A stochastic volatility Libor model and its robust calibration

Monte Carlo Methods …, 2009

In this paper we propose an extension of the Libor market model with a high-dimensional specially structured system of square root volatility processes, and give a road map for its calibration. As such the model is well suited for Monte Carlo simulation of derivative interest rate instruments. As a key issue, we require that the local covariance structure of the market model is preserved in the stochastic volatility extension. In a case study we demonstrate that the extended Libor model allows for stable calibration to the cap-strike matrix. The calibration algorithm is FFT based, so fast and easy to implement.

Journal of Industrial and Management Optimization, 2006

In this paper we extend the standard LIBOR market model to accommodate the pronounced phenomenon of implied volatility smiles/skews. We adopt a multiplicative stochastic factor to the volatility functions of all relevant forward rates. The stochastic factor follows a square-root diffusion process, and it can be correlated to the forward rates. For any swap rate, we derive an approximate process under its corresponding forward swap measure. By utilizing the analytical tractability of the approximate process, we develop a closed-form formula for swaptions in term of Fourier transforms. Extensive numerical tests are carried out to support the swaptions formula. The extended model captures the downward volatility skews by taking negative correlations between forward rates and their volatilities, which is consistent with empirical findings.

SSRN Electronic Journal, 2000

In 1997 three papers that introduced very similar lognormal diffusion processes for interest rates appeared virtuously simultaneously. These models, now commonly called the 'LIBOR models' are based on either lognormal diffusions of forward rates as in Brace, and or lognormal diffusions of swap rates, as in Jamshidian . The consequent research interest in the calibration of the LIBOR models has engendered a growing empirical literature, including many papers by Brigo and Mercurio, and Riccardo Rebonato (www.fabiomercurio.it and www.damianobrigo.it and www.rebonato.com). The art of model calibration requires a reasonable knowledge of option pricing and a thorough background in statistics − techniques that are quite different to those required to design noarbitrage pricing models. Researchers will find the book by and the forthcoming book by invaluable aids to their understanding.

European journal of operational research, 2005

In this paper we consider several parametric assumptions for the instantaneous covariance structure of the LIBOR market model, whose role in the modern interest-rate derivatives theory is becoming more and more central. We examine the impact of each different parameterization on the evolution of the term structure of volatilities in time, on terminal correlations and on the joint calibration to the caps and swaptions markets. We present a number of cases of calibration in the Euro market. In particular, we consider calibration via a parameterization establishing a controllable one to one correspondence between instantaneous covariance parameters and swaptions volatilities, and assess the benefits of smoothing the input swaption matrix before calibrating.

SSRN Electronic Journal, 2019

Implied volatility skew and smile are ubiquitous phenomena in the financial derivative market especially after the Black Monday 1987 crash. Various stochastic volatility models have been proposed to capture volatility skew and smile in derivative pricing and hedging. Almost 30 years after the advent of the first type of stochastic volatility model calibrating them to the market volatility surface still remains challenging, especially when stochastic interest rate has to be also taken into account for long-dated options. Many techniques have been applied to tackle this problem, including Fast Fourier Transform, singular perturbation expansion, heat kernel expansion, Markovian projection, to name a few. Although they have achieved some success in deriving either a close-form solution for a specific type of model or asymptotic solution in more general, none of them can really solve the calibration problem satisfying our need in term of both efficiency and accuracy. Monte Carlo method is a flexible numerical pricing method but has not been considering for calibration because of its slow convergence. However, with the great advance in computational power, in particular, parallel computation and the invention of other variance reduction techniques fast and accurate calibration using Monte Carlo becomes possible. This paper presents a Monte Carlo calibration method for stochastic volatility models with stochastic interest rate, which reduces simulation dimension by conditional expectation and further improves speed by vectorization. Numerical experiments show that both the calibration speed and accuracy of this generic method are satisfactory for almost all applications.

Abstract This thesis is focused on the financial model for interest rates called the LIBOR Market Model, which belongs to the family of market models and it has as main objects the forward LIBOR rates. We will see it from its theoretical approach to its calibration to data provided by the market. In the appendixes, we provide the theoretical tools needed to understand the mathematical manipulations of the model, largely deriving from the theory of stochastic differential equations.

2013

The aim of this paper is to study stochastic volatility models and their calibration to real market data. This task is formulated as the optimization problem and several optimization techniques are compared and used in order to minimize the difference between the observed market prices and the model prices. At first we demonstrate the complexity of the calibration process on the popular Heston model and we show how well the model can fit a particular set of market prices. This is ensured by using a deterministic grid which eliminates the initial guess sensitivity specific to this problem. The same level of errors can be reached by employing optimization techniques introduced in the paper, while also preserving time efficiency. We further apply the same calibration procedures to the recent fractional stochastic volatility model, which is a jump-diffusion model of market dynamics with approximative fractional volatility. The novelty of this paper is especially in showing how the proposed calibration procedures work for even more complex SV model, such as the introduced long-memory fractional model.

International Journal of Theoretical and …, 2004

We propose a two-regime stochastic volatility extension of the LIBOR market model that preserves the positive features of the recently introduced (Joshi and Rebonato 2001) stochastic-volatility LIBOR market model (ease of calibration to caplets and swaptions, efficient pricing of ...

Applied Mathematical Finance, 2016