Pricing Contingent Claims on a Risky Asset and Stochastic Interest Rates: A New Discrete Time Approach

This paper presents a new discrete time approach to pricing contingent claims on a risky asset and stochastic interest rates. The term structure of interest rates is modeled so that arbitrage-free bond prices depend on an observable initial forward rate curve rather than an exogenously specified market price of risk. A restricted binomial process is employed to model both interest rates and an asset price. As a result, a complete market valuation formula obtains. By choosing the parameters of the discrete joint distribution such that, in the limit, the discrete model converges to the continuous one, a model is obtained that requires the estimation of only three parameters. The approach is parsimonious with respect to alternative models in the literature and can be used to price contingent claims on any two state variables. The procedure is used to numerically analyze the effects of the volatility of interest rates on the determination of mortgage contract rates.     

A New Prepayment Model (with Default): An Occupation-time Derivative Approach

A new prepayment model is developed, which improves the modeling of the borrowers decision process by incorporating an occupation-time derivative in the valuation framework of a fixed-rate mortgage. This option-theoretic mortgage valuation model is based on stochastic house-price and interest-rate models, and requires a particularly subtle technique to incorporate a new type of occupation-time derivative, where the barrier (which activates the derivative) is in the value process and not the underlying process (as it is in standard occupation-time derivatives). This new model simulates a delay in prepayment by the borrower (beyond the time simple ruthless prepayment dictates), thus increasing the value of the mortgage to the lender, compared to the value gained using more basic models. This allows for a more advanced borrower decision process, where a rational exercise structure is retained in a modified form. Empirical evidence supports this theory, which should be beneficial for accurate mortgage-backed security pricing. The results in this paper explore thoroughly the effect on the mortgage value of a delay in prepayment by the borrower on the embedded options held and on the insurance component.

An analysis of option pricing under systematic consumption risk using GARCH

An issue in the pricing of contingent claims is whether to account for consumption risk. This is relevant for contingent claims on stock indices, such as the FTSE 100 share price index, as investor’s desire for smooth consumption is often used to explain risk premiums on stock market portfolios, but is not used to explain risk premiums on contingent claims themselves. This paper addresses this fundamental question by allowing for consumption in an economy to be correlated with returns. Daily data on the FTSE 100 share price index are used to compare three option pricing models: the Black–Scholes option pricing model, a GARCH (1, 1) model priced under a risk-neutral framework, and a GARCH (1, 1) model priced under systematic consumption risk. The findings are that accounting for systematic consumption risk only provides improved accuracy for in-the-money call options. When the correlation between consumption and returns increases, the model that accounts for consumption risk will produce lower call option prices than observed prices for in-the-money call options. These results combined imply that the potential consumption-related premium in the market for contingent claims is constant in the case of FTSE 100 index options.     

Pricing fixed rate mortgages: Some empirical evidence

We develop a simple model based on the hypothesis that yields in the secondary mortgage market provide a basis for pricing new loans in the primary mortgage market. The model is then expanded to include potential interest rate variations due to lender characteristics and whether the loans meet securitization requirements. The empirical results, using a two-year sample of single-family mortgage rates, conform to the predictions of the model. In particular, we find that the interest rates on FRMs in the primary market move in a one-to-one relationship with secondary market yields. We also find significantly lower interest rates on these mortgages that can be sold in the secondary market versus those that cannot, thus indicating the value of the ability to securitize mortgages.     

The Pricing of Futures and Options Contracts on the Value Line Index

This paper considers the problems peculiar to the Value Line Index, because of its use of geometric averaging, as regards the pricing of options and futures on that index. The Value Line Composite Index (VLCI) is an equally weighted geometric average index of nearly 1700 stocks. The VLCI futures market has existed since 1982 while the VLCI options market was established in 1985. This paper provides valuation formulas and analyzes the economic properties of these contracts. Because of the geometric averaging in the VLCI, its contingent claims have special properties. For example, the futures price may fall short of the spot price and the value of a VLCI call option may decline when the volatility of the index is increased. VLCI futures are shown to provide a direct means for duplicating an equally weighted portfolio of the underlying stocks.     

