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Abstract:
The term structure of interest rates gives the relationship between the yield on an investment and the term to maturity of the investment. Since the term structure is typically measured using default-free, continuously-compounded, annualized zero-coupon yields, it is not directly observable from the published coupon bond prices and yields. This paper focuses on how to estimate the default-free term structure of interest rates from bond data using three methods: the bootstrapping method, the McCulloch cubic-spline method, and the Nelson and Siegel method. Nelson and Siegel method is shown to be more robust than the other two methods. The results of this paper can be implemented using user-friendly Excel spreadsheets.

interest rates, term structure, bonds, fixed income, Excel

Abstract:
In our experience, most finance students are unnecessarily confused by the roles that duration and convexity play in the traditional textbook plot of bond price versus bond yield. The slope of the bond price-yield plot does not define bond duration, and the curvature of the bond price-yield plot does not define bond convexity. We demonstrate several common misunderstandings regarding duration and convexity, and we offer a new bond return-yield plot for illustrating the roles of duration and convexity.

Duration, Convexity, Bond return, Bond price, yield

Abstract:
This paper gives a practical, and easy-to-follow introduction to arbitrage-free pricing using martingales with a discrete two-period information structure. Using simple heuristic derivations, we illustrate the concepts of Arrow-Debreu prices, complete and incomplete markets, risk-neutral measure, stochastic discount factor (or pricing kernel), and Radon-Nikodym derivative. We use the discrete-time setup to give a clear and intuitive demonstration of the fundamental theorem of asset pricing, which states that absence of arbitrage is equivalent to the existence of an equivalent martingale measure under which discounted prices are martingales. We also introduce arbitrage-free pricing using martingales in continuous-time, and show the correspondence of the continuous-time results with the discrete-time results. Further, we provide two additional theorems in continuous-time, given as the Girsanov theorem and the Feynman-Kac theorem. The first theorem is used for performing a change of measure, and the second theorem is used for deriving a PDE from an expectation, and vice-versa.

arbitrage-free pricing, valuation, martingales, Arrow-Debreu prices, riskneutral measure, forward measure, stochastic discount factor, pricing kernel, Radon-Nikodym derivative, Girsanov theorem, Feynman Kac theorem

Abstract:
This paper reviews the recent literature on CAPM and APT, and reaches a surprising conclusion. While APT died a silent death, CAPM's progeny is alive and well! We provide a short review of the recent literature on the conditional CAPMs, intertemporal CAPMs, and higher-order Co-Moments-based CAPMs. Some of these multifactor extensions of CAPM not only have higher explanatory power than the three-factor Fama and French (FF) model, but also are not rejected in the empirical tests, while the FF model is rejected.

APT, CAPM, Multifactor Models, Beta, Asset Pricing

Abstract:
Is the Arbitrage Pricing Theory dead? This paper addresses this question by deriving a multibeta representation theorem, which can price assets using arbitrary reference variables that are not the true factors. Under this theorem, the upper bound on pricing deviations depends upon the correlations not only between the reference variables and the factors but also between the reference variables and the residual risks. A new concept of a well-diversified variable is introduced, which though free of residual risk, may be less than perfectly correlated with the true factors. Well-diversified variables correlated with the factors play a key role in the pricing of assets, since these variables can replace the factors without any loss in pricing accuracy under all linear asset pricing theories. A set of corollaries to the multibeta representation theorem are derived, which are fatal for both the APT as well as the equilibrium APT. Both these theories lead to pricing of arbitrary reference variables, which allows rejection of these theories based on casual empiricism. Since Merton's ICAPM does not suffer from such arbitrary pricing, it represents a viable theoretical framework for the empirical multifactor asset pricing models, such as the 3-factor Fama and French model.

