scholarly journals Numerical methods for an optimal multiple stopping problem

2016 ◽  
Vol 16 (05) ◽  
pp. 1650016 ◽  
Author(s):  
Imène Ben Latifa ◽  
Joseph Fréderic Bonnans ◽  
Mohamed Mnif
Keyword(s):  
Numerical Scheme ◽  
Value Function ◽  
Numerical Solutions ◽  
Conditional Expectations ◽  
Dynamic Programing ◽  
Optimal Quantization ◽  
Swing Option ◽  
Multiple Stopping ◽  
Viscosity Sense ◽  
The Value Function

This paper deals with numerical solutions to an optimal multiple stopping problem. The corresponding dynamic programing (DP) equation is a variational inequality satisfied by the value function in the viscosity sense. The convergence of the numerical scheme is shown by viscosity arguments. An optimal quantization method is used for computing the conditional expectations arising in the DP equation. Numerical results are presented for the price of swing option and the behavior of the value function.

THE SWING OPTION ON THE STOCK MARKET

2005 ◽  
Vol 08 (01) ◽  
pp. 123-139 ◽  
Author(s):  
MARTIN DAHLGREN ◽  
RALF KORN
Keyword(s):  
Optimal Stopping ◽  
Value Function ◽  
Additional Constraint ◽  
Numerical Implementation ◽  
Optimal Stopping Problem ◽  
Existence Of A Solution ◽  
Swing Option ◽  
Time Distance ◽  
The Value Function

The valuation of a Swing option for stocks under the additional constraint of a minimum time distance between two different exercise times is considered. We give an explicit characterization of its pricing function as the value function of a multiple optimal stopping problem. The solution of this problem is related to a system of variational inequalities. We prove existence of a solution to this system and discuss the numerical implementation of a valuation algorithm.

Quicker Convergence for Iterative Numerical Solutions to Stochastic Problems: Probabilistic Interpretations, Ordering Heuristics, and Parallel Processing

1990 ◽  
Vol 4 (4) ◽  
pp. 493-521 ◽  
Author(s):  
Albert G. Greenberg ◽  
Robert J. Vanderbei
Keyword(s):  
Markov Chain ◽  
Parallel Processing ◽  
Invariant Measure ◽  
Value Function ◽  
Numerical Solutions ◽  
Jacobi Method ◽  
Optimal Stopping Problem ◽  
Current Estimate ◽  
The Value Function ◽  
General Method

Gauss-Seidel is a general method for solving a system of equations (possibly nonlinear). It makes repeated sweeps through the variables; within a sweep as each new estimate for a variable is computed, the current estimate for that variable is replaced with the new estimate immediately, instead of on completion of the sweep. The idea is to use new data as soon as it is computed. Gauss- Seidel is often efficient for computing the invariant measure of a Markov chain (especially if the transition matrix is sparse), and for computing the value function in optimal control problems. In many applications the computation can be significantly improved by appropriately ordering the variables within each sweep. A simple heuristic is presented here for computing an ordering that quickens convergence. In parallel processing, several variables must be computed simultaneously, which appears to work against Gauss-Seidel. Simple asynchronous parallel Gauss-Seidel methods are presented here. Experiments indicate that the methods retain the benefit of a good ordering, while further speeding up convergence by a factor of P if P processors participate.In this paper, we focus on the optimal stopping problem. A probabilistic interpretation of the Gauss-Seidel (and the Jacobi) method for computing the value function is given, which motivates our ordering heuristic. However, the ordering heuristic and parallel processing methods apply in a broader context, in particular, to the important problem of computing the invariant measure of a Markov chain.

scholarly journals A variational inequality arising from optimal surrender of variable annuity with lookback benefit

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Junkee Jeon ◽  
Minsuk Kwak
Keyword(s):  
Variational Inequality ◽  
Value Function ◽  
Numerical Solutions ◽  
Integration Method ◽  
Comparative Statics ◽  
One Dimensional ◽  
Math Econ ◽  
The One ◽  
Recursive Integration ◽  
The Value Function

AbstractWe introduce a variable annuity (VA) contract with a surrender option and lookback benefit, that is, the benefit of the VA contract is linked to the maximum process of the policyholder’s account value. In contrast to the constant guarantee model provided in Bernard et al. (Insur. Math. Econ. 55:116–128, 2014), it is optimal for the policyholder of the VA contract with lookback benefit to surrender the VA contract when the policyholder’s account value is below or equal to the optimal surrender boundary. Thus, from the perspective of the insurer to construct a portfolio of VA contracts, utilizing the VA contracts with lookback benefit along with VA contracts with constant guarantee provides the diversification of early surrenders. The valuation of this contract can be described as a two-dimensional parabolic variational inequality. By converting this into the one-dimensional problem, we obtain the integral equations for the value function and the free boundary. The recursive integration method is applied to obtain the numerical solutions. We also provide comparative statics of the optimal surrender boundaries with respect to various parameters.

