Christian Schittenkopf

GARCH vs. stochastic volatility: Option pricing and risk management

This paper examines the out-of-sample performance of two common extensions of the Black-Scholes framework, namely a GARCH and a stochastic volatility option pricing model. The models are calibrated to intraday FTSE 100 option prices. We apply two sets of performance criteria, namely out-of-sample valuation errors and Value-at-Risk oriented measures. When we analyze the fit to observed prices, GARCH clearly dominates both stochastic volatility and the benchmark Black Scholes model. However, the predictions of the market risk from hypothetical derivative positions show sizable errors. The fit to the realized profits and losses is poor and there are no notable differences between the models. Overall we therefore observe that the more complex option pricing models can improve on the Black Scholes methodology only for the purpose of pricing, but not for the Value-at-Risk forecasts. (authors abstract)

Two strategies to avoid overfitting in feedforward networks

Abstract We present a new network topology to avoid overfitting in two-layered feedforward networks. We use two additional linear layers and principal component analysis to reduce the dimension of both inputs and internal representations and to transmit the essential information. Thereby neurons with small variance in the output are removed, which results in better generalization properties. Our network and learning rules can also be seen as a procedure to reduce the number of free parameters without using second order information of the error function. As a second strategy we derive a penalty term, which drives the network to keep the variances of the hidden layer outputs small. Experimental results show that thereby the transmitted information is limited, which reduces the noise and gives better generalization. The variances of the outputs of the hidden neurons are used again as a pruning criterion.

Financial volatility trading using recurrent neural networks

We simulate daily trading of straddles on financial indexes. The straddles are traded based on predictions of daily volatility differences in the indexes. The main predictive models studied are recurrent neural nets (RNN). Such applications have often been studied in isolation. However, due to the special character of daily financial time-series, it is difficult to make full use of RNN representational power. Recurrent networks either tend to overestimate noisy data, or behave like finite-memory sources with shallow memory; they hardly beat classical fixed-order Markov models. To overcome data nonstationarity, we use a special technique that combines sophisticated models fitted on a larger data set, with a fixed set of simple-minded symbolic predictors using only recent inputs. Finally, we compare our predictors with the GARCH family of econometric models designed to capture time-dependent volatility structure in financial returns. GARCH models have been used to trade volatility. Experimental results show that while GARCH models cannot generate any significantly positive profit, by careful use of recurrent networks or Markov models, the market makers can generate a statistically significant excess profit, but then there is no reason to prefer RNN over much more simple and straightforward Markov models. We argue that any report containing RNN results on financial tasks should be accompanied by results achieved by simple finite-memory sources combined with simple techniques to fight nonstationarity in the data.

Forecasting Time-dependent Conditional Densities: A Semi-non- parametric Neural Network Approach

In _nancial econometrics the modelling of asset return series is closely related to the estimation of the corresponding conditional densities[ One reason why one is interested in the whole conditional density and not only in the conditional mean is that the conditional variance can be interpreted as a measure of time!dependent volatility of the return series[ In fact\ the mod! elling and the prediction of volatility is one of the central topics in asset pricing[ In this paper we propose to estimate conditional densities semi! non!parametrically in a neural network framework[ Our recurrent mixture density networks realize the basic ideas of prominent GARCH approaches but they are capable of modelling any continuous conditional density also allowing for time!dependent higher!order moments[ Our empirical analysis of daily FTSE 099 data demonstrates the importance of distributional assumptions in volatility prediction and shows that the out!of!sample fore! casting performance of neural networks slightly dominates those of GARCH models[ Copyright 1999 John Wiley + Sons\ Ltd[

Risk-neutral density extraction from option prices: improved pricing with mixture density networks

One of the central goals in finance is to find better models for pricing and hedging financial derivatives such as call and put options. We present a new semi-nonparametric approach to risk-neutral density extraction from option prices, which is based on an extension of the concept of mixture density networks. The central idea is to model the shape of the risk-neutral density in a flexible, nonlinear way as a function of the time horizon. Thereby, stylized facts such as negative skewness and excess kurtosis are captured. The approach is applied to a very large set of intraday options data on the FTSE 100 recorded at LIFFE. It is shown to yield significantly better results in terms of out-of-sample pricing accuracy in comparison to the basic and an extended Black-Scholes model. It is also significantly better than a more elaborate GARCH option pricing model which includes a time-dependent volatility process. From the perspective of risk management, the extracted risk-neutral densities provide valuable information for value-at-risk estimations.

