A Closed Form Formula for Valuing Mortgages

We develop a closed form formula for the value of a fixed-rate residential mortgage that includes the provision that the borrower can prepay at any time with no penalty. The value of the mortgage equals the expectation, under the risk neutral probability measure, of the future cash flows. We model future cash flows by estimating an empirical model of prepayment behavior. A second change of measure leads to a closed form expression for the expectation. The closed form values explain most of the time series variation in MBS prices. The closed form formula significantly shortens the time to calculate mortgage values and durations and can be a useful tool for portfolio management and hedging.     

An Asymptotic Expansion Approach to Pricing Financial Contingent Claims

We propose a new methodology for the valuation problem of financial contingent claims when the underlying asset prices follow a general class of continuous Itô processes. Our method can be applicable to a wide range of valuation problems including contingent claims associated with stocks, foreign exchange rates, the term structure of interest rates, and even their combinations. We illustrate our method by discussing the Black-Scholes economy when the underlying asset prices follow the continuous diffusion processes, which are not necessarily time-homogeneous. The standard Black-Scholes model on stocks and the Cox-Ingersoll-Ross model on the spot interest rate are simple examples. Then we shall give a series of examples on the valuation formulae including plain vanilla options, average options, and other contingent claims. We shall also give some numerical evidence of the accuracy of the approximations we have obtained for practical purposes. Our approach can be rigorously justified by an infinite dimensional mathematics, the Malliavin-Watanabe-Yoshida theory recently developed in stochastic analysis.     

Empirical Performance of Alternative Option Pricing Models

Substantial progress has been made in developing more realistic option pricing models. Empirically, however, it is not known whether and by how much each generalization improves option pricing and hedging. We fill this gap by first deriving an option model that allows volatility, interest rates and jumps to be stochastic. Using S&P 500 options, we examine several alternative models from three perspectives: (1) internal consistency of implied parameters/volatility with relevant time-series data, (2) out-of-sample pricing, and (3) hedging. Overall, incorporating stochastic volatility and jumps is important for pricing and internal consistency. But for hedging, modeling stochastic volatility alone yields the best performance.     

Anatomy of an ARM: The Interest-Rate Risk of Adjustable-Rate Mortgages

This article analyzes the dynamics of the commonly used indices for adjustable rate mortgages and systematically compares the effects of their time-series properties on the interest-rate sensitivity of adjustable-rate mortgages. Our ARM valuation methodology allows us simultaneously to capture the effects of index dynamics, discrete coupon adjustment, mortgage prepayment, and both lifetime and periodic caps and floors. We can, moreover, either calculate an optimal prepayment strategy for mortgage holders or use an empirical prepayment function. We find that the different dynamics of the major ARM indices lead to significant variation in the interest-rate sensitivities of loans based on different indices. We also find that changing assumptions about contract features, such as loan caps and coupon reset frequency, has a significant, and in some cases unexpected, impact on our results.     

Valuing Commercial Mortgages: An Empirical Investigation of the Contingent-Claims Approach to Pricing Risky Debt

This paper empirically investigates a contingent-claims model of commercial mortgage pricing. We find that the magnitude of the observed default premia for a sample of nonprepayable fixed rate bullet mortgages can be explained by the contingent-claims model. In addition, the model explains a significant proportion of the period-to-period changes in the default premia. However, given an assumed negative correlation between building value changes and interest rate changes, the model's risk structure tends to increase less steeply with increasing maturity than the observed risk structure.     

Reduced Form Mortgage Pricing as an Alternative to Option-Pricing Models

This paper extends the traditional hazard technique of estimating prepayment and default by allowing their baselines to be stochastic processes, rather than known paths of time, as is typically assumed. By working in the reduced form, this method offers an alternative to the empirical valuation of mortgages more easily implemented than the standard structural form approach of options pricing.     

An Extended Set of Risk Neutral Valuation Relationships for the Pricing of Contingent Claims

This article establishes an extended set of risk neutral valuation relationships (RNVR's), assuming a representative agent who has an extended power utility function, of the HARA class of utility functions. The utility function of the representative agent displays either increasing, constant or decreasing proportional risk aversion. Aggregate consumption and the random payoff of the underlying asset are bivariate three-parameter lognormal distributed. As an application of the RNVR's, closed-form solutions for the price of a call written on a stock are derived. These include an extra parameter, the threshold parameter, not contained in the Black-Scholes formula.     

Pricing and Hedging of Contingent Claims in Term Structure Models with Exogenous Issuing of New Bonds

If calibrated to an observed term structure of interest rates that only covers a finite range of times-to-maturity an HJM-model of the term structure of interest rates will eventually die out in finite time as bonds reach maturity. This poses problems for the pricing and hedging of certain contingent claims. Therefore, we extend the HJM-model in such a way that it lives on an arbitrary time horizon and possesses term structures that cover a constant finite interval of times-to-maturity. We consider the pricing and hedging of contingent claims in this framework.     

A Unified Approach for Pricing Contingent Claims on Multiple Term Structures

This paper provides a unified approach for pricing contingent claims on multiple term structures using a foreign currency analogy. All existing option pricing applications are seen to be special cases of this unified approach. This approach is used to price options on financial securities subject to credit risk.     

