Multi-factor Term Structure Models Research Articles

A Semi-Static Replication Method for Bermudan Swaptions under an Affine Multi-Factor Model

We present a semi-static replication algorithm for Bermudan swaptions under an affine, multi-factor term structure model. In contrast to dynamic replication, which needs to be continuously updated as the market moves, a semi-static replication needs to be rebalanced on just a finite number of instances. We show that the exotic derivative can be decomposed into a portfolio of vanilla discount bond options, which mirrors its value as the market moves and can be priced in closed form. This paves the way toward the efficient numerical simulation of xVA, market, and credit risk metrics for which forward valuation is the key ingredient. The static portfolio composition is obtained by regressing the target option’s value using an interpretable, artificial neural network. Leveraging the universal approximation power of neural networks, we prove that the replication error can be arbitrarily small for a sufficiently large portfolio. A direct, a lower bound, and an upper bound estimator for the Bermudan swaption price are inferred from the replication algorithm. Additionally, closed-form error margins to the price statistics are determined. We practically study the accuracy and convergence of the method through several numerical experiments. The results indicate that the semi-static replication approaches the LSM benchmark with basis point accuracy and provides tight, efficient error bounds. For in-model simulations, the semi-static replication outperforms a traditional dynamic hedge.

A comparison of multi-factor term structure models for interbank rates

A comparison of multi-factor term structure models for interbank rates

Static Markowitz mean-variance portfolio selection model with long-term bonds

We propose a static Markowitz mean-variance portfolio selection model suitable for long-term zero-coupon bonds. The model uses a multi-factor term structure model of Vasicek (Ornstein-Uhlenbeck) type to compute the portfolio's expected return and its variance in the model. German Government zero-coupon bonds with short to very long time to maturity are considered; the data spans August 2002 to December 2020. The main investment assumption is the re-investment of cash flows of zero-coupon bonds with maturities less than the planning horizon at the current spot interest rate. Solutions for the zero-coupon holding vector and the tangency portfolio are obtained in closed form. Model parameters are estimated under an assumption of modeling ambiguity which takes the form of Knightian uncertainties at the level of the latent factors, allowing the use of a Kalman filter. Different investment strategies are examined on various risk portfolios. Results show that one- and two-factor Vasicek models produce attractive out-of-sample portfolio predictions in terms of the Sharpe ratio especially on long-term investments. It is also noted that a small number of risky bonds can adequately produce very attractive portfolio risk-return profiles.

Interpolating the Yield Curve

This article proposes an interpolation (and an extrapolation to the short rate) of the yield curve between a given number of observed yields on benchmark bonds that is compatible with an arbitrage-free model of the term structure of interest rates. The model is a multivariate time-homogeneous Gaussian yield factor model. It is shown that the interpolation that minimizes the integral of the yield variance, called the maximum stability interpolation, corresponds to the case that all eigenvalues of the mean-reversion matrix of the benchmark yields are equal. Explicit equations for calculation of the maximum stability interpolation are given. <b>TOPICS:</b>Factor-based models, derivatives <b>Key Findings</b> • Explicit equations are given for interpolation of the yield curve. • The equations are based on an equilibrium multifactor term structure model. • A criterion for maximum stability of the interpolation is introduced.

Error bounds for the perturbation solution of the transition density under a multi-factor CIR term structure model with weak mean-reversion effect

We consider a multi-factor Cox-Ingersoll-Ross (CIR) model of the term structure of interest rates with weak mean-reversion effect. We use perturbation theory to analyze its conditional characteristic function illustrated by a system of Riccati equations and derive the error bounds for the perturbation approximations. Using the Fourier inversion theorem, we clarify that the perturbation approximation of the conditional characteristic function can be applied to estimate the transition density and likelihood function. We provide their error bounds and accuracy orders. Finally, we discuss the performance of the perturbation approximation in estimating the transition density via simulation.

다요인 이자율 기간구조 모형의 SMM 추정을 통한 실증 분석

다요인 이자율 기간구조 모형의 SMM 추정을 통한 실증 분석

Generalised Additive-Gaussian Models of the Term Structure

This draft paper outlines a term structure model that models the short rate as a sum of n Gaussian processes (a 'Gn' model). It is shown that the given specification involves no redundant parameters and gives rise to a non-trivial multifactor term structure model. A closed-form expression, involving only the Gaussian cumulative distribution function, is derived. This appears unique in the literature for short rate models with more than two factors. We then delineate the class of term structure models that can be transformed to the Gn model and can therefore use the closed-form option expression.

