The value of mortgage prepayment and default options

We use an implicit alternating direction numerical procedure to estimate the value of a fixed‐rate mortgage (FRM) with embedded default and prepayment options. The value of FRMs depends on interest rates, the house value, and mortgage maturity. Our numerical results suggest that the joint option value of prepayment and default is considerably high, even at loan origination. We extend the model to include prepayment penalties in FRM valuation. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:840–861, 2009     

Does Prepayment Risk of Mortgages Affect Excess Returns of Bank Stocks?

This paper explores the relationship between the prepayment risk embedded in conventional, fixed-rate residential mortgages and excess returns for bank stocks. There are two interesting findings in this study. First, commercial banks traded in the Nasdaq market are more meanvariance efficient than the other seven groups of industrial stocks. Second, the prepayment risk factor is significant for these banks. The prepayment risk mainly reflects a call option embedded in a mortgage plus foreclosure costs associated with a mortgage put option. This risk is measured by a remaining part of mortgage rates after excluding the influence of real estate market, maturity, and default risks on mortgage rates. The results of this study suggest that the prepayment risk factor does significantly affect excess returns for bank stocks in the period with high levels of mortgage refinancing activities. JEL Classification G210     

INTENSITY-BASED VALUATION OF RESIDENTIAL MORTGAGES: AN ANALYTICALLY TRACTABLE MODEL

This paper presents an analytically tractable valuation model for residential mortgages. The random mortgage prepayment time is assumed to have an intensity process of the form h _t = h ₀( t ) +γ ( k − r _t )⁺ , where h ₀( t ) is a deterministic function of time, r _t is the short rate, and γ and k are scalar parameters. The first term models exogenous prepayment independent of interest rates (e.g., a multiple of the PSA prepayment function). The second term models refinancing due to declining interest rates and is proportional to the positive part of the distance between a constant threshold level and the current short rate. When the short rate follows a CIR diffusion, we are able to solve the model analytically and find explicit expressions for the present value of the mortgage contract, its principal-only and interest-only parts, as well as their deltas. Mortgage rates at origination are found by solving a non-linear equation. Our solution method is based on explicitly constructing an eigenfunction expansion of the pricing semigroup, a Feynman-Kac semigroup of the CIR diffusion killed at an additive functional that is a linear combination of the integral of the CIR process and an area below a constant threshold and above the process sample path (the so-called area functional). A sensitivity analysis of the term structure of mortgage rates and calibration of the model to market data are presented.     

Risk-neutral valuation of the non-recourse protection in reverse mortgages: A case study for Korea

Reverse mortgages are increasingly seen as an alternative source of retirement income among Koreans. All reverse mortgage loans in Korea are sold with a non-recourse protection, limiting the borrowers' exposure to house price appreciation risk. This paper performs risk-neutral valuation for the non-recourse protection in the Korean reverse mortgage market. Specifically, we adopt a multivariate DCC-GARCH model that incorporates different forms of correlations between the economic variables. Risk-neutralization is accomplished using the minimum relative entropy method. Our valuation results reveal several limitations of the fee structure currently used by reverse mortgage providers. Recommendations to improve the fee structure are provided.     

CONVEX RISK MEASURES FOR GOOD DEAL BOUNDS

We study convex risk measures describing the upper and lower bounds of a good deal bound, which is a subinterval of a no‐arbitrage pricing bound. We call such a convex risk measure a good deal valuation and give a set of equivalent conditions for its existence in terms of market. A good deal valuation is characterized by several equivalent properties and in particular, we see that a convex risk measure is a good deal valuation only if it is given as a risk indifference price. An application to shortfall risk measure is given. In addition, we show that the no‐free‐lunch (NFL) condition is equivalent to the existence of a relevant convex risk measure, which is a good deal valuation. The relevance turns out to be a condition for a good deal valuation to be reasonable. Further, we investigate conditions under which any good deal valuation is relevant.     

