Claudio Fontana

A general HJM framework for multiple yield curve modelling

We propose a general framework for modelling multiple yield curves which have emerged after the last financial crisis. In a general semimartingale setting, we provide an HJM approach to model the term structure of multiplicative spreads between FRA rates and simply compounded OIS risk-free forward rates. We derive an HJM drift and consistency condition ensuring absence of arbitrage and, in addition, we show how to construct models such that multiplicative spreads are greater than one and ordered with respect to the tenor’s length. When the driving semimartingale is an affine process, we obtain a flexible and tractable Markovian structure. Finally, we show that the proposed framework allows unifying and extending several recent approaches to multiple yield curve modelling.

Information, no-arbitrage and completeness for asset price models with a change point

We consider a general class of continuous asset price models where the drift and the volatility functions, as well as the driving Brownian motions, change at a random time τ. Under minimal assumptions on the random time and on the driving Brownian motions, we study the behavior of the model in all the filtrations which naturally arise in this setting, establishing martingale representation results and characterizing the validity of the NA1 and NFLVR no-arbitrage conditions.

Affine multiple yield curve models

We provide a general and tractable framework under which all multiple yield curve modeling approaches based on affine processes, be it short rate, Libor market, or HJM modeling, can be consolidated. We model a numeraire process and multiplicative spreads between Libor rates and simply compounded OIS rates as functions of an underlying affine process. Besides allowing for ordered spreads and an exact fit to the initially observed term structures, this general framework leads to tractable valuation formulas for caplets and swaptions and embeds all existing multi-curve affine models. The proposed approach also gives rise to new developments, such as a short rate type model driven by a Wishart process, for which we derive a closed-form pricing formula for caplets. The empirical performance of two specifications of our framework is illustrated by calibration to market data.

Choices under Risk

This chapter presents the foundations of decision making problems in a risky environment. By introducing suitable axioms on the preference relation, the existence of an expected utility function representation is proved. We discuss the notions of risk aversion, risk premium and certainty equivalent and characterize stochastic dominance criteria for comparing random variables. The chapter ends with a discussion of mean-variance preferences and their relation with stochastic dominance and expected utility.

Multi-Period Models: Empirical Tests

This chapter is devoted to an extensive overview of the empirical evidence on classical asset pricing theory. In particular, the attention is focused on the empirical properties of the observed prices and returns and on several anomalies reported in the literature, including the excess volatility phenomenon, the predictability of asset returns, the equity premium puzzle, the risk free rate puzzle and other related asset pricing puzzles.

General Equilibrium Theory and No-Arbitrage

This chapter deals with general equilibrium theory in a risky environment, where agents interact in a financial market. The chapter starts by presenting the notion of Pareto optimality and its implications in terms of risk sharing. The concept of rational expectations equilibrium is introduced and characterized in the context of a two-period economy. Different financial market structures are considered, with a particular attention to the important case of complete markets. The last part of the chapter is devoted to the fundamental theorem of asset pricing, which relates the absence of arbitrage opportunities to the existence of a strictly positive linear pricing functional. The relation of this important result to the existence of an equilibrium of an economy and its implications for the valuation of financial assets are also discussed.

Multi-Period Models: Portfolio Choice, Equilibrium and No-Arbitrage

In this chapter, we extend the analysis developed in the previous chapters to the case of dynamic multi-period economies. The chapter starts by studying the optimal investment-consumption problem of an individual agent in a multi-period setting, by relying on the dynamic programming approach. Under suitable assumptions on the utility function, closed-form solutions are derived. The chapter then proceeds by extending the general equilibrium theory established in Chap. 4 to a dynamic setting, introducing the notion of dynamic market completeness and analysing the aggregation property of the economy. The fundamental theorem of asset pricing is then established in a multi-period setting and its relation to the equilibrium of the economy is also discussed. Later in the chapter, the most important asset pricing relations presented in Chap. 5 are extended to a dynamic context and specialized for several classes of utility functions. The chapter ends by considering multi-period economies with an infinite time horizon and the possibility of asset price bubbles.

Factor Asset Pricing Models: CAPM and APT

In this chapter, on the basis of the general equilibrium theory developed in Chap. 4, we present some of the most important asset pricing models, including the Consumption Capital Asset Pricing Model (CCAPM), the Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT). The relations of these asset pricing models with the absence of arbitrage opportunities are also discussed. In this chapter, the theoretical presentation of the models and of their implications is accompanied by an overview of the empirical evidence reported in the literature. In particular, several asset pricing anomalies and puzzles are described and discussed.

Portfolio, Insurance and Saving Decisions

In this chapter, in the context of a simple two-period economy, we study the optimal portfolio problem of a risk averse agent, first in the case of a single risky asset and then in the more general case of multiple risky assets. We present several comparative statics results as well as closed-form solutions. We then derive the mean-variance portfolio frontier and present its most important properties, also including a risk free asset. The chapter closes by considering optimal insurance problems and optimal consumption-saving problems in a two-period economy.

Solutions of Selected Exercises

In this appendix, we provide the detailed solutions to a selection of the exercises proposed at the end of the chapters. The solutions to the exercises that are not solved in this appendix can be found in the solutions manual.