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Valuing Prepayment and Default in a Fixed-Rate Mortgage: A Bivariate Binomial Options Pricing Technique
Abstract
This paper uses a bivariate binomial options pricing technique to value the prepayment and default options in a fixed-rate mortgage. The American style options are dependent on two stochastic variables: (1) house price, (2) one year spot rate. The paper uses the standard lognormal process for house price and the CIR square-root process for interest rates. By forcing the two underlying state variables to undergo transformations, two new uncorrelated variables with constant volatilities are established. With constant volatilities, a computationally simple bivariate binomial tree is formed which greatly reduces the complexity of working with two state variables and is pedagogicallyuseful. Using this procedure, the price of any real estate contingent claim whose value is dependent on the one year spot rate and house price can be determined. Results are compared with those from a finite difference model.
... In addition, the variance of S equals 2 , where is the instantaneous volatility of the property price, and is the time period between the contract signing date, 0 , and the maturity date of the contract, . Following [15,16], this study also assumes that the service flow is proportional to the value of the underlying property, S. ...
... The parameters are chosen in line with those reported in the literature. In particular, see the discussion in [15,16]. ...
... The parameters are chosen in line with those reported in the literature; in particular, see the discussion in [15,16]. ...
This paper evaluates the deposit and purchase pricing of purchase contracts in a risk-neutral framework. First, we determine the fair deposit price of a single-installment purchase contract based on theoretical modeling and numerical analysis. Second, the buyer’s threshold pricing in dual-installment and multi-installment contracts is investigated under the framework of compound options. Lastly, the pricing behavior of deposits and purchases is further analyzed using a simultaneous equations modeling framework.
... There exist few articles (e.g., works by D. Cossin et al. [24]) on the loan prepayment option but a related subject, the prepayment option in fixed-rate mortgage loan, has been widely covered in several papers by J.E. Hilliard and J.B. Kau [34] and more recent works by Chen et al. [21]. To approximate the PDE satisfied by the value of the prepayment option, they defined two state variables (interest rate and house price). ...
... There exist few articles (e.g., works by D. Cossin et al. [24]) on the loan prepayment option but a close subject, the prepayment option in a fixed-rate mortgage loan, has been widely covered in several papers by J.E. Hilliard and J.B. Kau [34] and more recent works by Chen et al. [21]. To approximate the PDE satisfied by the prepayment option, they define two state variables (interest rate and house price). ...
... There exist few articles (e.g., works by Cossin et al. [24]) on the loan prepayment option but a close subject, the prepayment option in a fixed-rate mortgage loan, has been covered in several papers by Hilliard and Kau [34] and more recent works by Chen et al. [21]. To approximate the PDE satisfied by the prepayment option, they define two state variables (interest rate and house price). ...
This PhD thesis investigates the pricing of a corporate loan according to the credit risk, the liquidity cost and the embedded prepayment option. A loan contract issued by a bank for its corporate clients is a financial agreement that often comes with more flexibility than a retail loan contract. These options are designed to meet clients' expectations and can include e.g., a prepayment option (which entitles the client, if he desires so, to pay all or a fraction of its loan earlier than the maturity). The prepayment is the main option and it will be study in this thesis. In order to decide whether the exercise of the option is worthwhile the borrower compares the remaining payments with the outstanding amount of the loan. If the remaining payments exceed the nominal value then it is optimal for the borrower to refinance his debt at a lower rate. For a bank, the prepayment option is essentially a reinvestment risk, i.e. the risk that the borrower decides to repay earlier his/her loan and that the bank cannot reinvest his/her excess of cash in a new loan with same characteristics.The valuation problem of the prepayment option can be modelled as an embedded compound American option on a risky debt owned by the borrower. We choose in this thesis to price a loan and its prepayment option by resolving the associated PDE instead of binomial trees (time-consuming) or Monte Carlo techniques (slow to converge).
... Kau et al. (1995) used the arbitrage principle in options pricing theory to establish a valuation model of the mortgage with prepayment and default risks, which used the CIR process to characterize the risk-free rate. Hilliard et al. (1998) considered the implied prepayment and default options of fixed-rate mortgages with full prepayment and default risks using the option pricing method, where the house price process was characterized by Ornstein-Uhlenbeck (OU) process. Bhattacharya et al. (2019) used Bayesian methods to establish a full prepayment and default model of borrowers and performed posterior analysis and predictive reasoning through Monte Carlo methods based on Markov chains. ...
... This uncertainty leads to prepayment and default risks for the lenders and difficulties in valuing mortgages. Prepayment and default on mortgage loans have been examined in a large number of literature, for example, Schwartz and Torous (1992), Kau et al. (1995), Hilliard et al. (1998), Ambrose and Sanders (2003), Tsai et al. (2009), Jones andChen (2016), Hall and Maingi (2021). As described above, prepayment is divided into full prepayment and partial prepayment. ...
We investigate the valuation problem of a mortgage contract with full prepayment and default risks using the reduced-form model with regime switching. The hazard rates of full prepayment and default are specified as linear functions of the risk-free interest rate and house prices, respectively, which are characterized by Ornstein-Uhlenbeck processes with regime switching. To derive the explicit valuation formula, we derive the distribution of the number of transitions for two-state Markov processes in finite time and the conditional joint probability density function of transition times using the uniformization technique. Finally, we analyze the effect of parameters on the valuation of the mortgage.
... While the academic literature is rich in research on prepayment models for residential and commercial mortgages À À À some of the most relevant articles include Richard and Roll (1989), Brown (1992), Harmon (1996), McConnell and Singh (1991), Sanyal (1994), Hilliard et al. (1998), Chen et al. (2009), Daniel (2008), Cossin and Lu (2004), Papin and Turinici (2015), Schwartz and Torous (1992), and Schwartz and Torous (1989) À À À relatively little has been published on modeling institutional loan prepayments. A prominent article dealing strictly with US corporate loan prepayments is McGuire (2008). ...
... A¯rst empirical model for rational prepayments based on the prepayment option's evaluation was introduced by Richard and Roll (1989). Hilliard et al. (1998), andChen et al. (2009) expanded the analyses by valuing the prepayment option via an approach based on a bivariate partial di®erential equation with reference rates following a Cox Ingersol Ross (CIR) equilibrium model while property values followed a Geometric Brownian Motion process. Statistical models currently representing the bulk of the industry installations were introduced in institutional proprietary applications and described in Schwartz and Torous (1992), and Schwartz and Torous (1989). ...
This paper aims to evaluate an inference of bank internal PDs (Default Probabilities) on subsequent prepayments of variable rate institutional loans. Since variable rate loans hardly present an economic motivation for early prepayments in that they would not offer a cheaper refinancing alternative, we test the conjecture of a correlation between improvements in obligors’ creditworthiness (as reflected by negative changes in Bank Internal PDs) and subsequent loan prepayments, as obligors might be tempted to renegotiate more advantageous terms (lower credit spreads) with their lenders. The analysis is purported to serve as an early warning mechanism for banks pursuing the Basel III internal rating-based (IRB) approach for unexpected inflows of liquidity in the near future. We use Machine Learning (ML) ensemble methods to forecast potential prepayments and perform a conditional prepayment analysis to make an inference on the prepayment amounts and the prepayment timing distributions while controlling for macroeconomic and corporate idiosyncratic characteristics.
... To investigate the determinants of the interest rate differential, we choose a set of plausible parameter values that resemble those chosen by Hull and White [1990] and Hilliard, Kau, and Slawson [1998]: 1) the initial spot interest rate is equal to 6% per year, that is, r(0) = 0.06; 5 2) the reversion speed of interest rates per year is equal to 20% (i.e., a = 0.2); 3) the long-run steady state of spot interest rates is equal to 6% per year (i.e., b = 0.06); 4) the parameter c is equal to 0.05 such that the volatility of the growth rate of spot interest rates evaluated at the current interest rate is equal to c r( ) . % 0 20 4 = per year; and 5) the transaction cost parameter α = 0.2. ...
