This series of studies are focused on two important issues about option pricing. The first one concerns the implementation of continuous time stochastic volatility models,
including procedures for estimation and pricing. The second is about the loss functions which are important both in parameter estimation and model evaluation for pricing options.
For the estimation of stochastic volatility models, a partially observed GARCH model is proposed to approximate the likelihood function. Beyond the purpose of estimation, this type of models indeed bridges the gap between the continuous time stochastic volatility models and the discrete time GARCH models. Furthermore, inferences including estimation and pricing can be done under the same schema.
The most fascinating is that the method provides an approach for pricing options based on an approximated conditional distribution of the future price given the prices up to now, instead of the current price and a “filtered” volatility. And this approach obviously leads to some interesting results, for example higher option prices accompanied by descent paths of prices and lower prices by ascent paths.
Clearly the properties of the partially observed GARCH models shall be worth further investigations. It is not clear if the filtered variance via this method will be an (asymptotic) unbiased estimator of the true variance. If not, it may help explain partly why option prices are generally higher than that by the Black-Scholes formula with volatility
estimated by past prices, which in turn is about the behaviors of market participants when they cannot observe some state variables such as volatilities.
Loss functions are important and critical for option pricing, even though no theory can be found to formally address it. However, as pointed out in the third chapter, it is essential that the loss function should not be composed of option prices only since there are certainly much more other elements to constitute the information contents together.
From a practical point of view, the Black-Scholes implied volatility or other
nonparametric volatility index is certainly to provide as a basis for constructing loss functions. Then mean reversion as a major characteristic of volatilities should be quantified for the introduction of loss functions. That is, a precise definition of mean reversion may be the next step to a more compact and meaningful loss functions.
Some elementary statistical concepts are incorporated into all of the studies. For example, error sum of squares used as loss functions should be based on homogeneity of variance, and taking expectations over unobserved random variables instead of just “estimating” it.
For financial engineering as “engineering”, the introduction of these concepts may help increase the proportionality of science.
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