Asset pricing, APT, Arbitrage pricing theory, Multibeta CAPM, ICAPM, Factor models, Fama and French model

Abstract:
This paper shows how to price American interest rate options under the exponential jumps-extended Vasicek model, or the Vasicek-EJ model. We modify the Gaussian jump-diffusion tree of Amin [1993] and apply to the exponential jumps-based short rate process under the Vasicek-EJ model. The tree is truncated at both ends to allow fast computation of option prices. We also consider the time-inhomogeneous version of this model, denoted as the Vasicek-EJ model that allows exact calibration to the initially observable bond prices. We provide an analytical solution to the deterministic shift term used for calibrating the short rate process to the initially observable bond prices, and show how to generate the jump-diffusion tree for the Vasicek-EJ model. Our simulations show fast convergence of European option prices obtained using the jump-diffusion tree, to those obtained using the Fourier inversion method for options on zero-coupon bonds (or caplets), and the cumulant expansion method for options on coupon bonds (or swaptions).

Bond options, Interest Rate Trees, Jumps, Vasicek Model, American options

Abstract:
Dynamic Term Structure Modeling, the second book in the trilogy on the Fixed Income Valuation Course, shows how to value interest rate derivatives and credit derivatives using a variety of affine, quadratic, HJM, and LIBOR market models. Using a new taxonomy, this book classifies all term structure models as either fundamental models, or preference-free single-plus, double-plus, and triple-plus models. Filled with in-depth insights and expert advice, this book shows how to price basic interest rate and credit derivative products, such as Treasury and Eurodollar futures, bond options, interest rate options (e.g., caps, floors, and swaptions), forward rate agreements, interest rate swaps, credit default swaps, credit spread options, and others. With intuitive explanations and fully developed examples, this book provides new transforms for building efficient trees under state-dependent volatility models, stochastic volatility models, and jump-diffusion models, for pricing American options, and describes fast computational methods such as the Fourier inversion method (including the FFT) and the cumulant expansion method, for valuing interest rate derivatives and credit derivatives, under a variety of affine, quadratic, and LIBOR market models. This book is also accompanied by an informative CD-ROM, which contains various Excel®/VBA® spreadsheets. This software allows for valuation of interest rate derivatives by building interest rate trees for low-dimensional affine models, as well as computing solutions using quasi-analytical formulas for higher-dimensional affine, quadratic, and LIBOR market models. Though most of the programs require coding in advanced scientific languages, such as C and C++, the final output is presented in user-friendly Excel/VBA spreadsheets.

term structure models, caps, swaptions, credit default swaps, credit derivatives

Abstract:
This paper tests empirically whether convexity is return enhancing (the traditional view based upon parallel term structure shifts), or return diminishing (the equilibrium view suggesting convexity is priced). Results of empirical tests over different time periods show bond convexity to be either insignificantly or negatively related to ex-ante bond returns. These results are consistent with the critique of the traditional duration model by Ingersoll, Skelton, and Weil [1978] and suggest that bond convexity may be priced. Further, the magnitude of bond convexity is shown to be related directly to the immunization risk inherent in a bond portfolio, consistent with the implications of Fong and Vasicek's [1983, 1984] M-Square model.

Convexity, risk, immunization, bonds, M-Square, hedging

Abstract:
How do the managers of financial institutions hedge against the effects of non-parallel yield curve shifts? This paper addresses this important issue by reviewing the important findings in the area of interest rate risk management over the past two decades. We discuss four classes of models in the fixed income literature that deal with hedging the risk of large, non-parallel yield curve shifts. These models are given as M-Absolute/M-Square models, duration vector models, key rate duration models, and principal component duration models. These models can be used for designing various passive strategies such as portfolio immunization, bond index replication, and duration gap management; and hybrid strategies (i.e., active/passive) such as targeted yield curve shifts speculation (based on change in either the height, and/or the slope, and/or the curvature of the yield curve) and contingent immunization.

interest rate, yield curve, fixed income, duration, immunization, portfolio strategy

Abstract:
In a recent paper, Beliaeva and Nawalkha [2008] present a multidimensional transform for generating path-independent trees for pricing American options under low-dimensional stochastic volatility models. This approach has advantages over both the GARCH tree method of Ritchken and Trevor [1999] and the Monte Carlo regression method of Longstaff and Schwartz [2001]. In this paper, we give an explicit demonstration of this approach using the specific example of the Heston [1993] stochastic volatility model. This approach obtains highly accurate American option prices within a fraction of a second using the control variate method.

