Continuous-time Trading Research Articles

On theoretical foundations of mostly model-free cross-coupled simultaneously long-short stock trading controllers

In the last decades, feedback trading and especially simultaneously long-short (SLS) trading attracted a lot of attention in the control community. The main reason for this is the so-called robust positive expectation property, which states that under specific conditions like, e.g., continuous time trading and survival the SLS controller achieves for almost all (a.a.) parameter settings, positive expected gains not only for several price models but also in mostly model-free frameworks. In finance, topics like optimal pairs trading are of interest for research and implementation. In the intersection of pairs trading and SLS so-called cross-coupled SLS (CC-SLS) rules are analyzed. Typically the CC-SLS rule is analyzed either empirically or with specific model assumptions. In the work at hand, under specific assumptions we do not only show that a new type of cross-coupled SLS (CC*-SLS) rule achieves the robust positive expectation property but also that this CC*-SLS is in expectation for a.a. parameter settings superior to the standard SLS rule used stock by stock. Furthermore, we perform simulations to illustrate the controllers’ behavior.

Dynamically complete markets under Brownian motion

This paper investigates how continuous-time trading renders complete a financial market in which the underlying risk process is a Brownian motion. A sufficient condition, that the instantaneous dispersion matrix of the relative dividends is non-degenerate, has been established in the literature for single-commodity, pure-exchange economies with many heterogenous agents where the securities’ dividends as well as the agents’ utilities and endowments include flows during the trading horizon which are analytic functions. In sharp contrast, the present analysis is based upon a different mathematical argument that assumes neither analyticity nor a particular underlying economic environment. The novelty of our approach lies in deriving closed-form expressions for the dispersion coefficients of the securities’ prices. To this end, we assume only that the pricing kernels and dividends satisfy standard growth and smoothness restrictions (mild enough to allow even for options). In this sense, our sufficiency conditions apply irrespectively of preferences, endowments or other structural elements (for instance, whether or not the budget constraints include only pure exchange).

Speed and learning in high-frequency auctions

Faster trading improves liquidity in periodic call auction markets, in contrast to continuous-time markets. We build a model where high-frequency traders (HFTs) engage in duels to trade on stale quotes. More frequent periodic auctions increase the likelihood that a single HFT arrives in any given auction and subsequently acts as a monopolist on information. Higher trading speed increases the expected number of arbitrageurs participating in auctions, promoting competition between snipers and improving liquidity. We find that faster trading and longer auction intervals are substitute instruments to reduce bid-ask spreads. Relative to continuous-time trading, periodic batch auctions reduce HFT informational rents.

A Finite Agent Equilibrium in an Incomplete Market and its Strong Convergence to the Mean-Field Limit

We investigate the problem of equilibrium price formation in an incomplete securities market. Each financial firm (agent) tries to minimize its cost via continuous-time trading with a securities exchange while facing the systemic and idiosyncratic noises as well as the stochastic order-flows from its over-the-counter clients. We have shown, in the accompanying paper (Fujii-Takahashi (2020)), that the solution to a certain forward backward stochastic differential equation of conditional McKean-Vlasov type gives a good approximate of the equilibrium price which clears the market in the large population limit. In this work, we prove the existence of a unique market clearing equilibrium among the heterogeneous agents of finite population size. We show the strong convergence to the corresponding mean-field limit under suitable conditions. In particular, we provide the stability relation between the market clearing price for the heterogeneous agents and that for the homogeneous mean-field limit.

Costs and benefits of dynamic trading in a lemons market

We study a dynamic market with asymmetric information that creates the lemons problem. We compare efficiency of the market under different assumptions about the timing of trade. We identify positive and negative aspects of dynamic trading, describe the optimal market design under regularity conditions and show that continuous-time trading can always be improved upon if sellers are present at t=0. Instead, continuous trading is optimal if sellers arrive stochastically over time.

Costs and Benefits of Dynamic Trading in a Lemons Market

We study a dynamic market with asymmetric information that creates the lemons problem. We compare efficiency of the market under different assumptions about the timing of trade. We identify positive and negative aspects of dynamic trading, describe the optimal market design under regularity conditions and show that continuous-time trading can be always improved upon.

Utility maximization under risk constraints and incomplete information for a market with a change point

ABSTRACTIn this article, we consider an optimization problem of expected utility maximization of continuous-time trading in a financial market. This trading is constrained by a benchmark for a utility-based shortfall risk measure. The market consists of one asset whose price process is modelled by a Geometric Brownian motion where the market parameters change at a random time. The information flow is modelled by initially and progressively enlarged filtrations which represent the knowledge about the price process, the Brownian motion and the random time. We solve the maximization problem and give the optimal terminal wealth depending on these different filtrations for general utility functions by using martingale representation results for the corresponding filtration.

