Neural Networks for Extracting the Asset Price Dynamics Implicit in Market Prices of Stock Index Options
How to extract the asset price dynamics implicit in market prices of stock index options is an important research topic. The main factor in deciding the market price of a stock index option is the expectation of investors on future movement of the stock index level. Therefore, market prices of stock index options imply the stochastic process of future stock index level. The implied information reflects the sentiment of investors. Hence, it is important for investors who use the options as a speculation or hedging tool and investment banks that develop over-the-counter options on the stock index. However, not only the observable option prices are discrete, insufficient, and noisy, but also the implied asset price dynamics is nonstationary. Therefore, extracting an implied asset price dynamics accurately is not an easy task. Multilayer feedforward neural networks are applied to overcome the difficulties coming from market reality. Assume that the movement of stock index level follows an Ito process. A multilayer feedforward neural network can be trained to learn the implied volatility function . The neural network has the strike price and maturity as inputs and implied volatility of Black-Scholes-Merton formula as the target. The trained neural network represents a continuous function mapping of implied volatility with respect to strike price and maturity. After successful training, the neural network is able to provide an accurate implied volatility value given a strike price and a maturity. For an option contract, its theoretical price is determined by the Black-Scholes-Merton formula with the implied volatility generated by the neural network. With the hybrid framework, the options pricing model not only fits market prices of options accurately but also is a continuous and smooth function of the strike price and maturity. Based on Fokker-Plank equation, an implied Ito process can be derived from the first and second partial derivative of the option price function with respect to the strike price and maturity. Consequently, the implied Ito process can be extracted by applying numerical differentiation to the hybrid options pricing model. Market prices of S&P 500 Index options, SPX, from the Berkeley Options Database (BODB) are used to investigate performances of the proposed method. Monte Carlo simulation is implemented to perform the in-sample and out-of-sample tests. Empirical tests show that the extracted Ito process is consistent with market data accurately and predicts future option price structure precisely. It indicates that little information has been lost in the derived Ito process. As a result, the derived Ito process is useful for pricing exotic options, calculating value-at-risk of options positions, and versatile financial applications.
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