INCORPORATING RISK AND AMBIGUITY AVERSION INTO A HYBRID MODEL OF DEFAULT

It is well known that purely structural models of default cannot explain short‐term credit spreads, while purely intensity‐based models lead to completely unpredictable default events. Here we introduce a hybrid model of default, in which a firm enters a “distressed” state once its nontradable credit worthiness index hits a critical level. The distressed firm then defaults upon the next arrival of a Poisson process. To value defaultable bonds and credit default swaps (CDSs), we introduce the concept of robust indifference pricing. This paradigm incorporates both risk aversion and model uncertainty. In robust indifference pricing, the optimization problem is modified to include optimizing over a set of candidate measures, in addition to optimizing over trading strategies, subject to a measure dependent penalty. Using our model and valuation framework, we derive analytical solutions for bond yields and CDS spreads, and find that while ambiguity aversion plays a similar role to risk aversion, it also has distinct effects. In particular, ambiguity aversion allows for significant short‐term spreads.     

Convergence of utility indifference prices to the superreplication price in a multiple‐priors framework

This paper formulates a utility indifference pricing model for investors trading in a discrete time financial market under nondominated model uncertainty. Investor preferences are described by possibly random utility functions defined on the positive axis. We prove that when the investors's absolute risk aversion tends to infinity, the multiple‐priors utility indifference prices of a contingent claim converge to its multiple‐priors superreplication price. We also revisit the notion of certainty equivalent for multiple‐priors and establish its relation with risk aversion.     

Relaxed Credit Standards in the U.S. Housing Boom: Changes in Risk Characteristics of Mortgage Recipients*

This paper examines components of credit risk and how their ability to predict the interface between households and mortgage market changed under the relaxed lending standards prevailing during the U.S. housing boom of the 2000s. Using data from the Federal Reserve Board's 1998 and 2007 Survey of Consumer Finances, the paper evaluates changes between 1998 and 2007 in the significance of credit risk characteristics in explaining three variables regarding the purchase of a primary residence by households in low‐, moderate‐, and high‐income groups: the loan‐to‐value ratio of the purchase mortgage, the likelihood of purchase, and the price paid. The study also analyzes the extent to which the period saw increases in the values of those three variables. The findings strongly suggest a decline in the ability of credit risk characteristics in predicting the interface between households and mortgage market over the period.     

INDIFFERENCE PRICES AND IMPLIED VOLATILITIES

We consider a general local‐stochastic volatility model and an investor with exponential utility. For a European‐style contingent claim, whose payoff may depend on either a traded or nontraded asset, we derive an explicit approximation for both the buyer's and seller's indifference prices. For European calls on a traded asset, we translate indifference prices into an explicit approximation of the buyer's and seller's implied volatility surfaces. For European claims on a nontraded asset, we establish rigorous error bounds for the indifference price approximation. Finally, we implement our indifference price and implied volatility approximations in two examples.     

住房贷款证券化中的提前偿付预测

提前偿付风险极大地影响着住房贷款证券化的定价和运作 ,因此国外对该方面的研究投入了大量的精力。当前我国银行也被抵押贷款逐年增加的提前偿付现象而困扰 ,这直接使银行面临再投资风险 ,并影响我国即将开展的证券化运作。借鉴国外提前偿付风险的特点和有关的预测控制模型 ,希望对我国该方面的研究带来积极的意义     

FROM SMILE ASYMPTOTICS TO MARKET RISK MEASURES

The left tail of the implied volatility skew, coming from quotes on out‐of‐the‐money put options, can be thought to reflect the market's assessment of the risk of a huge drop in stock prices. We analyze how this market information can be integrated into the theoretical framework of convex monetary measures of risk. In particular, we make use of indifference pricing by dynamic convex risk measures, which are given as solutions of backward stochastic differential equations, to establish a link between these two approaches to risk measurement. We derive a characterization of the implied volatility in terms of the solution of a nonlinear partial differential equation and provide a small time‐to‐maturity expansion and numerical solutions. This procedure allows to choose convex risk measures in a conveniently parameterized class, distorted entropic dynamic risk measures, which we introduce here, such that the asymptotic volatility skew under indifference pricing can be matched with the market skew. We demonstrate this in a calibration exercise to market implied volatility data.     