... These predictions, however, are based on very simplified assumptions that may be relaxed in the following ways. First, we may allow for stochastic movements of housing prices and stochastic movements of spot interest rates (see, e.g., Leung and Sirmans [1990]; Kau et al. [1995]; Hilliard, Kau, and Slawson [1998]; Ambrose and Buttimer [2000]); we can then simultaneously value the option to refinance and the option to default. Second, we may include other factors, such as the loan-to-value ratio, tax consequences, or other types of transaction costs (see, e.g., Kau, Keenan, and Kim [1993]; Stanton [1995]) in our framework and then examine whether our results still hold. ...
A fixedrate borrower has the option to pay off the loan after paying certain penalties. The borrower will do so only if the spot interest rate falls sufficiently below the contract rate. Assuming that the spot interest rate follows a meanreverting process, this article finds that the fixedrate borrower has a more valuable prepayment option such that the interest rate differential between a fixedand a floatingrate loan expands when the term of the loan lasts longer, either the steadystate or initial spot interest rate is lower, penalties associated with prepayment are lower, or interest rates increase at an unpredictable rate.
... The following benchmark parameter values resemble those set by Cox et al. (1985) and Hilliard et al. (1998): (i) The current interest rate is equal to 10% per year, i.e., ...
... This article makes several simplified assumptions, which may be relaxed in the future. First, we may also allow stochastic movements of housing prices (see, e.g., Ambrose and Buttimer, 2000; Hilliard et al., 1998; Kau et al., 1994; Leung and Sirmans, 1990), in addition to stochastic movements of spot interest rates. We can then simultaneously value the option to refinance, as well as the option to default. ...
Abstract Anassumable,loan that carries a lower-than-market rate of interest has value because ofits advantageous rate. An investor who employs a traditional loan can, however, avoid the downside ,risk of interest ,rates because ,the investor has the option to refinance the loan when,future market interest rates decrease. This option value can explain why,only one-third to two-thirds of the premium,associated with assumption financing is capitalized into the home,selling prices. The premium,associated with the assumable loan will be eroded more by the option value to refinance when the term of
... At the same time the only advantage of a binomial model is its simplicity. If other processes (such as an interest rate process) are simulated by a binomial tree too (see Kau, Hilliard and Slawson [28]), then one should be careful with the state space mash because of the conditional stability of this explicit numerical algorithm. It is a well known fact of numerical analysis that the time step must be quadratically smaller than step sizes of the other variables in explicit approximations of diffusion processes. ...
... We considered different specifications of our model in the view of this new classification. Our general model is not tied to a particular numerical procedure as some existing models (e.g., Kau, Hilliard and Slawson [28]). As an example we showed that Stanton's model [40] is in fact just a variant of a splitting-up, numerical method applied to our model. ...
The models in the literature for valuation of mortgage contracts subject to prepayment can generally be classified into one of two categories: the option-based models and the empirical mortgage-to-mortgage-refinancing models. Using risk-neutral martingale methods together with the intensity-based approach borrowed from credit risk modeling, this paper develops a framework that not only generalizes but considerably extends both ap-proaches. In the case of option-based specifications based on non-optimal liability, our prepayment model is the first which is not tied to some par-ticular numerical procedure. In the case where the state process is a diffusion, the mortgage equation is a semi-linear parabolic PDE. To illus-trate this point we consider the continuous time limit of Stanton's model [40] and show that the model is the first order version of splitting-up nu-merical method applied to our PDE. Throughout the paper we point out various new ways to develop mortgage modeling.
... At the expiration date of the mortgage, the expected present value of cash flows is zero (V t, ω = 0), at each state. According to (16), the American call price, at each node, could be stated as: ...
... The volatility is expressed in term of percentage points. Credit spread and transaction costs level are assumed constant, respectively equal to γ = 2% yearly and X = 0.05%.16 See note Chart 2. ...
We analyze the variables determining the exercise of the prepayment option in mortgages for the Italian banking industry. We first examine the main studies on the valuation of mortgages and mortgage-backed securities (Sharp et al., 2008; Boudoukh et al., 1995; Black et al., 1990). We then apply a specific model to measure the prepayment option in the case of a refinancing option exercise. We consider a fixed-rate mortgage and try to estimate the exposure of a selected bank to interest rate risk. ALM studies have recognized the importance of this issue for financial intermediaries, especially for banks (Kalotay et al., 2007; Staikouras, 2006). Our main findings highlight that an increase in volatility causes an increase in option prices and a decrease in the mortgage market value. Moreover, the reduction of the positive gradient of the yield curve determines an increase in the prepayment option price. In the case of a yield curve with a marked positive gradient, the prepayment option price exhibits high sensitivity to volatility changes; and the mortgage value market reacts similarly. Finally, we observe that the prepayment option price can be considered a decreasing function of the credit spread and of the transaction costs.
... Prepayment is usually priced as an American call option on a defaultable debt owned by the borrower. Hilliard and Kau (1998) value prepayment and default options in a ¿xed-rate mortgage using a bivariate binomial lattice, with real-estate value and spot interest rate as the two underlying state variables. The same options are valued by Kau et. ...
... al. (1992) via a two-state explicit ¿nite-difference technique. The model presented in this paper differs from Hilliard and Kau (1998) in many ways. First, business borrowers prepay with different incentives second, the two underlying state variables are forward credit spread and forward interest rate, developed by Heath, Jarrow andMorton (1990, 1992) D For its role in analyzing CMBS, see Sanders (1999). ...
This paper presents a pricing model of commercial and industrial (C&I) loan prepayment option. Modeling of prepayment is essential in pricing mortgage contracts since prepayment truncates the timing and amount of expected cash flows. Lenders normally charge a penalty for prepayment, for example interest on unused principal. We first propose that prepayment penalties should be viewed as fees in arrears reflecting the value of a call option at origination. We then rationalize business prepayment behavior with both refinancing and non-refinancing incentives. Business borrowers make rational decisions to refinance when interest rates fall or their credit standing improves. Business borrowers also prepay with internal equity to optimize capital structure and reduce debt. Prepayment value in C&I loans reflects the option value depending on the market conditions, borrower's creditworthiness and the need to reduce debt. In the rest of the paper, we propose a contingent claims approach to the pricing C&I loan prepayment contract, given loans being defaultable, and transaction costs and fees being proportional to the loan amount. An implementation example is provided with prepayment option value being determined via a double non-recombining tree. The two underlying state variables are forward credit spread and forward interest rate specified in the HJM framework and their volatilities are exponentially dampened and proportional to the spot rates. We find that prepayment option value concentrates on the first few periods, and locking out these periods would reduce the option value signicantly. We also find that prepayment option value on a variablerate loan is insensitive to the forward interest rate volatility but very sensitive to the forward credit spread volatility.
... The option framework is widely used in studying various aspects of real estate transactions, such as termination, prepayment, and default on mortgages (Ambrose and Buttimer, 2000;Deng et al., 2000;Hilliard et al., 1998); land and property development (Capozza and Li, 2001;Cheng et al., 2021;Grenadier, 1996;Yao and Pretorius, 2014); and presale contract analysis (Buttimer et al., 2008;Chan et al., 2012;Edelstein et al., 2012;Lai et al., 2004). However, option theory is rarely applied to understand market participants' behaviour in the context of the residential real estate presale market. ...
