Alternative Defaultable Term Structure Models

The objective of this paper is to consider defaultable term structure models in a general setting beyond standard risk-neutral models. Using as numeraire the growth optimal portfolio, defaultable interest rate derivatives are priced under the real-world probability measure. Therefore, the existence of an equivalent risk-neutral probability measure is not required. In particular, the real-world dynamics of the instantaneous defaultable forward rates under a jump-diffusion extension of a HJM type framework are derived. Thus, by establishing a modelling framework fully under the real-world probability measure, the challenge of reconciling real-world and risk-neutral probabilities of default is deliberately avoided, which provides significant extra modelling freedom. In addition, for certain volatility specifications, finite dimensional Markovian defaultable term structure models are derived. The paper also demonstrates an alternative defaultable term structure model. It provides tractable expressions for the prices of defaultable derivatives under the assumption of independence between the discounted growth optimal portfolio and the default-adjusted short rate. These expressions are then used in a more general model as control variates for Monte Carlo simulations of credit derivatives. Nicola Bruti-Liberati: In memory of our beloved friend and colleague.     

The optimal trade-off between interest rate risk and annual return of bond ladders

Bond laddering is a popular fixed-income investment strategy. The main purpose of this paper is to develop a methodology for determining private investors’ most interest rate risk (IRR)-return-efficient investment horizon for bond ladders (BLs), which are virtually free of credit risk. Two IRR measures of a continuously rolling and homogenous BL (CRHBL) are analytically derived under the assumption that interest rates are martingales. The first measure is the modified duration, which assumes a flat term structure of interest rates. However, this assumption is not fully supported by the empirical data and, thus, an additional IRR measure is proposed. Under each of these two measures, the ratios between the annual return in excess of the demand deposit rate and IRR of CRHBLs with different investment horizons are calculated. As expected, CRHBLs with rather low IRR are most risk-return-efficient. The results for the theoretical CRHBLs also apply to “real-world” discrete BLs. Thus, the proposed methodology can help private investors construct IRR-return-efficient discrete BLs.     

“Social Security—Adequacy,Equity, and Progressiveness: A Review of Criteria Based on Experience In Canada and the United States”, Robert L. Brown,January 2000

Abstract

We consider the valuation of credit default swaps (CDSs) under an extended version of Merton’s structural model for a firm’s corporate liabilities. In particular, the interest rate process of a money market account, the appreciation rate, and the volatility of the firm’s value have switching dynamics governed by a finite-state Markov chain in continuous time. The states of the Markov chain are deemed to represent the states of an economy. The shift from one economic state to another may be attributed to certain factors that affect the profits or earnings of a firm; examples of such factors include changes in business conditions, corporate decisions, company operations, management strategies, macroeconomic conditions, and business cycles. In this article, the Esscher transform, which is a well-known tool in actuarial science, is employed to determine an equivalent martingale measure for the valuation problem in the incomplete market setting. Systems of coupled partial differential equations (PDEs) satisfied by the real-world and risk-neutral default probabilities are derived. The consequences for the swap rate of a CDS brought about by the regimeswitching effect of the firm’s value are investigated via a numerical example for the case of a two-state Markov chain. We perform sensitivity analyses for the real-world default probability and the swap rate when different model parameters vary. We also investigate the accuracy and efficiency of the PDE approach by comparing the numerical results from the PDE approach to those from the Monte Carlo simulation.     

Pricing defaultable bonds: a middle-way approach between structural and reduced-form models

In this paper we present a valuation model that combines features of both the structural and reduced-form approaches for modelling default risk. We maintain the cause and effect or ‘structural’ definition of default and assume that default is triggered when a state variable reaches a default boundary. However, in our model, the state variable is not interpreted as the assets of the firm, but as a latent variable signalling the credit quality of the firm. Default in our model can also occur according to a doubly stochastic hazard rate. The hazard rate is a linear function of the state variable and the interest rate. We use the Cox et al. (A theory of the term structure of interest rates. Econometrica, 1985, 53(2), 385–407) term structure model to preclude the possibility of negative probabilities of default. We also horse race the proposed valuation model against structural and reduced-form default risky bond pricing models and find that term structures of credit spreads generated using the middle-way approach are more in line with empirical observations.     

