Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues

15

Beyond the Gaussian hypothesis: potential troubles with derivatives valuation

This chapter has two parts:

• a review of alternatives to the Gaussian hypothesis, developed within other processes or models than the ones considered in the previous chapters;
• some views about criticisms and troubles that may arise from the various valuation methods of derivatives.

As you will see, these topics are somewhat related.

15.1 ALTERNATIVES TO THE GAUSSIAN HYPOTHESIS

The stochastic component of the processes considered up to now has always been based upon the normality of the returns distribution. Yet, several clues of non-normality may be observed from market time series, that encourage looking after alternative, more realistic hypotheses to the Gaussian distribution. The main issue will be that working on more complex distributions than the normal – requiring more than two parameters (the mean and the standard deviation) – implies quantifying the extra parameters that are themselves sources of error measurement, in such a way that the final result may well happen to be less reliable than the more simplistic, but more robust, hypothesis of normality…

The importance of the assumptions about the probability distribution of an asset price over time is evident in the case of derivatives valuation, as instruments which the value is affected by the future evolution of the underlying price. It is also relevant for spot instruments, within the framework of the Portfolio Theory (cf. Chapter 4, ...