A growing body of literature has shown, again and again, that investors suffer from behavioural biases. They make cognitive errors by processing the available information in a selective and biased manner and rely on additional emotional heuristics to take mental shortcuts. Instead of making optimal decisions, they seek satisfactory solutions. Overall, they can muster bounded rationality only. Real estate investors are as human and biased as investors in other financial assets. Previous work has found that the intransparency of markets, low liquidity and the heterogeneity of assets paired with significant emotional involvement of buyers and sellers increase behavioural biases. That is why it is particularly important to understand the behavioural bias of real estate investors in general and owner-occupiers in particular. In this dissertation, I study the bounded rationality of three different types of real estate investors and link these behavioural biases to real estate market dynamics and investment performances. The first two studies focus on the phenomenon of anchoring—a frequently-observed type of cognitive error where investors select reference prices and evaluate offers or market movements relative to existing price points (the anchors). In the first study, I reconfirm that prior purchase prices unduly influence sellers in the resale market: Sellers facing nominal losses try to sell at higher prices than those selling at a nominal gain. Evidence of anchoring bias in list and transaction prices has been found in the housing markets. I contribute to the prior literature by showing that the power of the anchors on transaction prices is a function of information availability and general market conditions. The effect is economically significant only when data from comparable transactions are scarce. Second, the effect is relatively weak in a bust period of the market cycle but it grows when markets recover. The second study sheds light on the anchoring effect in the presale property market. Here, presale homebuyers only pay deposits when signing the purchase contracts and have the option to strategically default in case property values fall sufficiently before the delivery of their units. Effectively, the mental reference points are expected to differ from the anchors used in the resale market if contract holders rationally consider the deposits as option premiums (or sunk costs). I find that presale contract holders still anchor to the full contract prices rather than the outstanding payments. A presale contract is more likely to be rescinded if the property’s market price at settlement is lower than its contract price. In contrast, I do not find a sharp increase in the rescission rate when the market price drops further below the outstanding payment at settlement. Moreover, for those presale homebuyers who settle the out-of-money contracts, I find they are more likely to substantially increase their holding periods to recover from the implied losses. One might think that institutional investors are less prone to suffer from bounded rationality than individual investors. After all, they are well-trained professionals with better access to market information. Nevertheless, earlier papers have shown that institutional investors are still subject to cognitive errors like price anchoring. The third segment of this dissertation studies how institutional investors rely on additional emotional heuristics in their decision-making, which are more challenging to correct than cognitive errors. I research investment decisions by local and out-of-town investors who have different access to local information and might be subject to familiarity bias—an emotional heuristic denoting that investors prefer to over-concentrate on familiar investments. I find that the familiarity bias can dominate the information advantage of local investors when adverse shocks depress the value of home-market assets. Using the public non-REIT firm acquisitions as geography-specific demand shocks, I find equity REITs perform worse if they hold more properties in counties where the acquired non-REIT firms are located. However, due to familiarity bias, institutional home investors are less likely to short the affected REITs than institutional non-home investors despite the continuous decrease in the REITs’ rental income up to at least one year after the shocks, which leads to implied investment losses.
... The option framework is widely used in studying various aspects of real estate transactions, such as termination, prepayment, and default on mortgages (Hilliard et al., 1998;Ambrose and Buttimer, 2000;Deng et al., 2000); -land and property development (Capozza and Li, 2001;Cheng et al. 2021;Grenadier, 1996;Yao and Pretorius, 2014); and presale contract analysis (Lai et al., 2004;Buttimer et al., 2008;Chan et al., 2012;Edelstein et al., 2012). However, option theory is rarely applied to understand market participants' behaviour in the context of the residential real estate presale market. ...
This study documents that over 10% of the presale contracts in the Hong Kong housing market between 1996 and 2014 were rescinded, resulting in a loss of HKD 436.67 million per year. We then investigate potential determinants of contracts rescission from a novel perspective of option theory. We find out-of-the-money presale contracts (with market price being lower than the outstanding payment at settlement) have a 12.2% higher rescission rate. The rescission rate is also higher when presale homebuyers bear more of the price risk as proxied by option delta and time-induced risk as proxied by time-to-maturity. Moreover, we find rescission rates drop significantly after the Hong Kong government's housing market macroprudential measures. Our findings shed light on understanding the mechanism of presale contracts rescission, homebuyers’ strategic default behaviour, and the role of housing market regulation in mitigating rescissions.
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... Thereafter, some improvements have been developed within this framework. A striking example of this is the study (Hilliard et al. 1998), which produces a less cumbersome numerical procedure working with a bivariate binomial lattice model and the study (Ambrose and Buttimer 2000), which splits the default option into two parts: a right to stop making payments temporarily and a right to give up the property. ...
This study explores the hedging coefficients of the financial options to default and to prepay embedded into mortgage contracts based on the change in spot rate, underlying house price and its volatility. In the computations, the finite-dimensional Malliavin calculus is applied since the distribution of both options is unknown and their payoffs are non-differentiable. Naturally, the hedging coefficients are obtained as a product of option’s payoff and an independent weight, which permits the user to derive estimations for the hedging coefficients by running a crude Monte Carlo (MC) algorithm. The simulations reveal that the financial options to default and to prepay are both more sensitive to a change in spot rate than a change in underlying house price and its volatility. There are two potential usages of the hedging coefficients: first, they allow the user to determine the effects of spot rate and underlying house price change and its volatility on the default and prepayment options, and second, borrowers and lenders can replicate and hedge their main portfolio by using the balance between these coefficients and the default and prepayment options.
... Option theoretic approach was first applied to mortgage contracts by Asay (1979). Since then, many researchers have refined the application of option theoretic pricing to mortgage contracts: Kau et al. (1994) use a finite-difference approach to value the joint prepayment and default option embedded in a mortgage contract; Hilliard et al. (1996) employ two stochastic processes to simulate house price and interest rates, and use a binomial tree for valuing prepayment and default option; Kau et al. (1992) study the impact of house price volatility and loan-to-value ratio on mortgage default by applying appropriate boundary conditions; Foster and Order (1985) suggest using current rates for calculating present value of a mortgage contract, Schwartz and Torous (1989) apply heuristic functions to represent default frequency, and Stanton (1995) applies conditions based on economic factors to model default frequency. In the option theoretic framework, a mortgage contract is a portfolio of amortizing bond with monthly coupon and principal amortization payments, a call option that gives the holder of mortgage contract (home owner) the right to buy back the mortgage by paying the outstanding principal balance, and a put option that gives the mortgage holder the right to terminate the mortgage contract by defaulting on mortgage payments. ...
Mortgage default has been characterized as ruthless or predominantly driven by the relation of house price to mortgage value by proponents of pure option theoretic model (Kau et al. Journal of Money, Credit and Banking, 24(3), 279–299 1992, Ambrose et al. Journal of Money, Credit, and Banking, 29(3), 314–325 1997), and as non-ruthless by researchers who argue that transaction costs and other idiosyncratic factors determine when mortgage holders default (Foster and Order Housing Finance Review, 3(4), 351–372 1984; Vandell Journal of Housing Research, 6(2), 245–264 1995). This work uses Fannie Mae loan performance data to present evidence supporting the hypothesis that a significant number of mortgage defaults are non-ruthless. It uses a new approach to track mortgages that are likely to default by tracking 90-day delinquent mortgages and studying which ones eventually default. It evaluates the joint put-call option embedded in a mortgage contract using a Monte-Carlo simulation for the underlying stochastic variables. It identifies key differences between ruthless and non-ruthless mortgage defaults and illustrates the propensity of non-ruthless mortgage defaulters to become current on their 90-day delinquent mortgages. These observations provide valuable insights for policymakers and creditors in their task of structuring debt-relief programs for delinquent mortgage holders. It augments the analysis of mortgage defaults by considering the impact of loan-to-value ratio at mortgage origination.
... Section 489 (1) (2) of the German Civil Code entitles private borrowers to terminate a fixed-rate mortgage loan, in whole or in part, at any time observing a notice period of six over the last decades (e. g. Chen et al. 2009;Ding et al. 2016;Hilliard et al. 1998;Kalotay et al. 2004;Kau et al. 1995). As the value of prepayment options depends very much on the exercise strategy of the borrowers, the strand of literature that deals with the valuation of prepayment options is strongly related to papers that develop and analyse exercise strategies for these embedded options (e. g. ...
Section 489 of the German Civil Code anchors a prepayment option in all fixed-rate retail loans with a term of more than 10.5 years. The primary purpose of this paper is to develop an approach for valuing legal prepayment options (LPOs), embedded in 15-year German mortgage loans, at their origination. The analysis is based on 11,201 pairs of 10-and 15-year German mortgage rates that cover the period from June 2001 until February 2018 in steps of one month. In order to value the LPOs, trajectories of 10-year German mortgage rates are simulated by means of the exponential Vasicek model. The exercise strategy of the borrowers is a main driver of the value of LPOs. Our simulation results reveal that the following exercise strategy maximizes the average value of the LPOs under investigation (from the perspective of the day of their origination): On average, borrowers should exercise their LPOs, embedded in 15-year German mortgage loans, and refinance either if the present value of interest savings is at least 1.2 % of the outstanding loan amount or if the prevailing refinancing rate is, first, below the 15-year mortgage rate and, second, close to its presumed floor of 0.1 %.
