Raquel M. Gaspar

Machine Learning Vasicek Model Calibration with Gaussian Processes

In this article, we calibrate the Vasicek interest rate model under the risk neutral measure by learning the model parameters using Gaussian processes for machine learning regression. The calibration is done by maximizing the likelihood of zero coupon bond log prices, using mean and covariance functions computed analytically, as well as likelihood derivatives with respect to the parameters. The maximization method used is the conjugate gradients. The only prices needed for calibration are zero coupon bond prices and the parameters are directly obtained in the arbitrage free risk neutral measure.

On Swap Rate Dynamics: To Freeze or Not to Freeze?

We explore the implications of a common market and academic practice which is known as freezing the drift when dealing with swap interest rate dynamics. In mathematical terms this can be better understood as imposing a low variance martingale (LVM) assumption. We look into the LVM Assumption implications, both on the shape and dynamics for default-free yield curves. We show that the LVM Assumption is equivalent to consider future yield curves are nothing but deterministic translations of the initial curve. For the particular case of the Nelson Siegel yield curve calibration, we show the LVM Assumption requires a deterministic parameters evolution and, thus, imposes the need to constantly recalibrate the model. Finally, based upon ECB historical data on evolution of the default-free Euro area yield curve, we illustrate periods in which the LVM may be applicable and others in which is not.

Interest Rate Theory and Geometry

In this paper we provide an overview of some basic topics in interest rate theory, from the point of view of arbitrage free pricing. We cover short rate models, affine term structure models, inversion of the yield curve and the Musiela parameterization. We treat geometric interest rate theory in some detail, and we also review the potential approach to positive interest rates.

Investment Analysis of Autocallable Contingent Income Securities

Autocallable contingent income securities, or autocalls, are a relatively new type of structured finance security whose payout is contingent on the performance of an underlying asset and that give investors an opportunity to earn high yields in a low interest environment. We collect data on autocalls issued in the US and describe their contractual properties and the properties of their underlying assets at issuance. We find that autocalls are issued on underlying assets displaying high volatility, negative skewness and high prices. We then model a typical autocall under different assumptions about the price of the underlying asset and (i) analyze the rationale behind the characteristics of the underlying asset at issuance, and (ii) discuss valuation of autocalls in the various models. While the literature consistently finds that structured products are overpriced, we find that incorporating stochastic volatility into the pricing model can help explain some of the overpricing routinely reported in prior studies.

Portfolio Insurance — A Comparison of Alternative Strategies

This study makes a comparison between the most popular strategies of Portfolio Insurance based on Monte Carlo simulation. This work aims to define the best strategy at comparing different strategies and provide a contribution to solving some divergences in literature. Most of the previous comparisons do not take into consideration all the strategies discussed in this study and this analysis intends to add some relevant findings.The OBPI, CPPI and SLPI strategies are evaluated in terms of moments of the distribution, performance ratios (Sharpe ratio, Sortino ratio, Omega ratio and Upside Potential ratio) and stochastic dominance in different market conditions represented by an underlying asset that follows a geometric Brownian motion. In order to have a perception of a real situation in financial markets, the strategies are later also applied to three major stock indices (S&P 500, DJ EuroStoxx 50 and Nikkei 225).We find that CPPI 1 and SLPI strategies should be preferred in all scenarios according to the higher performance ratios, the higher expected returns and other measures. The choice between them is based on the preferences of the investor or manager, but we also find that the CPPI 1 strategy stochastically dominates, on second and third order, the others strategies in bear market scenarios. From our results we can state that a value of 100% for the floor should be preferred in terms of performance ratios, expected returns and other measures. This comparison allows improving the efficiency of decision making of an investor or manager in a Portfolio Insurance investment.

Implied Volatility and Forward Price Term Structures

This paper discusses the relation between forward price models (FPM) and the so called implied volatility term structure (VTS). We start by considering the case of pure deterministic forward price volatilities and suppose both forward contracts and at-the-money (ATM) options, on a same underlying, are liquidly traded in the market. We, then, derive no arbitrage conditions between the functional form of the ATM implied VTS and the functional form of forward price volatilities. We conclude the first part by characterizing a parametric family of ATM implied VTS that is compatible with a finite dimensional realization (FDR) of forward prices. Finally, we consider the possibility of stochastic forward price volatilities and derive a no arbitrage drift condition that must hold for the dynamics of ATM implied VTS.

Brownian Bridge and Other Path-dependent Gaussian Processes Vectorial Simulation

The iterative simulation of the Brownian bridge is well known. In this article, we present a vectorial simulation alternative based on Gaussian processes for machine learning regression that is suitable for interpreted programming languages implementations. We extend the vectorial simulation of path-dependent trajectories to other Gaussian processes, namely, sequences of Brownian bridges, geometric Brownian motion, fractional Brownian motion, and Ornstein–Ulenbeck mean reversion process.

Historical VaR for Bonds - A New Approach

Bonds historical returns cannot be used directly to compute VaR because the maturities of returns implied by the historical prices do not have the relevant maturities to compute VaR. Given the so-called pull-to-par in bonds, with return volatilities necessarily decreasing with diminishing time-to-maturity, direct use of historical returns would lead to overestimation of the true VaR. Market practice deals with the problem of computing VaR for portfolios of bonds or mixed portfolios with cumbersome methods of cash-flow mappings.In this paper we propose a new approach. We develop a technique to adjust bonds historical prices and extract from them an adjusted history of returns, that can be used directly to compute historical VaR for bonds or bond portfolios. We illustrate the method using one concrete traded market zero-coupon bond, but the simplicity of the method makes this enough to the reader to understand how it would work with coupon bonds, portfolios of bonds or mixed portfolios.

Counterparty and Liquidity Risk — An Analysis of the Negative Basis

In this study we analyse the equivalence between credit default swap (CDS) spreads and corporate bond yield spreads from March 2007 to March 2011 for investment graded corporate entities in the eurozone. We find evidence of cointegration between the two markets and that CDS prices tends to lead corporate yield spreads. We find support for significant effects of counterparty and funding risks in the basis, measured as the difference between CDS and corporate yield spreads, with negative impact and that liquidity also matters in this context.

Liquidity Risk Premia: An Empirical Analysis of European Corporate Bond Yields

In this study we highlight the importance of liquidity risk, especially in periods of market stress, and advocate in favour of an explicit consideration of a liquidity premium when using mark-to-model methodologies to value financial assets. For European corporate bonds, we show that the liquidity premium, calculated as the difference between the yield spread of corporate bonds and the spread of credit default swaps, grew significantly during the recent market turmoil not only in absolute terms but also in relative terms. Although liquidity premiums were far from stable during the time frame of analysis- from 1 January 2005 to 31 December 2009 - on average roughly 40% of corporate yield spreads can be interpreted in terms of liquidity premia. We propose direct matching between the CDS and the underlying reference assets when computing liquidity premia. This differs from what seems to be the industry standard, which is simply to use indices when trying to infer market implied liquidity premia. Although computationally more demanding, the method we use is sounder from a theoretical point of view and produces richer results and analysis. With this method we are able present an analysis of liquidity risk premia per sector of activity. Classification- JEL: