Optimal Stopping Time Problem Research Articles

Is Reinforcement Learning Good at American Option Valuation?

This paper investigates algorithms for identifying the optimal policy for pricing American Options. The American Option pricing is reformulated as a Sequential Decision-Making problem with two binary actions (Exercise or Continue), transforming it into an optimal stopping time problem. Both the least square Monte Carlo simulation method (LSM) and Reinforcement Learning (RL)-based methods were utilized to find the optimal policy and, hence, the fair value of the American Put Option. Both Classical Geometric Brownian Motion (GBM) and calibrated Stochastic Volatility models served as the underlying uncertain assets. The novelty of this work lies in two aspects: (1) Applying LSM- and RL-based methods to determine option prices, with a specific focus on analyzing the dynamics of “Decisions” made by each method and comparing final decisions chosen by the LSM and RL methods. (2) Assess how the RL method updates “Decisions” at each batch, revealing the evolution of the decisions during the learning process to achieve optimal policy.

The new bond on the block — Designing a carbon-linked bond for sustainable investment projects

Over the last decade, the green bond market experienced strong growth rates fueled by the need to combat climate change. However, the discourse on enhancing the effectiveness of green bonds primarily revolves around regulatory measures, often overlooking the possibility of designing inherent incentives. We show that a green bond with a coupon structure positively related to the carbon price development stimulates (early) investment in an emission-reducing project and creates higher net present values (NPVs) when applied in project financing. In our simulation-based framework, we model carbon prices using a geometric Brownian motion, and create a general optimal stopping time problem regarding the start of the project. The green bond in our setting carries the risk of default, also mitigated by its carbon price-linked coupon structure.

Pricing American Options under Azzalini Ito-McKean Skew Brownian Motions

In this paper, the pricing of American options whose asset price dynamics follow Azzalini Itô-McKean skew Brownian motions is considered. The corresponding optimal stopping time problem is then formulated, and the main properties of its value function are provided. We show that if the payoff function is positive and decreasing, then the value function and its partial derivatives are continuous and locally bounded, and therefore several variational inequalities are derived. Furthermore, the Feyman-Kac formula is calculated. Finally, under this more general as well as very versatile setting, the Black-Scholes option pricing model is nested as a special case.

Deep combinatorial optimisation for optimal stopping time problems: application to swing options pricing.

A new method for stochastic control based on neural networks and using randomisation of discrete random variables is proposed and applied to optimal stopping time problems. The method models directly the policy and does not need the derivation of a dynamic programming principle nor a backward stochastic differential equation. Unlike continuous optimization where automatic differentiation is used directly, we propose a likelihood ratio method for gradient computation. Numerical tests are done on the pricing of American and swing options. The proposed algorithm succeeds in pricing high dimensional American and swing options in a reasonable computation time, which is not possible with classical algorithms.

Procrastination, self-imposed deadlines and other commitment devices

In this paper we model a decision maker who must exert costly effort to complete a single task by a fixed deadline. Effort costs evolve stochastically in continuous time. The decision maker optimally waits to exert effort until costs are less than a given threshold, the solution to an optimal stopping time problem. We derive the solution to this model for three cases: (1) exponential decision makers, (2) Naïve hyperbolic discounters and (3) sophisticated hyperbolic discounters. Absent deadlines, we show that sophisticated hyperbolic decision makers behave as if they were time consistent but instead have a smaller reward for completing the task, while naïfs never complete the task. In the presence of deadlines, sophisticated decision makers may, counterintuitively, have a threshold which is decreasing as they approach the deadline. An extensive numerical study shows that, unlike exponential or naïfs who always prefer longer deadlines, sophisticated decision makers will often self-impose a binding deadline as a form of commitment, while naïve decision makers will not, and we show how this varies with changes in underlying cost, preference and self-control parameters.

Optimal Edge Computing for Infrastructure-Assisted UAV Systems

The ability of Unmanned Aerial Vehicles (UAV) to autonomously operate is constrained by the severe limitations of their on-board resources. The limited processing capacity and energy storage of these devices inevitably makes the real-time analysis of complex signals – the key to autonomy – challenging. In urban environments, the UAVs can leverage the communication and computation resources of the surrounding city-wide Internet of Things infrastructure to enhance their capabilities. For instance, the UAVs can interconnect with edge computing resources and offload computation tasks to improve response time to sensor input and reduce energy consumption. However, the complexity of the urban topology and large number of devices and data streams competing for the same network and computation resources create an extremely dynamic environment, where poor channel conditions and edge server congestion may penalize the performance of task offloading. This paper develops a framework enabling optimal offloading decisions as a function of network and computation load parameters and current state. The optimization is formulated as an optimal stopping time problem over a semi-Markov process. We solve the optimization problem using Dynamic Programming and Deep Reinforcement learning at different levels of abstraction and prior knowledge of the system underlying stochastic processes. We validate our results in a realistic scenario, where a UAV performs a building inspection task while connected to an edge server.

Two-Stage Bayesian Sequential Change Diagnosis

In this paper, we formulate and solve a two-stage Bayesian sequential change diagnosis (SCD) problem. Different from the one-stage sequential change diagnosis problem considered in the existing work, after a change has been detected, we can continue to collect low-cost samples so that the post-change distribution can be identified more accurately. The goal of a two-stage SCD rule is to minimize the total cost including delay, false alarm probability, and misdiagnosis probability. To solve the two-stage SCD problem, we first convert the problem into a two-ordered optimal stopping time problem. Using tools from optimal multiple stopping time theory, we obtain the optimal SCD rule. Moreover, to address the high computational complexity issue of the optimal SCD rule, we further propose a computationally efficient threshold-based two-stage SCD rule. By analyzing the asymptotic behaviors of the delay, false alarm, and misdiagnosis costs, we show that the proposed threshold SCD rule is asymptotically optimal as the per-unit delay costs go to zero.

Primal–dual quasi-Monte Carlo simulation with dimension reduction for pricing American options

The pricing of American options is one of the most challenging problems in financial engineering due to the involved optimal stopping time problem, which can be solved by using dynamic programming (DP). But applying DP is not always practical, especially when the state space is high dimensional. However, the curse of dimensionality can be overcome by Monte Carlo (MC) simulation. We can get lower and upper bounds by MC to ensure that the true price falls into a valid confidence interval. During the recent decades, progress has been made in using MC simulation to obtain both the lower bound by least-squares Monte Carlo method (LSM) and the upper bound by duality approach. However, there are few works on pricing American options using quasi-Monte Carlo (QMC) methods, especially to compute the upper bound. For comparing the sample variances and standard errors in the numerical experiments, randomized QMC (RQMC) methods are usually used. In this paper, we propose to use RQMC to replace MC simulation to compute both the lower bound (by the LSM) and the upper bound (by the duality approach). Moreover, we propose to use dimension reduction techniques, such as the Brownian bridge, principal component analysis, linear transformation and the gradients based principle component analysis. We perform numerical experiments on American–Asian options and American max-call options under the Black–Scholes model and the variance gamma model, in which the options have the path-dependent feature or are written on multiple underlying assets. We find that RQMC in combination with dimension reduction techniques can significantly increase the efficiency in computing both the lower and upper bounds, resulting in better estimates and tighter confidence intervals of the true price than pure MC simulation.

A Class of Itô Diffusions with Known Terminal Value and Specified Optimal Barrier

In this paper, we study the optimal stopping-time problems related to a class of Itô diffusions, modeling for example an investment gain, for which the terminal value is a priori known. This could be the case of an insider trading or of the pinning at expiration of stock options. We give the explicit solution to these optimization problems and in particular we provide a class of processes whose optimal barrier has the same form as the one of the Brownian bridge. These processes may be a possible alternative to the Brownian bridge in practice as they could better model real applications. Moreover, we discuss the existence of a process with a prescribed curve as optimal barrier, for any given (decreasing) curve. This gives a modeling approach for the optimal liquidation time, i.e., the optimal time at which the investor should liquidate a position to maximize the gain.

