Abstract:
Chapter 2 discussed the basic principles underlying of the two major option pricing formulae. It clearly showed that two totally different approaches were followed in each case, and yet both arrived at approximately the same value for the price of an option. Both these approaches made certain assumptions in their derivation of the formulae in order to simplify the final expressions, and to produce a more workable solution. They both however made substantial use of statistical probability in order to determine the likelihood of a certain event occurring. Chapter 3 gave a detailed derivation of both the Black and Scholes and the Binomial tree pricing formulae, as well as the associated criticism and advantages of the respective approaches. Value at risk, or VaR, was used in determining the statistical probability of a certain portfolio consisting of a specified option losing more than a certain percentage of its value over a given period of time. The resulting number obtained can be used to judge the riskiness of a portfolio in the given market conditions. All of these formulae are used on a daily basis by financial professionals in the daily operations of a magnitude of different institutions in order to value financial portfolios, the risk associated with these portfolios and the probability of certain events occurring within the portfolios in order to make better decisions and increase the profitability of these institutions, without actually knowing the underlying principles. - As- such these --formulae merely become a number crunching business, and interpretation of these numbers, without realising the pitfalls associated with the approaches in establishing these formulae. The random walk theory for unrestricted movement assumes that at t=0, the rates are at the origin. This can be interpreted as 0%, and instinctively any person would agree that 0% is not possible in any fixed income environment, due to the time value attached to money. Choosing the ruling rate as the origin would be more practical in determining the origin, but care must be taken in assigning probabilities to the up and down movements. At the onset of the problems amongst the emerging markets during 1998, the probability of rates increasing once it reached 17,00% was much higher than that of the rates decreasing. However, barely a month later when the rates had reached its peak at more than 21,00% and were declining again, the probability of the rates increasing once it reached 17,00% again was much lower than that of it decreasing further. This would
have a significant effect on the probability generating function, and hence also an effect on the mean and variance thus derived. The probability curve of the rates during these times were also not represented by a standard normal curve, and as such the heteroscedacity of the curve had a major influence on the pricing of options. During extreme periods both the random walk theory and the Wiener process would be totally skewed, and unreliable answers would be derived from this approach. By 'adjusting the expression for a non-standard distribution, these problems can be eliminated and an accurate approach once again obtained using this process. Problems that could occur when using this approach to solve inaccuracies would amongst others include the following: The incorrect distribution function is being applied for the specific set of conditions prevailing in the market. This is due to the fact that under these abnormal conditions the distribution function can change over a very short period of time. Incorrect skews being applied to the distribution function due to fast changing market conditions. When to revert back to the normal distribution function. It then becomes a question not of an improper analytical approach, but incorrect timing approach. Since markets mostly perform according to the standardised normal distribution function the Wiener approach hold true for most applications.