VOLATILITY AND COVARIATION ESTIMATION WHEN MICROSTRUCTURE NOISE AND TRADING TIMES ARE ENDOGENOUS

This paper considers practically appealing procedures for estimating intraday volatility measures of financial assets. The underlying microstructure model accommodates the inherent properties of ultra high‐frequency data with the assumption of continuous efficient price processes. In this model, microstructure noise and trading times are endogenous but do not only depend on the prices. Using the (observed) last traded prices of the assets, we develop a new approach that enables to approximate the values of the efficient prices at some random times. Based on these approximated values, we build an estimator of the integrated volatility and give its asymptotic theory. We also give a consistent estimator of the integrated covariation when two assets (asynchronous by construction of the model) are observed.     

DYNAMIC DEFAULTABLE TERM STRUCTURE MODELING BEYOND THE INTENSITY PARADIGM

The two main approaches in credit risk are the structural approach pioneered by Merton and the reduced‐form framework proposed by Jarrow and Turnbull and by Artzner and Delbaen. The goal of this paper is to provide a unified view on both approaches. This is achieved by studying reduced‐form approaches under weak assumptions. In particular, we do not assume the global existence of a default intensity and allow default at fixed or predictable times, such as coupon payment dates, with positive probability. In this generalized framework, we study dynamic term structures prone to default risk following the forward‐rate approach proposed by Heath, Jarrow, and Morton. It turns out that previously considered models lead to arbitrage possibilities when default can happen at a predictable time. A suitable generalization of the forward‐rate approach contains an additional stochastic integral with atoms at predictable times and necessary and sufficient conditions for an appropriate no‐arbitrage condition are given. For efficient implementations, we develop a new class of affine models that do not satisfy the standard assumption of stochastic continuity. The chosen approach is intimately related to the theory of enlargement of filtrations, for which we provide an example by means of filtering theory where the Azéma supermartingale contains upward and downward jumps, both at predictable and totally inaccessible stopping times.     

Periodic strategies in optimal execution with multiplicative price impact

We study the optimal execution problem with multiplicative price impact in algorithmic trading, when an agent holds an initial position of shares of a financial asset. The interselling decision times are modeled by the arrival times of a Poisson process. The criterion to be optimized consists in maximizing the expected net present value of the gains of the agent, and it is proved that an optimal strategy has a barrier form, depending only on the number of shares left and the level of the asset price.     

MODEL‐INDEPENDENT NO‐ARBITRAGE CONDITIONS ON AMERICAN PUT OPTIONS

We consider the pricing of American put options in a model‐independent setting: that is, we do not assume that asset prices behave according to a given model, but aim to draw conclusions that hold in any model. We incorporate market information by supposing that the prices of European options are known. In this setting, we are able to provide conditions on the American put prices which are necessary for the absence of arbitrage. Moreover, if we further assume that there are finitely many European and American options traded, then we are able to show that these conditions are also sufficient. To show sufficiency, we construct a model under which both American and European options are correctly priced at all strikes simultaneously. In particular, we need to carefully consider the optimal stopping strategy in the construction of our process.     

OPTIMAL INVESTMENT WITH AN UNBOUNDED RANDOM ENDOWMENT AND UTILITY-BASED PRICING

This paper studies the problem of maximizing the expected utility of terminal wealth for a financial agent with an unbounded random endowment, and with a utility function which supports both positive and negative wealth. We prove the existence of an optimal trading strategy within a class of permissible strategies—those strategies whose wealth process is a super-martingale under all pricing measures with finite relative entropy. We give necessary and sufficient conditions for the absence of utility-based arbitrage, and for the existence of a solution to the primal problem. We consider two utility-based methods which can be used to price contingent claims. Firstly we investigate marginal utility-based price processes (MUBPP's). We show that such processes can be characterized as local martingales under the normalized optimal dual measure for the utility maximizing investor. Finally, we present some new results on utility indifference prices, including continuity properties and volume asymptotics for the case of a general utility function, unbounded endowment and unbounded contingent claims.     

The Ethics of Insider Trading Revisited

Following Manne (1966, Insider Trading and the Stock Market (New York, Free Press)) we introduce a distinction between insider trading and market manipulation on the one hand and corporate insiders versus misappropriators on the other hand. This gives rise to four types of alleged inside transactions. We argue that the literature on insider trading has often targeted inside transactions type II, III and IV but that these arguments do not necessarily hold for type I transactions. We look for consequentionalist as well as non-consequentionalist arguments against type I transactions and demonstrate that these are hard to find. Throughout the article we refer extensively to the economic literature on insider trading in order to overcome a relative divide between the economic, legal, and philosophical discussion on insider trading.