Empirical evaluation of asset pricing models: Arbitrage and pricing errors in contingent claims

Hansen and Jagannathan (1997) have developed two measures of pricing errors for asset-pricing models: the maximum pricing error in all static portfolios of the test assets and the maximum pricing error in all contingent claims of the assets. In this paper, we develop simulation-based Bayesian inference for these measures. While the literature reports that the time-varying extensions substantially reduce pricing errors of classic models on the standard test assets, our analysis shows that the reduction is much smaller based on the second measure. Those time-varying models have large pricing errors on the contingent claims of the test assets because their stochastic discount factors are often negative and admit arbitrage opportunities.     

Legal Restrictions in Personal Loan Markets

This study examines the pricing of personal loans in the form of second mortgages to determine whether state-specific default laws have an effect on the availability and cost of that debt. We examine the pricing of loans to higher risk borrowers and whether borrowers in states that limit lender ability to seek default remedies pay higher credit costs. Our results indicate that, for the most part, lenders rationally price loans to higher risk borrowers. However, when we focus on borrowers with low credit scores, the results indicate that mean actual loan rates are higher than those predicted by our model. The results also indicate that state-specific default laws have an effect on the price of credit. Finally, the results show that there is a greater degree of error in the pricing of second mortgage loans to borrowers with low credit scores than to borrowers with high credit scores.     

Subprime Lenders and Mortgage Market Completion

Without a subprime market, some borrowers by virtue of poor credit history, unstable income, and other characteristics are unable to qualify for a mortgage. With a subprime market, there is a more complete credit supply schedule with the market pricing for poorer credit quality in the mortgage rate. By completing the capital market, subprime lenders reduce borrowing constraints. The result is a social welfare gain. Low-credit applicants otherwise denied funding are able to qualify by paying higher interest rates in exchange for offering more equity or lower loan-to-value ratios. This prediction is consistent with the subprime applicants financing or refinancing their mortgages at relatively low loan-to-value ratios.     

Econometric models of the Eleventh District Cost of Funds Index

The Eleventh District Cost of Funds Index (COFI) is a popular index for pricing adjustable-rate mortgages. COFI is calculated from the interest expenses incurred by thrifts when raising funds. It is a mixture of current and past interest rates on many different financial instruments. COFI can be modelled well with simple econometric models. Commonly used, simple COFI models are compared using a method developed by Hendry (1989). Some of these models, which appear to fit the data well, have nonrobust parameters, significant serial correlation, and heteroscedastic errors. These poorly specified models may lead to systematic mispricing of COFI mortgages. Once a robust econometric model is chose, the lagged adjustment of COFI to movements in interest rates can be incorporated into mortgage pricing models.     

An Improved Fixed-Rate Mortgage Valuation Methodology with Interacting Prepayment and Default Options

This paper considers in detail a realistic mortgage valuation model (including the potential for early prepayment and the risk of default), based on stochastic house-price and interest-rate models. As well as the development of a highly accurate numerical scheme to tackle the resulting partial differential equations, this paper also exploits singular perturbation theory (a mathematically rigorous procedure, based on the idea of the smallness of the volatilities), whereby mortgage valuation can be accurately approximated by very simple closed-form solutions. Determination of equilibrium contract rates, previously requiring many computational hours is reduced to just a few seconds, rendering this a highly useful portfolio management tool; these approximations compare favorably with the full numerical solutions. The method is of wide applicability in US or other mortgage markets and is demonstrated for UK fixed-rate mortgages, including insurance and coinsurance.     

Pricing mortgages: An interpretation of the models and results

Mortgages, like all debt securities, can be viewed as risk-free assets plus or minus contingent claims that can be usefully viewed as options. The most important options are: prepayment, which is a call option giving the borrower the right to buy back the mortgage at par; and default, which is a put option giving the borrower the right to sell the house in exchange for the mortgage. This article reviews and interprets the large and growing body of literature that applies recent results of option pricing models to mortgages. We also provide a critique of the models and suggest directions for future research.The Ohio State University and the National Bureau of Economic ResearchThe Urban Institute University of California at Los Angeles     

A Proportional Hazards Model of Commercial Mortgage Default with Originator Bias

A proportional hazards model with competing risks is specified and is extended to correct for the possibility of originator bias. The model is used to examine the ability of option-theoretic models of mortgage pricing to forecast commercial mortgage defaults. Among the findings, those especially of interest include the influence of contemporaneous loan-to-value and debt-service-coverage ratios on commercial mortgage default probabilities. The paper also finds that option-theoretic models of mortgage pricing are quite capable of producing default estimates that fit the actual default rates well, especially when the model is corrected for originator bias.     

The Valuation of Multivariate Contingent Claims in Discrete Time Models

There are several examples in the literature of contingent claims whose payoffs depend on the outcomes of two or more stochastic variables. Familiar cases of such claims include options on a portfolio of options, options whose exercise price is stochastic, and options to exchange one asset for another. This paper derives risk neutral valuation relationships (RNVRs) in a discrete time setting that facilitate the pricing of such complex contingent claims in two specific cases: joint lognormally distributed underlying variables and constant proportional risk aversion on the part of investors, and joint normally distributed underlying variables and constant absolute risk aversion preferences, respectively. This methodology is then applied to the valuation of several interesting complex contingent claims such as multiperiod bonds, multicurrency option bonds, and investment options.     

A Finite Difference Approach to the Valuation of Path Dependent Life Insurance Liabilities

This paper sets up a model for the valuation of traditional participating life insurance policies. These claims are characterized by their explicit interest rate guarantees and by various embedded option elements, such as bonus and surrender options. Owing to the structure of these contracts, the theory of contingent claims pricing is a particularly well-suited framework for the analysis of their valuation.The eventual benefits (or pay-offs) from the contracts considered crucially depend on the history of returns on the insurance company's assets during the contract period. This path-dependence prohibits the derivation of closed-form valuation formulas but we demonstrate that the dimensionality of the problem can be reduced to allow for the development and implementation of a finite difference algorithm for fast and accurate numerical evaluation of the contracts. We also demonstrate how the fundamental financial model can be extended to allow for mortality risk and we provide a wide range of numerical pricing results.     

Risky debt,jump processes,and safety covenants

The usual assumptions in the continuous-time contingent claims pricing of risky debt are (1) the firm is in default only when the value of its remaining assets falls short of the currently due promised payment and (2) the firm value follows continuous diffusion-process dynamics. It is the joint relaxation of these two simplifying assumptions that motivate this paper in its study of the valuation of risky debt and safety covenants when the firm value follows (possibly) discontinuous sample paths. Explicit solutions are derived and compared to the work of Black and Cox (1976).     

The valuation of multivariate contingent claims under transformed trinomial approaches

This study develops a transformed-trinomial approach for the valuation of contingent claims written on multiple underlying assets. Our model is characterized by an extension of the Camara and Chung (J Futur Mark 26: 759–787, 2006) transformed-binomial model for pricing options with one underlying asset, and a discrete-time version of the Schroder (J Finance 59(5): 2375–2401, 2004) model. However, unlike the Schroder model, our model can facilitate straightforward valuation of American-style multivariate contingent claims. The major advantage of our transformed-trinomial approach is that it can easily tackle the volatility skew observed within the markets. We go on to use numerical examples to demonstrate the way in which our transformed-trinomial approach can be utilized for the valuation of multivariate contingent claims, such as binary options.     

