Discrete-time Market Research Articles

Market consistent bid-ask option pricing under Dempster-Shafer uncertainty

We refer to the discrete-time market model under ambiguity introduced in [Cinfrignini, A., Petturiti, D. and Vantaggi, B., Dynamic bid–ask pricing under Dempster-Shafer uncertainty. J. Math. Econ., 2023a, 107, 102871], formed by a frictionless risk-free bond and a non-dividend paying stock with bid-ask spread. For a European derivative, we generalize the classical binomial pricing formula by allowing for bid-ask prices and investigate the properties of the ensuing replicating strategies. Next, for an American derivative, we propose a backward bid-ask pricing procedure and prove that the resulting discounted price processes are the bid-ask Choquet-Snell envelopes of the discounted payoff process, respectively. Moreover, for an American call option, we prove a generalization of the well-known Merton's theorem [Merton, R.C., Theory of rational option pricing. Bell J. Econ. Manage. Sci., 1973, 4, 141–183] holding for both the bid and the ask price processes. Finally, we introduce a market consistent calibration procedure and show the use of the calibrated model in bid-ask option pricing.

Option Pricing Using a Skew Random Walk Binary Tree

We develop a binary tree pricing model with underlying asset price dynamics following Itô–McKean skew Brownian motion. Our work was motivated by the Corns–Satchell, continuous-time, option pricing model. However, the Corns–Satchell market model is incomplete, while our discrete-time market model is defined in the natural world, extended to the risk-neutral world under the no-arbitrage condition where derivatives are priced under uniquely determined risk-neutral probabilities, and is complete. The skewness introduced in the natural world is preserved in the risk-neutral world. Furthermore, we show that the model preserves skewness under the continuous-time limit. We provide empirical applications of our model to the valuation of European put and call options on exchange-traded funds tracking the S&P Global 1200 index.

On statistical indistinguishability of complete and incomplete discrete time market models

On statistical indistinguishability of complete and incomplete discrete time market models

Representation for martingales living after a random time with applications

Our financial setting consists of a market model with two flows of information. The smallest flow $ \mathbb{F} $ is the 'public' flow of information which is available to all agents, while the larger flow $ \mathbb{G} $ has additional information about the occurrence of a random time $ \tau $. This random time can model the default time in credit risk or death time in life insurance. Hence the filtration $ \mathbb{G} $ is the progressive enlargement of $ \mathbb{F} $ with $ \tau $. In this framework, when $ \tau $ is a finite honest time, we describe explicitly how $ \mathbb G $-local martingales can be represented in terms of $ \mathbb{F} $-local martingales and parameters of $ \tau $. This representation complements Choulli, Daveloose and Vanmaele [12] to the case when martingales live 'after $ \tau $'. Under some mild assumptions on the pair $ (\mathbb{F}, \tau) $, we fully elaborate the application of these results to the explicit parametrization of all deflators under $ \mathbb{G} $. The results are illustrated in the case of a jump-diffusion model and a discrete-time market model.

Neural network approximation for superhedging prices

AbstractThis article examines neural network‐based approximations for the superhedging price process of a contingent claim in a discrete time market model. First we prove that the α‐quantile hedging price converges to the superhedging price at time 0 for α tending to 1, and show that the α‐quantile hedging price can be approximated by a neural network‐based price. This provides a neural network‐based approximation for the superhedging price at time 0 and also the superhedging strategy up to maturity. To obtain the superhedging price process for , by using the Doob decomposition, it is sufficient to determine the process of consumption. We show that it can be approximated by the essential supremum over a set of neural networks. Finally, we present numerical results.

Discrete-time mean-CVaR portfolio selection and time-consistency induced term structure of the CVaR

We investigate a discrete-time mean-risk portfolio selection problem, where risk is measured by the conditional value-at-risk (CVaR). A substantial challenge is the combination of a time-inconsistent objective with an incomplete and dynamic model for the financial market. We are able to solve this problem analytically by embedding the original, time-inconsistent problem into a family of time-consistent expected utility maximization problems with a piecewise linear utility function. The optimal investment strategy is a fully adaptive feedback policy and the cumulated amount invested in the risky assets is of a characteristic V-shaped pattern as a function of the current wealth. For the incomplete, discrete-time market considered herein, the mean-CVaR efficient frontier is a straight line in the mean-CVaR plane and thus economically meaningful. This contrasts the complete, continuous-time setting where the mean-CVaR efficient frontier is degenerate or does not exist at all. We further solve an inverse investment problem, where we investigate how mean-CVaR preferences need to adapt in order for the pre-committed optimal strategy to remain optimal at any point in time. Our result shows that a pre-committed mean-CVaR investor behaves like a naive mean-CVaR investor with a time-increasing confidence level for the CVaR, who revises the investment decision at every point in time. Finally, an empirical application of our results suggests that risk measured by the CVaR might help to understand the long-standing equity premium puzzle.

