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Dissertation writing help - A Parsimonious Model for the Joint Evolution of Yield Curves

Custom Dissertation Writing Service - Interest Rate Smile Surface under the Objective Measure

Counterparty credit risk management and validation of out-of-the-money hedges require risk factor evolution models that are capable of reproducing essential statistical properties of historical time-series. Most evolution models designed for pricing applications are unlikely to satisfy these criteria due to constraints imposed by the choice of a risk-neutral measure, namely the choice of drift and forward covariance. In this work we provide a roadmap to extend the historical simulation framework proposed by Rebonato et al. [2005] to interest rate implied volatilities. Historical perturbations of a set of interest rate implied volatilities are rather complex at least due to the high dimensionality of the system, which makes it extremely difficult to come up with an evolution model that captures all observed features (e.g. see Andersen and Andreasen [2000], Piterbarg [2003], Rebonato et al. [2009]). We propose a parsimonious and compact approach for generating a large number of arbitrage-free scenarios of caplet volatility surfaces consistent with the historic observations of yield curves and corresponding implied volatilities. The approach is a fusion of concepts introduced by Nelson and Siegel [1987] and Rebonato et al. [2005], and involves sampling of filtered historical quantities. It can be broken down into the following steps:

- We apply a well fitting, arbitrage-free and compact paremeterisation based on the coupling of disjoint SABR processes proposed by Rebonato et al. [2009].
- A history of parameters is obtained by fitting the implied volatility surface to each available observation.
- The time series of parameters are studied and a model for evolving these parameters is proposed. We use a one-stage AR(1) and two stage AR(1)-GARCH(1,1) filters.
- A windowed historical sampling mechanism proposed by Rebonato et al. [2005] is used to generate consistent paths of synthetic implied volatilities that remain arbitrage-free.

The approach is readily extendable to swaption implied volatilities through an approximation due to Rebonato [2002]. Having built synthetic scenarios we study the resulting multivariate distributions and run a number of tests to show that the scenarios generated by the proposed model are indeed plausible. As it typically happens in a complex multi-dimensional setup the lack of available historical data is the key constraint to increasing the power of such tests.