. such that $$sup_{pi}J(pi)= J(pi^{*}), text {where } J(pi)= mathbb{E}[log(X^{pi}(T))].$$ The insider market is called emph{viable} if this value is finite. We study under what inside information flow $mathbb{H}$ the insider market is viable or not. For example, assume that for all $t<T$ the insider knows the value of $B(t+epsilon_t)$, where $t + epsilon_t geq T$ converges monotonically to $T$ from above as $t$ goes to $T$ from below. Then (assuming that the insider has a perfect memory) at time $t$ she has the inside information $mathcal{H}_t$, consisting of the history $mathcal{F}_t$ of $B(s); 0 leq s leq t$ plus all the values of Brownian motion in the interval $[t+epsilon_t, epsilon_0]$, i.e. we have the enlarged filtration begin{equation}label{eq0.2} mathbb{H}={mathcal{H}_t}_{tin[0.T]},quad mathcal{H}_t=mathcal{F}_tveesigma(B(t+epsilon_t+r),0leq r leq epsilon_0-t-epsilon_t), forall tin [0,T]. end{equation} Using forward integrals, Hida-Malliavin calculus and Donsker delta functionals we show that if $$int_0^Tfrac{1}{varepsilon_t}dt=infty,$$ then the insider market is not viable.
The problem of portfolio allocation in the context of stocks evolving in random environments, that is with volatility and returns depending on random factors, has attracted a lot of attention. The problem of maximizing a power utility at a terminal time with only one random factor can be linearized thanks to a classical distortion transformation. In the present paper, we address the problem with several factors using a perturbation technique around the case where these factors are perfectly correlated reducing the problem to the case with a single factor. We illustrate our result with a particular model for which we have explicit formulas. A rigorous accuracy result is also derived using a verification result for the HJB equation involved. In order to keep the notations as explicit as possible, we treat the case with one stock and two factors and we describe an extension to the case with two stocks and two factors.
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