Lapse rate modeling: a rational expectation approach

The surrender option embedded in many life insurance products is a clause that allows policyholders to terminate the contract early. Pricing techniques based on the American Contingent Claim (ACC) theory are often used, though the actual policyholders' behavior is far from optimal. Inspired by many prepayment models for mortgage backed securities, this paper builds a Rational Expectation (RE) model describing the policyholders' behavior in lapsing the contract. A market model with stochastic interest rates is considered, and the pricing is carried out through numerical approximation of the corresponding two-space-dimensional parabolic partial differential equation. Extensive numerical experiments show the differences in terms of pricing and interest rate elasticity between the ACC and RE approaches as well as the sensitivity of the contract price with respect to changes in the policyholders' behavior.     

The probability of prepayment

Prepayment is a risk of holding a mortgage or derivative security. Incorrect pricing of prepayment risk leads to increased volatility and uncertainty in mortgage security markets. This article prices prepayment risk within an underlying callable bonds model. To price mortgages accurately, a probability of prepayment is required. A mortgage is a callable bond with a package of an option to prepay currently and a sequence of options to prepay up to the date of maturity. This sequence is summarized by a compound option. The probability of prepayment is determined by the prices of the current call and this compound option. These option prices depend on market interest rates and age, and on the contract terms of the originated mortgage.     

The valuation at origination of fixed-rate mortgages with default and prepayment

This paper develops a model to rationally price fixed-rate mortgages, using the arbitrage principles of option pricing theory. The paper incorporates amortization, prepayment and default in valuing the mortgage. Having completely specified the model, numerical procedures value the different features of the mortgage contract under a variety of economic conditions. The necessity of having both the interest rate and the house price as explanatory variables, due to the interaction of default and prepayment, is demonstrated. The numerical solutions presented center around mortgage pricing at origination. Thus, variations in the equilibrium contract rate are examined for differing economic conditions and changes in the contract. Finally, by presenting a complete model, the paper yields insights for the existence of common institutional practices.     

Mortgage prepayment with an uncertain holding period

A discrete-time-option pricing model is developed to value a mortgage and its embedded prepayment option when the effective life of the mortgage is a random variable with a probability distribution of known parameters. The model can be applied when the borrower's ex ante expectation of his tenure follows any probability distribution bounded to the left at zero. The Gamma distribution is used to illustrate the model.The pricing model is further applied to determine the conditions under which financially motivated prepayment is optimal. The results show that the certainty model understates the Interest Rate Differential needed to justify prepayment (IRD) for short Expected Holding Period (EHP) borrowers and overstates the IRD for long EHP borrowers. When the EHP is relatively long, the certainty model provides relatively good estimates of IRD during the beginning years of the mortgage life. Under most other conditions, the estimates of the certainty holding period model are biased.     

The refinancing threshold pricing model: An economic approach to valuing MBS

Investigation of MBS prepayment data indicates that mortgagors have different interest rate levels, or thresholds, at which they exercise their option to prepay their mortgage. In order to properly value an MBS with heterogeneous mortgagors, Merrill Lynch has developed the Refinancing Threshold Pricing Model (RTP). The RTP model focuses on the refinancing decision of the mortgagor when pricing the mortgage pool. The model divides each pool into groups of mortgagors who share similar refinancing costs. Using market data, the RTP model endogenously determines both the implied costs that mortgagors face, as well as the proportion of the MBS pool in each refinancing cost group. In addition to determining pool value, the RTP model also calculates MBS duration, dP/dY and convexity. Comparison between RTP model values and actual market data reveals a strong correlation. The RTP has a wide range of applications, including valuing 15-year and 30-year conventional MBS; pricing interest-only (IO)/principal-only (PO) derivative MBS; determining new versus seasoned MBS price spreads; and valuing specific MBS pools.The information set forth was obtained from sources we believe reliable, but we do not guarantee its accuracy. Neither the information, nor any opinion expressed constitutes a solicitation by us for the purchase or sale of any securities or commodities. Merrill Lynch, Pierce, Fenner & Smith, Inc. or its affiliates may have either a long or short position in, and may buy and sell for its own account or the accounts of others, these securities.     

Valuation of GNMA Mortgage-Backed Securities

GNMA mortgage-backed pass-through securities are supported by pools of amortizing, callable loans. Additionally, mortgagors often prepay their loans when the market interest rate is above the coupon rate of their loans. This paper develops a model for pricing GNMA securities and uses it to examine the impact of the amortization, call, and prepayment features on the prices, risks and expected returns of GNMA's. The amortization and prepayment features each have a positive effect on price, while the call feature has a negative impact. All three features reduce a GNMA security's interest rate risk and, consequently, its expected return.     