Local volatility function models under a benchmark approach

Without requiring the existence of an equivalent risk-neutral probability measure this paper studies a class of one-factor local volatility function models for stock indices under a benchmark approach. It is assumed that the dynamics for a large diversified index approximates that of the growth optimal portfolio. Fair prices for derivatives when expressed in units of the index are martingales under the real-world probability measure. Different to the classical approach that derives risk-neutral probabilities the paper obtains the transition density for the index with respect to the real-world probability measure. Furthermore, the Dupire formula for the underlying local volatility function is recovered without assuming the existence of an equivalent risk-neutral probability measure. A modification of the constant elasticity of variance model and a version of the minimal market model are discussed as specific examples together with a smoothed local volatility function model that fits a snapshot of S&P500 index options data.     

Insurance Considering a New Stochastic Model for the Discount Factor

In many empirical situations (e.g.: Libor), the rate of interest will remain fixed at a certain level (random instantaneous rate i i ) for a random period of time ( t i ) until a new random rate should be considered, i i + 1 , that will remain for t i + 1 , waiting time until the next change in the rate of interest. Three models were developed using the approach cited above for random rate of interest and random waiting times between changes in the rate of interest. Using easy integral transforms (Laplace & Fourier) we will be able to calculate the moments of the probability function of the discount factor, V ( t ), and even its c.d.f. The approach will also be extended to the calculation of the expected value (net premium) and variance of a term insurance and we will get its c.d.f., something not very common in actuarial literature due to its complexity, but very useful when the law of large numbers cannot be applied and consequently use normal approximations.     

A Two-Factor Model for Low Interest Rate Regimes

This paper derives a two-factor model for the term structure of interest rates that segments the yield curve in a natural way. The first factor involves modelling a non-negative short rate process that primarily determines the early part of the yield curve and is obtained as a truncated Gaussian short rate. The second factor mainly influences the later part of the yield curve via the market index. The market index proxies the growth optimal portfolio (GOP) and is modelled as a squared Bessel process of dimension four. Although this setup can be applied to any interest rate environment, this study focuses on the difficult but important case where the short rate stays close to zero for a prolonged period of time. For the proposed model, an equivalent risk neutral martingale measure is neither possible nor required. Hence we use the benchmark approach where the GOP is chosen as numeraire. Fair derivative prices are then calculated via conditional expectations under the real world probability measure. Using this methodology we derive pricing functions for zero coupon bonds and options on zero coupon bonds. The proposed model naturally generates yield curve shapes commonly observed in the market. More importantly, the model replicates the key features of the interest rate cap market for economies with low interest rate regimes. In particular, the implied volatility term structure displays a consistent downward slope from extremely high levels of volatility together with a distinct negative skew. 1991 Mathematics Subject Classification: primary 90A12; secondary 60G30; 62P20 JEL Classification: G10, G13     

Can negative interest rates really affect option pricing? Empirical evidence from an explicitly solvable stochastic volatility model

The profound financial crisis generated by the collapse of Lehman Brothers and the European sovereign debt crisis in 2011 have caused negative values of government bond yields both in the USA and in the EURO area. This paper investigates whether the use of models which allow for negative interest rates can improve option pricing and implied volatility forecasting. This is done with special attention to foreign exchange and index options. To this end, we carried out an empirical analysis on the prices of call and put options on the US S&P 500 index and Eurodollar futures using a generalization of the Heston model in the stochastic interest rate framework. Specifically, the dynamics of the option’s underlying asset is described by two factors: a stochastic variance and a stochastic interest rate. The volatility is not allowed to be negative, but the interest rate is. Explicit formulas for the transition probability density function and moments are derived. These formulas are used to estimate the model parameters efficiently. Three empirical analyses are illustrated. The first two show that the use of models which allow for negative interest rates can efficiently reproduce implied volatility and forecast option prices (i.e. S&P index and foreign exchange options). The last studies how the US three-month government bond yield affects the US S&P 500 index.     

