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Lorenz Schneider

Publications and source records attributed to Lorenz Schneider.

7 recordsLinked to original sources

Electricity Price Prediction Using Multi-Kernel Gaussian Process Regression Combined with Kernel-Based Support Vector Regression

This paper presents a new hybrid model for predicting German electricity prices. The algorithm is based on a combination of Gaussian Process Regression (GPR) and Support Vector Regression (SVR). Although GPR is a competent model for learning stochastic patterns within data and for interpolation, its performance for out-of-sample data is not very promising. By choosing a suitable data-dependent covariance function, we can enhance the performance of GPR for the German hourly power prices being tested. However, since the out-of-sample prediction is dependent on the training data, the prediction is vulnerable to noise and outliers. To overcome this issue, a separate prediction is calculated using SVR, which applies margin-based optimization. This method is advantageous when dealing with non-linear processes and outliers, since only certain necessary points (support vectors) in the training data are responsible for regression. The individual predictions are then linearly combined using uniform weights. When tested on historic German power prices, this approach outperforms the publicly available benchmarks, namely the LASSO estimated autoregressive regression model, deep neural network provided in the recent research by [1].

cs.LG↗

Euclidean Affine Functions and Applications to Calendar Algorithms

We study properties of Euclidean affine functions (EAFs), namely those of the form $f(r) = (α\cdot r + β)/δ$, and their closely related expression $\mathring{f}(r) = (α\cdot r + β)\%δ$, where $r$, $α$, $β$ and $δ$ are integers, and where $/$ and $\%$ respectively denote the quotient and remainder of Euclidean division. We derive algebraic relations and numerical approximations that are important for the efficient evaluation of these expressions in modern CPUs. Since simple division and remainder are particular cases of EAFs (when $α= 1$ and $β= 0$), the optimisations proposed in this paper can also be appplied to them. Such expressions appear in some of the most common tasks in any computer system, such as printing numbers, times and dates. We use calendar calculations as the main application example because it is richer with respect to the number of EAFs employed. Specifically, the main application presented in this article relates to Gregorian calendar algorithms. We will show how they can be implemented substantially more efficiently than is currently the case in widely used C, C++, C# and Java open source libraries. Gains in speed of a factor of two or more are common.

cs.DS↗

Seasonal Stochastic Volatility and the Samuelson Effect in Agricultural Futures Markets

We introduce a multi-factor stochastic volatility model for commodities that incorporates seasonality and the Samuelson effect. Conditions on the seasonal term under which the corresponding volatility factor is well-defined are given, and five different specifications of the seasonality pattern are proposed. We calculate the joint characteristic function of two futures prices for different maturities in the risk-neutral measure. The model is then presented under the physical measure, and its state-space representation is derived, in order to estimate the parameters with the Kalman filter for time series of corn, cotton, soybean, sugar and wheat futures from 2007 to 2017. The seasonal model significantly outperforms the nested non-seasonal model in all five markets, and we show which seasonality patterns are particularly well-suited in each case. We also confirm the importance of correctly modelling the Samuelson effect in order to account for futures with different maturities. Our results are clearly confirmed in a robustness check carried out with an alternative dataset of constant maturity futures for the same agricultural markets.

q-fin.PR↗

Seasonal Stochastic Volatility and Correlation together with the Samuelson Effect in Commodity Futures Markets

We introduce a multi-factor stochastic volatility model based on the CIR/Heston volatility process that incorporates seasonality and the Samuelson effect. First, we give conditions on the seasonal term under which the corresponding volatility factor is well-defined. These conditions appear to be rather mild. Second, we calculate the joint characteristic function of two futures prices for different maturities in the proposed model. This characteristic function is analytic. Finally, we provide numerical illustrations in terms of implied volatility and correlation produced by the proposed model with five different specifications of the seasonality pattern. The model is found to be able to produce volatility smiles at the same time as a volatility term-structure that exhibits the Samuelson effect with a seasonal component. Correlation, instantaneous or implied from calendar spread option prices via a Gaussian copula, is also found to be seasonal.

q-fin.PR↗

From the Samuelson Volatility Effect to a Samuelson Correlation Effect: Evidence from Crude Oil Calendar Spread Options

We introduce a multi-factor stochastic volatility model based on the CIR/Heston stochastic volatility process. In order to capture the Samuelson effect displayed by commodity futures contracts, we add expiry-dependent exponential damping factors to their volatility coefficients. The pricing of single underlying European options on futures contracts is straightforward and can incorporate the volatility smile or skew observed in the market. We calculate the joint characteristic function of two futures contracts in the model in analytic form and use the one-dimensional Fourier inversion method of Caldana and Fusai (JBF 2013) to price calendar spread options. The model leads to stochastic correlation between the returns of two futures contracts. We illustrate the distribution of this correlation in an example. We then propose analytical expressions to obtain the copula and copula density directly from the joint characteristic function of a pair of futures. These expressions are convenient to analyze the term-structure of dependence between the two futures produced by the model. In an empirical application we calibrate the proposed model to volatility surfaces of vanilla options on WTI. In this application we provide evidence that the model is able to produce the desired stylized facts in terms of volatility and dependence. In a separate appendix, we give guidance for the implementation of the proposed model and the Fourier inversion results by means of one and two-dimensional FFT methods.

q-fin.PR↗

A Note on "A Family of Maximum Entropy Densities Matching Call Option Prices"

In Neri and Schneider (2012) we presented a method to recover the Maximum Entropy Density (MED) inferred from prices of call and digital options on a set of n strikes. To find the MED we need to numerically invert a one-dimensional function for n values and a Newton-Raphson method is suggested. In this note we revisit this inversion problem and show that it can be rewritten in terms of the Langevin function for which numerical approximations of its inverse are known. The approach is very similar to that of Buchen and Kelly (BK) with the difference that BK only requires call option prices. Then, in continuation of our first paper, we presented another approach which uses call prices only and recovers the same density as BK with a few advantages, notably, numerical stability. This second paper provides a detailed analysis of convergence and, in particular, gives various estimates of how far (in different senses) the iterative algorithm is from the solution. These estimates rely on a constant m > 0. The larger m is the better the estimates will be. A concrete value of m is suggested in the second paper, and this note provides a sharper value.

q-fin.PR↗

A Family of Maximum Entropy Densities Matching Call Option Prices

We investigate the position of the Buchen-Kelly density in a family of entropy maximising densities which all match European call option prices for a given maturity observed in the market. Using the Legendre transform which links the entropy function and the cumulant generating function, we show that it is both the unique continuous density in this family and the one with the greatest entropy. We present a fast root-finding algorithm that can be used to calculate the Buchen-Kelly density, and give upper boundaries for three different discrepancies that can be used as convergence criteria. Given the call prices, arbitrage-free digital prices at the same strikes can only move within upper and lower boundaries given by left and right call spreads. As the number of call prices increases, these bounds become tighter, and we give two examples where the densities converge to the Buchen-Kelly density in the sense of relative entropy when we use centered call spreads as proxies for digital prices. As pointed out by Breeden and Litzenberger, in the limit a continuous set of call prices completely determines the density.

q-fin.PR↗