Szymon Borak

Time Series Modelling with Semiparametric Factor Dynamics

High-dimensional regression problems, which reveal dynamic behavior, are typically analyzed by time propagation of a few number of factors. The inference on the whole system is then based on the low-dimensional time series analysis. Such high-dimensional problems occur frequently in many different fields of science. In this article we address the problem of inference when the factors and factor loadings are estimated by semiparametric methods. This more flexible modeling approach poses an important question: Is it justified, from an inferential point of view, to base statistical inference on the estimated times series factors? We show that the difference of the inference based on the estimated time series and “true” unobserved time series is asymptotically negligible. Our results justify fitting vector autoregressive processes to the estimated factors, which allows one to study the dynamics of the whole high-dimensional system with a low-dimensional representation. We illustrate the theory with a simulation study. Also, we apply the method to a study of the dynamic behavior of implied volatilities and to the analysis of functional magnetic resonance imaging (fMRI) data.

Convenience Yields for CO2 Emission Allowance Futures Contracts

In January 2005 the EU-wide CO2 emissions trading system (EU-ETS) has formally entered into operation. Within the new trading system, the right to emit a particular amount of CO2 becomes a tradable commodity - called EU Allowances (EUAs) - and affected companies, traders and investors will face new strategic challenges. In this paper we investigate the nature of convenience yields for CO2 emission allowance futures. We conduct an empirical study on price behavior, volatility term structure and correlations in different CO2 EUA contracts. Our findings are that the market has changed from initial backwardation to contango with significant convenience yields in future contracts for the Kyoto commitment period starting in 2008. A high fraction of the yields can be explained by the price level and volatility of the spot prices. We conclude that the yields can be interpreted as market expectation on the price risk of CO2 emissions allowance prices and the uncertainty of EU allocation plans for the Kyoto period.

Models for Heavy-tailed Asset Returns

Many of the concepts in theoretical and empirical finance developed over the past decades – including the classical portfolio theory, the Black-Scholes-Merton option pricing model or the RiskMetrics variance-covariance approach to VaR – rest upon the assumption that asset returns follow a normal distribution. But this assumption is not justified by empirical data! Rather, the empirical observations exhibit excess kurtosis, more colloquially known as fat tails or heavy tails. This chapter is intended as a guide to heavy-tailed models. We first describe the historically oldest heavy-tailed model – the stable laws. Next, we briefly characterize their recent lighter-tailed generalizations, the so-called truncated and tempered stable distributions. Then we study the class of generalized hyperbolic laws, which – like tempered stable distributions – can be classified somewhere between infinite variance stable laws and the Gaussian distribution. Finally, we provide numerical examples.

FFT-based Option Pricing

The Black-Scholes formula, one of the major breakthroughs of modern finance, allows for an easy and fast computation of option prices. But some of its assumptions, like constant volatility or log-normal distribution of asset prices, do not find justification in the markets. More complex models, which take into account the empirical facts, often lead to more computations and this time burden can become a severe problem when computation of many option prices is required, e.g. in calibration of the implied volatility surface. To overcome this problem Carr and Madan (1999) developed a fast method to compute option prices for a whole range of strikes. This method and its application are the theme of this chapter. In Section 1.3, we briefly discuss the Merton, Heston and Bates models concentrating on aspects relevant for the option pricing method. In the following section, we present the method of Carr and Madan which is based on the fast Fourier transform (FFT) and can be applied to a variety of models. We also consider brie∞y some further developments and give a short introduction to the FFT algorithm. In the last section, we apply the method to the three analyzed models, check the results by Monte Carlo simulations and comment on some numerical issues.

DSFM fitting of implied volatility surfaces

Implied volatility is one of the key issues in modern quantitative finance, since plain vanilla option prices contain vital information for pricing and hedging of exotic and illiquid options. European plain vanilla options are nowadays widely traded, which results in a great amount of high-dimensional data especially on an intra day level. The data reveal a degenerated string structure. Dynamic semiparametric factor models (DSFM) are tailored to handle complex, degenerated data and yield low dimensional representations of the implied volatility surface (IVS). We discuss estimation issues of the model and apply it to DAX option prices.

A semiparametric factor model for electricity forward curve dynamics

In this paper we introduce the dynamic semiparametric factor model (DSFM) for electricity forward curves. The biggest advantage of our approach is that it not only leads to smooth, seasonal forward curves extracted from exchange traded futures and forward electricity contracts, but also to a parsimonious factor representation of the curve. Using closing prices from the Nordic power market Nord Pool we provide empirical evidence that the DSFM is an efficient tool for approximating forward curve dynamics.

Financial Time Series Models

This section deals with financial time series analysis. The statistical properties of asset and return time series are inuenced by the media (daily news on the radio, television and newspapers) that informs us about the latest changes in stock prices, interest rates and exchange rates. This information is also available to traders who deal with immanent risk in security prices. It is therefore interesting to understand the behavior of asset prices. Economic models on the pricing of securities are mostly based on theoretical concepts which involve the formation of expectations, utility functions and risk preferences. In this section we concentrate on answering the empirical questions. Firstly, given a data set we aim to specify an appropriate model reecting the main characteristics of the empirically observable stock price process and we wish to know whether the assumptions underlying the model are fulfilled in reality or whether the model has to be modified. A new model on the stock price process could possibly effect the function of the markets. To this end we apply statistical tools to empirical data and start with considering the concepts of univariate analysis before moving on to multivariate time series.

Models for the Interest Rate and Interest Rate Derivatives

Pricing interest rate derivatives fundamentally depends on the underlying term structure. The often made assumptions of constant risk free interest rate and its independence of equity prices will not be reasonable when considering interest rate derivatives. Just as the dynamics of a stock price are modeled via a stochastic process, the term structure of interest rates is modeled stochastically. As interest rate derivatives have become increasingly popular, especially among institutional investors, the standard models for the term structure have become a core part of financial engineering. It is therefore important to practice the basic tools of pricing interest rate derivatives. For interest rate dynamics, there are one-factor and two-factor short rate models, the Heath Jarrow Morton framework and the LIBOR Market Model.

Portfolio Credit Risk

Financial institutions are interested in loss protection and loan insurance. Thus determining the loss reserves needed to cover the risk stemming from credit portfolios is a major issue in banking. By charging risk premiums a bank can create a loss reserve account which it can exploit to be shielded against losses from defaulted debt. However, it is imperative that these premiums are appropriate to the issued loans and to the credit portfolio risk inherent to the bank. To determine the current risk exposure it is necessary that financial institutions can model the default probabilities for their portfolios of credit instruments appropriately. To begin with, these probabilities can be viewed as independent but it is apparent that it is plausible to drop this assumption and to model possible defaults as correlated events.

Does hedging with implied volatility factors improve the hedging efficiency of barrier options

The price of a barrier option depends on the shape of the entire implied volatility surface which is a high-dimensional dynamic object. Barrier options are hence exposed to non-trivial volatility risk. We extract the key risk factors of implied volatility surface fluctuations by means of a semiparametric factor model. Based on the factors we define a practical hedging procedure within a local volatility framework. The hedging performance is evaluated using DAX index options.