Guarantees in With-Profit and Unitized With-Profit Life Insurance Contracts: Fair Valuation Problem in Presence of the Default Option

The purpose of the article is to apply contingent claim theory to the valuation of the type of participating life insurance policies commonly sold in the UK. The article extends the techniques developed by Haberman, Ballotta, and Wang (2003) to allow for the default option. The default option is a feature of the design of these policies, which recognizes that the insurance company's liability is limited by the market value of the reference portfolio of assets underlying the policies that have been sold. The valuation approach is based on the classical contingent claim pricing “machinery,” underpinned by Monte Carlo techniques for the computation of fair values. The article addresses in particular the issue of a fair contract design for a complex type of participating policy and analyzes in detail the feasible set of policy design parameters that would lead to a fair contract and the trade‐offs between these parameters.     

The financial reporting of fair value based on managerial inputs versus market inputs: evidence from mortgage servicing rights

This research examines whether the fair value of mortgage servicing rights (MSRs) based on managerial inputs (Level 3) better reflects the cash flow and risk characteristics of the underlying assets than the fair value of MSRs based on market inputs (Level 2). Using mortgage servicing fees as a proxy for the underlying cash flows, we find that the valuation multiples for MSRs based on Level 3 inputs are more positively associated with the persistence of future servicing fees compared with the fair value of MSRs based on Level 2 inputs. We also document that only the valuation multiples based on Level 3 fair values are negatively associated with proxies for risk factors. Our results suggest that, although unobservable inputs are subject to managerial discretions, managers can generate higher quality fair value estimates than market inputs due to their information advantage, especially when the market for the underlying asset is inactive.     

Mortgage Contracts,Strategic Options and Stochastic Collateral

This paper offers a game-theoretic model for both the analysis and valuation of mortgage contracts in the context of an economy with complete information and complete contingent claims markets. We analyze the equilibrium strategy of the lender, who holds an option over the magnitude of mortgage credit extended per dollar of collateral offered, and the mortgagor, who holds options to default or prepay, in a class of intertemporal mortgage contracts collateralized by property evolving according to a random process which is common knowledge to both parties to the mortgage contract. Using continuous–time arbitrage valuation principles, we derive the value of the mortgage contract to both parties and show, through both analytical solutions and numerical simulations, that Markov perfect equilibria exist in which, among other properties, a lower flow of housing services accruing to the borrower, per dollar of initial house value, and a correspondingly lower rate of effective depreciation, will elicit a larger volume of funds offered by a lender; the amount of credit offered, the values of the contract to both lender and mortgagor, and the expected losses to both parties from costly bankruptcy are highly sensitive to the perceived volatility of the value of the property collateralizing the mortgage, even in an economy with complete markets or risk neutrality on the parts of lender and borrower; the upper limit on mortgage credit offered by a rational lender may be a small fraction of the current fair market value of the property, regardless of the contractual yield offered by the borrower, and will decrease, at each such yield, as bankruptcy costs or housing service flows increase; and under significant but plausible values for bankruptcy and costs of liquidating property under foreclosure, the flow of mortgage credit can become negatively related to the spread of the mortgage yield over the riskless rate, with the lender preferring a lower contractual yield to a higher one.     

Pricing Effects of the Decision to Sell or Hold Adjustable Rate Mortgage Loans in a Portfolio

Residential mortgage originators can transfer loans to ultimate lenders quickly and efficiently using the secondary mortgage market. Some adjustable rate mortgage (ARM) lenders use this outlet consistently while others hold whole loans in their portfolios on a long-term basis. Selling and holding lenders should respond to different economic factors when setting yields on ARM loans originated because their long-term positions in the loans are so diverse. This paper develops and tests a model of differential pricing behavior for selling and holding strategies. Empirical results support the notion that lenders use different factors to price loans and that these factors are related to the risks faced by the originating lender given its origination strategy. Additional findings suggest that institutional and firm-specific pricing tendencies exist in the primary mortgage market for adjustable rate debt.