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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. Index Terms--Hedging, mixture density networks, options, pricing, risk-neutral densities.

We simulate daily trading of straddles on the financial indexes DAX and FTSE 100. The straddles are traded based on predictions of daily volatility differences in the underlying indexes. The main predictive models studied in this paper are recurrent neural networks (RNNs). In the past, applications of RNNs in the financial domain were often studied in isolation. We argue against such a practice by showing that, 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 the noisy data, or behave like finite-memory sources with a relatively shallow memory. In fact, they can hardly beat (rather simple) classical fixed-order Markov models. To overcome the inherent nonstationarity in the data, 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, thereby avoiding older (and potentially misleading) data. 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 in the past to trade volatility. Experimental results show that while GARCH models are not able to generate any significantly positive profit, by careful use of recurrent networks or Markov models, the market makers can generate a statistically significant excess profit. However, on this type of problems, there is no reason to prefer RNNs 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. Index Terms--Financial indexes, Markov models, options, prediction suffix trees, recurrent neural networks, straddle, volatility.