Discrete-time Financial Market Research Articles

Portfolio management under drawdown constraint in discrete-time financial markets

AbstractConsidering a representative agent in the market, we study the long-term optimal investment problem in a discrete-time financial market, introducing a set of restrictions in the admissible strategies. The drawdown constraints limit the size of possible losses of the portfolio and impose a floor-based performance measure. The optimal growth rate is characterized, and under suitable hypotheses it is proved that an optimal strategy exists. The approach to solving this problem is based on dynamic programming techniques and a fixed point argument adapted from the theory of Markov decision processes.

Optimal long-term investment in illiquid markets when prices have negative memory

In a discrete-time financial market model with instantaneous price impact, we find an asymptotically optimal strategy for an investor maximizing her expected wealth. The asset price is assumed to follow a process with negative memory. We determine how the optimal growth rate depends on the impact parameter and on the covariance decay rate of the price.

Utility Maximization with Proportional Transaction Costs Under Model Uncertainty

We consider a discrete time financial market with proportional transaction costs under model uncertainty and study a numéraire-based semistatic utility maximization problem with an exponential utility preference. The randomization techniques recently developed in Bouchard, Deng, and Tan [Bouchard B, Deng S, Tan X (2019) Super-replication with proportional transaction cost under model uncertainty. Math. Finance 29(3):837–860.], allow us to transform the original problem into a frictionless counterpart on an enlarged space. By suggesting a different dynamic programming argument than in Bartl [Bartl D (2019) Exponential utility maximization under model uncertainty for unbounded endowments. Ann. Appl. Probab. 29(1):577–612.], we are able to prove the existence of the optimal strategy and the convex duality theorem in our context with transaction costs. In the frictionless framework, this alternative dynamic programming argument also allows us to generalize the main results in Bartl [Bartl D (2019) Exponential utility maximization under model uncertainty for unbounded endowments. Ann. Appl. Probab. 29(1):577–612.] to a weaker market condition. Moreover, as an application of the duality representation, some basic features of utility indifference prices are investigated in our robust setting with transaction costs.

Convergence of utility indifference prices to the superreplication price in a multiple‐priors framework

AbstractThis paper formulates a utility indifference pricing model for investors trading in a discrete time financial market under nondominated model uncertainty. Investor preferences are described by possibly random utility functions defined on the positive axis. We prove that when the investors's absolute risk aversion tends to infinity, the multiple‐priors utility indifference prices of a contingent claim converge to its multiple‐priors superreplication price. We also revisit the notion of certainty equivalent for multiple‐priors and establish its relation with risk aversion.

Arbitrage-free modeling under Knightian uncertainty

We study the Fundamental Theorem of Asset Pricing for a general financial market under Knightian Uncertainty. We adopt a functional analytic approach which requires neither specific assumptions on the class of priors $$\mathcal {P}$$ nor on the structure of the state space. Several aspects of modeling under Knightian Uncertainty are considered and analyzed. We show the need for a suitable adaptation of the notion of No Free Lunch with Vanishing Risk and discuss its relation to the choice of an appropriate technical filtration. In an abstract setup, we show that absence of arbitrage is equivalent to the existence of approximate martingale measures sharing the same polar set of $$\mathcal {P}$$ . We then specialize our results to a discrete-time financial market in order to obtain martingale measures.

A Complement to the Grigoriev Theorem for the Kabanov Model

We provide an equivalent characterisation of absence of arbitrage opportunity NA for the Bid and Ask financial market model analog to the Dalang--Morton--Willinger theorem formulated for discrete-time financial market models without friction. This result completes the Grigoriev theorem for conic models in the two dimensional case by showing that the set of all terminal liquidation values is closed under NA..

On the quasi-sure superhedging duality with frictions

We prove the superhedging duality for a discrete-time financial market with proportional transaction costs under model uncertainty. Frictions are modelled through solvency cones as in the original model of Kabanov (Finance Stoch. 3:237–248, 1999) adapted to the quasi-sure setup of Bouchard and Nutz (Ann. Appl. Probab. 25:823–859, 2015). Our approach allows removing the restrictive assumption of no arbitrage of the second kind considered in Bouchard et al. (Math. Finance 29:837–860, 2019) and showing the duality under the more natural condition of strict no arbitrage. In addition, we extend the results to models with portfolio constraints.

