| We propose a neural network option pricing model that can fit listed option prices accurately, and be used to recover the implied asset price distribution and asset price dynamics.; The observable market option prices are noisy and insufficient. To overcome the problem, two option pricing models constructed using multilayer feedforward neural networks are investigated. The first one uses a neural network to learn the implied volatility function of Black-Scholes-Merton model. To price an option, this neural network must work together with Black-Scholes-Merton formulas. The other one uses a neural network to learn the function mapping between the option price and observable affecting factors. This neural network is a complete option pricing model and can function independently of any option pricing formula.; Based on a theory derived by Breeden and Litzenberger, the implied risk-neutral probability density surface can be extracted from the second partial derivative of the option price function with respect to the strike price. While both neural network option pricing models fit observed option prices well, only the first model is suitable for extracting a risk-neutral probability density surface. Risk-neutral valuation method is used to perform in-sample and out-of-sample tests.; Based on the Fokker-Plank equation, an implied Ito process can be derived from the first and second partial derivatives of the option price function with respect to the strike price and the maturity. Similarly, only the first neural network option pricing model is suitable for deriving an Ito process. Monte Carlo simulation is used to perform in-sample and out-of-sample tests.; The pricing errors from the extracted risk-neutral probability density surface and Ito process are only slightly larger than that directly from the neural network option pricing model. The small difference indicates that little information has been lost in the extracted risk-neutral probability density surface and Ito process. As a result, exotic options can be priced with the extracted information. |
