A flexible matrix Libor model with smiles

We present a flexible approach for the valuation of interest rate derivatives based on Affine Processes. We extend the methodology proposed in Keller-Ressel et al. (2009) by changing the choice of the state space. We provide semi-closed-form solutions for the pricing of caps and floors. We then show that it is possible to price swaptions in a multifactor setting with a good degree of analytical tractability. This is done via the Edgeworth expansion approach developed in Collin-Dufresne and Goldstein (2002). A numerical exercise illustrates the flexibility of Wishart Libor model in describing the movements of the implied volatility surface

Smiles all around: FX joint calibration in a multi-Heston model

We introduce a novel multi-factor Heston-based stochastic volatility model, which is able to reproduce consistently typical multi-dimensional FX vanilla markets, while retaining the (semi)-analytical tractability typical of affine models and relying on a reasonable number of parameters. A successful joint calibration to real market data is presented together with various in- and out-of-sample calibration exercises to highlight the robustness of the parameters estimation. The proposed model preserves the natural inversion and triangulation symmetries of FX spot rates and its functional form, irrespective of choice of the risk-free currency. That is, all currencies are treated in the same way.Comment: Journal of Banking and Finance. Accepte

Stochastic volatility

Given the importance of return volatility on a number of practical financial management decisions, the efforts to provide good real- time estimates and forecasts of current and future volatility have been extensive. The main framework used in this context involves stochastic volatility models. In a broad sense, this model class includes GARCH, but we focus on a narrower set of specifications in which volatility follows its own random process, as is common in models originating within financial economics. The distinguishing feature of these specifications is that volatility, being inherently unobservable and subject to independent random shocks, is not measurable with respect to observable information. In what follows, we refer to these models as genuine stochastic volatility models. Much modern asset pricing theory is built on continuous- time models. The natural concept of volatility within this setting is that of genuine stochastic volatility. For example, stochastic-volatility (jump-) diffusions have provided a useful tool for a wide range of applications, including the pricing of options and other derivatives, the modeling of the term structure of risk-free interest rates, and the pricing of foreign currencies and defaultable bonds. The increased use of intraday transaction data for construction of so-called realized volatility measures provides additional impetus for considering genuine stochastic volatility models. As we demonstrate below, the realized volatility approach is closely associated with the continuous-time stochastic volatility framework of financial economics. There are some unique challenges in dealing with genuine stochastic volatility models. For example, volatility is truly latent and this feature complicates estimation and inference. Further, the presence of an additional state variable - volatility - renders the model less tractable from an analytic perspective. We examine how such challenges have been addressed through development of new estimation methods and imposition of model restrictions allowing for closed-form solutions while remaining consistent with the dominant empirical features of the data.Stochastic analysis

Applications of physics to finance and economics: returns, trading activity and income

This dissertation reports work where physics methods are applied to financial and economical problems. The first part studies stock market data (chapter 1 to 5). The second part is devoted to personal income in the USA (chapter 6). We first study the probability distribution of stock returns at mesoscopic time lags (return horizons) ranging from about an hour to about a month. For mesoscopic times the bulk of the distribution (more than 99% of the probability) follows an exponential law. At longer times, the exponential law continuously evolves into Gaussian distribution. After characterizing the stock returns at mesoscopic time lags, we study the subordination hypothesis. The integrated volatility V_t constructed from the number of trades process can be used as a subordinator for a Brownian motion. This subordination is able to describe approximatly 85% of the stock returns for time lags that start at 1 hour but are shorter than one day. Finally, we show that the CIR process describes well enough the empirical V_t process, such that the corresponding Heston model is able to describe the log-returns x_t process, with approximately the maximum quality that the subordination allows. Finally, we study the time evolution of the personal income distribution. We find that the personal income distribution in the USA has a well-defined two-income-class structure. The majority of population (97-99%) belongs to the lower income class characterized by the exponential Boltzmann-Gibb(``thermal'') distribution, whereas the higher income class (1-3% of population) has a Pareto power-law (``superthermal'') distribution. We show that the ``thermal'' part is stationary in time.Comment: 24 pages and 45 figures. PhD thesis presented to the committee members on May 10th 2005. This thesis is based on 3 published papers with one chapter (chapter 5) with new unpublished result

Design and Estimation of Quadratic Term Structure Models

We consider the design and estimation of quadratic term structure models. We start with a list of stylized facts on interest rates and interest rate derivatives, classified into three layers: (1) general statistical properties, (2) forecasting relations, and (3) conditional dynamics. We then investigate the implications of each layer of property on model design and strive to establish a mapping between evidence and model structures. We calibrate a two­factor model that approximates these three layers of properties well, and illustrate how the model can be applied to pricing interest rate derivatives.quadratic model; term structure; positive interest rates; humps; expectation hy­pothesis; GMM; caps and floors.

A Multivariate Stochastic Levy Correlation Model with Integrated Wishart Time Change and Its Application in Option Pricing

We develop a new multivariate Levy correlation model which is formulated by evaluating Levy processes subordinate to the integral of a Wishart process. This new model captures not only stochastic mean, stochastic volatility, and stochastic skewness, but also stochastic correlation of cross-sectional asset returns while still being analytical tractable. It is a multivariate extension of the time changed Levy process introduced by Carr, Geman, Madan and Yor, which can capture the individual dynamics as well as the interdependencies among several assets. In this dissertation, two different methods are employed to simulate paths of the instantaneous rate of time change matrix A(t), followed by a Wishart process. The simulation paths successfully display desirable clustering and persistence features. In addition, we analyze the behavior of the joint log return distribution generated in this new model and show that the model provides a rich dependence structure. The option pricing problem involves computing the closed form of the characteristic functions, which are usually not easily obtained in the multivariate correlated case. In this thesis, we derive explicit forms of both marginal and joint conditional characteristic functions by applying the `Matrix Riccati Linearization' technique creatively. Our work is distinguished from existing multivariate stochastic volatility models, with the advantage that it can deal with stochastic skewness effects introduced by Carr and Wu. Finally, we derive pricing methods for multi asset options as well as single asset options by using both simulation and Fast Fourier transform methods. More important, this model can be well calibrated to the real market. We chose options on two major FX currencies to perform the calibration and remarkable consistency has been observed

Topics in volatility models

In this thesis I will present my PhD research work, focusing mainly on financial modelling of asset’s volatility and the pricing of contingent claims (financial derivatives), which consists of four topics: 1. Several changing volatility models are introduced and the pricing of European options is derived under these models; 2. A general local stochastic volatility model with stochastic interest rates (IR) is studied in the modelling of foreign exchange (FX) rates. The pricing of FX options under this model is examined through the use of an asymptotic expansion method, based on Watanabe-Yoshida theory. The perfect/partial hedging issues of FX options in the presence of local stochastic volatility and stochastic IRs are also considered. Finally, the impact of stochastic volatility on the pricing of FX-IR structured products (PRDCs) is examined; 3. A new method of non-biased Monte Carlo simulation for a stochastic volatility model (Heston Model) is proposed; 4. The LIBOR/swap market model with stochastic volatility and jump processes is studied, as well as the pricing of interest rate options under that model. In conclusion, some future research topics are suggested. Key words: Changing Volatility Models, Stochastic Volatility Models, Local Stochastic Volatility Models, Hedging Greeks, Jump Diffusion Models, Implied Volatility, Fourier Transform, Asymptotic Expansion, LIBOR Market Model, Monte Carlo Simulation, Saddle Point Approximation