The Robust Superreplication Problem: A Dynamic Approach

In the frictionless discrete time financial market of Bouchard and Nutz [Ann. Appl. Probab., 25 (2015), pp. 823--859] we consider a trader who is required to hedge $\xi$ in a risk-conservative way relative to a family of probability measures ${\cal P}$. We first describe the evolution of $\pi_t(\xi)$---the superhedging price at time $t$ of the liability $\xi$ at maturity $T$---via a dynamic programming principle, show that $\pi_t(\xi)$ can be seen as a concave envelope of $\pi_{t+1}(\xi)$ evaluated at today's prices, and prove its dual characterization. Under suitable assumptions, we show that the robust superreplication price is equal to the classical $P$-superhedging price for an extreme prior $P\in {\cal P}$. Then we consider an optimal investment problem for the trader who is rolling over her robust superhedge and phrase this as a robust maximization problem, where the expected utility of intertemporal consumption is optimized subject to a robust superhedging constraint. This utility maximization is carried out under a subset ${\cal P}^u$ of ${\cal P}$ representing the trader's subjective views on market dynamics. Under suitable assumptions on the trader's utility functions, we show that optimal investment and consumption strategies exist and further specify when, and in what sense, these may be unique.

Multiple-priors optimal investment in discrete time for unbounded utility function

This paper investigates the problem of maximizing expected terminal utility in a discrete-time financial market model with a finite horizon under nondominated model uncertainty. We use a dynamic programming framework together with measurable selection arguments to prove that under mild integrability conditions, an optimal portfolio exists for an unbounded utility function defined on the half-real line.

No-arbitrage and optimal investment with possibly non-concave utilities: a measure theoretical approach

We consider a discrete-time financial market model with finite time horizon and investors with utility functions defined on the non-negative half-line. We allow these functions to be random, non-concave and non-smooth. We use a dynamic programming framework together with measurable selection arguments to establish both the characterisation of the no-arbitrage property for such markets and the existence of an optimal portfolio strategy for such investors.

ON THE NUMERICAL ASPECTS OF OPTIMAL OPTION HEDGING WITH TRANSACTION COSTS

We present a numerical study of non-self-financing hedging of European options under proportional transaction costs. We describe an algorithmic approach based on a discrete time financial market model that extends the classical binomial model. We review the analytical basis for our algorithm and present a variety of empirical results using real market data. The performance of the algorithm is evaluated by comparing to a Black–Scholes delta hedge with transaction costs incorporated. We also evaluate the impact of recalibrating the hedging strategy one or more times during the life of the option using the most recent market data. These results are compared to a recalibrated Black–Scholes delta hedge modified for transaction costs.

Super-replication of European Options with Convex Payoff under Proportional Transaction Costs

The paper examines the super-replication of contingent claims in a discrete time financial market with proportional transaction costs. The sole assumption on stock price dynamics is that the returns are bounded. The class of path-dependent European options with nonnegative convex payoff functions is considered. It is proved that the pricing of this type of options can be studied through the pricing of a suitable binomial model. As a consequence, it is shown that the pricing algorithm, which is essentially a dynamic programming procedure on a tree, can be used when the set of possible scenarios is not finite.

Fundamental Theorem of Asset Pricing Under Transaction Costs and Model Uncertainty

We prove the fundamental theorem of asset pricing for a discrete time financial market where trading is subject to proportional transaction costs and the asset price dynamic is modeled by a family of probability measures, possibly nondominated. Using a backward-forward scheme, we show that when the market consists of a money market account and a single stock, no-arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of consistent price systems. We also show that when the market consists of multiple dynamically traded assets and satisfies efficient friction, strict no-arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of strictly consistent price systems.

