Svetlana Borovkova

Limit theorems for functionals of mixing processes with applications to U-statistics and dimension estimation

In this paper we develop a general approach for investigating the asymptotic distribution of functional Xn = f((Zn+k)k∈z) of absolutely regular stochastic processes (Zn)n∈z. Such functional occur naturally as orbits of chaotic dynamical systems, and thus our results can be used to study probabilistic aspects of dynamical systems. We first prove some moment inequalities that are analogous to those for mixing sequences. With their help, several limit theorems can be proved in a rather straightforward manner. We illustrate this by re-proving a central limit theorem of Ibragimov and Linnik. Then we apply our techniques to U-statistics Matrix Equation with symmetric kernel h : R × R → R. We prove a law of large numbers, extending results of Aaronson, Burton, Dehling, Gilat, Hill and Weiss for absolutely regular processes. We also prove a central limit theorem under a different set of conditions than the known results of Denker and Keller. As our main application, we establish an invariance principle for U-processes (Un(h))h, indexed by some class of functions. We finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system.

A Closed Form Approach to the Valuation and Hedging of Basket and Spread Options

The lognormal diffusion model for the returns on the underlying asset is a key assumption of Black-Scholes and many other derivatives models. One advantage of the lognormal is that it is impossible for the price to become negative in the model; another is that asset prices exhibit positive skewness, as is apparent in real world securities. However, when an option is written on a spread or on a portfolio containing both long and short positions in assets that individually follow lognormal diffusions, these properties may not be appropriate. In this paper, Borovkova, Permana, and v.d. Weide propose modeling such portfolios using negative and/or shifted lognormal distributions. This allows a much better fit to the empirical distributions for such cases, while still retaining the closed-form solutions of the Black-Scholes framework for option values and Greek letter risks. The effectiveness and simplicity of the technique are demonstrated by Monte Carlo simulations with a variety of basket options.

Modelling electricity prices by the potential jump-diffusion

In liberalized electricity markets prices exhibit features, such as price spikes, rarely seen in other commodity markets. Models for electricity spot price, such as mean-reverting jump-diffusions and regime-switching models are only partially successful in modelling price spikes. In this paper we introduce a new approach to electricity price modelling: a potential function jump-diffusion model, which allows for a continuously varying mean-reversion rate and provides a flexible way to model price spikes.

News, volatility and jumps: the case of natural gas futures

We investigate the impact of Thompson Reuters News Analytics (TRNA) news sentiment on the price dynamics of natural gas futures traded on the New York Mercantile Exchange (NYMEX). We propose a Local News Sentiment Level (LNSL) model, based on the Local Level model of Durbin and Koopman (2001), to construct a running series of news sentiment on the basis of the 5-minute time grid. Additionally, we construct several return and variation measures to proxy for the fine dynamics of the front month natural gas futures prices. We employ event studies and Granger causality tests to assess the effect of news on the returns, price jumps and the volatility. We find significant relationships between news sentiment and the dynamic characteristics of natural gas futures returns. For example, we find that the arrival of news in non-trading periods causes overnight returns, that news sentiment is Granger caused by volatility and that strength of news sentiment is more sensitive to negative than to positive jumps. In addition to that, we find strong evidence that news sentiment severely Granger causes jumps and conclude that market participants trade as some function of aggregated news. We apply several state-of-the-art volatility models augumented with news sentiment and conduct an out-of-sample volatility forecasting study. The first class of models is the generalized autoregressive conditional heteroskedasticity models (GARCH) of Engle (1982) and Bollerslev (1986) and the second class is the high-frequency-based volatility (HEAVY) models of Shephard and Sheppard (2010) and Noureldin et al. (2011). We adapt both models to account for asymmetric volatility, leverage and time to maturity effects. By augmenting all models with a news sentiment variable, we test the hypothesis whether including news sentiment in volatility models results in superior volatility forecasts. We find significant evidence that this hypothesis holds.