... We employ three types of options namely default and two prepayment clauses, that are defeasance and prepayment penalty which are widely used in commercial and residential mortgages respectively. The option pricing method which is similar to Hilliard et al. (1998) is embedded into the model and Longsta↵ and Schwartz (2001) where the simulation method is used to calculate the option prices. ...
Since the 2008 financial crisis, the mortgage market has been renovating its tools and instruments in order to avoid a new crisis. One such innovative instrument is the participating mortgage, in which the lender gains part of the net operating income and/or future appreciation. In this paper, we establish a financing model for participating mortgages, incorporating early termination options such as default and two prepayment clauses, defeasance and prepayment penalty. Later, we illustrate a detailed sensitivity analysis of the model. The values of early termination options depend on the choice of parameters in the model, as well as the term structure of short term rates. Finally, we show that a participation rate of 11.24% results in zero mortgage interest rate using the parameters in our simulation.
... that involved a prepayment option appeared in the study of mortgages, see for instance (Hillard and Slawson, 1998) and (Chen et al., 2009) for recent contributions to that literature. In that view, the prepayment option is a function depending on the interest rate and house price. ...
We investigate the prepayment option related to a corporate loan. The default intensity of the firm is supposed to follow a Cox-Ingersoll-Ross (CIR) process and the short interest rate is assumed constant. A liquidity term that represents the funding costs of the bank is introduced and modeled as a continuous time discrete state Markov jump process. The prepayment option is an American option with the payoff being an implicit function of the parameters of the problem. We give a verification result that allows to compute the price of the option. Numerical results are completely consistent with the theory; it is seen that the exercise domain may entirely disappear during such a liquidity crisis meaning that it is not optimal for the borrower to prepay. The method allows to quantify and interpret these findings.
... Deriving the risk-neutral process for house price by changing to a risk neutral probability measure involves replacing the expected drift term μ − δ by μ − δ − λσ , where λ represents the market price of risk associated to the uncertainty of the house price [8]. Using risk neutrality arguments, μ − λσ is equal to the risk-free interest rate r t . ...
In this paper we consider the valuation of fixed-rate mortgages including prepayment and default options, where the underlying stochastic factors are the house price and the interest rate. The mathematical model to obtain the value of the contract is posed as a free boundary problem associated to a partial differential equation (PDE) model. The equilibrium contract rate is determined by using an iterative process. Moreover, appropriate numerical methods based on a Lagrange–Galerkin discretization of the PDE, an augmented Lagrangian active set method and a Newton iteration scheme are proposed. Finally, some numerical results to illustrate the performance of the numerical schemes, as well as the qualitative and quantitative behaviour of solution and the optimal prepayment boundary are presented.
... A great deal of research has focused on mortgage valuation (Kau, Keenan, Muller, and Epperson (1992); Kau (1995); Hilliard, Kau and Slawson (1998); and Downing, Stanton, and ...
... There exist few articles (e.g., works by Cossin et al. [14]) on the loan prepayment option but a close subject, the prepayment option in a fixed-rate mortgage loan, has been covered in several papers by Hilliard and Kau [25] and more recent works by Chen et al. [12] that express the prepayment option as a function depending on two state variables : interest rate and house price. Their approach is based on a bivariate binomial option pricing technique with a stochastic interest rate and a stochastic house value (CIR processes). ...
We investigate in this paper a perpetual prepayment option related to a corporate loan. The short interest rate and default intensity of the firm are supposed to follow Cox-Ingersoll-Ross (CIR) processes. A liquidity term that represents the funding costs of the bank is introduced and modeled as a continuous time discrete state Markov chain. The prepayment option needs specific attention as the payoff itself is a derivative product and thus an implicit function of the parameters of the problem and of the dynamics. We prove verification results that allows to certify the geometry of the exercise region and compute the price of the option. We show moreover that the price is the solution of a constrained minimization problem and propose a numerical algorithm building on this result. The algorithm is implemented in a two-dimensional code and several examples are considered. It is found that the impact of the prepayment option on the loan value is not to be neglected and should be used to assess the risks related to client prepayment. Moreover, the Markov chain liquidity model is seen to describe more accurately clients’ prepayment behavior than a model with constant liquidity.
... There exist few articles (e.g., works by Cossin and Lu [1]) on the loan prepayment option but a close subject, the prepayment option in a fixed-rate mortgage loan, has been widely covered in several papers by Hilliard et al. [4] and more recent works by Chen et al. [5]. To approximate the PDE satisfied by the prepayment option, they define two state variables (interest rate and house price). ...
We investigate in this paper a perpetual prepayment option related to a corporate loan. The default intensity of the firm is supposed to follow a CIR process. We assume that the contractual
margin of the loan is defined by the credit quality of the borrower and the liquidity cost that reflects the funding cost of the bank. Two frameworks are discussed: firstly a loan margin without liquidity cost and secondly a multiregime framework with a liquidity cost dependent on the regime. The
prepayment option needs specific attention as the payoff itself is an implicit function of the parameters of the problem and of the dynamics. In the unique regime case, we establish quasianalytic formulas for the payoff of the option; in both cases we give a verification result that allows for the computation of the price of the option. Numerical results that implement the findings are also presented and are completely consistent with the theory; it is seen that when liquidity parameters are very different (i.e., when a liquidity crisis occurs) in the high liquidity cost regime, the exercise domain may entirely disappear, meaning that it is not optimal for the borrower to prepay during such a liquidity crisis. The method allows for quantification and interpretation of these findings.
... Given that longevity risk has important implication for the determination of reverse mortgage loan limits, a simple application of the standard option pricing model to reserve mortgages without consideration of uncertain mortality is likely to result in an inaccurate evaluation of maximum mortgage loan amount. As a result, this paper develops a breakeven framework for evaluating mortgage loan limits that attempts to integrate longevity risk as well as cross-over risk into the option-based valuation 3 See for example Kau et al. (1992, 1995), Hilliard et al. (1998), Epperson et al. (1985), Brunson et al. (2001) , Deng (1997), Deng et al. (2000), Ciochetti and Vandell (1999), and Archer et al. (2002). Kau and Keenan (1995) provide a comprehensive survey on such category of earlier literature. ...
Since the loan limit of a reverse mortgage is a major concern for the borrower as well as the lender, this paper attempts to develop an option-based model to evaluate the loan limits of reverse mortgages. Our model can identify several crucial determinants for reverse mortgage loan limits, such as initial housing price, expected housing price growth, house volatility, mortality distribution, and interest rates. We also pay special attention to the important implication of information asymmetry between lenders and elderly borrowers with respect to the beliefs on housing market risk. In reverse mortgage markets, elderly borrowers typically hold far less or no information about the characteristics associated with lenders’underlying property pools compared to the lenders. Such information asymmetry leads these two categories of market participants to generate different perspectives on the risk of the collateralized properties, which can be identified to be important in determining the maximum loan amounts in this analysis. We further find that the maximum loan amount of a reverse mortgage decreases in the correlation between the systematic return and the idiosyncratic return on its underlying housing property but increases with the number of the pooled reverse mortgages.
In the last two decades, academicians and professionals intending to study in any area of real estate and finance not only must master advanced financial mathematics concepts and mathematical/econometric models but also should be able to implement those concepts computationally to improve real estate markets' efficiency. This comprehensive thesis mainly aims to combine the theory of financial mathematics with an emphasis on real-life applications in keeping with the way, both investors and policymakers, in today's real estate markets. Unlike most studies on real estate markets, housing markets and mortgages, the thesis covers both non-parametric statistical modeling methods (Multivariate Adaptive Regression Splines (MARS) and Generalized Linear Models (GLM)) and stochastic calculus (Stochastic Differential Equations (SDE), Malliavin calculus theory) with Monte Carlo simulations, Capital Asset Pricing Model (CAPM) and Fama French three-factor model with its extensions. The thesis offers thorough models in the subject of housing markets and provides hedging strategies of default and prepayment options embedded into mortgage contracts. Along with with the theoretical aspects, the thesis presents numerous applications for pricing, investment decision, risk management via hedging strategies, and portfolio management. The numerical illustrations are on determining the housing market price drivers and the effect of large investors, house price forecasting by using the US housing market data, determining hedging strategies for both mortgage default and prepayment options by computing the hedging coefficients via using Monte Carlo simulations and analyzing the T-REITs returns performance in various aspects.
