Mathematical Finance最新文献

Linear factor models, where the factors are affine processes, play a key role in Finance, since they allow for quasi-closed form expressions of the term structure of risks. We introduce the class of noncausal affine linear factor models by considering factors that are affine in reverse time. These models are especially relevant for pricing sequences of speculative bubbles. We show that they feature nonaffine dynamics in calendar time, while still providing (quasi) closed form term structures and derivative pricing formulas. The framework is illustrated with term structure of interest rates and European call option pricing examples.

We investigate the optimal portfolio deleveraging (OPD) problem with permanent and temporary price impacts, where the objective is to maximize equity while meeting a prescribed debt/equity requirement. We take the real situation with cross impact among different assets into consideration. The resulting problem is, however, a nonconvex quadratic program with a quadratic constraint and a box constraint, which is known to be NP-hard. In this paper, we first develop a successive convex optimization (SCO) approach for solving the OPD problem and show that the SCO algorithm converges to a KKT point of its transformed problem. Second, we propose an effective global algorithm for the OPD problem, which integrates the SCO method, simple convex relaxation, and a branch-and-bound framework, to identify a global optimal solution to the OPD problem within a prespecified ε-tolerance. We establish the global convergence of our algorithm and estimate its complexity. We also conduct numerical experiments to demonstrate the effectiveness of our proposed algorithms with both real data and randomly generated medium- and large-scale OPD instances.

This study examines the continuous-time optimal stopping problem with an infinite horizon under Markov processes. Existing research focuses on finding explicit solutions under certain assumptions of the reward function or underlying process; however, these assumptions may either not be fulfilled or be difficult to validate in practice. We developed a continuous-time Markov chain (CTMC) approximation method to find the optimal solution, which applies to general reward functions and underlying Markov processes. We demonstrated that our method can be used to solve the optimal stopping problem with a random delay, in which the delay could be either an independent random variable or a function of the underlying process. We established a theoretical upper bound for the approximation error to facilitate error control. Furthermore, we designed a two-stage scheme to implement our method efficiently. The numerical results show that the proposed method is accurate and rapid under various model specifications.

We introduce a general framework for Markov decision problems under model uncertainty in a discrete-time infinite horizon setting. By providing a dynamic programming principle, we obtain a local-to-global paradigm, namely solving a local, that is, a one time-step robust optimization problem leads to an optimizer of the global (i.e., infinite time-steps) robust stochastic optimal control problem, as well as to a corresponding worst-case measure. Moreover, we apply this framework to portfolio optimization involving data of the . We present two different types of ambiguity sets; one is fully data-driven given by a Wasserstein-ball around the empirical measure, the second one is described by a parametric set of multivariate normal distributions, where the corresponding uncertainty sets of the parameters are estimated from the data. It turns out that in scenarios where the market is volatile or bearish, the optimal portfolio strategies from the corresponding robust optimization problem outperforms the ones without model uncertainty, showcasing the importance of taking model uncertainty into account.

This paper shows how to use results of statistical learning theory and stochastic algorithms to have a better understanding of the convergence of Reinforcement Learning (RL) once it is formulated as a fixed point problem. This can be used to propose improvement of RL learning rates. First, our analysis shows that the classical asymptotic convergence rate $��O�(�1�/�\sqrt{�N�}�)�$ is pessimistic and can be replaced by $��O�\left(�{�(�\log ��\left(�N�\right)��/�N�)�}^{\beta }�\right)�$ with $��\frac{1}{2}�\le �\beta �\le �1�$, and $�N$ the number of iterations. Second, we propose a dynamic optimal policy for the choice of the learning rate used in RL. We decompose our policy into two interacting levels: the inner and outer levels. In the inner level, we present the PASS algorithm (for “PAst Sign Search”) which, based on a predefined sequence of learning rates, constructs a new sequence for which the error decreases faster. The convergence of PASS is proved and error bounds are established. In the outer level, we propose an optimal methodology for the selection of the predefined sequence. Third, we show empirically that our selection methodology of the learning rate outperforms significantly standard algorithms used in RL for the three following applications: the estim

This article clarifies the relationship between pricing kernel monotonicity and the existence of opportunities for stochastic arbitrage in a complete and frictionless market of derivative securities written on a market portfolio. The relationship depends on whether the payoff distribution of the market portfolio satisfies a technical condition called adequacy, meaning that it is atomless or is comprised of finitely many equally probable atoms. Under adequacy, pricing kernel nonmonotonicity is equivalent to the existence of a strong form of stochastic arbitrage involving distributional replication of the market portfolio at a lower price. If the adequacy condition is dropped then this equivalence no longer holds, but pricing kernel nonmonotonicity remains equivalent to the existence of a weaker form of stochastic arbitrage involving second-order stochastic dominance of the market portfolio at a lower price. A generalization of the optimal measure preserving derivative is obtained, which achieves distributional replication at the minimum cost of all second-order stochastically dominant securities under adequacy.

We propose a systematic algorithmic reverse-stress testing methodology to create “worst case” scenarios for regulatory stress tests by accounting for losses that arise from distressed portfolio liquidations. First, we derive the optimal bank response for any given shock. Then, we introduce an algorithm which systematically generates scenarios that exploit the key vulnerabilities in banks' portfolio holdings and thus maximize contagion despite banks' optimal response to the shock. We apply our methodology to data of the 2016 European Banking Authority (EBA) stress test, and design worst case scenarios for the portfolio holdings of European banks at the time. Using spectral clustering techniques, we group 10,000 worst-case scenarios into twelve geographically concentrated families. Our results show that even though there is a wide range of different scenarios within these 12 families, each cluster tends to affect the same banks. An “Anna Karenina” principle of stress testing emerges: Not all stressful scenarios are alike, but every stressful scenario stresses the same banks. These findings suggest that the precise specification of a scenario is not of primal importance as long as the most vulnerable banks are targeted and sufficiently stressed. Finally, our methodology can be used to uncover the weakest links in the financial system and thereby focus supervisory attention on these, thus building a bridge between macroprudential and microprudential stress tests.

Based on a rough path foundation, we develop a model-free approach to stochastic portfolio theory (SPT). Our approach allows to handle significantly more general portfolios compared to previous model-free approaches based on Föllmer integration. Without the assumption of any underlying probabilistic model, we prove a pathwise formula for the relative wealth process, which reduces in the special case of functionally generated portfolios to a pathwise version of the so-called master formula of classical SPT. We show that the appropriately scaled asymptotic growth rate of a far reaching generalization of Cover's universal portfolio based on controlled paths coincides with that of the best retrospectively chosen portfolio within this class. We provide several novel results concerning rough integration, and highlight the advantages of the rough path approach by showing that (nonfunctionally generated) log-optimal portfolios in an ergodic Itô diffusion setting have the same asymptotic growth rate as Cover's universal portfolio and the best retrospectively chosen one.

We present a pathwise approach to continuous-time finance based on causal functional calculus. Our framework does not rely on any probabilistic concept. We introduce a definition of continuous-time self-financing portfolios, which does not rely on any integration concept and show that the value of a self-financing portfolio belongs to a class of nonanticipative functionals, which are pathwise analogs of martingales. We show that if the set of market scenarios is generic in the sense of being stable under certain operations, such self-financing strategies do not give rise to arbitrage. We then consider the problem of hedging a path-dependent payoff across a generic set of scenarios. Applying the transition principle of Rufus Isaacs in differential games, we obtain a pathwise dynamic programming principle for the superhedging cost. We show that the superhedging cost is characterized as the solution of a path-dependent equation. For the Asian option, we obtain an explicit solution.