Convergence of Optimal Expected Utility for a Sequence of Discrete-Time Market

We examine Kreps’ (2019) conjecture that optimal expected utility in the classic Black–Scholes–Merton (BSM) economy is the limit of optimal expected utility for a sequence of discrete-time economies that “approach” the BSM economy in a natural sense: The nth discrete-time economy is generated by a scaled n-step random walk based on an unscaled random variable ζ with mean zero, variance one, and bounded support. We confirm Kreps’ conjecture if the consumer’s utility function U has asymptotic elasticity strictly less than one, and we provide a counterexample to the conjecture for a utility function U with asymptotic elasticity equal to 1, for ζ such that E [ζ3]>0.

No-arbitrage under additional information for thin semimartingale models

This paper completes the studies undertaken in Aksamit et al. (2017, 2018) [[1,2]] and Choulli and Deng (2017) [[8]], where we quantify the impact of a random time on the no-unbounded-profit-with-bounded-risk concept (called NUPBR hereafter) for quasi-left-continuous models and discrete-time market models respectively. Herein, we focus on NUPBR for semimartingale models that live on thin predictable sets only and when the extra information about the random time is added progressively over time. This leads to the probabilistic setting of two filtrations where one filtration contains the other and makes the random time a stopping time. For this framework, we explain how far NUPBR is affected when one stops the model by an arbitrary random time, or when one incorporates in a progressive way an honest time into the model. Furthermore, we show how to construct explicitly some local martingale deflators in the largest filtration for a particular class of models. As a consequence, by combining the current results on the thin case and those of Aksamit et al. (2017, 2018) [[1,2]], we elaborate universal results for general semimartingale models.

Discrete-time market models from the small investor point of view and the first fundamental-type theorem

Abstract In this paper, we discuss the no-arbitrage condition in a discrete financial market model which does not hold the same interest rate assumptions. Our research was based on, essentially, one of the most important results in mathematical finance, called the Fundamental Theorem of Asset Pricing. For the standard approach a risk-free bank account process is used as numeraire. In those models it is assumed that the interest rates for borrowing and saving money are the same. In our paper we consider the model of a market (with d risky assets), which does not hold the same interest rate assumptions. We introduce two predictable processes for modelling deposits and loans. We propose a new concept of a martingale pair for the market and prove that if there exists a martingale pair for the considered market, then there is no arbitrage opportunity. We also consider special cases in which the existence of a martingale pair is necessary and the sufficient conditions for these markets to be arbitrage free.

SUPER-HEDGING AMERICAN OPTIONS WITH SEMI-STATIC TRADING STRATEGIES UNDER MODEL UNCERTAINTY

We consider the super-hedging price of an American option in a discrete-time market in which stocks are available for dynamic trading and European options are available for static trading. We show that the super-hedging price [Formula: see text] is given by the supremum over the prices of the American option under randomized models. That is, [Formula: see text], where [Formula: see text] and the martingale measure [Formula: see text] are chosen such that [Formula: see text] and [Formula: see text] prices the European options correctly, and [Formula: see text] is the price of the American option under the model [Formula: see text]. Our result generalizes the example given in Hobson & Neuberger (2016) that the highest model-based price can be considered as a randomization over models.

No-arbitrage for informational discrete time market models

This paper focuses on the stability of no-arbitrage, for discrete time market models, under additional uncertainty generated by a random time . At the practical level, this random time represents the death time, the default time of a firm, or any occurrence time of an event that might affect the market somehow. We address the no-arbitrage issue for the resulting new flow of information (filtration) which makes the random time either a nontrivial stopping time (progressive enlargement) or a known time from the beginning (initial enlargement). Our main conclusions are twofold. On the one hand, for a fixed initial market S, we completely and precisely characterize the interplay between S and such that the no-arbitrage is preserved for the new market model. On the other hand, we give the necessary and sufficient conditions on to ensure the preservation of the no-arbitrage under the additional uncertainty of for any market. Two concrete examples are presented to illustrate the results.

Explicit Description of HARA Forward Utilities and Their Optimal Portfolios

This paper deals with forward performances of HARA type. Precisely, for a market model in which stock price processes are modeled by a locally bounded $d$-dimensional semimartingale, we elaborate a complete and explicit characterization for this type of forward utilities. In particular, under some mild technical assumptions of integrability, we prove that the risk aversion process of a power-type forward utility is constant. Furthermore, the optimal portfolios for all HARA forward utilities are explicitly described. Our approach is based on the minimal Hellinger martingale densities that are obtained from the important statistical concept of Hellinger process. These martingale densities were introduced recently and appeared herein tailor-made for these forward utilities. After outlining our parametrization method for the HARA forward, we provide illustrations on discrete-time market models.