Cash management using multi-stage stochastic programming

We consider a cash management problem where a company with a given financial endowment and given future cash flows minimizes the Conditional Value at Risk of final wealth using a lower bound for the expected terminal wealth. We formulate the optimization problem as a multi-stage stochastic linear program (SLP). The company can choose between a riskless asset (cash), several default- and option-free bonds, and an equity investment, and rebalances the portfolio at every stage. The uncertainty faced by the company is reflected in the development of interest rates and equity returns. Our model has two new features compared to the existing literature, which uses no-arbitrage interest rate models for the scenario generation. First, we explicitly estimate a function for the market price of risk and change the underlying probability measure. Second, we simulate scenarios for equity returns with moment-matching by an extension of the interest rate scenario tree.     

A generalization of reset options pricing formulae with stochastic interest rates

A generalization of reset call options with predetermined dates is derived in the case of time-dependent volatility and time-dependent interest rate by applying martingale method and change of nume?aire or change of probability measure. An analytical pricing formula for the reset call option is also obtained when the interest rate follows an extended Vasicek’s model. Numerical results show that the correlated coefficient between the stock price and interest rate is almost unacted on the price of reset call option with short maturity and Monte Carlo method is inefficient. Monte Carlo method should be only used if there is no closed-formed solution for option pricing.     

Bond price representations and the volatility of spot interest rates

A common approach to modeling the term structure of interest rates in a single-factor economy is to assume that the evolution of all bond prices can be described by the current level of the spot interest rate. This article investigates the restrictions that this assumption imposes. Specifically, we show that this Markovian restriction, together with the no-arbitrage requirement, curtails the relationship of forward rates and their volatilities relative to spot-rate volatilities. Among such Markovian models, only a few provide simple analytical relationships between bond prices and the spot interest rate. This article identifies the class of spot-rate volatility specifications that permit simple analytical linkages to be derived between bond prices and interest rates. Included in the class are the volatility structures used by Vasicek and by Cox, Ingersoll, and Ross. Surprisingly, no other volatility structures permit simple analytical representations.     

A contingent claims analysis of the interest rate risk characteristics of corporate liabilities

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.     

Multi-factor Affine Term Structure Model with Single Regime Shift: Real Term Structure under Zero Interest Rate

In this paper, we extend the one-factor, single regime shift, affine term structure model with time-dependent regime-shift probability to a multi-factor model. We model the nominal interest rate and the expected inflation rate, and estimate the term structure of the real interest rate in the Japanese government bond market using inflation-indexed bond data under zero interest rates. Incorporating the economic structure that the Bank of Japan terminates the zero interest rate when the expected inflation rate gets out of deflationary regime, we estimate the yield curve of the real interest rate for less than 10 years, consistent with the expectation of the market participants in the Japanese government bond market, where inflation-indexed bonds are traded for only around 10 years.     

International money and stock market contingent claims

We develop a unified approach with closed-form solutions for pricing bonds, stocks, currencies and their derivatives. The specification assumes a fundamental risk factor represented by a stochastic positive definite matrix following a Wishart autoregressive (WAR) process. By assuming a volatility-in-mean specification for the domestic stock returns and the relative changes of exchange rates, and a domestic stochastic discount factor exponential affine with respect to the fundamental risk, it is possible to derive closed form solutions for the term structures of interest rates and for the risk-neutral probabilities while keeping the flexibility of the model. In particular:

i) The domestic and foreign term structures are jointly affine and correspond to Wishart quadratic term structures, which can ensure the positivity of interest rates;

ii) In this framework where the stock price follows a model with stochastic volatility, we obtain explicit or quasi-explicit formulas for futures and forward contracts, swaps and options. This extends results by

Heston (1993)

and

Ball and Roma (1994)

.

