Narrow Results

We introduce a new class of lattice models based on a continuous time Markov chain approximation scheme for affine processes, whereby the approximating process itself is affine. A key property of this class of lattice models is that the location of the time nodes can be chosen in a payoff dependent way and one has the flexibility of setting them only at the relevant dates. Time stepping invariance relies on the ability of computing node-to-node discounted transition probabilities in analytically closed form. The method is quite general and far reaching and it is introduced in this article in the framework of the broadly used single-factor, affine short rate models such as the Vasiček and CIR models. To illustrate the use of affine lattice models in these cases, we analyze in detail the example of Bermuda swaptions.

Given a Heath–Jarrow–Morton (HJM) interest rate model and a parametrized family of finite dimensional forward rate curves , this paper provides a technique for projecting the infinite dimensional forward rate curve r_t given by onto the finite dimensional manifold . The Stratonovich dynamics of the projected finite dimensional forward curve are derived and it is shown that, under the regularity conditions, the given Stratonovich differential equation has a unique strong solution. Moreover, this projection leads to an efficient algorithm for implicit parametric estimation of the infinite dimensional HJM model. The feasibility of this method is demonstrated by applying the generalized method of moments.

We consider a slight perturbation of the Hull-White short rate model and the resulting modified forward rate equation. We identify the model coefficients by using the martingale property of the normalized bond price. The forward rate and the system parameters are then estimated by using the maximum likelihood method.

We consider a heat kernel approach for the development of stochastic pricing kernels. The kernels are constructed by positive propagators, which are driven by time-inhomogeneous Markov processes. We multiply such a propagator with a positive, time-dependent and decreasing weight function, and integrate the product over time. The result is a so-called weighted heat kernel that by construction is a supermartingale with respect to the filtration generated by the time-inhomogeneous Markov processes. As an application, we show how this framework naturally fits the information-based asset pricing framework where time-inhomogeneous Markov processes are utilized to model partial information about random economic factors. We present examples of pricing kernel models which lead to analytical formulae for bond prices along with explicit expressions for the associated interest rate and market price of risk. Furthermore, we also address the pricing of fixed-income derivatives within this framework.

Internal-rate-of-return (IRR) settled swaptions are the main interest rate volatility instruments in the European interest rate markets. Industry practice is to use an approximation formula to price IRR swaptions based on Black model, which is not arbitrage-free. We formulate a unified market model to incorporate both swaptions and constant maturity swaps (CMS) pricing under a single, self-consistent framework. We demonstrate that the model is able to calibrate to market quotes well, and is also able to efficiently price both IRR-settled and swap-settled swaptions, along with CMS products. We use the model to illustrate the difference in implied volatilities for IRR-settled payer and receiver swaptions, the pricing of zero-wide collars and in-the-money (ITM) swaptions, the implication on put-call parity, and the issue of negative vega. These findings offer important insights to the ongoing reform in the European swaption market.

We present some results on Bernstein processes, which are Brownian diffusions that appear in Euclidean Quantum Mechanics. We express the distributions of these processes with the help of those of Bessel processes. We then determine two solutions of the dual equation of the heat equation with potential. These results first appeared in the first author’s PhD thesis (Rouen, 2013).

Most reported stochastic volatility (SV) model for interest rates only deals with an AR specification for the latent factor process. We show in this paper the technical details for specifying the SV model for interest rates that includes an ARMA structure, a jump component and additional exogenous variables for the latent factor process. We demonstrate the efficacy of this approach with an application to the US short-term interest rate data. We find that the elasticity parameter of the variance is closer to 0.5, i.e., similar to that of the Cox–Ingersoll–Ross (1985) model of interest rates. This is quite a contrast to the finding Chan et al. [Chan, KC, GA Karolyi, F Longstaff and A Sanders (1992). The volatility of short-term interest rates: An empirical comparison of alternative models of term structure of interest rates, Journal of Finance, 47, 1209–1227]. who found the elasticity to be close to 1.5.

We consider a heat kernel approach for the development of stochastic pricing kernels. The kernels are constructed by positive propagators, which are driven by time-inhomogeneous Markov processes. We multiply such a propagator with a positive, time-dependent and decreasing weight function, and integrate the product over time. The result is a so-called weighted heat kernel that by construction is a supermartingale with respect to the filtration generated by the time-inhomogeneous Markov processes. As an application, we show how this framework naturally fits the information-based asset pricing framework where time-inhomogeneous Markov processes are utilized to model partial information about random economic factors. We present examples of pricing kernel models which lead to analytical formulae for bond prices along with explicit expressions for the associated interest rate and market price of risk. Furthermore, we also address the pricing of fixed-income derivatives within this framework.

We consider a heat kernel approach for the development of stochastic pricing kernels. The kernels are constructed by positive propagators, which are driven by time-inhomogeneous Markov processes. We multiply such a propagator with a positive, time-dependent and decreasing weight function, and integrate the product over time. The result is a so-called weighted heat kernel that by construction is a supermartingale with respect to the filtration generated by the time-inhomogeneous Markov processes. As an application, we show how this framework naturally fits the information-based asset pricing framework where time-inhomogeneous Markov processes are utilized to model partial information about random economic factors. We present examples of pricing kernel models which lead to analytical formulae for bond prices along with explicit expressions for the associated interest rate and market price of risk. Furthermore, we also address the pricing of fixed-income derivatives within this framework.