How long will payments under a residential mortgage loan continue? This question is central to pricing mortgages and mortgage-backed securities, now one of the largest sectors of the bond market. In this chapter, I review the research, identify major themes, and attempt a synthesis: what do we know and what do we not know?
Loss mitigation is the process by which lenders attempt to minimize losses associated with foreclosure. As competition increases in the mortgage industry, lenders and servicers are under great pressure to adopt loss mitigation tactics rather than simply use foreclosure as the means of dealing with borrowers in default. This study presents a mortgage-pricing model that fully specifies all borrower options with respect to default, including the ability to reinstate the mortgage out of default. We document the impact of various loss mitigation programs, including forbearance and antideficiency judgments, as well as the value of credit on borrower default behavior.
Prepayment penalties are ubiquitous in the commercial mortgage market yet reviled and highly restricted by law and regulation in the residential mortgage market. Considering the perspectives of both the borrower and the lender, we attempt a balanced cost-benefit analysis of this controversial contract feature for residential mortgage loans. We will address the following questions: What is the economic value of the prepayment penalty feature? Why is it more prevalent in the subprime than the prime market segment? Do borrowers obtain an offsetting economic benefit when they contract for a loan containing a prepayment penalty? What is the average cost of a prepayment penalty to borrowers, and how often is this cost actually incurred? In general, although we find a significant reduction in interest rates for loans containing a prepayment penalty, the expected costs outweigh the benefits by a considerable margin. © 2008 METROPOLITAN INSTITUTE AT VIRGINIA TECH. ALL RIGHTS RESERVED.
This paper presents a review and synthesis of the past fifteen years of research on mortgage termination risk. Understanding termination risk is fundamental to explaining the workings of the now $13 trillion mortgage market. The review covers theoretical, empirical, and methodological work, including both residential and commercial market segments. In addition to synthesizing existing findings, open questions for further research are identified. The overall objective is to introduce this important topic to journal readers who may not be actively involved in the mortgage field and to stimulate thinking about open issues that deserve additional research.
According to Canadian tax law one way of transforming from non-tax deductible (personal mortgage) to tax deductible interest expenses is to borrow against home equity to make investments. A re-advanceable mortgage is a product in which the mortgagor immediately re-borrows principal as it is paid. We assume the re-borrowed funds are invested into a single risky asset and study the risk associated with this strategy to provide an accurate description of the mortgagor’s position. Our model accommodates stochastically changing interest rates and housing prices. We find that immediately borrowing from the increase in housing price decreases the expected mortgage payoff time, but also significantly increases the risk of taking a much longer time to pay off the mortgage. The risk of this strategy is very sensitive to changes in housing price volatility. Additionally, both the expected mortgage payoff time and the risk of taking a long time to pay off the mortgage increase with interest rate volatility. To reduce both the expected mortgage payoff time and the risk of a longer payoff time, the asset chosen should be (i) negatively correlated with housing price; and (ii) positively correlated with interest rates. From the case study using historical data we find that business cycle has an effect on the strategy’s performance and a reasonable decision rule would be to (i) implement this strategy at the beginning or in the middle of an expansionary period; and (ii) delay its implementation otherwise. Results of this study are relevant to homeowners, financial planners and policymakers.
We develop an equilibrium model of land valuation, expected returns and development. The model extends the standard real options framework by combining cash flow uncertainty with uncertain buyer reservation prices. These reservation prices also serve as a proxy for general economic conditions. The model allows for the realism of lumpy uncertain demand and absorption of space over time where renters/buyers arrive and depart according to a Poisson process with a stochastic intensity that is a function of the uncertain variables. Buyers and renters can arrive and depart before construction has commenced, while additional buyers/renters can arrive during and after construction. This structure allows for presales/preleasing of space to mitigate developer’s risk. Given various parameter sets, we numerically solve the model and derive numerous outputs including: the equilibrium value of undeveloped land, the equilibrium value of developed space, the optimal construction commencement trigger as a function of the presales, price and expected absorption levels, the expected time to construction, and the expected return on land investment
The dominant consideration in the valuation of mortgage-backed securities (MBS) is modeling the prepayments of the pool of underlying mortgages. Current industry practice is to use historical data to project future prepayments. In this paper we introduce a new approach and show how it can be used to value both pools of mortgages and mortgage-backed securities issued by the two government sponsored enterprises (Fannie Mae and Freddie Mac). We distinguish between prepayments that do not depend on interest rates and refinancings that do. Turnover and curtailment are modeled using a vector of prepayment speeds, while refinancings are modeled using a pure option-based approach. We describe the full spectrum of refinancing behavior using a notion of refinancing efficiency. There are financial engineers who refinance at just the right time, leapers who do it too early, and laggards who wait too long. We partition the initial mortgage pool into “efficiency buckets, ” whose sizes are calibrated to market prices. The composition of the seasoned pool is then determined by the excess refinancings over baseline prepayments. Leapers are eliminated first, then financial engineers, and finally laggards. As the mortgage pool ages, its composition gradually shifts towards laggards, and this automatically accounts for what is commonly referred to as “prepayment burnout. ” Our approach has two distinguishing features: (1) our primary focus is on understanding the market value of a mortgage, in contrast with standard models that strive (often unsuccessfully) to predict future cash flows, and (2) we use two separate yield curves, one for discounting mortgage cash flows and the other for MBS cash flows. 1 An Option-Theoretic Prepayment Model for Mortgages and Mortgage-Backed Securities
This paper is concerned with a finite-horizon optimal investment and consumption problem in continuous-time regime-switching models. The market consists of one bond and n≥1 correlated stocks. An investor distributes his/her wealth among these assets and consumes at a non-negative rate. The market parameters (the interest rate, the appreciation rates and the volatilities of the stocks) and the utility functions are assumed to depend on a continuous-time Markov chain with a finite number of states. The objective is to maximize the expected discounted total utility of consumption and the expected discounted utility from terminal wealth. We solve the optimization problem by applying the stochastic control methods to regime-switching models. Under suitable conditions, we prove a verification theorem. We then apply the verification theorem to a power utility function and obtain, up to the solution of a system of coupled ordinary differential equations, an explicit solution of the value function and the optimal investment and consumption policies. We illustrate the impact of regime-switching on the optimal investment and consumption policies with numerical results and compare the results with the classical Merton problem that has only a single regime.
This paper constructs the discriminant analysis model and logistic model to recognize the major influencing factor about the mortgage prepayment, and puts the support vector machine theory into the empirical study. The result of the three forecast models show the support vector machine is more superior
We model competing risks of mortgage termination where the borrower faces a repeated choice to continue to pay, refinance the loan, move or default. Most previous empirical work on mortgage prepayment has ignored the distinction between prepayments triggered by refinancing and moving, combining them into a single prepayment rate. We show that financial considerations are the primary drivers of the refinance choice while homeowner characteristics have more influence on the move decision. We demonstrate that these differences are statistically significant and that combining these two distinct choices into a single measure of prepayment shifts coefficients toward zero and produces inaccurate predictions of aggregate termination rates. For example, a combined model underestimates the effect of the market price of the loan on refinancing; it misses entirely the opposite effects of borrower income on moving and refinancing. Our results suggest that existing prepayment models are inconsistent predictors of mobility-driven prepayment and underestimate the effect of market conditions and borrower characteristics on refinancing and housing decisions. Our findings have great significance to mortgage investors because mobility-driven prepayments are likely to be a more significant source of prepayments in thenext decade.
This paper has debated between fractional and two part models in the context of default and prepayment behaviour, applications
where a large proportion of zero observations are typically found. The predictive capacity in this paper is to provider the
direction of consideration and very good fit result to the behaviours of default and prepayment for bank. From the perspective
of practitioners, it could point of view the characteristics of borrowers of each variable would seem to be choosing in this
paper. This paper also provides the simply looking at the extent to which policy decision of bank conclusions would differ
depending upon the characteristics of customers’ behaviour consideration.
This paper alters the traditional model of valuation by introducing trinomial trees, and determines the difference between the traditional plain-vanilla, adjustable-rate mortgages and cash rebate mortgages. By presenting an economic model for studying the combined effects of mortgage value by prepayment, delinquency, default, and cure speed, the model builds a simulation program to generate different cash flow scenarios. The results indicate that the value of cash rebate mortgages is higher than that of standard mortgages, although they are more sensitive to embedded options. If the probability of exercising an option is higher, then the value of cash rebate mortgages will drop at a faster rate than that of standard mortgages.