Heston, options, stochastic volatility, American options, trees

Abstract:
This paper presents efficient binomial and trinomial trees for the Cox, Ingersoll, and Ross (CIR) and the constant-elasticity-of-variance (CEV) short rate models. We correct an error in the original square root transform of Nelson and Ramaswamy [1990], and modify their transform by truncating the tree exactly at the zero-boundary. This not only allows us to create computationally more efficient trees for the CIR square-root process, but also for the entire class of CEV models of the short rate. Our simulations show fast convergence and significantly improved performance of the truncated-tree approach over the Nelson-Ramaswamy approach.

Trees, Binomial, Trinomial, American options, CIR, Cox Ingersoll and Ross, Constant Elasticity of Variance, Short rate, caps, interest rate

Abstract:
Are the managers of financial institutions ready for the small but increasingly significant risk of inflation in the near future, due to the unprecedented fiscal and monetary responses of the U.S. government to prevent an economic collapse? This paper addresses this important issue by reviewing important findings in the area of interest rate risk management. We discuss five classes of models in the fixed income literature that deal with hedging the risk of large, non-parallel yield curve shifts. These models are given as M-Absolute/M-Square models, duration vector models, key rate duration models, principal component duration models, and extensions of these models for fixed income derivatives, for valuing and hedging bonds, loans, demand deposits, and other fixed income instruments. These models can be used for designing various hedging strategies such as portfolio immunization, bond index replication, duration gap management, and contingent immunization, to protect against changes in the height, slope, and curvature of the yield curve. We argue that the current regulatory models proposed by the U.S. Federal Reserve, the Office of Thrift Supervision, and the Bank of International Settlements, may understate the true interest rate risk exposure of financial institutions, if sharp increases in interest rates lead to higher default risk and quickening of the pace of deposit withdrawals.

interest rate risk, duration, convexity, key rate, inflation

Abstract:
This paper presents jump extensions to the Cox, Ingersoll, and Ross (CIR) and the constant-elasticity-of-variance (CEV) models of the short rate, with analytical solutions for the case of exponential jumps, and efficient lattice-based solutions for both exponential jumps and lognormal jumps. We demonstrate how to superimpose a recombining multinomial jump tree on the diffusion tree, creating the mixed jump-diffusion trees for CIR and CEV models extended with jumps. Finally we also present the preference-free versions of these models that allow these models to be fully calibrated to an initially observed forward rate curve, making them consistent with the HJM [1992] paradigm. Our simulations show fast convergence of the trees to the respective analytical solutions.

Interest rate models, Term structure models, Jumps, CIR, CEV, Trees

Abstract:
This paper provides arbitrage and equilibrium foundations of the traditional duration risk measure (see Macaulay [1938] and Hicks [1939]), by relating it to the Heath, Jarrow and Morton (HJM) [1992] term structure theory and Merton's intertemporal CAPM [1973]. Under the new approach the duration model is shown to be consistent with a subset of arbitrage-free forward rate processes of HJM, some of which preclude the occurrence of negative interest rates by allowing interest rate level dependent volatilities. Conditions are derived under which the convexity risk measure may or may not be priced. Finally, we demonstrate that when Merton's [1973] ICAPM is identified with the above HJM [1992] forward rate processes, the appropriate equilibrium measure of the systematic risk of a default-free security is its duration, and not its bondbeta as derived by Jarrow [1978], and others, under more restrictive assumptions. This paper addresses all of the arbitrage-based and equilibrium-based criticisms of the duration risk measure given by Ingersoll [1978], Sharpe [1983], and others.