Arbitrage‐free XVA

We develop a framework for computing the total valuation adjustment (XVA) of a European claim accounting for funding costs, counterparty credit risk, and collateralization. Based on no‐arbitrage arguments, we derive backward stochastic differential equations associated with the replicating portfolios of long and short positions in the claim. This leads to the definition of buyer's and seller's XVA, which in turn identify a no‐arbitrage interval. In the case that borrowing and lending rates coincide, we provide a fully explicit expression for the unique XVA, expressed as a percentage of the price of the traded claim, and for the corresponding replication strategies. In the general case of asymmetric funding, repo, and collateral rates, we study the semilinear partial differential equations characterizing buyer's and seller's XVA and show the existence of a unique classical solution to it. To illustrate our results, we conduct a numerical study demonstrating how funding costs, repo rates, and counterparty risk contribute to determine the total valuation adjustment.     

How do U.S. and Canadian consumers value credence attributes associated with beef labels after the North American BSE crisis of 2003?

A consumer survey conducted in 2006 (n = 419), and therefore after the first confirmed bovine spongiform encephalopathy (BSE) cases in North America in 2003, employs attribute‐based choice experiments for a cross‐country comparison of consumers' valuation of credence attributes associated with beef steak labels; specifically a guarantee that beef was tested for BSE, a guarantee that the steaks were produced without genetically modified organisms (GMOs) and a guarantee that beef steaks were produced without growth hormones and antibiotics. Considering consumers' socio‐economic characteristics, the results suggest that consumers in Montana (U.S.) and Alberta (Canada) are significantly heterogeneous in their valuation of the above attributes, although consumers' relative valuation of these process attributes does not appear to have changed since the 2003 BSE crisis in each region. Alberta consumers place a significant valuation on beef tested for BSE, which is striking because Canada's current legal environment does not permit testing and labelling of such beef by private industry participants. Montana consumers' valuation was found highest for a guarantee that the steaks were produced without GMO. Effective supply‐chain responses to consumers' valuation of credence attributes, for example, in the form of labelling, should therefore take consumers' heterogeneity into account.     

Robust XVA

We introduce an arbitrage‐free framework for robust valuation adjustments. An investor trades a credit default swap portfolio with a risky counterparty, and hedges credit risk by taking a position in defaultable bonds. The investor does not know the exact return rate of her counterparty's bond, but she knows it lies within an uncertainty interval. We derive both upper and lower bounds for the XVA process of the portfolio, and show that these bounds may be recovered as solutions of nonlinear ordinary differential equations. The presence of collateralization and closeout payoffs leads to important differences with respect to classical credit risk valuation. The value of the super‐replicating portfolio cannot be directly obtained by plugging one of the extremes of the uncertainty interval in the valuation equation, but rather depends on the relation between the XVA replicating portfolio and the closeout value throughout the life of the transaction. Our comparative statics analysis indicates that credit contagion has a nonlinear effect on the replication strategies and on the XVA.     

The valuation of inflation‐indexed and FX convertible bonds

This paper derives a valuation model of inflation‐indexed convertible bonds that incorporates the firm's stock price, inflation indexing and the firm's credit risk. The pricing of inflation‐indexed convertible bonds traded on the Tel‐Aviv Stock Exchange (TASE) was empirically tested by using a comprehensive database. The study is the first to empirically test the pricing of convertibles in emerging markets. It was found that the theoretical values for the bonds are, on average, 1.94% higher than the observed market prices. Unlike previous studies, it was found that the underpricing increases with the moneyness of the convertible. It was found that as the maturity lengthens, the underpricing increases. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:634–655, 2008     

An Examination of Participation in Homebuyer Education and Counseling Services

Homebuyer education and counseling aims to help potential homebuyers understand benefits and risks of homeownership, choose a home and an appropriate mortgage, and build the financial know‐how needed for sustainable homeownership and financial health. U.S. Department of Housing and Urban Development's (HUD's) First‐Time Homebuyer Education and Counseling Demonstration found that women and individuals with greater education were generally more likely to participate in services. Enrollees referred to in‐person services are more likely to participate if they are early in the homebuying process or if they planned to purchase a home without a coborrower. They may perceive benefits of services to outweigh the costs of scheduling, traveling to, and attending in‐person services. Enrollees offered remote services (phone, online) are more likely to participate if they are “better off,” having higher mortgage literacy and credit scores. Results may inform agencies' messaging, outreach, and approach to providing services and meeting consumers' needs, federal policy, and interpretation of future impact estimates.     