Purpose
The purpose of this paper is to provide additional evidence that housing prices significantly impact aggregate refinancing and thus directly influence mortgage termination.
Design/methodology/approach
Regression analysis is applied to examine refinancing activity in US cities.
Findings
The evidence shows that positive appreciation in housing prices provides the borrower with positive incentives to refinance in response to the associated increased borrowing capacity when mortgage rates have declined. On the other hand, depreciation in housing prices may depress refinancing.
Research limitations/implications
Housing price movements, not only collateral constraints on refinancing but also the disincentive to engage in cash‐out refinancing caused by depreciation as well as the incentive for cash‐out refinancing brought by appreciation, should be included in modeling total termination risks of mortgage‐backed securities.
Originality/value
In contrast to previous studies, this paper provides empirical support for both the incentive and the disincentive to engage in cash‐out refinancing produced by housing price changes, in addition to support for the traditional collateral constraint effect of housing prices on refinancing.
The main objective of this study is to develop a quasi-analytical solution for the valuation of commercial mortgages. We consider the existence of a single source of risk—the risk of defaulting on a mortgage—and therefore the existence of a single state variable: the value of the mortgaged property. The value of the mortgage corresponds to the present value of the future payments on the loan, minus the value of the embedded American default option. The major difficulty in designing such a model is calculating the value of this option, since it is necessary to determine the lowest property price below which it must be immediately exercised, i.e. the critical value of the property. Under such a framework, the partial differential equation for the value of the default option is that presented by Black and Scholes [J. Polit. Econ., 1973, 81, 637–659]. The boundary conditions applicable to the value of the default option are naturally different, given the possibility of early default. To obtain a quasi-closed-form solution, instead of a continuous-time framework, a discrete-time setting is considered for the differential equation: the ‘Method of Lines’. The partial differential equation is transformed into an ordinary non-homogenous differential equation, replacing the derivative of the function (price of the option in relation to time) with a finite difference, and keeping the derivatives of the mortgaged property unchanged. In order to improve the performance of the model, Richardson's extrapolation is applied, allowing for an increase in the speed of convergence of the model results towards reasonable values. Compared with the alternative numerical valuation techniques, the proposed quasi-closed-form solution provides much quicker mortgage valuation results and enhances the ability to perform the corresponding sensitivity analysis. The current work constitutes one of the first attempts to address the development of analytical solutions to mortgage valuation using contingent claims analysis.
We show how the dynamics of economic instability and financial crises articulated by Keynes in the General Theory and developed by Minsky as the Financial Instability Hypothesis can be formalized using Weidlich’s sociodynamics of opinion formation. The model addresses both the lending sentiment of a lender in isolation as well as the impact on that lending sentiment of the behavior of other lenders. The risk associated with lending is incorporated through a stochastic treatment of loan dynamics that treats prepayment and default as competing risks. With this model we are able to generate endogenously the rapid changes in lending opinion that attend slow changes in lending profitability and find these dynamics to be consistent with the rise and collapse of the non-Agency mortgage-backed securities market in 2007/2008. As the parameters of this model correspond to well-known phenomena in cognitive and social psychology, we can both explain why economic instability has proved robust to advances in risk measurement and suggest how policy for reducing economic instability might be formulated in an experimentally sound manner.Highlights► Lending sociodynamics formalizes Minsky’s Financial Instability Hypothesis. ► Competing default and prepayment risks mask systemic risk when assets appreciate. ► Lending sociodynamics explains rapid rise and spectacular collapse of MBS market. ► Lending sociodynamics suggests approach to reducing economic instability.
We examine the effect of parameter instability on the valuation of mortgages and mortgage-backed securities. In particular, we price 1997 issue mortgages subject to the 1998 bond market rally events assuming an empirically derived prepayment model constructed on data reflecting the 1993 experience and compare results to those that would have been obtained had actual parameter drift been captured. Using carefully constructed micro-data, we find that the refinancing propensity was greater in 1998 for a 1997 issue given the same incentives, compared to the 1993 performance of a 1992 issue. The associated change in cash flow patterns produces a 1.7% change in asset values. Results are consistent with technological improvement and greater efficiency in the mortgage market and may help explain the large losses often sustained by mortgage market participants during bond market rallies.
Efficient execution is a significant task faced by mortgage bankers attempting to profit from the secondary market. The challenge of efficient execution is to sell or securitize a large number of heterogeneous mortgages in the secondary market in order to maximize expected revenue under a risk tolerance. This paper develops a stochastic optimization model to perform efficient execution that considers secondary marketing functionality including loan-level efficient execution, guarantee fee buy-up or buy-down, servicing retain or release, and excess servicing fee. Since efficient execution involves random cash flows, lenders must balance between expected revenue and risk. We employ a CVaR risk measure in this efficient execution model that maximizes expected revenue under a constraint. By solving the efficient execution problem under different risk tolerances specified by a CVaR constraint, an efficient frontier could be found. The model is formulated as a mixed 0-1 linear programming problem. A case study shows that realistic instances of the efficient execution problem can be solved in acceptable time (approximately one minute) with CPLEX-90 solver on a PC. CVaR Keywords: efficient execution, secondary mortgage market, mortgage-backed security (MBS), Fannie Mae, conditional value-at-risk (CVaR).
We propose a new mortgage contract that endogenizes the risk of house price declines and thus minimizes default risk resulting from changes in the underlying asset value while still retaining contract rates near the cost of a standard fixed-rate mortgage. Our new mortgage recognizes that the lender is the most economically efficient bearer of house price risk. By reducing the role of the legal system in mitigating house price risk, the new mortgage reduces the negative externalities and social costs arising from defaults resulting from house price risk. In other words, the new mortgage minimizes the need to use the legal foreclosure system to deal with the economic risk of house price declines.
Mortgages are complicated instruments, and the pace of financial innovation complicates them further. Unlike standard single family mortgages, the terms on commercial loans are heterogeneous, and the motivations of property owners are not well understood. Single family lenders are increasingly adopting the prepayment penalties found in the commercial market. This paper fills in a gap in the study of mortgage finance by determining the optimal refinancing strategy under common forms of prepayment penalties. Our paper uses a bivariate binomial options pricing technique to value call options within various mortgages with different types of prepayment penalties. Using simulations, we calculate the spread between contract and prevailing interest rates needed to trigger an optimal refinance, and find that it is often substantially larger for mortgages with declining penalties than for mortgages with static penalties. The larger required spreads caused by the enhanced value of delay explain much of the slow prepayment behavior discovered in the multifamily market by Follain, Ondrich, and Sinha (1997). Therefore, borrower decisions which appear slow (or irrational) in a static setting may be quite rational in a dynamic setting. Our results have important implications for both the interpretation of regression results based on historical data, and for the pricing of newly originated mortgages. Declining prepayment penalties were used extensively for multifamily mortgages in the 1970's and 1980's. While their use has declined in the 1990's in favor of yield maintainance agreements, empirical studies which use 1970's and 1980's data need to incorporate the effects of these penalty structures or risk specification bias. At the same time that their use is declining in the multifamily market it is growing in the single-family market. Understanding the behavior of these mortgages will be crucial to their pricing.
Call protection in debt contracts is ubiquitous in the bond market, customary in the commercial mortgage market, yet reviled and highly restricted in the residential mortgage market. We examine the reasons for these differences and use Monte Carlo techniques to calculate the economic value of call protection (prepayment penalties) in residential mortgage contracts. Results explain why prepayment penalties are more prevalent in the subprime, compared to the prime, market segment. From the simulation results we also compute the reduction in equilibrium contract rates call protection allows and then verify those predictions using empirical using data on actual contracts, extending the existing literature. Results strongly suggest that call protection has significant economic value to both borrowers and lenders and that, hence, the trend toward greater restrictions on this contract feature is bad public policy.
