Implied Volatilities of Caps: a Gaussian approach.

Implied volatilities of interest rate derivatives present some distinctive features, like the inverse relation with the underlying rates and the humped or decreasing shape of their term structure. The objective of this paper is to analyze and explain such features in a Gaussian framework. We will use an approximate relation which separates in a simple and natural way the effects on the implied volatility of the level and of the uncertainty of the interest rates. This is a useful tool for understanding the features of different models and to interpret some characteristics of the market.Implied volatility, forward rates, HJM models, calibration

Pricing caps with HJM models: the benefits of humped volatility

In this paper we compare different multifactor HJM models with humped volatility structures, to each other and to models with strictly decreasing volatility. All the models are estimated on Euribor and swap rates panel data. We develop the analysis in two steps: first we study the in-sample properties of the estimated models, then we study the pricing performance on caps. We find the humped volatility specification to greatly improve the model estimation and to provide sufficiently accurate cap prices, although the models has been calibrated on interest rates data and not on cap prices. Moreover we find the two factor humped volatility model to outperform the three factor models in pricing capsFinance, interest rates, humped volatility, Kalman filter, cap and floor pricing

A multiobjective approach using consistent rate curves to the calibration of a Gaussian Heath-Jarrow-Morton model

In this paper we propose an alternate calibration algorithm, by using a consistent family of yield curves, that fits a Gaussian Heath-Jarrow-Morton model jointly to the implied volatilities of caps and zero-coupon bond prices. The algorithm is capable for finding several Pareto optimal points as is expected for a general nonlinear multicriteria optimization problem. The calibration approach is evaluated in terms of in-sample data fitting as well as stability of parameter estimates. Furthermore, the efficiency is tested against a non-consistent traditional method by using simulated and US market data.HJM models, consistent forward rate curves, multiobjective calibration

On the variance and skewness of the swap rate in a stochastic volatility interest rate model

This paper provides new insight in the distribution of the (forward par) swap rate in a stochastic volatility model for the dynamics of the forward rate curve. First the swap rate dynamics are obtained in a multi-curve environment with deterministic spread. Then, the variance of the swap rate is derived making use of a result on the distribution of random variables generated by extended square-root diffusion processes. Also, the skewness is derived by Itô calculus. These results give rise to moment-matching swaption price formulas which are expected to permit a fast approximate calibration of the model

On Correlation Effects and Default Clustering in Credit Models

We establish Markovian models in the Heath, Jarrow and Morton paradigm where the credit spreads curves of multiple firms and the term structure of interest rates can be represented analytically at any point in time in terms of a finite number of state variables. The models make no restrictions on the correlation structure between interest rates and credit spreads. In addition to diffusive and jump-induced default correlations, default events can impact credit spreads of surviving firms. This feature allows a greater clustering of defaults. Numerical implementations highlight the importance of taking interest rate-credit spread correlations, credit-spread impact factors and the full credit spread curve information into account when building a unified model framework that prices any credit derivative.

Optimal Portfolio Allocation under Asset and Surplus VaR Constraints

In this paper we propose an approach to Asset Liability Management of various institutions, in particular insurance companies, based on a dual VaR constraint for the asset and the surplus. A key ingredient of this approach is a flexible modelling of the term structure of interest rates leading to an explicit formula for the returns of bonds. VaR constraints on the asset and on the surplus also take tractable forms, and graphical illustrations of the impact and of the sensitivity of these constraints are easily explicited in terms of various parameters: share of stocks, duration and convexity of the bonds on the asset and liability sides, expected return and volatility of the asset...Asset Liability Management, interest rates, Asset VaR constraint, Surplus VaR constraint, Optimal Portfolio.

Modeling interest rate dynamics: an infinite-dimensional approach

We present a family of models for the term structure of interest rates which describe the interest rate curve as a stochastic process in a Hilbert space. We start by decomposing the deformations of the term structure into the variations of the short rate, the long rate and the fluctuations of the curve around its average shape. This fluctuation is then described as a solution of a stochastic evolution equation in an infinite dimensional space. In the case where deformations are local in maturity, this equation reduces to a stochastic PDE, of which we give the simplest example. We discuss the properties of the solutions and show that they capture in a parsimonious manner the essential features of yield curve dynamics: imperfect correlation between maturities, mean reversion of interest rates and the structure of principal components of term structure deformations. Finally, we discuss calibration issues and show that the model parameters have a natural interpretation in terms of empirically observed quantities.Comment: Keywords: interest rates, stochastic PDE, term structure models, stochastic processes in Hilbert space. Other related works may be retrieved on http://www.eleves.ens.fr:8080/home/cont/papers.htm

What Should You Pay to Cap your ARM?—A Note on Capped Adjustable Rate Mortgages

In this paper, an Adjustable Rate Mortgage (ARM) and a Fixed Rate Mortgage (FRM) are formalized and studied in a simple continuous-time setting under the assumption of a simple one-factor Affine Term Structure (ATS). Through an application of existing results from ATS theory, it is shown that when the short rate reaches a certain pre-determined boundary, the constant payment stream on a new FRM equals the payments on an existing ARM. Hereby, this paper provides a theoretical build-in cap on the formalized ARM. The finite boundary for the short-rate suggests that certain caps on ARMs should (in theory) be offered free of charge

Debt and deficit fluctuations and the structure of bond markets

This paper tests for the market environment within which US fiscal policy operates, that is we test for the incompleteness of the US government bond market. We document the stochastic properties of US debt and deficits and then consider the ability of competing optimal tax models to account for this behaviour. We show that when a government pursues an optimal tax policy and issues a full set of contingent claims, the value of debt has the same or less persistence than other variables in the economy and declines in response to higher deficit shocks. By contrast, if governments only issue one-period risk free bonds (incomplete markets), debt shows more persistence than other variables and it increases in response to expenditure shocks. Maintaining the hypothesis of Ramsey behavior, US data conflicts.Optimal fiscal policy, complete vs incomlete markets, tax smoothing, government debt, persistence of debt