## Abstract

We present a fully non-parametric method for extracting risk neutral densities (RNDs) from observed option prices. The aim is to obtain a continuous, smooth, monotonic, and convex pricing function that is twice differentiable. Thus, irregularities such as negative probabilities that afflict many existing RND estimation techniques are reduced. Our method employs neural networks to obtain a smoothed pricing function, and a central finite difference approximation to the second derivative to extract the required gradients.

This novel technique was successfully applied to a large set of FTSE 100 daily European exercise (ESX) put options data and as an Ansatz to the corresponding set of American exercise (SEI) put options. The results of paired t-tests showed significant differences between RNDs extracted from ESX and SEI option data, reflecting the distorting impact of early exercise possibility for the latter. In particular, the results for skewness and kurtosis suggested different shapes for the RNDs implied by the two types of put options. However, both ESX and SEI data gave an unbiased estimate of the realised FTSE 100 closing prices on the options’ expiration date. We confirmed that estimates of volatility from the RNDs of both types of option were biased estimates of the realised volatility at expiration, but less so than the LIFFE tabulated at-the-money implied volatility.

This novel technique was successfully applied to a large set of FTSE 100 daily European exercise (ESX) put options data and as an Ansatz to the corresponding set of American exercise (SEI) put options. The results of paired t-tests showed significant differences between RNDs extracted from ESX and SEI option data, reflecting the distorting impact of early exercise possibility for the latter. In particular, the results for skewness and kurtosis suggested different shapes for the RNDs implied by the two types of put options. However, both ESX and SEI data gave an unbiased estimate of the realised FTSE 100 closing prices on the options’ expiration date. We confirmed that estimates of volatility from the RNDs of both types of option were biased estimates of the realised volatility at expiration, but less so than the LIFFE tabulated at-the-money implied volatility.

Original language | English |
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Pages (from-to) | 121-128 |

Number of pages | 8 |

Journal | Physica A: Statistical Mechanics and Its Applications |

Volume | 382 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2007 |