When the outstanding balance exceeds the housing value before the loan is settled, the insurer suffers an exposure to crossover risk induced by three risk factors: interest rates, house prices and mortality rates. With consideration of housing price risk, interest rate risk and longevity risk, we provide a three-dimensional lattice method that simultaneously captures the evolution of housing prices and short-term interest rates to calculate the fair valuation of reverse mortgages numerically. For a reverse mortgage insurer, the premium structure of reverse mortgage insurance is determined by setting the present value of the total expected claim losses equal to the present value of the premium charges. However, when the actual loss is higher than the expected loss, the insurer will incur an unexpected loss. To offset the potential loss, we also design two types of crossover bonds to transfer the unexpected loss to bond investors. Therefore, through the crossover bonds, reverse mortgage insurers can partially transfer crossover risk onto bond holders.
Purpose
– This paper seeks to examine the role that regulation and regulatory agencies played in the creating of the subprime mortgage market, and the subsequent crash of the mortgage market. The paper has two goals. First, it seeks to document the degree to which the US housing markets, and the US housing finance market, were regulated prior to the crash. Second, it seeks to show that regulatory bodies set policies which created both incentives and explicit requirements for Fannie Mae and Freddie Mac, as well as depository institutions, to enter the subprime market.
Design/methodology/approach
– The paper examines the regulatory environment of the subprime market. It uses regulatory filings and other documents as primary sources.
Findings
– The popular perception that the subprime mortgage market arose because housing finance was largely unregulated is incorrect. In point of fact, the housing finance market was very heavily regulated. Indeed, the paper shows that the creation of the subprime market was a formal goal of the federal government, and that federal regulatory agencies explicitly required participation by the Government Sponsored Enterprises (GSEs).
Originality/value
– The paper's primary implication is that incentive conflicts within the US housing finance system significantly contributed to the mortgage crisis. These incentive conflicts were not just within private firms, but also extend to the GSEs and regulatory agencies. Regulatory agencies not only failed to anticipate the crisis; they actively encouraged the policies which created it. As a result, the primary focus of reform efforts should be on identifying and eliminating such conflicts.
Setting project financing parameters, such as the loan to valuation ratio, loan interest rate, repayment schedules, and fees, requires detailed modelling of the resulting credit risk in a non-recourse setting. Structured credit risk models, based on the early work of Merton, have been developed in continuous time which can assist with project financing structuring. These models require a level of mathematical sophistication that may not always be available to those undertaking project financing analysis. This note provides an overview of a discrete time binomial approach to structural credit risk modelling, which enables project financing analysts a more accessible tool to evaluate project loan structures.
This paper suggests a modification to the explicit finite difference method for valuing derivative securities. The modification ensures that, as smaller time intervals are considered, the calculated values of the derivative security converge to the solution of the underlying differential equation. It can be used to value any derivative security dependent on a single state variable and can be extended to deal with many derivative security pricing problems where there are several state variables. The paper illustrates the approach by using it to value bonds and bond options under two different interest rate processes. © 1990, School of Business Administration, University of Washington. All rights reserved.
We study the functioning of secured and unsecured interbank markets in the presence of credit risk. The model generates empirical predictions that are in line with developments during the 2007–09 financial crisis. Interest rates decouple across secured and unsecured markets following an adverse shock to credit risk. The scarcity of underlying collateral may amplify the volatility of interest rates in secured markets. We use the model to discuss various policy responses to the crisis.
The long history of the theory of option pricing began in 1900 when the French mathematician Louis Bachelier deduced an option pricing formula based on the assumption that stock prices follow a Brownian motion with zero drift. Since that time, numerous researchers have contributed to the theory. The present paper begins by deducing a set of restrictions on option pricing formulas from the assumption that investors prefer more to less. These restrictions are necessary conditions for a formula to be consistent with a rational pricing theory. Attention is given to the problems created when dividends are paid on the underlying common stock and when the terms of the option contract can be changed explicitly by a change in exercise price or implicitly by a shift in the investment or capital structure policy of the firm. Since the deduced restrictions are not sufficient to uniquely determine an option pricing formula, additional assumptions are introduced to examine and extend the seminal Black-Scholes theory of option pricing. Explicit formulas for pricing both call and put options as well as for warrants and the new "down-and-out" option are derived. The effects of dividends and call provisions on the warrant price are examined. The possibilities for further extension of the theory to the pricing of corporate liabilities are discussed.
In this article, we suggest an efficient method of approximating a general, multivariate log-normal distribution by a multivariate
binomial process. There are two important features of such multivariate distributions. First, the state variables may have
volatilities that change over time. Second, the two or more relevant state variables involved may covary with each other in
a specified manner, with a time-varying covariance structure. We discuss the asymptotic properties of the resulting processes
and show how the methodology can be used to value a complex, multiple exerciseable option whose payoff depends on the prices
of two assets.
In this paperwe examine the household's option to prepay or call a standard fixed-rate mortgage. Results based on simulation indicate that the value of this option is sensitive to the expected path of interest rates, the variation around that path, risk aversion and refinancing costs. Unfortunately, efforts to estimate the interest rate process (by us and by previous authors) have met with only limited success, and uncertainty exists regarding the degree of risk aversion and the magnitude of refinancing costs.Thus we conclude that the application of contingent-claims methodology to options on bonds is conceptually more difficult and operationally less reliable than is the analogous application to options on stocks.Despite these reservations concerning the use of our model as a technique for absolute valuation, preliminary findings on the effects of changes in mortgage contract design on the value of the prepayment option are encouraging. For example, our estimate of the relative values of the call options on 30- and 15-year mortgages and on level-payment and graduated-payment mortgages appear to be reasonably robust with respect to specifications of the interestrate process and the other parameters.These findings suggest that our model may be of considerable use within the context of relative or comparative valuation.
This paper uses an intertemporal general equilibrium asset pricing model to study the term structure of interest rates. In this model, anticipations, risk aversion, investment alternatives, and preferences about the timing of consumption all play a role in determining bond prices. Many of the factors traditionally mentioned as influencing the term structure are thus included in a way which is fully consistent with maximizing behavior and rational expectations. The model leads to specific formulas for bond prices which are well suited for empirical testing. See also the authors’ paper [ibid. 53, 363–384 (1985; Zbl 0576.90006)].
We extend the procedures developed by Nelson and Ramaswamy (RFS, 1990) and Hull and White (JFQA, 1990) to accomodate more generalized diffusions and two possible correlated state variables thus yielding a bivariate vinomial options pricing technique. The advantage this technique offers is the ability to price American style options, thereby accomodating early exercise, despite the existence of two correlated underlying state variables. We illustrate the technique with an application to American futures options where the futures price and the short-rate of interest are stochastic. Our technique is computationally efficient and can be further generalized for multiple state variables, albeit with an accompanying rise in computational expense.
Discrepancies between the Black-Scholes value of Japanese equity warrants and their observed prices are explained in part by the stochastic volatility of changes in prices of the underlying stocks. We fit GARCH and EGARCH models to the stochastic volatility and briefly compare their performance to the CEV model. A hopscotch algorithm is used to value the warrants in the presence of the stochastic stock price volatility. The stochastic volatility-hopscotch warrant values still differ substantially from corresponding prices; in contrast, away-from-the-money short-term Nikkei 225 options valued with the same stochastic volatility models are close to observed prices. A regression model is used to fit the differences between warrant values and prices as a function of proxies for market impediments.
We consider the problem of pricing an American contingent claim whose payoff depends on several sources of uncertainty. Using classical assumptions from the Arbitrage Pricing Theory, the theoretical price can be computed as the maximumover all possible early exercise strategies of the discounted expected cash flows under the modified risk-neutral information process. Several efficient numerical techniques exist for pricing American securities depending on one or few (up to 3) risk sources. They are either lattice-based techniques or finite difference approximations of the Black-Scholes diffusion equation. However, these methods cannot be used for high-dimensional problems, since their memory requirement is exponential in the number of risk sources. In this paper, we present an efficient numerical technique that combines Monte Carlo simulation with a particular partitioning method of the underlying assets space, which we call Stratified State Aggregation (SSA). Using this technique we...