Fast Swaption Pricing in Gaussian Term Structure Model

We propose a fast and accurate numerical method to price European swaptions in a multi-factor Gaussian term structure model that can speed up the calibration to the volatility surface. Pricing an interest rate option in such a multi-factor term structure model involves evaluating a multi-dimensional integration of the underlying claim's payoff on a domain where the payoff is positive. In our method, we approximate the exercise boundary of the state variables as a hyperplane tangent to the maximum probability point on the boundary and simplify the multi-dimensional integration into an analytic form. The maximum probability point can be found using the gradient descent method. We show that the quality of our method is superior to the result of previous studies by directly comparing them to the exact price obtained from the numerical integration.

PRICING SWAPTIONS UNDER MULTIFACTOR GAUSSIAN HJM MODELS

Several approximations have been proposed in the literature for the pricing of European-style swaptions under multifactor term structure models. However, none of them provides an estimate for the inherent approximation error. Until now, only the Edgeworth expansion technique of Collin-Dufresne and Goldstein is able to characterize the order of the approximation error. Under a multifactor HJM Gaussian framework, this paper proposes a new approximation for European-style swaptions, which is able to set bounds on the magnitude of the approximation error and is based on the conditioning approach initiated by Curran and Rogers and Shi. All the proposed pricing bounds will arise as a simple by-product of the Nielsen and Sandmann setup, and will be shown to provide a better accuracy–efficiency trade-off than all the approximations already proposed in the literature.

Estimating affine multifactor term structure models using closed-form likelihood expansions

We develop and implement a technique for closed-form maximum likelihood estimation (MLE) of multifactor affine yield models. We derive closed-form approximations to likelihoods for nine Dai and Singleton (2000) affine models. Simulations show our technique very accurately approximates true (but infeasible) MLE. Using US Treasury data, we estimate nine affine yield models with different market price of risk specifications. MLE allows non-nested model comparison using likelihood ratio tests; the preferred model depends on the market price of risk. Estimation with simulated and real data suggests our technique is much closer to true MLE than Euler and quasi-maximum likelihood (QML) methods.

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.

Empirical performance of multifactor term structure models for pricing and hedging Eurodollar futures options

This article compares two one-factor, two two-factor, two three-factor models in the HJM class and Black's [Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics, 3, 167–179.] implied volatility function in terms of their pricing and hedging performance for Eurodollar futures options across strikes and maturities from 1 Jan 2000 to 31 Dec 2002. We find that three-factor models perform the best for 1-day and 1-week prediction, as well as for 5-day and 20-day hedging. The moneyness bias and the maturity bias appear for all models, but the three-factor models produce lower bias. Three-factor models also outperform other models in hedging, in particular for away-from-the-money and long-dated options. Making Black's volatility a square root or exponential function performs similar to one-factor HJM models in pricing, but not in hedging. Correctly specified and calibrated multifactor models are thus important and cannot be replaced by one-factor models in pricing or hedging interest rate contingent claims.

Intertemporal Investment Strategies Under Inflation Risk

This paper studies intertemporal investment strategies under inflation risk by extending the intertemporal framework of Merton (1973) to include a stochastic price index. The stochastic price index gives rise to a two-tier evaluation system: agents maximize their utility of consumption in real terms while investment activities and wealth evolution are evaluated in nominal terms. We include inflation-indexed bonds in the agents’ investment opportunity set and study their effectiveness in hedging against inflation risk. A new multifactor term structure model is developed to price both inflation-indexed bonds and nominal bonds, and the optimal rules for intertemporal portfolio allocation, both with and without inflation-indexed bonds are obtained in closed form. The theoretical model is estimated using data of US bond yield, both real and nominal, and S&P 500 index. The estimation results are employed to construct the optimal investment strategy for an actual real market situation. Wachter (2003) pointed out that without inflation risk, the most risk averse agents (with an infinite risk aversion parameter) will invest all their wealth in the long term nominal bond maturing at the end of the investment horizon. We extend this result to the case with inflation risk and conclude that the most risk averse agents will now invest all their wealth in the inflation-indexed bond maturing at the end of the investment horizon.

A multi-factor Markovian HJM model for pricing American interest rate derivatives

This article presents a numerically efficient approach for constructing an interest rate lattice for multi-state variable multi-factor term structure models in the Makovian HJM [Econometrica 70 (1992) 77] framework based on Monte Carlo simulation and an advanced extension to the Markov Chain Approximation technique. The proposed method is a mix of Monte Carlo and lattice-based methods and combines the best from both of them. It provides significant computational advantages and flexibility with respect to many existing multi-factor model implementations for interest rates derivatives valuation and hedging in the HJM framework.

Interest Rate Caps “Smile” Too! But Can the LIBOR Market Models Capture the Smile?

ABSTRACTUsing 3 years of interest rate caps price data, we provide a comprehensive documentation of volatility smiles in the caps market. To capture the volatility smiles, we develop a multifactor term structure model with stochastic volatility and jumps that yields a closed‐form formula for cap prices. We show that although a three‐factor stochastic volatility model can price at‐the‐money caps well, significant negative jumps in interest rates are needed to capture the smile. The volatility smile contains information that is not available using only at‐the‐money caps, and this information is important for understanding term structure models.

