Option Pricing in Multivariate Stochastic Volatility Models of OU Type

We present a multivariate stochastic volatility model with leverage, which is flexible enough to recapture the individual dynamics as well as the interdependencies between several assets while still being highly analytically tractable. First we derive the characteristic function and give conditions that ensure its analyticity and absolute integrability in some open complex strip around zero. Therefore we can use Fourier methods to compute the prices of multi-asset options efficiently. To show the applicability of our results, we propose a concrete specification, the OU-Wishart model, where the dynamics of each individual asset coincide with the popular Gamma-OU BNS model. This model can be well calibrated to market prices, which we illustrate with an example using options on the exchange rates of some major currencies. Finally, we show that covariance swaps can also be priced in closed form.Comment: 28 pages, 5 figures, to appear in SIAM Journal on Financial Mathematic

Modelling Electricity Swaps with Stochastic Forward Premium Models

We present a new model for pricing electricity swaps. Two general factors affect contracts but unique risk elements affect each contract. General factors are average swap prices and deterministic trend-seasonal components, and unique elements are forward premiums. Innovations follow MNIG distributions. We estimate the model with data from the European Energy Exchange. The model outperforms four competitors, both in in-sample valuation and in out-of-sample forecasting, and in fitting the term structure of volatilities by market segments. Competitor models are (i) diffusion spot prices, (ii) jump-diffusion spot prices with time dependent volatility, (iii) HJM-based and (iv) Levy multifactor model with NIG distributions. Value-at-Risk measures based on normality strongly underestimate tail risk but our model gives estimates that are more exact.Juan Ignacio Peña and Rosa Rodriguez acknowledge financial support from the Ministry of Economics and Competitiveness, respectively, through grants ECO2012-35023, ECO2016-77807-P, and ECO2012-3655

Modeling VIX And VIX Derivatives With Mean Reverting Models And Parameter Estimation Using Filter Methods

In this thesis, we study the mean reverting property of the VIX time series, and use the VIX process as the underlying. We employ various mean reverting processes, including the Ornstein-Uhlenbeck (OU) process, the Cox-Ingersoll-Ross (CIR) process and the OU processes driven by Levy processes (Levy OU) to fit historical data of VIX, and calibrate the VIX option prices. The first contribution of this thesis is to use the Levy OU process to model the VIX process, in order to explain the observed high kurtosis. To price the option using the Levy OU process, we develop a FFT method. The second contribution is to build a joint framework to consistently model the VIX and VIX derivatives together on the entire time series of market data. We choose multi-factor mean-reverting models, in which we model the VIX process as a linear combination of latent factors. To estimate the models, we use Euler approximation to find a discrete approximation for the VIX process. Based on this approximate, we consider various filter methods, namely, the Unscented Kalman Filter (UKF), constrained UKF, mixed Gaussian UKF and Particle Filter (PF) for estimation. The performances of these models are compared and discussed. Radon Nikodym derivatives of the risk-neutral measure are discussed with respect to the physical measure for the jumps. A simple dynamic trading strategy was tested on these models

金融証券市場に関する研究 : 価格、意思決定および実証分析

首都大学東京, 2015-09-30, 博士（経営学）, 甲第553号首都大学東

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