Trading under the proof-of-stake protocol – A continuous-time control approach

We develop a continuous-time control approach to optimal trading in a Proof-of-Stake (PoS) blockchain, formulated as a consumption-investment problem that aims to strike the optimal balance between a participant's (or agent's) utility from holding/trading stakes and utility from consumption. We present solutions via dynamic programming and the Hamilton–Jacobi–Bellman (HJB) equations. When the utility functions are linear or convex, we derive close-form solutions and show that the bang-bang strategy is optimal (i.e., always buy or sell at full capacity). Furthermore, we bring out the explicit connection between the rate of return in trading/holding stakes and the participant's risk-adjusted valuation of the stakes. In particular, we show when a participant is risk-neutral or risk-seeking, corresponding to the risk-adjusted valuation being a martingale or a sub-martingale, the optimal strategy must be to either buy all the time, sell all the time, or first buy then sell, and with both buying and selling executed at full capacity. We also propose a risk-control version of the consumption-investment problem; and for a special case, the “stake-parity” problem, we show a mean-reverting strategy is optimal.     

A Leland model for delta hedging in central risk books

Using a tractable extension of the model of Leland (1985), we study how a delta-hedging strategy can realistically be implemented using market and limit orders in a centralized, automated market-making desk that integrates trading and liquidity provision for both options and their underlyings. In the continuous-time limit, the optimal limit-order exposure can be computed explicitly by a pointwise maximization. It is determined by the relative magnitudes of adverse selection, bid–ask spreads, and volatilities. The corresponding option price—from which the option can be replicated using market and limit orders—is characterized via a nonlinear PDE. Our results highlight the benefit of tactical liquidity provision for contrarian trading strategies, even for a trading desk that is not a competitive market maker. More generally, the paper also showcases how reduced-form models are competitive with “brute force” numerical approaches to market microstructure. Both the estimation of microstructure parameters and the simulation of the optimal trading strategy are made concrete and reconciled with real-life high frequency data.     

Optimal trade execution in order books with stochastic liquidity

In financial markets, liquidity changes randomly over time. We consider such random variations of the depth of the order book and evaluate their influence on optimal trade execution strategies. If the stochastic structure of liquidity changes satisfies certain conditions, then the unique optimal trading strategy exhibits a conventional structure with a single wait region and a single buy region, and profitable round‐trip strategies do not exist. In other cases, optimal strategies can feature multiple wait regions and optimal trade sizes that can be decreasing in the size of the position to be liquidated. Furthermore, round‐trip strategies can be profitable depending on bid–ask spread assumptions. We illustrate our findings with several examples including the Cox–Ingersoll–Ross model for the evolution of liquidity.     

MULTIPLICATIVE APPROXIMATION OF WEALTH PROCESSES INVOLVING NO‐SHORT‐SALES STRATEGIES VIA SIMPLE TRADING

A financial market model with general semimartingale asset–price processes and where agents can only trade using no‐short‐sales strategies is considered. We show that wealth processes using continuous trading can be approximated very closely by wealth processes using simple combinations of buy‐and‐hold trading. This approximation is based on controlling the proportions of wealth invested in the assets. As an application, the utility maximization problem is considered and it is shown that optimal expected utilities and wealth processes resulting from continuous trading can be approximated arbitrarily well by the use of simple combinations of buy‐and‐hold strategies.     

Optimal consumption and investment with liquid and illiquid assets

I consider an optimal consumption/investment problem to maximize expected utility from consumption. In this market model, the investor is allowed to choose a portfolio that consists of one bond, one liquid risky asset (no transaction costs), and one illiquid risky asset (proportional transaction costs). I fully characterize the optimal consumption and trading strategies in terms of the solution of the free boundary ordinary differential equation (ODE) with an integral constraint. I find an explicit characterization of model parameters for the well‐posedness of the problem, and show that the problem is well posed if and only if there exists a shadow price process. Finally, I describe how the investor's optimal strategy is affected by the additional opportunity of trading the liquid risky asset, compared to the simpler model with one bond and one illiquid risky asset.     

Discrete-time risk sensitive portfolio optimization with proportional transaction costs

In this paper we consider a discrete-time risk sensitive portfolio optimization over a long time horizon with proportional transaction costs. We show that within the log-return i.i.d. framework the solution to a suitable Bellman equation exists under minimal assumptions and can be used to characterize the optimal strategies for both risk-averse and risk-seeking cases. Moreover, using numerical examples, we show how a Bellman equation analysis can be used to construct or refine optimal trading strategies in the presence of transaction costs.     

Instrument rules in monetary policy under heterogeneity in currency trade

We embed different instrument rules into Galí and Monacelli’s new Keynesian model for a small open economy that is augmented with technical trading in currency trade to examine the prerequisites for monetary policy. Specifically, conditions for a determinate and least squares learnable REE are in focus. When a contemporaneous data specification of the rule is used in policy-making, the degree of trend following in currency trade does not affect these conditions, except in case of an extensive use of trend following, whereas a forward expectations specification makes it less likely to have a determinate and learnable REE when the degree of trend following is increasing. We allow for interest rate inertia in the analysis.     