Asymptotics of bond yields and volatilities for extended CIR models under the real-world measure

ABSTRACT

The Cox–Ingersoll–Ross CIR short rate model is a mean-reverting model of the short rate which, for suitably chosen parameters, permits closed-form valuation formulae of zero-coupon bonds and options on zero-coupon bonds. This article supplies proofs of the formulae for the expected present value of payoffs under the real-world probability measure, known as actuarial valuation. Importantly, we give formulae for asymptotic levels of bond yields and volatilities for extended CIR models when suitable conditions are imposed on the model parameters.     

Valuing Fixed-Income Options and Mortgage-Backed Securities with Alternative Term Structure Models

To value mortgage-backed securities and options on fixed-income securities, it is necessary to make assumptions regarding the term structure of interest rates. We assume that the multi-factor fixed parameter term structure model accurately represents the actual term structure of interest rates, and that the values of mortgage-backed securities and discount bond options derived from such a term structure model are correct. Differences in the prices of interest rate derivative securities based on single-factor term structure models are therefore due to pricing bias resulting from the term structure model. The price biases that result from the use of single-factor models are compared and attributed to differences in the underlying models and implications for the selection of alternative term structure models are considered.     

Negative real interest rates

Standard textbook general equilibrium term structure models such as that developed by Cox, Ingersoll, and Ross [1985b. “A Theory of the Term Structure of Interest Rates.” Econometrica 53 (2): 385–407], do not accommodate negative real interest rates. Given this, the Cox, Ingersoll, and Ross [1985b. “A Theory of the Term Structure of Interest Rates.” Econometrica 53 (2): 385–407] ‘technological uncertainty variable’ is formulated in terms of the Pearson Type IV probability density. The Pearson Type IV encompasses mean-reverting sample paths, time-varying volatility and also allows for negative real interest rates. The Fokker–Planck (i.e. the Chapman–Kolmogorov) equation is then used to determine the conditional moments of the instantaneous real rate of interest. These enable one to determine the mean and variance of the accumulated (i.e. integrated) real rate of interest on a bank (or loan) account when interest accumulates at the instantaneous real rate of interest defined by the Pearson Type IV probability density. A pricing formula for pure discount bonds is also developed. Our empirical analysis of short-dated Treasury bills shows that real interest rates in the UK and the USA are strongly compatible with a general equilibrium term structure model based on the Pearson Type IV probability density.     

Pricing the term structure of inflation risk premia: Theory and evidence from TIPS

In this paper, we study inflation risk and the term structure of inflation risk premia in the United States' nominal interest rates through the Treasury Inflation Protection Securities (TIPS) with a multi-factor, modified quadratic term structure model with correlated real and inflation rates. We derive closed form solutions to the real and nominal term structures of interest rates that drastically facilitate the estimation of model parameters and improve the accuracy of the valuation of nominal rates and TIPS prices. In addition, we contribute to the literature by estimating the term structure of inflation risk premia implied from the TIPS market. The empirical evidence using data from the period of January 1998 through October 2007 indicates that the expected inflation rate, contrary to data derived from the consumer price indices, is very stable and the inflation risk premia exhibit a positive term structure.     

Risk management of non-maturing liabilities

Risk management of non-maturing liabilities is a relatively unstudied issue of significant practical importance. Non-maturing liabilities include most of the traditional deposit accounts like demand deposits, savings accounts and short time deposits and form the basis of the funding of depository institutions. Therefore, the asset and liability management of depository institutions depends crucially on an accurate understanding of the liquidity risk and interest rate risk profile of these deposits.In this paper we propose a stochastic three-factor model as general quantitative framework for liquidity risk and interest rate risk management for non-maturing liabilities. It consists of three building blocks: market rates, deposit rates and deposit volumes. We give a detailed model specification and present algorithms for simulation and calibration. Our approach to liquidity risk management is based on the term structure of liquidity, a concept which forecasts for a specified period and probability what amount of cash is available for investment. For interest rate risk management we compute the value, the risk profile and the replicating bond portfolio of non-maturing liabilities using arbitrage-free pricing under a variance-minimizing measure. The proposed methodology is demonstrated by means of a case study: the risk management of savings accounts.