Time-Varying Mortgage Prepayment Penalties

Recent empirical findings reveal that prepayment decisions of commercial property owners are slower than predicted by the pure options-pricing model (OPM). Borrower decisions appearing slow, however, may be quite rational when prepayment penalties of a time-varying nature are incorporated into the OPM. This article uses a competing risks OPM, adjusted for each of four different categories of prepayment penalties, to analyze borrower prepayment behavior. We find the value of delaying prepayment is often higher for mortgages with declining-rate penalties than for mortgages with static-rate penalties, frequently requiring a substantially higher interest rate spread to trigger a refinance. Multifamily loan prepayment records reveal the type of prepayment pattern that the adjusted OPM indicate should occur, reducing the gap between empirical findings and theoretical predictions. The results have implications for the specification of regressions fit to historical data, for the pricing of newly originated commercial mortgages, and for pricing in the single-family market where prepayment penalties are reemerging.     

Semi-analytical MBS Pricing

This paper presents a multi-factor valuation model for fixed-rate callable mortgage backed securities (MBS). The model yields semi-analytic solutions for the value of MBS in the sense that the MBS value is found by solving a system of ordinary differential equations. Instead of modelling the conditional prepayment rate (CPR), as is customary, the pool size is the primary modelling object. It is shown that the value of a single MBS payment due at time t _n can be found by computing two expectations of the pool size at time t _n–1 and t _n respectively. This is a general result independent of any interest rate model. However, if the pool size is specified in a way that makes the expectations solvable using transform methods, semi-analytic pricing formulas are achieved. The affine and quadratic pricing frameworks are combined to get flexible and sophisticated prepayment functions. We show that the model has no problem of generating negative convexity as the spot rate falls, and still be close to a similar non-callable bond when the spot rate rises.     

Private information and incentives: Implications for mortgage contract terms and pricing

We examine the linkage in equilibrium among (1) contract design; (2) expected prepayment and default likelihoods; and (3) the pricing of mortgage contracts by focusing upon the effects of the borrower's private information at the time of contracting. We examine the implications of these perspectives upon the empirical analysis of prepayment behavior and use the framework to examine the predominance of long-term mortgage contracts in the United States. We consider examples that explore the trade-offs between fixed and adjustable rate instruments, assumable and due-on-sale loans, and contract interest rates and initial discounts (points).     

Mortgage Termination: An Empirical Hazard Model with a Stochastic Term Structure

Pricing for mortgage and mortgage-backed securities is complicated due to the stochastic and interdependent nature of prepayment and default risks. This paper presents a unified economic model of the contingent claims and competing risks of mortgage termination by prepayment and default. I adopt a proportional hazard framework to analyze these competing and interdependent risks in a model with time-varying covariates. The paper incorporates a stochastic interest rate model into the hazard function for prepayment. The empirical results reported in the paper provide new evidence about the ruthlessness of default and prepayment behavior and the sensitivity of these decisions to demographic as well as financial phenomena. The results also illustrate that evaluating the interest rate contingent claims with a stochastic term structure has effects on predicting not only the mortgage prepayment behavior but also the mortgage default behavior.     

An Early Assessment of Residential Mortgage Performance in China

The residential mortgage market becomes a financial engine for the booming residential housing development and sustained economic growth in China. Our study provides the first rigorous empirical analysis on the earlier performance of residential mortgage market in China based on a unique micro dataset of mortgage loan history collected from a major residential mortgage lender in China. We found that while the option theory fails to explain prepayment and default behavior in the residential mortgage market in China, other non-option theory related financial economic factors play major roles in determining the prepayment and default risks in China. We also found that borrower’s characteristics are significant in determining prepayment behavior, hence may be used as an effective tool for screening potential high risk borrowers in the loan origination process. Adopting a risk-based pricing in residential mortgage lending in China can improve the efficiency of the market, and enhance the credit availability to the most needed households, i.e., the younger households, blue-collar workers, lower income households, and help them become homeowners.     

Housing-Price Cycles and Prepayment Rates of U.S. Mortgage Pools

Empirical mortgage prepayment models generally have trouble explaining differences in mortgage-prepayment speeds among pools with similar interest rates on the underlying mortgages. In this article, we model some of the sources of termination heterogeneity across mortgage pools, particularly the role of regional variations in housing prices in generating atypical prepayment speeds. Using a sample of Freddie Mac mortgage pools from 1991 to 1998, we compare two classes of empirical models: a rational option-pricing model using a backward-solving pricing algorithm and an empirical hazard model. In both empirical estimation strategies, we find evidence that differences in house-price dynamics across regions are an important source of between-pool heterogeneity. This finding is then shown to be robust to alternative ways of parameterizing pool heterogeneity in mortgage termination models.     