DYNAMIC INDIFFERENCE VALUATION VIA CONVEX RISK MEASURES

The (subjective) indifference value of a payoff in an incomplete financial market is that monetary amount which leaves an agent indifferent between buying or not buying the payoff when she always optimally exploits her trading opportunities. We study these values over time when they are defined with respect to a dynamic monetary concave utility functional, that is, minus a dynamic convex risk measure. For that purpose, we prove some new results about families of conditional convex risk measures. We study the convolution of abstract conditional convex risk measures and show that it preserves the dynamic property of time-consistency. Moreover, we construct a dynamic risk measure (or utility functional) associated to superreplication in a market with trading constraints and prove that it is time-consistent. By combining these results, we deduce that the corresponding indifference valuation functional is again time-consistent. As an auxiliary tool, we establish a variant of the representation theorem for conditional convex risk measures in terms of equivalent probability measures.     

BOUNDING WRONG‐WAY RISK IN CVA CALCULATION

A credit valuation adjustment (CVA) is an adjustment applied to the value of a derivative contract or a portfolio of derivatives to account for counterparty credit risk. Measuring CVA requires combining models of market and credit risk to estimate a counterparty's risk of default together with the market value of exposure to the counterparty at default. Wrong‐way risk refers to the possibility that a counterparty's likelihood of default increases with the market value of the exposure. We develop a method for bounding wrong‐way risk, holding fixed marginal models for market and credit risk and varying the dependence between them. Given simulated paths of the two models, a linear program computes the worst‐case CVA. We analyze properties of the solution and prove convergence of the estimated bound as the number of paths increases. The worst case can be overly pessimistic, so we extend the procedure by constraining the deviation of the joint model from a baseline reference model. Measuring the deviation through relative entropy leads to a tractable convex optimization problem that can be solved through the iterative proportional fitting procedure. Here, too, we prove convergence of the resulting estimate of the penalized worst‐case CVA and the joint distribution that attains it. We consider extensions with additional constraints and illustrate the method with examples.     

Network valuation in financial systems

We introduce a general model for the balance‐sheet consistent valuation of interbank claims within an interconnected financial system. Our model represents an extension of clearing models of interdependent liabilities to account for the presence of uncertainty on banks' external assets. At the same time, it also provides a natural extension of classic structural credit risk models to the case of an interconnected system. We characterize the existence and uniqueness of a valuation that maximizes individual and total equity values for all banks. We apply our model to the assessment of systemic risk and in particular for the case of stress testing. Further, we provide a fixed‐point algorithm to carry out the network valuation and the conditions for its convergence.     

Measuring consumer's willingness to pay for organic and Fair Trade products

Previous research has extensively studied consumer's environmental and social concerns. However, measuring the value of the environmental or social dimension of a product remains a challenge. This paper proposes to partially fill this gap by measuring the double ‘Fair Trade and organic’ labels' value using an experimental method – the Becker‐DeGroot‐Marschak's mechanism. Two ‘organic and Fair Trade’ and two conventional chocolate products were tested on a sample of 102 consumers. Results show that organic and Fair Trade labels increase consumers' willingness to pay, and allow the identification of three consumers clusters. The first cluster represents people insensitive to the label. For the second cluster, the ‘organic and Fair Trade’ labels' influence on the improving image of the products is positive and important. And finally, for the third cluster, the valuation of the ‘organic and Fair Trade’ label is determined by the product's taste. Our research contributes to a better understanding of consumers' valuation of Fair Trade and organic labels, leading to our conclusions, which offer managerial implications with respect to this market (importance of taste and usefulness of double labels).