An Empirical Investigation of the Arbitrage Pricing Theory

Empirical tests are reported for Ross' [48] arbitrage theory of asset pricing. Using data for individual equities during the 1962–72 period, at least three and probably four priced factors are found in the generating process of returns. The theory is supported in that estimated expected returns depend on estimated factor loadings, and variables such as the own standard deviation, though highly correlated (simply) with estimated expected returns, do not add any further explanatory power to that of the factor loadings.     

Structural Models of Corporate Bond Pricing: An Empirical Analysis

This article empirically tests five structural models of corporatebond pricing: those of Merton (1974), Geske (1977), Longstaffand Schwartz (1995), Leland and Toft (1996), and Collin-Dufresneand Goldstein (2001). We implement the models using a sampleof 182 bond prices from firms with simple capital structuresduring the period 1986–1997. The conventional wisdom isthat structural models do not generate spreads as high as thoseseen in the bond market, and true to expectations, we find thatthe predicted spreads in our implementation of the Merton modelare too low. However, most of the other structural models predictspreads that are too high on average. Nevertheless, accuracyis a problem, as the newer models tend to severely overstatethe credit risk of firms with high leverage or volatility andyet suffer from a spread underprediction problem with saferbonds. The Leland and Toft model is an exception in that itoverpredicts spreads on most bonds, particularly those withhigh coupons. More accurate structural models must avoid featuresthat increase the credit risk on the riskier bonds while scarcelyaffecting the spreads of the safest bonds.     

Libor Market Models versus Swap Market Models for Pricing Interest Rate Derivatives: An Empirical Analysis

We empirically compare Libor and Swap Market Models for the pricing of interest rate derivatives, using panel data on prices of US caplets and swaptions. A Libor Market Model can directly be calibrated to observed prices of caplets, whereas a Swap Market Model is calibrated to a certain set of swaption prices. For both models we analyze how well they price caplets and swaptions that were not used for calibration. We show that the Libor Market Model in general leads to better prediction of derivative prices that were not used for calibration than the Swap Market Model. Also, we find that Market Models with a declining volatility function give much better pricing results than a specification with a constant volatility function. Finally, we find that models that are chosen to exactly match certain derivative prices are overfitted; more parsimonious models lead to better predictions for derivative prices that were not used for calibration.     

Libor Market Models versus Swap Market Models for Pricing Interest Rate Derivatives: An Empirical Analysis

We empirically compare Libor and Swap Market Models for thepricing of interest rate derivatives, using panel data on pricesof US caplets and swaptions. A Libor Market Model can directlybe calibrated to observed prices of caplets, whereas a SwapMarket Model is calibrated to a certain set of swaption prices.For both models we analyze how well they price caplets and swaptionsthat were not used for calibration. We show that the Libor MarketModel in general leads to better prediction of derivative pricesthat were not used for calibration than the Swap Market Model.Also, we find that Market Models with a declining volatilityfunction give much better pricing results than a specificationwith a constant volatility function. Finally, we find that modelsthat arechosen to exactly match certain derivative prices areoverfitted; more parsimonious models lead to better predictionsfor derivative prices that were not used for calibration. JELClassification: G12, G13, E43.     

An Empirical Comparison of Alternative Models of the Short-Term Interest Rate

We estimate and compare a variety of continuous-time models of the short-term riskless rate using the Generalized Method of Moments. We find that the most successful models in capturing the dynamics of the short-term interest rate are those that allow the volatility of interest rate changes to be highly sensitive to the level of the riskless rate. A number of well-known models perform poorly in the comparisons because of their implicit restrictions on term structure volatility. We show that these results have important implications for the use of different term structure models in valuing interest rate contingent claims and in hedging interest rate risk.     

The use of digital footprints in the US mortgage market

We study the rise of digital footprint (DF) users in the U.S. residential mortgage market. The proportion of lenders that use a borrower's DF has witnessed remarkable growth from 6% in 2013 to 34% in 2018 in a short span – according to our analysis. We show that the use of DFs can significantly reduce a lender's overall risk and that the use of DFs can result in considerable societal benefits, by reducing the overall discriminatory forces. In sum, we provide evidence that the informational advantage associated with using a borrower's DF far outweighs that of existing traditional lending methods.     

Convertible Preferred Stock Valuation: Tests of Alternative Models

We undertake a comprehensive test of several contingent claim valuation models adapted to callable, convertible preferred stocks employing a sample of 24 issues and over 27,000 daily price observations. To our knowledge, no large-scale tests of these models have been published. The most complete model tested is an extension of the 1970s developments of Ingersoll and of Brennan and Schwartz, allowing for realistic contract features including delayed callability and nonconstant call prices. The mean and the mean absolute pricing errors are approximately –0.18 percent and 5.4 percent, respectively, and this model fits the data substantially better than the simpler alternatives that ignore such features. Thus, the added computational complexity required for the most complete model examined is evidently merited. Moreover, to the extent that the most complete model accurately mirrors reality, the evidence suggests that investors rationally account for many of the complex features imbedded in typical contracts.