Analyses of Mortgage-Backed Securities Based on Unobservable Prepayment Cost Processes

We propose a prepayment model of mortgage based on a structural approach in order to analyze prepayment risk of mortgage-backed securities (MBS). We introduce a continuous process named prepayment cost process. Specifically, each mortgager's prepayment time is defined by the first time when her or his prepayment cost process falls below zero, but prepayment cost processes are supposed to be unobservable in the market. We also introduce a risk unique to each loan pool of mortgages, called a loan pool risk (LPR), and we regard LPR as a systematic risk other than interest rate. Using the model, we discuss the conditional distribution of prepayment times and a risk-neutral valuation of pass-through MBS. It is shown that each mortgager's conditional non-prepayment probability and the posterior distribution of LPR play quite important roles in our study.This research is partially supported by Grant-in-Aid for Young Scientists (B) No. 16710108 from the Ministry of Education, Culture, Sports, Science and Technology.     

Market Risk of Mortgage-Backed Securities with Consistent Measures

Due to the complex prepayment behavior, mortgage contracts and their derivatives are generally priced using Monte Carlo simulations. The typical approach used by the industry, which involves simulating interest rates under the risk-neutral measure and applying a physically measured prepayment function, is subject to the problem of internal inconsistency. This is the first paper that directly investigates the potential impact of this issue. Following the general equilibrium setting by Cox, Ingersoll and Ross, we incorporate the market risk price parameter to derive the physical interest rate process from an observed yield curve. This allows us to model mortgage values under the consistent physical measures of interest rates and prepayment functions. By analyzing a default-free Ginnie Mae MBS, we find that the mixed measures lead to slower prepayment rate estimates and overpriced mortgage securities by approximately 5%. Further, there can be substantial biases in the duration and convexity measures depending on market condition and the particular security of interest. The internal inconsistency also leads to biased predictions of both expected and stressed returns for different investment horizons. Depending on the particular security, the bias in expected and stressed returns can be either positive or negative. These biases in risk estimates can introduce misallocation of risk-based capital and/or failure in hedging the market risk of a mortgage-related portfolio.

The weekend effect on the distribution of stock prices: Implications for option pricing

Evidence of weekend effects on the distribution of security returns suggests that returns are generated by a process operating closer to trading time rather than calendar time. In contrast, accumulation of interest over the weekend follows a calendar-time process. Since both the variance of returns and the interest rate are important parameters of the Black-Scholes option pricing model, this paper suggests that the model be stated to account for this by utilizing a trading-time variance and a calendar-time interest rate. Empirical evidence indicates that this allows the model to better explain market option prices.     

SECONDARY MORTGAGE MARKET PURCHASE COMMITMENT YIELDS

We provide empirical evidence that quoted secondary market mortgage yields conform to the predictions of option theory. We compare Fannie Mae and Freddie Mac origination yields offered in the secondary market from 1985 to 2003 with the predictions of a two‐state binomial mortgage option valuation model. Our two‐state approach considers a mean‐reverting interest rate process as well as a stochastic housing price. Using predictions from option simulations, we find strong links between market practice and mortgage option prepayment and default factors over time. We also find cross‐sectional differences that are consistent with the institutional structure of the markets.     

Option Pricing under Stochastic Interest Rates: An Empirical Investigation

Using daily data of the Nikkei 225 index, call option prices and call money rates of the Japanese financial market,a comparison is made of the pricing performance of stock option pricing modelsunder several stochastic interest rate processes proposedby the existing term structure literature.The results show that (1) one option pricing modelunder a specific stochastic interest ratedoes not significantly outperformanother option pricing model under an alternative stochasticinterest rate, and (2) incorporating stochastic interest ratesinto stock option pricing does not contribute to the performanceimprovement of the original Black–Scholes pricing formula.     

Duration and the price behavior of fixed-rate level payment mortgages: An analytical investigation

In this research we use a continuous payment formula for duration to examine the price behavior of a fixed-rate level payment mortgage. In the case where the mortgage is held to maturity, duration increases monotonically as term-to-maturity increases, regardless of changes in the market rate of interest. In the case where the mortgage is prepaid prior to maturity, there exists a unique market interest rate below which duration is a monotonically increasing function of time of prepayment, but above which duration has a global maximum at some time of prepayment prior to the term-to-maturity.     

Analysis about the Black-Scholes asset price under the regime-switching framework

Assuming that the macroeconomic environment can be transformed into a two-district system, that is, the path of financial asset prices is uncertain, we track and study the motion of stocks and other asset price process under the conditional Black-Scholes model, and give the economical explanation of the mathematical formula. Further, we derive and analyze an option pricing formula for the Black-Scholes asset model under the condition that the risk-free interest rate is regime-switching too. The method in this article is applied to model the log rate of return of the Tencent stock in a two-district market environment. And the obtained parameter values are used to calculate the option price. In narrowing the gap with actual option prices, our method outperforms the classical option pricing model point by point. Compared with the general and pure mathematical model derived work and the empirical study work, our study does more work on the economic characteristics analysis and interpretation of the mathematical models, and plays a certain role in linking the results of mathematical models with empirical research.