Intensity-based framework and penalty formulation of optimal stopping problems

Financial derivatives commonly contain premature termination clauses, which are embedded rights held by the holder or writer. Well known examples of these stopping rights include the early exercise right in American options, the callable right in callable securities and the prepayment right in mortgage loans. In this paper, we show how to model the mortgagor's prepayment in mortgage loans and the issuer's call in the American warrant as an event risk using the intensity based approach, where the propensity of prepayment or calling is modeled by the intensity of a Poisson process. We illustrate that the corresponding pricing formulation resembles the penalty approximation approach commonly used in the solution of the linear complementarity formulation of an optimal stopping problem. We obtain several theoretical results on the prepayment strategies of mortgage loans and calling policies of American warrants. We also propose robust second order accurate numerical schemes for solving the penalty formulation of an optimal stopping problem.     

Optimal prepayment of Dutch mortgages*

Valuation of the prepayment option in Dutch mortgages is complicated. In the Netherlands, mortgagors are not allowed to prepay the full mortgage loan without a compensating penalty. Only a limited amount of the initial mortgage loan can be prepaid penalty‐free. We introduce a general model formulation for the valuation of limited callable mortgages, based on binomial trees. This model can be used for determining both the optimal prepayment strategy and the value of embedded prepayment options. For some mortgage types the prepayment option can be valued exactly, whereas other types require approximative methods for efficient valuation. The heuristic we propose here determines the prepayment option value efficiently and accurately for general mortgage types.     

Analyzing yield, duration and convexity of mortgage loans under prepayment and default risks

In this article, we construct a general model, which considers the borrower’s financial and non-financial termination behavior, to derive the closed-form formula of the mortgage value for analyzing the yield, duration and convexity of the risky mortgage. Since the risks of prepayment and default are reasonably expounded in our model, our formulae are more appropriate than traditional mortgage formulae. We also analyze the effects of the prepayment penalty and partial prepayment on the yield, duration and convexity of a mortgage, and provide lenders with an upper-bound for the mortgage default insurance rate. Our model provides portfolio managers a useful framework to more appropriately appraise the mortgage and more effectively hedge their mortgage holdings. From the results of sensitivity analyses, we find that higher interest-rate, prepayment and default risks will increase the mortgage yield and reduce the duration and convexity of the mortgage.     

Mortgage Valuation with Optimal Intertemporal Refinancing Strategies

This article extends previous work on mortgage valuation in two ways. First, I identify the prepayment boundary by solving for the borrower's optimal prepayment strategy over the expected tenure in the house. Previous work has treated the prepayment decision as a one-time decision, not as an element of a multiperiod strategy. Second, the model incorporates borrower heterogeneity in terms of expected tenure in the house. The results show that the optimal refinancing strategy differs significantly from a sequence of one-time decisions. A borrower following the optimal strategy is less aggressive in refinancing and pays more interest and less transaction costs than does a borrower following a myopic strategy. Estimated mortgage values are higher and interest rate sensitivity is lower when compared to values calculated using the traditional approach.     

A two-stage segment and prediction model for mortgage prepayment prediction and management

Predicting the risk of mortgage prepayments has been the focus of many studies over the past three decades. Most of these works have used single prediction models, such as logistic regressions and survival models, to seek the key influencing factors. From the point of view of customer relationship management (CRM), a two-stage model (i.e., the segment and prediction model) is proposed for analyzing the risk of mortgage prepayment in this research. In the first stage, random forests are used to segment mortgagors into different groups; then, a proportional hazard model is constructed to predict the prepayment time of the mortgagors in the second stage. The results indicate that the two-stage model predicts mortgage prepayment more accurately than the single-stage model (non-segmentation model).     

Valuation of fixed and variable rate mortgages: binomial tree versus analytical approximations

The recombining binomial tree approach, which has been initiated by Cox et?al. (J Financ Econ 7: 229?C263, 1979) and extended to arbitrary diffusion models by Nelson and Ramaswamy (Rev Financ Stud 3(3): 393?C430, 1990) and Hull and White (J Financ Quant Anal 25: 87?C100, 1990a), is applied to the simultaneous evaluation of price and Greeks for the amortized fixed and variable rate mortgage prepayment option. We consider the simplified binomial tree approximation to arbitrary diffusion processes by Costabile and Massabo (J Deriv 17(3): 65?C85, 2010) and analyze its numerical applicability to the mortgage valuation problem for some Vasicek and CIR-like interest rate models. For fixed rates and binomial trees with about thousand steps, we obtain very good results. For the Vasicek model, we also compare the closed-form analytical approximation of the callable fixed rate mortgage price by Xie (IAENG Int J Appl Math 39(1): 9, 2009) with its binomial tree counterpart. With respect to the binomial tree values one observes a systematic underestimation (overestimation) of the callable mortgage price (prepayment option price) analytical approximation. This numerical discrepancy increases at longer maturities and becomes impractical for a valuable estimation of the prepayment option price.     