Interest Rate Risk, Systematic Risk, Duration, Convexity, Bond beta, ICAPM, HJM

Abstract:
This paper derives simple stock valuation formulas for pricing zero-dividend and positive-dividend stocks using alternative wealth creation models given as: i) the EVA model of Stewart [1991], ii) the residual income model of Edwards and Bell [1961] and Ohlson [1990, 1991, 1995], and iii) the franchise factor model of Leibowitz and Kogelman [1990, 1992, 1994]. An advantage of these models over the wealth distribution models (such as, Gordon's [1963] dividend discount model) is that dividends are obtained endogenously under these models. We derive formulas both for dividend paying as well as zero-dividend paying stocks under multiple growth rates. These formulas are easy to use and allow a variety of assumptions that can be input by financial analysts for pricing stocks.

Stock Valuation; Dividend discount model, Franchise factor model, Residual income model, EVA model.

Abstract:
This paper generalizes the M-square and M-vector models (Fong and Fabozzi [1985] and Nawalkha and Chambers [1997]) by using a Taylor series expansion of the bond return function with respect to simple polynomial functions of the cash flow maturities. The classic M-vector computes the weighted averages of the distance between the maturity of each cash flow and the portfolio horizon, raised to integer powers (e.g., (t - H)^1, (t - H)^2, (t - H)^3, etc.). Implementation of the new approach involves computing the weighted averages of the distance between some polynomial function of the maturity of each cash flow and that of the portfolio horizon, raised to integer powers (e.g., (t^0.5 - H^0.5)^1, (t^0.5 - H^0.5)^2, (t^0.5 - H^0.5)^3, etc.). We test six different generalized M-vector models corresponding to six different polynomial functions. It is shown that polynomial functions of lower power (i.e., 0.25 or 0.5) provide significantly enhanced protection from interest rate risk, when higher-order generalized M-vector models are used.

immunization, duration, interest rate, risk management, fixed income

Abstract:
This article derives and tests a multiple-factor extension of the M-square model (see Fong and Vasicek [1984] and Fong and Fabozzi [1985]), termed as the M-vector model. Tests of the M-square model indicate that the model reduces the interest rate risk inherent in the traditional duration model by more than half. The M-vector model demonstrates near-perfect hedging performance, eliminating more than 95% of interest rate risk inherent in the traditional duration model.

Interest Rate Risk, Duration, Convexity, M-Square, M-Vector, Duration Vector

Abstract:
This paper provides a contingent claims analysis of the interest rate risk characteristics of corporate liabilities by identifying Merton's (1973) option pricing model with Vasicek's (1977) mean reverting term structure model. Only a non-zero positive range of duration values for the firms' assets is shown to be consistent with the previous empirical evidence on the interest rate sensitivity of corporate stocks and bonds. Chance's (1990) duration measure is shown to be biased downward under empirically realistic conditions. Theoretical conditions are derived under which the duration of a default-prone zero coupon bond can be either higher or lower than the duration of the corresponding default-free bond. The duration of the default-prone bond of a firm with high (low) interest rate sensitive assets is shown to be an increasing (decreasing) function of the bond's default-risk.

Interest rate risk, stocks, bonds, duration, Merton, Vasicek

Abstract:
A well known problem in finance is the absence of a closed form solution for volatility in common option pricing models. Several approaches have been developed to provide closed form approximations to volatility. This paper examines Chance's (1993, 1996) model, Corrado and Miller's (1996) model and Bharadia, Christofides and Salkin's (1996) model for approximating implied volatility. We develop a simplified extension of Chance's model that has greater accuracy than previous models. Our tests indicate dramatically improved results.

Abstract:
A well-known problem in finance is the absence of a closed form solution for volatility in common option pricing models. Several approaches have been developed to provide closed form approximations to volatility. This paper examines Chance's (1993, 1996) model, Corrado and Miller's (1996) model and Bharadia, Christofides and Salkin's (1996) model for approximating implied volatility. We develop a simplified extension of Chance's model that has greater accuracy than previous models. Our tests indicate dramatically improved results.