Discrete-type Approximations for Non-Markovian Optimal Stopping Problems: Part II

In this paper, we present a Longstaff-Schwartz-type algorithm for optimal stopping time problems based on the Brownian motion filtration. The algorithm is based on Leao et al. (??2019) and, in contrast to previous works, our methodology applies to optimal stopping problems for fully non-Markovian and non-semimartingale state processes such as functionals of path-dependent stochastic differential equations and fractional Brownian motions. Based on statistical learning theory techniques, we provide overall error estimates in terms of concrete approximation architecture spaces with finite Vapnik-Chervonenkis dimension. Analytical properties of continuation values for path-dependent SDEs and concrete linear architecture approximating spaces are also discussed.

Discrete-type approximations for non-Markovian optimal stopping problems: Part I

AbstractWe present a discrete-type approximation scheme to solve continuous-time optimal stopping problems based on fully non-Markovian continuous processes adapted to the Brownian motion filtration. The approximations satisfy suitable variational inequalities which allow us to construct $\varepsilon$ -optimal stopping times and optimal values in full generality. Explicit rates of convergence are presented for optimal values based on reward functionals of path-dependent stochastic differential equations driven by fractional Brownian motion. In particular, the methodology allows us to design concrete Monte Carlo schemes for non-Markovian optimal stopping time problems as demonstrated in the companion paper by Bezerra et al.

The optimal stopping problem revisited

We consider an optimal stopping time problem, related with many models found in real options problems. We present analytical solutions for a broad class of gain functions, considering quite general assumptions over the model. Also, an extensive and general sensitivity analysis is provided.

Quadratic reflected BSDEs and related obstacle problems for PDEs

In this paper, we study a kind of reflected backward stochastic differential equations (BSDEs) whose generators are of quadratic growth in z and linear growth in y. We first give an estimate of solutions to such reflected BSDEs. Then under the condition that the generators are convex with respect to z, we can obtain a comparison theorem, which implies the uniqueness of solutions for this kind of reflected BSDEs. Besides, the assumption of convexity also leads to a stability property in the spirit of above estimate. We further establish the nonlinear Feynman-Kac formula of the related obstacle problems for partial differential equations (PDEs) in our framework. At last, a numerical example is given to illustrate the applications of our theoretical results, as well as its connection with an optimal stopping time problem.

Finite Element Method for Pricing Swing Options under Stochastic Volatility

This paper studies the pricing of a swing option under the stochastic volatility. A swing option is an American -style contract with multiple exercise rights. As such, it is an optimal multiple-stopping time problem. In this paper, we reduce the problem to a sequence of optimal single stopping time problems. We propose an algorithm based on the finite element method to value the contract in a Black-Scholes - Merton frame-work. In many real- world applications, the volatility is typically not a constant. Stochastic volatility models are commonly used for modeling dynamic changes of volatility. Here we introduce an approach to handle this added complication and present numerical results to demonstrate that the approach is accurate and efficient.

Optimal Bankruptcy with a Continuous Debt Repayment

We investigate the optimal consumption and investment problem when a working debtor has an option to file for bankruptcy. By applying the duality approach, the closed-form solutions are obtained for the case of CRRA utility function. The optimal bankruptcy time is determined by the first hitting time when the financial wealth hits the wealth threshold derived from the optimal stopping time problem. Moreover, the numerical results show that the investment increases as the wealth approaches the threshold and the value gain from the bankruptcy option is vanished as wealth increases.