High-Frequency Limit of Nash Equilibria in a Market Impact Game with Transient Price Impact

We study the high-frequency limits of strategies and costs in a Nash equilibrium for two agents that are competing to minimize liquidation costs in a discrete-time market impact model with exponentially decaying price impact and quadratic transaction costs of size $\theta\ge0$. We show that, for $\theta=0$, equilibrium strategies and costs will oscillate indefinitely between two accumulation points. For $\theta>0$, however, strategies, costs, and total transaction costs will converge towards limits that are independent of $\theta$. We then show that the limiting strategies form a Nash equilibrium for a continuous-time version of the model with $\theta$ equal to a certain critical value $\theta^*>0$, and that the corresponding expected costs coincide with the high-frequency limits of the discrete-time equilibrium costs. For $\theta\neq\theta^*$, however, continuous-time Nash equilibria will typically not exist. Our results permit us to give mathematically rigorous proofs of numerical observations made in Schied and Zhang (2013). In particular, we provide a range of model parameters for which the limiting expected costs of both agents are decreasing functions of $\theta$. That is, for sufficiently high trading speed, raising additional transaction costs can reduce the expected costs of all agents.

Model-independent superhedging under portfolio constraints

In a discrete-time market, we study model-independent superhedging where the semi-static superhedging portfolio consists of three parts: static positions in liquidly traded vanilla calls, static positions in other tradable, yet possibly less liquid, exotic options, and a dynamic trading strategy in risky assets under certain constraints. By considering the limit order book of each tradable exotic option and employing the Monge–Kantorovich theory of optimal transport we establish a general superhedging duality, which admits a natural connection to convex risk measures. With the aid of this duality, we derive a model-independent version of the fundamental theorem of asset pricing. The notion “finite optimal arbitrage profit”, weaker than no-arbitrage, is also introduced. It is worth noting that our method covers a large class of delta and gamma constraints.

Model-Independent Superhedging Under Portfolio Constraints

In a discrete-time market, we study model-independent superhedging, while the semi-static superhedging portfolio consists of three parts: static positions in liquidly traded vanilla calls, static positions in other tradable, yet possibly less liquid, exotic options, and a dynamic trading strategy in risky assets under certain constraints. By considering the limit order book of each tradable exotic option and employing the Monge-Kantorovich theory of optimal transport, we establish a general superhedging duality, which admits a natural connection to convex risk measures. With the aid of this duality, we derive a model-independent version of the fundamental theorem of asset pricing. The notion fi nite optimal arbitrage pro fit, weaker than no-arbitrage, is also introduced. It is worth noting that our method covers a large class of Delta constraints as well as Gamma constraint.

Arbitrage in markets with bid-ask spreads

In this paper a finite discrete time market model with bid-ask spreads and a money account is considered in the setting of an arbitrary state space. The notions of an equivalent bid-ask martingale measure (EBAMM) and of supermartingale as well as submartingale consistent price systems are introduced. The fundamental theorem of asset pricing is proved using EBAMM as an equivalent condition for no-arbitrage. The Cox-Ross-Rubinstein model with bid-ask spreads is presented as an application of our results.

On asymptotic optimality of Merton's myopic portfolio strategies under time discretization

This paper studies the properties of discrete time stochastic optimal control problems associated with portfolio selection. We investigate if optimal continuous time strategies can be used effectively for a discrete time market after a straightforward discretization. We found that Merton’s strategy approximates the performance of the optimal strategy in a discrete time model with the sufficiently small time steps.

Optimal Investment with Nonconcave Utilities in Discrete-Time Markets

We consider an arbitrage-free, discrete-time and frictionless market. We prove that an investor maximizing the expected utility of his or her terminal wealth can always find an optimal investment strategy provided that his or her dissatisfaction of infinite losses is infinite and his or her utility function is nondecreasing, continuous, and bounded above. The same result is shown for cumulative prospect theory preferences, under additional assumptions.

Brak arbitrażu na rynkach z proporcjonalnymi kosztami transakcji

The aim of the paper was to present in a clear and unified way various results concerning the absence of arbitrage in the modelling of financial markets with proportional transaction costs. The absence of weak and strict arbitrage opportunities criteria in a finite time horizon discrete time market model are given. Sufficient conditions for the absence of simple arbitrage (i.e. arbitrage over simple investment strategies) in a continuous time market model are presented. Special attention is devoted to transactions without short selling.

Semi-Markov migration process in a stochastic market in credit risk

In the present the idea of stochastic Market environment comes into play to express the changes in general economy, which affects any industry in small or great amounts of turbulence. We model the evolution of the Market among its possible ν-states as an F-inhomogeneous semi-Markov process. This idea leads us to modeling the migration process of defaultable bonds as ν different F-inhomogeneous semi-Markov process. The survival probabilities of a defaultable bond in every credit grade are found. The asymptotic behaviour of the survival probabilities is established under certain conditions. Also, it is proved under what conditions the convergence is geometrically fast. The stochastic foundation of the general stochastic discrete-time Market is provided, by proving that the market is viable, if and only if, there exists an equivalent martingale measure, from which we construct the forward probability measure and under which the discounted default free bond price process for all possible states of the Market is a martingale. The term structure of credit spread and the change of real-world probability measure to forward probability measure are studied. In the form of a theorem it is proved that under certain conditions, changing the real probability measure to a forward probability measure, does not affect the inhomogeneous semi Markov process modeling the migration of defaultable bonds. That is, it is proved that it only changes the basic parameters and we provide a relation among the transition probabilities under the two measures. Finally, parameter estimation and calibration of the inhomogeneous semi-Markov chain in stochastic environment is being provided.