个人住房按揭贷款提前偿付风险研究

住房按揭贷款在国内银行的资产中正占据着越来越重要的地位,与此同时,按揭贷款的提前偿付行为也困扰着银行业的经营。本文分析了影响住房按揭贷款提前还款的主要因素,尝试建立一个符合我国目前状况的能够预测提前还款模型框架,同时提出我国商业银行个人住房按揭贷款提前还款风险管理的一些策略建议。     

Mortgagor Motivations in Prepayments for Adjustable Rate Mortgages

This paper provides the first rigorous analysis of residential adjustable mortgage prepayment using individual ARM mortgage data in Singapore. The prepayment rate for residential mortgages is low and is dominated more by macroeconomic factors than mortgage–specific factors. Specifically, the prepayment rate is increasing in residential property prices, but decreasing in income as proxied by GDP and volatility in mortgage rates. There is weak evidence to suggest that prepayment is increasing in the borrower's age, mortgage rate hikes, cash–availability variables and sentiments of the stock market, and decreasing in the price premium over valuation, payment–to–income ratio, loan–to–value ratio, loan term and floor level of the property.     

Structural prepayment risk behavior of the underlying mortgages for residential mortgage life insurance in a developing market

Since 1998 all residential mortgages in China have been adjustable rate mortgages (ARMs). However, the borrower’s motivation for prepayment is different from that in the US or other developed mortgage markets. In the US, mortgage insurance plays an imperative role in covering some of the risk typically faced by housing finance institutions. However, China’s residential mortgage life insurance (RMLI) market is in its infancy. It offers the insured mortgagor a life-insurance death benefit, arising from only illness or accident, settling the insured’s outstanding residential mortgage balance. Prepayments of some RMLIs’ underlying mortgages are observed, leading to a premature termination of both the residential mortgage and the insurance commitment to settle the outstanding mortgage balance even though the insured has not yet passed away. Because such prepayments significantly influence the pricing of the RMLI, it is imperative to know more about the prepayment rate of occurrence and the prepayment characteristics of the underlying residential mortgages in terms of observable macro economic factors, loan specific factors and borrower specific characteristics. Hence, this study investigates the prepayment risk behavior of the underlying mortgages for RMLIs, utilizing a pilot study of 1000 Shanghai residential mortgagors who took up RMLIs between January 1999 and December 2003. This study uses the Cox proportional hazard model to investigate RMLI-mortgage prepayment risk behavior. The resultant hazard rate is dependent on four primary factors: combined monthly income of the co-borrowers, growth in the gross domestic product, number of co-borrowers and initial loan-to-value ratio.     

Incomplete markets, ambiguity, and irreversible investment

The problem of irreversible investment with idiosyncratic risk is studied by interpreting market incompleteness as a source of ambiguity over the appropriate no-arbitrage discount factor. The maxmin utility over multiple priors framework is used to model and solve the irreversible investment problem. Multiple priors are modeled using the notion of κ‐ignorance. This set-up is used to analyze finitely lived options. For infinitely lived options the notion of constant κ‐ignorance is introduced. For these sets of density generators the corresponding optimal stopping problem is solved for general (in-)finite horizon optimal stopping problems driven by geometric Brownian motion. It is argued that an increase in the set of priors delays investment, whereas an increase in the degree of market completeness can have a non-monotonic effect on investment.     

Some Truths about Ostriches: Who Doesn't Prepay Their Mortgages and Why They Don't

Prepayment estimation is essential in forecasting expected mortgage cash flow patterns. Accordingly, mortgage and mortgage-backed security prices are highly dependent on prepayment assumptions. Yet borrower prepayment behavior appears to be highly irrational, in the sense that many borrowers prepay their mortgages when it is not optimal to do so and fail to prepay their mortgages when the prepayment option is substantially in the money. In this paper, we explore the latter phenomenon, using a large data set of loans originated during the relatively high-rate 1980s that have failed to prepay by year-end 1996. As a control group, we examine similar loans that did prepay during the refinancing boom of 1993. By coupling Case–Shiller house price index information at the zip-code level, we can analyze the effect of broader housing market trends, especially housing appreciation, on borrower prepayment behavior. Although housing prices did decline significantly during the late 1980s and early 1990s in many areas of the United States, we find evidence that only about 25% of non-refinancing households might have been constrained by declining collateral values. Household demographic characteristics may provide some further explanation for this apparently non-optimizing behavior; however, we do not have a complete explanation.     

Burn-in for a time-transformed exponential model

The problem of the optimal duration of a burn-in experiment is considered in the case of simultaneous testing n components with the conditionally independent time-transformed exponential life-times, given an unknown parameter. The explicit solution is derived by reformulation of the problem considered to an optimal stopping problem for a suitable defined three-dimensional Markov process and reduction to a free-boundary problem.     