Almost-sure hedging with permanent price impact

We consider a financial model with permanent price impact. Continuous time trading dynamics are derived as the limit of discrete rebalancing policies. We then study the problem of super-hedging a European option. Our main result is the derivation of a quasi-linear pricing equation. It holds in the sense of viscosity solutions. When it admits a smooth solution, it provides a perfect hedging strategy.

Discrete or Continuous Trading? HFT Competition and Liquidity on Batch Auction Markets

Frequent batch auction (FBA) markets do not necessarily improve liquidity relative to continuous-time trading. HFTs submit quotes that become stale if the market clears before new information becomes public, and may be sniped by other privately-informed HFTs. Less frequent auctions enhance learning by allowing for more public news: liquidity improves. For infrequent enough auctions, a frequency wall emerges as lower adverse selection is offset by lower liquidity demand by impatient traders. Beyond this wall, longer auction intervals do not improve liquidity anymore. However, higher HFT speed stimulates price competition between arbitrageurs, allowing for a lower spread.

Implementation Details for Frequent Batch Auctions: Slowing Down Markets to the Blink of an Eye

Our recent research (Budish, Cramton, and Shim 2013) proposes frequent batch auctions - uniform-price sealed-bid double auctions conducted at frequent but discrete time intervals - as a market design alternative to continuous-time trading in financial markets. This short paper discusses the implementation details of frequent batch auctions. We outline the process flow for frequent batch auctions, discuss a modification to the market design that accommodates market fragmentation and Reg NMS, and discuss the engineering and economic considerations relevant for determining the batch interval. Open questions are discussed throughout.

Radner Equilibria Under Ambiguous Volatility

The present paper considers a class of general equilibrium economies when the primitive uncertainty model features uncertainty about continuous-time volatility. This requires a set of mutually singular priors, which do not share the same null sets. For this setting we introduce an appropriate commodity space and the dual of linear and continuous price systems. All agents in the economy are heterogeneous in their preference for uncer- tainty. Each utility functional is of variational type. The existence of equi- librium is approached by a generalized excess utility fixed point argument. Such Arrow-Debreu allocations can be implemented into a Radner economy with continuous-time trading. Effective completeness of the market spaces al- ters to an endogenous property. Only mean unambiguous claims equivalently satisfying the classical martingale representation property build the marketed space.

Efficient discretization of stochastic integrals

Sharp asymptotic lower bounds on the expected quadratic variation of the discretization error in stochastic integration are given when the integrator admits a predictable quadratic variation and the integrand is a continuous semimartingale with nondegenerate local martingale part. The theory relies on inequalities for the kurtosis and skewness of a general random variable which are themselves seemingly new. Asymptotically efficient schemes which attain the lower bounds are constructed explicitly. The result is directly applicable to a practical hedging problem in mathematical finance; for hedging a payoff which is replicated by a continuous-time trading strategy, it gives an asymptotically optimal way to choose discrete rebalancing dates and portfolios with respect to transaction costs. The asymptotically efficient strategies in fact reflect the structure of the transaction costs. In particular, a specific biased rebalancing scheme is shown to be superior to unbiased schemes if the transaction costs follow a convex model. The problem is discussed also in terms of exponential utility maximization.

Continuous time trading of a small investor in a limit order market

We provide a mathematical framework to model continuous time trading of a small investor in limit order markets. We show how elementary strategies can be extended in a suitable way to general continuous time strategies containing orders with infinitely many different limit prices. The general limit buy order strategies are predictable processes with values in the set of nonincreasing demand functions. It turns out that our strategy set of limit and market orders is closed, but limit orders can turn into market orders when passing to the limit, and any element can be approximated by a sequence of elementary strategies.

Continuous-time trading and the emergence of probability

This paper establishes a non-stochastic analogue of the celebrated result by Dubins and Schwarz about reduction of continuous martingales to Brownian motion via time change. We consider an idealized financial security with continuous price path, without making any stochastic assumptions. It is shown that typical price paths possess quadratic variation, where "typical" is understood in the following game-theoretic sense: there exists a trading strategy that earns infinite capital without risking more than one monetary unit if the process of quadratic variation does not exist. Replacing time by the quadratic variation process, we show that the price path becomes Brownian motion. This is essentially the same conclusion as in the Dubins-Schwarz result, except that the probabilities (constituting the Wiener measure) emerge instead of being postulated. We also give an elegant statement, inspired by Peter McCullagh's unpublished work, of this result in terms of game-theoretic probability theory.