Abstract:
We derive a simple expression for the sensitivity of duration, convexity, and higher-order bond risk measures to changes in term structure shape parameters. Our analysis enables fixed income portfolio managers to capture the combined effects of term structure level, slope, and curvature shifts on any specific bond risk measure. These results are particularly important in volatile interest rate environments. We provide simple numerical examples.

bond risk measures, duration, convexity, term structure, M-square

Abstract:
This paper provides a continuous-time contingent claims framework for the duration vector models of Chambers, Carleton, and McEnally [1988], Prisman and Shores [1988], and others, by embedding these into Merton's [1973] stochastic interest rate option pricing model. Unlike the traditional comparative static duration approach, the duration vector model allows non-parallel term structure shifts resulting from height. slope, and curvature factors. The upper and lower bounds of the duration vectors of bond options and callable bonds are investigated. An empirical technique is suggested for the computation of the duration vectors of bond options and callable bonds by using the implied volatility information contained in the market prices of these contingent claims securities.

Interest rate risk, duration vector, options, calls, puts, callable bonds, puttable bonds, hedging, Merton

Abstract:
This paper reexamines the relationship between investors' preferences and the binomial option pricing model of Cox, Ross, and Rubinstein (CRR). It is shown that the independence of the binomial option pricing model from investors' preferences is a result of a special choice of binomial parameters made by CRR. For a more general choice of binomial parameters, risk neutrality cannot be obtained in discrete time. This analysis reveals the essential difference between the "risk neutral" valuation approach of Cox and Ross and the equivalent martingale approach of Harrison and Kreps in a discrete time framework.

binomial model, risk-neutrality, martingale valuation, options, trees

Abstract:
This paper gives a new taxonomy of dynamic term structure models that classifies all existing TSMs as either fundamental models or preference-free single-plus, double-plus, and triple-plus models. We exemplify the new taxonomy by considering preference-free versions of some well-known fundamental short rate models. Single-plus extensions of the fundamental models are shown to be both time-homogeneous and preference-free - two characteristics which do not simultaneously hold under any existing class of TSMs. Though the analytical apparatus for pricing fixed income securities is identical under fundamental models and single-plus models, the latter models are consistent with general non-linear forms of MPRs which may also depend upon an arbitrary set of state variables, leading to better estimates of risk-neutral parameters. The preference-free double-plus and triple-plus extensions of the fundamental models are similar to the Heath, Jarrow, and Morton [1992] models, in that time-inhomogeneous drifts and volatilities are used as "smoothing variables" to fit the initial bond prices and initial term structure of volatilities, respectively.

Term Structure, Interest Rate, Single-Plus, Double-Plus, Vasicek, CIR, USV

Abstract:
This paper presents a critical review of the different versions of the LIBOR market model (LMM). Based on the new taxonomy of the term structure models (see Nawalkha, Beliaeva, and Soto [2007a, 2007b]) the typical application of the LMM are shown to triple-plus type, exposing these to the dangers of “smoothing.” This paper also derives a double-plus version of Jarrow, Li, and Zhao’s (JLZ) [2007] LMM model with stochastic volatility and jumps, which is less exposed to the dangers of “smoothing” compared with the original triple-plus version of this model. Finally, this paper makes a persuasive case for considering high-dimensional double-plus term structure models in the affine and quadratic classes. Fast computational methods make these models powerful alternatives to the LMM for valuing and hedging interest rate derivatives.

LIBOR market model, affine and quadratic models, interest rate models, caps, swaptions

Abstract:
The traditional duration model has limited power for protecting against interest rate risk. A new risk measure, entitled M-Absolute, is designed to provide powerful and practical single-risk-measure immunization in particular circumstances. M-Absolute is similar to M-Square but is derived as a first-order interest-rate-risk hedging model. M-Absolute of a bond is defined as the weighted average of the absolute distances of the bond's cash flows from a horizon point. Even though it is a single-risk measure, M-Absolute can act effectively to reduce the impacts of several types of interest rate risks rather than hedge against only a single type of term structure shift. Empirical tests show that M-Absolute reduces the interest rate risk inherent in the traditional duration model by more than half. These results are independent of the particular time period chosen.