Regional economic models and continuous programming

This paper is concerned with an optimal investment allocation problem in a simple N-regional economic model. The problem is described as a class of optimal control problem, and formulated into a continuous linear programming problem. Both the primal and dual problems are considered. The procedure finds an optimal regional allocation of investment derived in terms of continuous programming.     

Steiner minimal trees and urban service networks

The optimal configuration of urban service networks has recently been shown to be a computationally difficult problem. However, there are efficient and effective techniques by which this optimal configuration of urban service networks can be approximated. In this paper, we analyze the L_p Steiner Network problem in the plane R² and demonstrate its applicability to the urban service network problem. We present a simple algorithm for estimating the L_p metric parameter for random points in the plane, then utilize it to find the L_p values for four different American cities. Finally, we apply the L_pSMT algorithm described within the text to one of the cities in order to demonstrate the effectiveness of our algorithm for determining optimal network configurations.     

Using a decision tree to analyze mortgage borrower decision behavior and values concerning interest rates and house prices

In reality, mortgage borrowers are more seriously concerned with the current mortgage boundary (i.e. option exercise) value than with the current option value (i.e. expected present value of the prospective option exercise value). Hence, by combining a simulation framework and a decision tree, the terminations of mortgage behavior can be classified forward but not backward as by the binomial lattice. After simulating 5000 projections for both Taiwan house prices and interest rates, as well as computing for current mortgage boundary values obtained by modifying Ambrose and Buttimer ( 2000 ) to step through the mortgage decision tree, the result shows that the prepayment is affected by rising interest rate volatility. Moreover, the delinquency and the reinstatement are affected by both rising interest rate and house price volatilities. However, due to the cost of delinquency and credit penalties, the foreclosure could not compete over the reinstatement when house prices and interest rates are in a high‐volatility situation. The reinstatement is encouraging for the borrowers. Copyright © 2010 John Wiley & Sons, Ltd.     

Valuation of stock loans with jump risk

A stock loan is a special loan with stocks as collateral, which offers the borrowers the right to redeem the stocks on or before the maturity (Xia and Zhou, 2007, Dai and Xu, 2011). We investigate pricing problems of both infinite- and finite-maturity stock loans under a hyper-exponential jump diffusion model. In the infinite-maturity case, we derive closed-form formulas for stock loan prices and deltas by solving the related optimal stopping problem explicitly. Moreover, we obtain a sufficient and necessary condition under which the optimal stopping time is finite with probability one. In the finite-maturity case, we provide analytical approximations to both stock loan prices and deltas by solving an ordinary integro-differential equation as well as a complicated non-linear system. Numerical experiments demonstrate that the approximation methods for both prices and deltas are accurate, fast, and easy to implement.     

Optimal pre-dispatch task assignment of volunteers in daily emergency response

In emergency response volunteer programs, volunteers in the vicinity of an emergency are alerted via their mobile phones to the scene of the event to perform a specific task. Tasks are usually assigned based on predetermined rules disregarding real-world uncertainties. In this paper, we consider some of these uncertainties and propose an optimization model for the dispatch of volunteers to emergencies, where all task assignments must be done before dispatch. This means that each volunteer must be given a task before knowing whether (s)he is available. The model becomes computationally demanding for large problem instances; therefore, we develop a simple greedy heuristic for the problem and ensure that it can produce high quality solutions by comparing it to the exact model. While the model is for a general emergency, we test it for the case of volunteers responding to out-of-hospital cardiac arrest (OHCA) incidents. We compare the results of the model to the dispatch strategies used in two ongoing volunteer programs in Sweden and in the Netherlands and use simulation to validate the results. The results show that the model most often outperforms the currently used strategies; however, the computational run times, even for the heuristic, are too high to be operationally useful for large problem instances. Thus, it should be possible to improve the outcome using optimization-based task assignments strategies, but a fast solution method is needed for such strategies to be practically useable.     

Applications of cluster analysis algorithm to geostatistical series

The common problems of cluster analysis are briefly reviewed and discussed. A cluster analysis algorithm for partitioning geostatistical series is extensively described and applied to examples of various visual and statistical types. The utilization of the algorithm in measuring the divergence between geostatistical distributions is shown. The usefulness of the algorithm is demonstrated in solving the problem of optimal location of m facilities. A method for reconstructing territories by simple geometric shapes via cluster analysis is also given.     

Comment on ‘the dynamics of job separation: The case of federal employees’

This paper recasts the model presented in ‘The Dynamics of Job Separation: The Case of Federal Employees’, as an optimal stopping problem. It shows that the individuals represented by the model are assumed to behave suboptimally and illustrates several policy analysis problems introduced by assuming such behaviour.     

A continuous-time sequential testing problem

A Bayesian testing problem for a simple hypothesis against a simple alternative is considered. However the observations on which the test is based are not immediately available, but occur atn i.i.d. random times. Assuming linear cost of time until a decision the task is to construct and to characterize an optimal stopping strategy for a time-sequential testing problem and to compute the Bayes risk. The paper provides a solution to this problem.