On Nonlinear, Stochastic Dynamics in Economic and Financial Time Series

The search for deterministic chaos in economic and financial time series has attracted much interest over the past decade. Evidence of chaotic structures is usually blurred, however, by large random components in the time series. In the first part of this paper, a sophisticated algorithm for estimating the largest Lyapunov exponent with confidence intervals is applied to artificially generated and real-world time series. Although the possibility of testing empirically for positivity of the estimated largest Lyapunov exponent is an advantage over other existing methods, the interpretability of the obtained results remains problematic. For instance, it is practically impossible to distinguish chaotic and periodic dynamics in the presence of dynamical noise even for simple dynamical systems. We conclude that the notion of sensitive dependence on initial conditions, as it has been developed for deterministic dynamics, can hardly be transferred into a stochastic context. Therefore, the second part of the paper aims to measure the dependencies of stochastic dynamics on the basis of a distributional characterization of the dynamics. For instance, the dynamics of financial return series are essentially captured by heteroskedastic models. We adopt a sensitivity measure proposed in literature and derive analytical expressions for the most important classes of stochastic dynamics. In practice, the sensitivity measure for the a priori unknown dynamics of a system can be calculated after estimating the conditional density of the systems state variable.

Volatility prediction with mixture density networks

Despite the lack of a precise definition of volatility in finance, the estimation of volatility and its prediction is an important problem. In this paper we compare the performance of standard volatility models and the performance of a class of neural models, i.e. mixture density networks (MDNs). First experimental results indicate the importance of long-term memory of the models as well as the benefit of using non-gaussian probability densities for practical applications. (authors abstract)

Finite automata-models for the investigation of dynamical systems

We describe a method to measure the complexity of a dynamical system. By complexity we mean the intrinsic information processing abilities which we believe to be visible only on an infinitesimal scale. The complexity measure is based on concepts from information theory and from the theory of formal languages.

Exploring the intrinsic information loss in single-humped maps by refining multi-symbol partitions

Abstract We study the intrinsic information flow for a class of chaotic maps by applying the concept of conditional entropy for an infinite alphabet. Our procedure consists in refining partitions and thereby revealing the information flow step by step. We prove that in the case of an infinite number of symbols the information about the initial conditions is lost linearly in time.

Testing nonlinear Markovian hypotheses in dynamical systems

Abstract We present a statistical approach for detecting the Markovian character of dynamical systems by analyzing their flow of information. Especially in the presence of noise which is mostly the case for real-world time series, the calculation of the information flow of the underlying system via the concept of symbolic dynamics is rather problematic since one has to use infinitesimal partitions. We circumvent this difficulty by measuring the information flow indirectly. More precisely, we calculate a measure based on higher order cumulants which quantifies the statistical dependencies between the past values of the time series and the point r steps ahead. As an extension of Theilers method of surrogate data (Theiler et al., 1992) this cumulant based information flow (a function of the look-ahead r ) is used as the discriminating statistic in testing the observed dynamics against a hierarchy of null hypotheses corresponding to nonlinear Markov processes of increasing order. This procedure is iterative in the sense that whenever a null hypothesis is rejected new data sets can be generated corresponding to better approximations of the original process in terms of information flow. Since we use higher order cumulants for calculating the discriminating statistic our method is also applicable to small data sets. Numerical results on artificial and real-world examples including non-chaotic, nonlinear processes, autoregressive models and noisy chaos show the effectiveness of our approach.