Coupon Trading and its Impacts on Consumer Purchase and Firm Profits

When a retailer distributes manufacturer coupons to consumers without perfectly identifying their product valuations, consumers may have incentives to trade coupons. We develop a model to capture the coupon trading phenomenon and compare three scenarios: (I) no coupon, (II) coupon without trading, and (III) coupon with trading. We find that coupon trading can increase the profits of either the retailer or the manufacturers, but not at the same time. The retailer benefits from coupon trading when the coupon market is competitive and consumer hassle cost is low, while the manufacturers benefit from coupon trading when the coupon market is uncompetitive and consumer hassle cost is high. In addition, coupon trading does not always increase total demand. Firms benefit from coupon trading by charging higher prices, which leads to a decreased total demand. As a result, consumers end up with a higher average cost under coupon trading. We also compare coupon trading with improved coupon targeting, and find that coupon trading may allow firms to gain higher profits than improved coupon targeting. Further, we extend the main model to a competitive setting where the products are substitutable, and find that the main results still hold. Finally, we employ numerical analysis to identify the optimal coupon face values in different scenarios, and the results suggest that coupon trading combined with incentive mechanisms may lead to Pareto improvement for the channel as a whole.     

On the Existence of Minimax Martingale Measures

Embedding asset pricing in a utility maximization framework leads naturally to the concept of minimax martingale measures. We consider a market model where the price process is assumed to be an ^d‐semimartingale X and the set of trading strategies consists of all predictable, X‐integrable, ^d‐valued processes H for which the stochastic integral (H.X) is uniformly bounded from below. When the market is free of arbitrage, we show that a sufficient condition for the existence of the minimax measure is that the utility function u : → is concave and nondecreasing. We also show the equivalence between the no free lunch with vanishing risk condition, the existence of a separating measure, and a properly defined notion of viability.     

Optimal investment,derivative demand,and arbitrage under price impact

This paper studies the optimal investment problem with random endowment in an inventory‐based price impact model with competitive market makers. Our goal is to analyze how price impact affects optimal policies, as well as both pricing rules and demand schedules for contingent claims. For exponential market makers preferences, we establish two effects due to price impact: constrained trading and nonlinear hedging costs. To the former, wealth processes in the impact model are identified with those in a model without impact, but with constrained trading, where the (random) constraint set is generically neither closed nor convex. Regarding hedging, nonlinear hedging costs motivate the study of arbitrage free prices for the claim. We provide three such notions, which coincide in the frictionless case, but which dramatically differ in the presence of price impact. Additionally, we show arbitrage opportunities, should they arise from claim prices, can be exploited only for limited position sizes, and may be ignored if outweighed by hedging considerations. We also show that arbitrage‐inducing prices may arise endogenously in equilibrium, and that equilibrium positions are inversely proportional to the market makers' representative risk aversion. Therefore, large positions endogenously arise in the limit of either market maker risk neutrality, or a large number of market makers.     

NEWS-GENERATED DEPENDENCE AND OPTIMAL PORTFOLIOS FOR n STOCKS IN A MARKET OF BARNDORFF-NIELSEN AND SHEPHARD TYPE

We consider Merton's portfolio optimization problem in a Black and Scholes market with non-Gaussian stochastic volatility of Ornstein–Uhlenbeck type. The investor can trade in n stocks and a risk-free bond. We assume that the dependence between stocks lies in that they partly share the Ornstein–Uhlenbeck processes of the volatility. We refer to these as news processes, and interpret this as that dependence between stocks lies solely in their reactions to the same news. The model is primarily intended for assets that are dependent, but not too dependent, such as stocks from different branches of industry. We show that this dependence generates covariance, and give statistical methods for both the fitting and verification of the model to data. Using dynamic programming, we derive and verify explicit trading strategies and Feynman–Kac representations of the value function for power utility.     

Closed-loop Nash competition for liquidity

We study a multiplayer stochastic differential game, where agents interact through their joint price impact on an asset that they trade to exploit a common trading signal. In this context, we prove that a closed-loop Nash equilibrium exists if the price impact parameter is small enough. Compared to the corresponding open-loop Nash equilibrium, both the agents' optimal trading rates and their performance move towards the central-planner solution, in that excessive trading due to lack of coordination is reduced. However, the size of this effect is modest for plausible parameter values.     

Bernoulli speculator and trading strategy risk

In a continuous‐time model of a complete information economy, we examine the case of a “pure” speculator who chooses to trade only on forward or futures contracts written on interest‐rate‐sensitive instruments. Assuming logarithmic utility, we assess whether his strategy exhibits the same structure as when he uses primitive assets only. It turns out that when interest rates follow stochastic processes, as in the model of Heath, Jarrow, and Morton (1992), where the instantaneous forward rate is driven by an arbitrary number of factors, the speculative trading strategy involving forwards exhibits an extra term vis‐a‐vis the one using futures or primitive assets. This extra term, different from a Merton–Breeden dynamic hedge, is novel and can be interpreted as a hedge against an “endogenous risk,” namely the interest‐rate risk brought about by the optimal trading strategy itself. Thus, only the strategy using futures (or the cash assets themselves) involves a single speculative term, even for the Bernoulli speculator. This result illustrates another major aspect of the marking to market feature that differentiates futures and forwards, and thus has some bearing on the issue of the optimal design of financial contracts. Real financial markets being, in fact, incomplete, the additional “endogenous” risk associated with forwards cannot be hedged perfectly. Since using futures eliminates the latter, risk‐averse agents will find them attractive in relation to forward contracts, other things being equal. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20: 507–523, 2000