Expectation Puzzles, Time-varying Risk Premia, and Dynamic Models of the Term Structure

Though linear projections of returns on the slope of the yield curve have contradicted the implications of the traditional expectations theory,' we show that these findings are not puzzling relative to a large class of richer dynamic term structure models. Specifically, we are able to match all of the key empirical findings reported by Fama and Bliss and Campbell and Shiller, among others, within large subclasses of affine and quadratic-Gaussian term structure models. Additionally, we show that certain risk-premium adjusted' projections of changes in yields on the slope of the yield curve recover the coefficients of unity predicted by the models. Key to this matching are parameterizations of the market prices of risk that let the risk factors affect the market prices of risk directly, and not only through the factor volatilities. The risk premiums have a simple form consistent with Fama's findings on the predictability of forward rates, and are shown to also be consistent with interest rate, feedback rules used by a monetary authority in setting monetary policy.

Arbitrage-free prediction of the implied volatility smile

This paper gives an arbitrage-free prediction for future prices of an arbitrary co-terminal set of options with a given maturity, based on the observed time series of these option prices. The statistical analysis of such a multi-dimensional time series of option prices corresponding to $n$ strikes (with $n$ large, e.g. $n\geq 40$) and the same maturity, is a difficult task due to the fact that option prices at any moment in time satisfy non-linear and non-explicit no-arbitrage restrictions. Hence any $n$-dimensional time series model also has to satisfy these implicit restrictions at each time step, a condition that is impossible to meet since the model innovations can take arbitrary values. We solve this problem for any n\in\NN in the context of Foreign Exchange (FX) by first encoding the option prices at each time step in terms of the parameters of the corresponding risk-neutral measure and then performing the time series analysis in the parameter space. The option price predictions are obtained from the predicted risk-neutral measure by effectively integrating it against the corresponding option payoffs. The non-linear transformation between option prices and the risk-neutral parameters applied here is \textit{not} arbitrary: it is the standard mapping used by market makers in the FX option markets (the SABR parameterisation) and is given explicitly in closed form. Our method is not restricted to the FX asset class nor does it depend on the type of parameterisation used. Statistical analysis of FX market data illustrates that our arbitrage-free predictions outperform the naive random walk forecasts, suggesting a potential for building management strategies for portfolios of derivative products, akin to the ones widely used in the underlying equity and futures markets.Comment: 18 pages, 2 figures; a shorter version of this paper has appeared as a Technical Paper in Risk (30 April 2014) under the title "Smile transformation for price prediction

Statistical Properties of Forward Libor Rates

historical forward rates are used to calibrate the lognormal forward rate model - as advocated by Hull and White (1999, 2000), Longstaff, Santa Clara and Schwartz (1999), Rebonato (1999a,b,c), Rebonato and Joshi (2001) and many others - a Libor yield curve needs to be fit to the available data on spot libor rates, forward rate agreements (FRAs) or futures, and swap rates. This paper compares the statistical properties of the time series of forward rates that are obtained using three different yield curve fitting techniques. Introduced by McCulloch (1975), Steely (1991) and Svensson (1994), each of the three techniques is well known for its application to the construction of bond yield curves. Our work focuses on the eigenstructure of estimated forward rate correlation matrices. These are shown to be dominated by the semi-parametric or parametric form that is used in the yield curve model. The spectral decomposition of forward rate correlation - and covariance - matrices is considered in some detail, and in particular we test the common principal component hypothesis of Flury (1988), which has been applied to the lognormal forward rate model by Alexander (2003). We conclude that, if historical data are used to calibrate the lognormal forward rate model, it is best to use Svensson forward rate correlation matrices. However, the empirical evidence is strongly in favour of the common principal component hypothesis, where the three principal eigenvectors in all correlation matrices of the same dimension are identical. Hence we further conclude that a parsimonious parameterisation of forward rate correlations is possible, and this allows for direct calibration of forward rate correlations to market data, so historical data are not necessary.Yield curve fitting, common principal component analysis, volatility, correlation, covariance, lognormal formal rate model

Hedging with interest rate caps compared with a policy of maintaining a balanced portfolio of loans (PLA) and averaging the borrowing costs

This paper compares two different strategies for managing interest rate exposure. One involves maintaining a borrowing portfolio using short and long term debt lines in order to maintain an average borrowing cost. The second involves using interest rate caps to manage exposure to interest rate risk. The two strategies are compared using a set of daily quarterly rates from three months out to 10 years (120 months) of BBSW zero rates, par rates and forward rates from June 2000 to September 2006. The data set of implied volatilities (Appendix I used for interest cap quoting and pricing) consists of volatilities for 1, 2, 3, 4, 5, 7 and 10 year maturities; the data set is made up of daily closing mid-quotes for the period. We examine whether interest rate caps would be a better alternative for minimising interest rate risk as compared to a structure that combines a portfolio of rolling short-term debt with one of rolling long-term debt lines. Using principal component analysis (PCA) we explore the behaviour of, and the number of factors driving volatilities. As caps are quoted in terms of implied volatilities, and we know BlackÃŒs (1976) model is very sensitive to changes in these volatilities. We use PCA to examine the factors driving cap price volatilities. We explore the best way of using caps to manage interest rate risk. This should help us understand what factors affect cap prices and how many factors might be used in the interest rate models used to price interest rate derivatives such as caps and floors. We use Sharpe ratios to assess the relative borrowing costs of different strategies in relation to the volatility of their outcomes. We examine whether interest rate caps would be a more efficient method for minimising interest rate risk as compared to the a portfolio of loans.Hedging with interest rate caps, Vegas, Sharpe ratios, Principle components analysis

Common Correlation and Calibrating the Lognormal Forward Rate Model

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, Gatarek & Musiela (1997) and Miltersen, Sandermann & Sondermann (1997) or lognormal diffusions of swap rates, as in Jamshidian (1997). 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 no-arbitrage pricing models. Researchers will find the book by Brigo and Mercurio (2001) and the forthcoming book by Rebonato (2002) invaluable aids to their understanding.

Stochastic Spot/Volatility Correlation in Stochastic Volatility Models and Barrier Option Pricing

Most models for barrier pricing are designed to let a market maker tune the model-implied covariance between moves in the asset spot price and moves in the implied volatility skew. This is often implemented with a local volatility/stochastic volatility mixture model, where the mixture parameter tunes that covariance. This paper defines an alternate model where the spot/volatility correlation is a separate mean-reverting stochastic variable which is itself correlated with spot. We also develop an efficient approximation for barrier option and one touch pricing in the model based on semi-static vega replication and compare it with Monte Carlo pricing. The approximation works well in markets where the risk neutral drift is modest.Comment: 23 pages, 11 figure