Optimal Exercise Strategies for Operational Risk Insurance via Multiple Stopping Times

In this paper we demonstrate how to develop analytic closed form solutions to optimal multiple stopping time problems arising in the setting in which the value function acts on a compound process that is modified by the actions taken at the stopping times. This class of problem is particularly relevant in insurance and risk management settings and we demonstrate this on an important application domain based on insurance strategies in Operational Risk management for financial institutions. In this area of risk management the most prevalent class of loss process models is the Loss Distribution Approach (LDA) framework which involves modelling annual losses via a compound process. Given an LDA model framework, we consider Operational Risk insurance products that mitigate the risk for such loss processes and may reduce capital requirements. In particular, we consider insurance products that grant the policy holder the right to insure k of its annual Operational losses in a horizon of T years. We consider two insurance product structures and two general model settings, the first are families of relevant LDA loss models that we can obtain closed form optimal stopping rules for under each generic insurance mitigation structure and then secondly classes of LDA models for which we can develop closed form approximations of the optimal stopping rules. In particular, for losses following a compound Poisson process with jump size given by an Inverse-Gaussian distribution and two generic types of insurance mitigation, we are able to derive analytic expressions for the loss process modified by the insurance application, as well as closed form solutions for the optimal multiple stopping rules in discrete time (annually). When the combination of insurance mitigation and jump size distribution does not lead to tractable stopping rules we develop a principled class of closed form approximations to the optimal decision rule. These approximations are developed based on a class of orthogonal Askey polynomial series basis expansion representations of the annual loss compound process distribution and functions of this annual loss.

A General Optimal Multiple Stopping Problem with an Application to Swing Options

In their paper, Carmona and Touzi [8] studied an optimal multiple stopping time problem in a market where the price process is continuous. In this article, we generalize their results when the price process is allowed to jump. Also, we generalize the problem associated to the valuation of swing options to the context of jump diffusion processes. We relate our problem to a sequence of ordinary stopping time problems. We characterize the value function of each ordinary stopping time problem as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman variational inequality.

Funding liquidity, debt tenor structure, and creditor’s belief: an exogenous dynamic debt run model

We propose a unified structural credit risk model incorporating both insolvency and illiquidity risks, in order to investigate how a firm’s default probability depends on the liquidity risk associated with its financing structure. We assume the firm finances its risky assets by mainly issuing short- and long-term debt. Short-term debt can have either a discrete or a more realistic staggered tenor structure. At rollover dates of short-term debt, creditors face a dynamic coordination problem. We show that a unique threshold strategy (i.e., a debt run barrier) exists for short-term creditors to decide when to withdraw their funding, and this strategy is closely related to the solution of a non-standard optimal stopping time problem with control constraints. We decompose the total credit risk into an insolvency component and an illiquidity component based on such an endogenous debt run barrier together with an exogenous insolvency barrier.

Dynkin game of convertible bonds and their optimal strategy

This paper studies the valuation and optimal strategy of convertible bonds as a Dynkin game by using the reflected backward stochastic differential equation method and the variational inequality method. We first reduce such a Dynkin game to an optimal stopping time problem with state constraint, and then in a Markovian setting, we investigate the optimal strategy by analyzing the properties of the corresponding free boundary, including its position, asymptotics, monotonicity and regularity. We identify situations when call precedes conversion, and vice versa. Moreover, we show that the irregular payoff results in the possibly non-monotonic conversion boundary. Surprisingly, the price of the convertible bond is not necessarily monotonic in time: it may even increase when time approaches maturity.

Optimal exit strategies for investment projects

We study the problem of an optimal exit strategy for an investment project which is unprofitable and for which the liquidation costs evolve stochastically. The firm has the option to keep the project going while waiting for a buyer, or liquidating the assets at immediate liquidity and termination costs. The liquidity and termination costs are governed by a mean-reverting stochastic process whereas the rate of arrival of buyers is governed by a regime-shifting Markov process. We formulate this problem as a multidimensional optimal stopping time problem with random maturity. We characterize the objective function as the unique viscosity solution of the associated system of variational Hamilton–Jacobi–Bellman inequalities. We derive explicit solutions and numerical examples in the case of power and logarithmic utility functions when the liquidity premium factor follows a mean-reverting CIR process.