A Non-Stationary Model for Statistical Arbitrage Trading

In this paper we present a non-stationary model for statistical arbitrage trading. This model represents the security price as the sum of an arithmetic Brownian motion and an Ornstein-Uhlenbeck process. A continuous time trading strategy is derived for the process and expressions for the expected value and the variance of the return are formulated. Analytic solutions to problem are obtained in the case where the non-stationary component has zero drift.

Continuous-time trading and the emergence of randomness

A new definition of events of game-theoretic probability zero in continuous time is proposed and used to prove results suggesting that trading in financial markets results in the emergence of properties usually associated with randomness. This paper concentrates on ‘qualitative’ results, stated in terms of order (or order topology) rather than in terms of the precise values taken by a price process. No stochastic assumptions are made, and the only assumption is that the price process is continuous.

Optimal Portfolio Liquidation with Execution Cost and Risk

We study the optimal portfolio liquidation problem over a finite horizon in a limit order book with bid-ask spread and temporary market price impact penalizing speedy execution trades. We use a continuous-time modeling framework, but in contrast with previous related papers (see, e.g., [L. C. G. Rogers and S. Singh, Math. Finance, 20 (2010), pp. 597–615] and [A. Schied and T. Schöneborn, Finance Stoch., 13 (2009), pp. 181–204]), we do not assume continuous-time trading strategies. We consider instead real trading that occur in discrete time, and this is formulated as an impulse control problem under a solvency constraint, including the lag variable tracking the time interval between trades. A first important result of our paper is to prove rigorously that nearly optimal execution strategies in this context actually lead to a finite number of trades with strictly increasing trading times, and this holds true without assuming ad hoc any fixed transaction fee. Next, we derive the dynamic programming quasi-variational inequality satisfied by the value function in the sense of constrained viscosity solutions. We also introduce a family of value functions which converges to our value function and is characterized as the unique constrained viscosity solutions of an approximation of our dynamic programming equation. This convergence result is useful for numerical purpose but is postponed until a companion paper [F. Guilbaud, M. Mnif, and H. Pham, Numerical Methods for an Optimal Order Execution Problem, preprint, 2010].

The Role of Signal Precision and Transaction Costs in Stock, Option and Volatility Trading

In this study we examine the rationale behind informed traders decision to speculate on the volatility of the common stock by trading in the derivative or in the cash markets. Using a continuous-time trading model, we demonstrate that the quality of the private information regarding the volatility parameter together with the relative transaction costs observed in the various segments of the cash and derivatives markets will determine informed agents’ trading habitats. We further show that in the presence of imprecise volatility signals, only the “most sophisticated” traders (those with highly precise volatility signals) will engage in pure volatility bets. Traders with less precise signals will choose a naked option strategy, while traders at the low spectrum of the precision scale will invest in the underlying stock. Thus, the low volume of pure volatility trades observed by Lakonishok et al. (2007) does not necessarily imply that only fringe traders have chosen to speculate on volatility. Rather, it may suggest that the majority of informed traders do not have very precise volatility signals.

The Role of Signal Precision and Transaction Costs in Stock, Option and Volatility Trading

In this study, we examine the rationale that informed traders use in choosing various financial instruments in order to speculate on the volatility of the underlying asset, here a common stock. Using a continuous-time trading model, we demonstrate that the quality of the private information regarding the volatility parameter together with the relative transaction costs observed in the various segments of the cash and derivatives markets will determine informed agents’ trading habitats. We further show that in the presence of imprecise volatility signals, only the “most sophisticated” traders (those with highly precise volatility signals) will engage in pure volatility bets. Traders with less precise signals will choose a naked option strategy, while traders at the low spectrum of the precision scale will invest in the underlying stock. Thus, the low volume of pure volatility trades observed by Lakonishok et al. (2007) does not necessarily imply that only fringe traders have chosen to speculate on volatility. Rather, it may suggest that the majority of informed traders do not have precise volatility signals.

The Role of Signal Precision and Transaction Costs in Stock, Option and Volatility Trading

In this study, we examine the rationale that informed traders use in choosing various financial instruments in order to speculate on the volatility of the underlying asset, here a common stock. Using a continuous-time trading model, we demonstrate that the quality of the private information regarding the volatility parameter together with the relative transaction costs observed in the various segments of the cash and derivatives markets will determine informed agents’ trading habitats. We further show that in the presence of imprecise volatility signals, only the “most sophisticated” traders (those with highly precise volatility signals) will engage in pure volatility bets. Traders with less precise signals will choose a naked option strategy, while traders at the low spectrum of the precision scale will invest in the underlying stock. Thus, the low volume of pure volatility trades observed by Lakonishok et al. (2007) does not necessarily imply that only fringe traders have chosen to speculate on volatility. Rather, it may suggest that the majority of informed traders do not have precise volatility signals.