In this chapter, we estimate parameters of model based implied probability distributions. The techniques we use include essentially the following:

Parametric approaches A-parametric density estimation and related techniques Entropy and Information measures

The parametric approach assumes a parametric risk-neutral distribution model with parameters estimated by best fit techniques. The a-parametric approach is distribution-free, defined by empirical risk-neutral distributions concurrent with properties of risk neutrality. Finally, entropy related techniques are explained and used to estimate the risk-neutral distribution that fits best a given data set and a prior personal probability distribution of future prices. Problems of particular importance include calculation of the implied volatility.

Trading in derivatives (options, credit derivatives and forwards, for example) is based on beliefs regarding future prices and losses. For example, the price of a European option whose maturity is at a future date *T* implies a belief regarding the underlying price of the option at its maturity. By the same token, an implied volatility expresses a current belief about the volatility of the underlying asset at time *T*.

To determine an implied distribution (or its implied volatility), data and beliefs (formulated as models or personal probability distributions) pertaining to future

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