Maximization of Nonconcave Utility Functions in Discrete-Time Financial Market Models

This paper investigates the problem of maximizing expected terminal utility in a (generically incomplete) discrete-time financial market model with finite time horizon. By contrast to the standard setting, a possibly nonconcave utility function U is considered, with domain of definition equal to the whole real line. Simple conditions are presented that guarantee the existence of an optimal strategy for the problem. In particular, the asymptotic elasticity of U plays a decisive role: Existence can be shown when it is strictly greater at −∞ than at +∞.

Arbitrage and Hedging in Model-Independent Markets with Frictions

We provide a fundamental theorem of asset pricing and a superhedging theorem for a model independent discrete time financial market with proportional transaction costs. We consider a probability-free version of the robust no arbitrage condition introduced by Schachermayer in [Math. Finance, 14 (2004), pp. 19--48] and show that this is equivalent to the existence of consistent price systems. Moreover, we prove that the superhedging price for a claim g coincides with the frictionless superhedging price of g for a suitable process in the bid-ask spread.

Optimal hedging in an extended binomial market under transaction costs

We develop an approach to optimal hedging of a contingent claim under proportional transaction costs in a discrete time financial market model which extends the binomial market model with transaction costs. Our model relaxes the binomial assumption on the stock price ratios to the case where the stock price ratio distribution has bounded support. Non-self-financing hedging strategies are studied to construct an optimal hedge for an investor who takes a short position in a European contingent claim settled by delivery. We develop the theoretical basis for our optimal hedging approach, extending results obtained in our previous work. Specifically, we derive a no-arbitrage option price interval and establish properties of the non-self-financing strategies and their residuals. Based on the theoretical foundation, we develop a computational algorithm for optimizing an investor relevant criterion over the set of admissible non-self-financing hedging strategies. We demonstrate the applicability of our approach using both simulated data and real market data.

Arbitrage and duality in nondominated discrete-time models

We consider a nondominated model of a discrete-time financial market where stocks are traded dynamically, and options are available for static hedging. In a general measure-theoretic setting, we show that absence of arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of martingale measures. In the arbitrage-free case, we show that optimal superhedging strategies exist for general contingent claims, and that the minimal superhedging price is given by the supremum over the martingale measures. Moreover, we obtain a nondominated version of the Optional Decomposition Theorem.

An Equilibrium Model of Market Efficiency with Bayesian Learning: Explicit Rates of Convergence to Rational Expectations Equilibrium

A simple discrete-time financial market model is introduced. The market participants consist of a collection of noise traders as well as a distinguished agent who uses the price information as it arrives to update her demand for the assets. It is shown that the distinguished agent's demand converges, both almost surely and in mean square, to a demand consistent with the rational expectations hypothesis, and the rate of convergence is calculated explicitly. Furthermore, the convergence of the standardised deviations from this limit is established. The rate of convergence, and hence the efficiency of this market, is an increasing function of both the risk-free interest rate and the relative number of noise traders in the market. An efficient market, therefore, measured in terms of a high proportion of informed traders, seems incompatible with the notion that efficient markets converge quickly.

Robust hedging with proportional transaction costs

A duality for robust hedging with proportional transaction costs of path-dependent European options is obtained in a discrete-time financial market with one risky asset. The investor’s portfolio consists of a dynamically traded stock and a static position in vanilla options, which can be exercised at maturity. Trading of both options and stock is subject to proportional transaction costs. The main theorem is a duality between hedging and a Monge–Kantorovich-type optimization problem. In this dual transport problem, the optimization is over all probability measures that satisfy an approximate martingale condition related to consistent price systems, in addition to an approximate marginal constraint.

Fundamental Theorem of Asset Pricing Under Transaction Costs and Model Uncertainty

We prove the Fundamental Theorem of Asset Pricing for a discrete time financial market where trading is subject to proportional transaction cost and the asset price dynamic is modeled by a family of probability measures, possibly non-dominated.Using a backward-forward scheme, we show that when the market consists of a money market account and a single stock, no-arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of consistent price systems. We also show that when the market consists of multiple dynamically traded assets and satisfies efficient friction, strict no-arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of strictly consistent price systems.