The value function for time-related decisions

2011 ◽  
Author(s):  
Anouk Festjens ◽  
Siegfried Dewitte ◽  
Enrico Diecidue ◽  
Sabrina Bruyneel
Keyword(s):  
Value Function ◽  
The Value Function

scholarly journals Pricing Perpetual American Put Options with Asset-Dependent Discounting

2021 ◽  
Vol 14 (3) ◽  
pp. 130
Author(s):  
Jonas Al-Hadad ◽  
Zbigniew Palmowski
Keyword(s):  
Value Function ◽  
Asset Price ◽  
Stopping Times ◽  
Martingale Measure ◽  
Put Options ◽  
American Put Options ◽  
Exact Calculations ◽  
Negative Exponential ◽  
American Put ◽  
The Value Function

The main objective of this paper is to present an algorithm of pricing perpetual American put options with asset-dependent discounting. The value function of such an instrument can be described as VAPutω(s)=supτ∈TEs[e−∫0τω(Sw)dw(K−Sτ)+], where T is a family of stopping times, ω is a discount function and E is an expectation taken with respect to a martingale measure. Moreover, we assume that the asset price process St is a geometric Lévy process with negative exponential jumps, i.e., St=seζt+σBt−∑i=1NtYi. The asset-dependent discounting is reflected in the ω function, so this approach is a generalisation of the classic case when ω is constant. It turns out that under certain conditions on the ω function, the value function VAPutω(s) is convex and can be represented in a closed form. We provide an option pricing algorithm in this scenario and we present exact calculations for the particular choices of ω such that VAPutω(s) takes a simplified form.

scholarly journals Robo-Advising: Learning Investors’ Risk Preferences via Portfolio Choices*

2020 ◽  
Author(s):  
Humoud Alsabah ◽  
Agostino Capponi ◽  
Octavio Ruiz Lacedelli ◽  
Matt Stern
Keyword(s):  
Opportunity Cost ◽  
Value Function ◽  
Risk Preference ◽  
Portfolio Decisions ◽  
Learning Framework ◽  
Portfolio Choices ◽  
Trading Decisions ◽  
Exploration Exploitation ◽  
The Value Function ◽  
Over Time

Abstract We introduce a reinforcement learning framework for retail robo-advising. The robo-advisor does not know the investor’s risk preference but learns it over time by observing her portfolio choices in different market environments. We develop an exploration–exploitation algorithm that trades off costly solicitations of portfolio choices by the investor with autonomous trading decisions based on stale estimates of investor’s risk aversion. We show that the approximate value function constructed by the algorithm converges to the value function of an omniscient robo-advisor over a number of periods that is polynomial in the state and action space. By correcting for the investor’s mistakes, the robo-advisor may outperform a stand-alone investor, regardless of the investor’s opportunity cost for making portfolio decisions.

Metric for Disassembly and Reuse Decisions: Formulation and Validation

2008 ◽  
Author(s):  
Vijitashwa Pandey ◽  
Deborah Thurston
Keyword(s):  
Maximum Likelihood ◽  
Value Function ◽  
Choice Theory ◽  
Decision Makers ◽  
Maximum Likelihood Estimates ◽  
Likelihood Method ◽  
Maximum Value ◽  
Long Range Planning ◽  
The Relationship ◽  
The Value Function

Design for disassembly and reuse focuses on developing methods to minimize difficulty in disassembly for maintenance or reuse. These methods can gain substantially if the relationship between component attributes (material mix, ease of disassembly etc.) and their likelihood of reuse or disposal is understood. For products already in the marketplace, a feedback approach that evaluates willingness of manufacturers or customers (decision makers) to reuse a component can reveal how attributes of a component affect reuse decisions. This paper introduces some metrics and combines them with ones proposed in literature into a measure that captures the overall value of a decision made by the decision makers. The premise is that the decision makers would choose a decision that has the maximum value. Four decisions are considered regarding a component’s fate after recovery ranging from direct reuse to disposal. A method on the lines of discrete choice theory is utilized that uses maximum likelihood estimates to determine the parameters that define the value function. The maximum likelihood method can take inputs from actual decisions made by the decision makers to assess the value function. This function can be used to determine the likelihood that the component takes a certain path (one of the four decisions), taking as input its attributes, which can facilitate long range planning and also help determine ways reuse decisions can be influenced.

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