BOND MARKET MODEL

We describe the Bond Market Model, a multi-factor interest rate term structure model, where it is possible to price with Black-like formulas the three classes of over-the-counter plain vanilla options. We derive the prices of caps/floors, bond options and swaptions. A comparison with Libor Market Model and Swap Market Model is discussed in detail, underlining advantages and limits of the different approaches.

Stochastic Volatility for Real

We combine classical ideas of separable volatility structures in the HJM framework with the latest techniques for calibration of stochastic volatility models and create a new class of efficient multi-factor term structure models with stochastic volatility. These models have the flexibility of as the Libor market models but the speed of the short rate models.

Bond Market Model

We describe the Bond Market Model, a multi-factor interest rate term structure model where it is possible to price with Black-like formulas the three classes of Over-The-Counter plain vanilla options. We derive the prices of caps, floors, bond options and swaptions. A comparison with Libor Market Model and Swap Market Model is discussed in detail, underlining advantages and limits of the different approaches.

On the interface of probabilistic and PDE methods in a multifactor term structure theory

Within the general framework of a multifactor term structure model, the fundamental partial differential equation (PDE) satisfied by a default-free zero-coupon bond price is derived via a martingale-oriented approach. Using this PDE, a result characterizing a model belonging to an exponential affine class is established using only a system of partial derivatives. It is also shown that the solution to the bond price PDE has the conditional expectation representation arising in martingale pricing through the application of a multi-dimensional version of Itô's lemma and a property of the stochastic integral.

Comments and Discussion

Comments and Discussion Benjamin M. Friedman and Lars E. O. Svensson Benjamin M. Friedman: This paper by Ben Bernanke, Vincent Reinhart, and Brian Sack is a welcome contribution to the new line of literature that examines, in a far more eclectic way than used to be done, how monetary policy works and what effects it has on the financial markets and the non-financial economy. The paper is valuable both because the question it asks is important—whether what the authors call nonstandard policy measures can be effective when ordinary monetary policy actions are constrained by the zero lower bound on short-term nominal interest rates—and because the analysis it brings to bear on this question is largely empirical. The paper also reaches what I regard as a sensible conclusion: that the potential efficacy of these nonstandard measures notwithstanding, it is wise for monetary policy both to aim at an inflation rate distinctly above zero, so as to allow for a buffer against disinflationary shocks, and to ease policy preemptively when necessary to avoid significant risk of hitting the zero lower bound on interest rates. Finally, the paper is important also because of who wrote it, but more on that below. The authors consider three conceptually distinct classes of nonstandard monetary policy measures: central bank communications, changes in the size of the central bank's balance sheet, and changes in the composition of the central bank's balance sheet. (The last of these is what would normally be called debt management policy if it were executed by a country's fiscal authority rather than by its central bank.) It is important to point out, however—and this is the focus of my one major concern about the paper—that even when the authors are addressing the effects of changes in the size or composition of the central bank's balance sheet, what they are actually analyzing in their empirical work is still communications. Most [End Page 79] of their empirical exercises focus not on policy actions but on official statements regarding intended future actions. Even when they examine such episodes as the U.S. Treasury's buyback of Treasury securities, during the happy but all-too-brief period in the late 1990s when the U.S. government ran surprisingly large budget surpluses instead of shamefully large deficits, their attention is more on what the Treasury said than on what it did. In principle, of course, credible statements about future actions should matter for the pricing of medium- and long-term assets in speculative markets. The ground for concern is rather in how this approach to addressing questions of monetary policy reinforces the increasingly exclusive focus in recent literature on what many economists engaged in this line of research (for example, Gauti Eggertsson and Michael Woodford in a Brookings Paper last year)1 call "expectations management." Surely everyone today believes that expectations matter, and therefore that whatever influences the public's expectations, including communication from the central bank, matters as well. I am also sympathetic to the authors' presumption that, when an economy has reached the zero lower bound on short-term interest rates, central bank communications may be even more important than normally. But the net impression, delivered both by the authors' discussion and by the battery of empirical tests they perform, nonetheless resonates too strongly, at least for my taste, with the idea, which they attribute to Eggertsson and Woodford, that "shaping the interest rate expectations of the public is essentially the only tool that central bankers have—not only when the ZLB binds, but under normal conditions as well." (I will not repeat here my criticism of this view, but interested readers can refer to my remarks as a discussant of the Eggertsson-Woodford paper.) This concern aside, I regard the authors' empirical analysis of the effect of Federal Reserve and Treasury statements on market interest rates as carefully crafted, and I find their conclusions easily credible. I especially admire their willingness to go beyond the standard event studies and use a multifactor term structure model to try to identify the role of "surprise" elements of Federal Reserve statements, as distinct from the policy actions that these...