A procedure is developed for the valuation of options when there are two underlying state variables. The approach involves an extension of the lattice binomial approach developed by Cox, Ross, and Rubinstein to value options on a single asset. Details are given on how the jump probabilities and jump amplitudes may be obtained when there are two state variables. This procedure can be used to price any contingent claim whose payoff is a piece-wise linear function of two underlying state variables, provided these two variables have a bivariate lognormal distribution. The accuracy of the method is illustrated by valuing options on the maximum and minimum of two assets and comparing the results for cases in which an exact solution has been obtained for European options. One advantage of the lattice approach is that it handles the early exercise feature of American options. In addition, it should be possible to use this approach to value a number of financial instruments that have been created in recent years.
This study develops a currency option pricing model under stochastic interest rates when interest rate parity holds, and it is assumed that domestic and foreign bond prices have local variances that depend only on time. We demonstrate how existing currency option models are simply derived from one framework. Empirical tests employing transactions option data reveal that a particularly simple form of the stochastic rate model is uniformly more accurate than a constant rate model for all boundaries and maturities tested.
We extend existing pricing models and develop a bivariate binomial option pricing technique that accommodates correlated state variables. This technique offers the ability to price American-style options, thereby accommodating early exercise, despite the existence of two correlated underlying state variables. Our technique is computationally efficient and can be further generalized for multiple-state variables, albeit with an accompanying rise in computational expense. © The Southern Finance Association and the Southwestern Finance Association.
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.
If options are correctly priced in the market, it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks. Using this principle, a theoretical valuation formula for options is derived. Since almost all corporate liabilities can be viewed as combinations of options, the formula and the analysis that led to it are also applicable to corporate liabilities such as common stock, corporate bonds, and warrants. In particular, the formula can be used to derive the discount that should be applied to a corporate bond because of the possibility of default.
This paper derives a forward-looking rational expectations house price model and empirically tests its ability to explain short-run fluctuations in real house prices. A novel approach to proxying the imputed rents of owner-occupied housing, as a function of observable housing market fundamentals, is combined with a housing market arbitrage relation to derive a present value model for real house prices. Tests of the rational expectations, nonlinear cross-equation restrictions reject the joint null hypothesis of rational expectations and the asset-based housing price model for quarterly, single-detached house prices in the city of Vancouver, British Columbia from 1979-1991. The model fails to fully capture observed house price dynamics in two real estate booms but tracks real house prices well in less volatile times, suggesting that prices may temporarily deviate from fundamental values in real estate price cycles. Copyright American Real Estate and Urban Economics Association.
The same option-based methodology now commonly used to value mortgages and their termination features also can be applied to calculate the probabilities that mortgage default will occur. This paper pursues that idea, and furthermore, enriches the idealized option-based approach by introducing both transaction costs and "suboptimal" termination. These latter features capture the individual considerations that cause a mortgage holder's actions to differ from what rationality would indicate based solely on the market value of the mortgage. These features are of considerable importance if the results of options-based models are to be made comparable to those calculations of default probabilities occurring in the empirical literature. Copyright American Real Estate and Urban Economics Association.
Existing models on the pricing of default and prepayment options in fixed-rate mortgages either use numerical methds or they do not consider refinancing or other transaction costs involved in default and prepayment. We provide in this paper an application of the Boyle [1] lattice model to price secured debt with two risky assets. This model is simple, efficient and capable of considering the major types of transaction costs involved in prepayment and default. Using our model, we estimate the option values under a range of assumptions about the underlying parameters. We also provide some comparisons of the lattice model estimates to other models in the literature. Copyright American Real Estate and Urban Economics Association.
This article presents a new model of mortgage prepayments, based on rational decisions by mortgage holders. These mortgage
holders face heterogeneous transaction costs, which are explicitly modeled. The model is estimated using a version of Hansen’s (1982) generalized method of moments, and is shown to capture many of the empirical features of mortgage prepayment. Estimation
results indicate that mortgage holders act as though they face transaction costs that far exceed the explicit costs usually
incurred on refinancing. They also wait an average of more than a year before refinancing, even when it is optimal to do so.
The model fits observed prepayment behavior as well as the recent empirical model of Schwartz and Torous (1989). Implications for pricing mortgage-backed securities are discussed.
I use a new technique to derive a closed-form solution for the price of a European call option on an asset with stochastic
volatility. The model allows arbitrary correlation between volatility and spot-asset returns. I introduce stochastic interest
rates and show how to apply the model to bond options and foreign currency options. Simulations show that correlation between
volatility and the spot asset's price is important for explaining return skewness and strike-price biases in the Black-Scholes
(1973) model. The solution technique is based on characteristic functions and can be applied to other problems
This article generalizes the Cox, Ross, and Rubinstein (1979) binomial option-pricing model, and establishes a convergence from discrete-time multivariate multinomial models to continuous-time
multidimensional diffusion models for contingent claims prices. The key to the approach is to approximate the $N$
-dimensional diffusion price process by a sequence of $N$
-variate, $(N+1)$
-nomial processes. It is shown that contingent claims prices and dynamic replicating portfolio strategies derived from the
discrete time models converge to their corresponding continuous-time limits.
A binomial approximation to a diffusion is defined as “computationally simple” if the number of nodes grows at most linearly
in the number of time intervals. It is shown how to construct computationally simple binomial processes that converge weakly
to commonly employed diffusions in financial models. The convergence of the sequence of bond and European option prices from
these processes to the corresponding values in the diffusion limit is also demonstrated. Numerical examples from the constant
elasticity of variance stock price and the Cox, Ingersoll, and Ross (1985) discount bond price are provided.
Option replication is discussed in a discrete-time framework with transaction costs. The model represents an extension of the Cox-Ross-Rubinstein binomial option pricing model to cover the case of proportional transaction costs. The method proceeds by constructing the appropriate replicating portfolio at each trading interval. Numerical values of these prices are presented for a range of parameter values. The paper derives a simple Black-Scholes type approximation for the option prices with transaction costs and demonstrates numerically that it is quite accurate for plausible parameter values. Copyright 1992 by American Finance Association.
One option-pricing problem which has hitherto been unsolved is the pricing of European call on an asset which has a stochastic volatility. This paper examines this problem. The option price is determined in series form for the case in which the stochastic volatility is independent of the stock price. Numerical solutions are also produced for the case in which the volatility is correlated with the stock price. It is found that the Black-Scholes price frequently overprices options and that the degree of overpricing increases with the time to maturity. Copyright 1987 by American Finance Association.
We introduce adaptive learning behavior into a general-equilibrium life-cycle economy with capital accumulation. Agents form forecasts of the rate of return to capital assets using least-squares autoregressions on past data. We show that, in contrast to the perfect-foresight dynamics, the dynamical system under learning possesses equilibria that are characterized by persistent excess volatility in returns to capital. We explore a quantitative case for theselearning equilibria. We use an evolutionary search algorithm to calibrate a version of the system under learning and show that this system can generate data that matches some features of the time-series data for U.S. stock returns and per-capita consumption. We argue that this finding provides support for the hypothesis that the observed excess volatility of asset returns can be explained by changes in investor expectations against a background of relatively small changes in fundamental factors.
Models now exist for valuing the default option embedded in a mortgage. Implicitly, these models generate all the information necessary to determine the probability of default, in any possible situation. Economists and policymakers may find such default probabilities considerably more interesting than the nonobservable dollar value of the default option. This paper provides the analytical procedure necessary to calculate such probabilities and presents a wide range of results. The decision to terminate a mortgage results in the loss of the options to default or prepay in the future. Because of this consideration, the price of a house must fall below the point of zero equity before a rational borrower defaults. We provide evidence that even in the absence of transaction costs this value of delay results in substantial levels of negative equity being observed without default occurring. In fact, by not accounting for the value of delay, most of the current empirical literature cited in this article substantially overestimates the role of transaction costs in the decision to default.
We examine the efficiency of residential housing markets by examining price, rent, and cost of capital indices generated from a transactions level data base for Alameda and San Francisco Counties in Northern California. We can reject both constant and nonconstant discount rate versions of the housing price present value relation in the short run. Long-run results are consistent with the housing price present value relation when we adjust the discount factor for changes in both tax rates and borrowing costs that characterize our 1970-1988 sample period. Our preferred explanation for the short-run rejection of, but long-run consistency with, the present value model is the high transaction costs that characterize the housing market. However, we cannot rule out the possibility that deviations of actual and present value price are due to asset market bubbles or nonrational expectations. © Academic Press.
ABSTRACTGNMA 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.