American Basket and Spread Option Pricing by a Simple Binomial Tree

The return on a portfolio is the weighted average of the returns on the individual assets in the portfolio. But the dynamics of portfolio returns are not so simple. The standard assumption that the underlying asset for an option follows geometric Brownian motion is convenient for individual stocks, but it runs into trouble for combinations of stocks, because a linear combination of lognormal returns does not have a lognormal distribution. Luckily, the true portfolio return distribution can be closely approximated by a generalized lognormal using the technique of matching moments, even when some weights are negative, as in a spread trade. This makes it easy to price European options on baskets of stocks or spreads. In this article, the authors show how to extend the same basic idea to construct a binomial tree for the portfolio return, which allows for efficient pricing of contracts with American exercise.

Implied volatility in oil markets

Modelling the implied volatility surface as a function of an options strike price and maturity is a subject of extensive research in financial markets. The implied volatility in commodity markets is much less studied, due to a limited liquidity and the complicated structure of commodity options. A new semi-parametric method is introduced for modelling the implied volatility surface and is applied to the option price data from oil markets. This approach combines the simplicity of a parametric method with the flexibility of a non-parametric approach. The method can successfully deal with a limited amount of option price data. Performance of the method is investigated by applying it to prices of exchange-traded crude oil and gasoline options, and the results are compared with those obtained by a purely parametric approach. Furthermore, the investigation of the relationship between volatilities implied from European and Asian options shows that Asian options in oil markets are significantly more expensive than theoretical arguments imply.

Analysis and Modelling of Electricity Futures Prices

We model electricity futures prices using a seasonal forward curve model, quantifying seasonalities by a deterministic seasonal forward premium. Stochastic features of the futures prices are contained in the stochastic forward premium: a quantity analogous to the well-known convenience yield. The model parameters are estimated from the historical data of IPE electricity futures prices and the spark spread, and electricity forward curves are deseasonalized to reveal their underlying stochastic structure. We apply principal component analysis to the deseasonalized forward curves and develop trading strategies using indicators based on these principal components.

Detecting Market Transitions and Energy Futures Risk Management Using Principal Components

Abstract An empirical approach to analysing the forward curve dynamics of energy futures is presented. For non-seasonal commodities—such as crude oil—the forward curve is well described by the first three principal components: the level, slope and curvature. A principal component indicator is described that detects transitions between the two fundamental market states remarkably well. For seasonal commodities—such as electricity and natural gas—it is shown how to extract the seasonal component from the forward curve. The principal component indicator can then be applied to the de-seasoned forward curve to detect significant price deviations that may support profitable trading strategies. A principal component approach to forward curve modelling is applied to computing portfolio value-at-risk. This approach is combined with a new two-step resampling procedure to improve value-at-risk estimates.

Commodity volatility modelling and option pricing with a potential function approach

We consider a novel approach to modelling of commodity prices and apply it to commodity option pricing and volatility estimation. This approach is particularly suited for prices with multiple attraction regions, such as crude oil and other energy and agricultural commodities. The price is modelled as a diffusion process governed by a potential function with minima at the attraction points. When applied to crude oil prices, the method captures characteristic behaviour of the prices remarkably well. Pricing of European options on spot and futures commodity contracts is developed within the potential model, and compared to the Black–Scholes framework. The approach provides a new way of estimating the volatility, which is particularly useful when option prices (and hence implied volatilities) are not readily available; this is often the case for commodity markets. European options on physical commodities and commodity futures are priced using the volatility forecasts obtained from the model. The performance of the model is evaluated on the basis of the hedging costs of an option. For options on crude oil, the method outperforms – in terms of hedging costs—the Black–Scholes approach with historical volatility.

A Potential-Field Approach to Financial Time Series Modelling

We present a new approach to the problem of time series modelling that captures the invariant distribution of time series data within the model. This is particularly relevant in modelling economic and financial time series, such as oil prices, that exhibit clustering around a few preferred market modes. We propose a potential function approach which determines the function that governs the underlying time series process. This approach extends naturally to modelling multivariate time series. We show how to estimate the potential function for dimensions one and higher and use it to model statistically the evolution of the time series. An illustration of the procedure shows that testing the resulting model against historical data of oil prices captures the essential price behavior remarkably well. The model allows the generation of copies of the observed time series as well as providing better predictions by reducing uncertainty about the future behavior of the time series.