Duration, Convexity, M-Absolute, M-Square, Interest Rate Risk, Hedging

Abstract:
This paper derives a multibeta representation theorem for pricing assets using arbitrary reference variables that are not necessarily the true factors. Under this theorem, the upper bound on pricing deviations depends upon the correlations not only between the reference variables and the factors but also between the reference variables and the residual risks. A new concept of a well-diversified variable is introduced, which though free of residual risk, may be less than perfectly correlated with the true factors. Well diversified variables correlated with the factors play a key role in the pricing of assets, since these variables can replace the factors without any loss in pricing accuracy under all linear asset pricing theories.

Asset pricing, APT, Multibeta CAPM, Factor models, Fama and French

Abstract:
This paper derives a multibeta representation theorem for pricing assets using arbitrary reference variables that are not necessarily the true factors. Under this theorem, the upper bound on pricing deviations depends upon the correlations not only between the reference variables and the factors but also between the reference variables and the residual risks. A new concept of a well-diversified variable is introduced, which though free of residual risk, may be less than perfectly correlated with the true factors. Well diversified variables correlated with the factors play a key role in the pricing of assets, since these variables can replace the factors without any loss in pricing accuracy under all linear asset pricing theories.

Asset pricing, APT, Multibeta CAPM, Factor models

Abstract:
This paper derives formulas for higher order duration measures, including D(1) (i.e. Macaulay duration), D(2) (i.e., slope duration), D(3) (curvature duration), etc. We develop a general iterative method to obtain formulas for any higher order measure D(m), for an arbitrary positive integer value for m. The higher order duration measures can be used to hedge against virtually any type of interest rate shift. Our algorithm can be easily adapted for spreadsheet applications.

Duration, interest rate hedging, immunization, term structure, bonds

Abstract:
Closed-form formulas for Macaulay duration, as given by Babcock and Chua, provide the user with a less cumbersome and more efficient procedure for calculating duration. Recent developments, however, have suggested alternative measures of bond portfolio immunization designed to overcome the severe restrictions that Macaulay duration places on permitted interest rate behavior. This note presents closed-form formulas for two such alternative measures - convexity and M-square and demonstrates how these measures can be used in an immunization strategy.

bond convexity, M-square, interest rates, immunization, interest rate risk

Abstract:
This paper considers Merton's (1973) model for partial equilibrium bond option pricing when stochastic bond price processes are involved. A log-normal process with a stochastic drift is suggested that allows the price of a pure discount bond to converge to its face value upon maturity. The stochastic process for the instantaneous short rate implicit in the bond price dynamics is identified. A necessary condition for our approach to be consistent with the arbitrage-free martingale pricing results of Harrison and Kreps (1979) is that the excess holding period bond return and the standard deviation of the bond return are both continuously differentiable with respect to time.

Bond options, Face value convergence, Contingent claims

Abstract:
This paper derives analytical solutions for valuing credit default swaps (CDS) using preference-free multifactor affine and quadratic models, under the recovery of face value (RFV) assumption. We use a preference-free framework, which is independent of the market prices of risk, and yet allows the short rate process and the spread process to be time-homogeneous. The solutions allow arbitrary number of factors for the short rate and the default intensity, and nest the solutions of Longstaff, Mithal, and Neis [2003], and Pan and Singleton [2005]. The multifactor framework used in this paper allows a better fit with default-free bond prices and CDS spreads.

credit default swaps, CDS, reduced form models, interest rate models, term structure models, affine, quadratic

Abstract:
This paper derives analytical solutions for valuing Eurodollar/Euribor futures using multifactor affine and quadratic models. We use a preference-free framework independent of the market prices of risk. The preference-free "single-plus" models allow the short rate process to be time-homogeneous. The preference-free "double-plus" models allow the short rate process to be time-inhomogeneous, such that the model prices are consistent with initially observable bond prices. Our solutions allow arbitrary number of factors for the short rate and nest virtually all other solutions given in the literature. We also solve the convexity-bias in closed-form under various multifactor affine and quadratic models.

Eurodollar futures, Euribor futures, Interest rate models, Term structure models, Affine, Quadratic, Convexity bias