Forecasting models for the dollar/rand spot rates.

[摘要] Owing to the complexity of hedging against the unfavourable price movements, derivatives cameinto being to solve this problem if used in an effective and appropriate manner. Movements inshare or stock prices, foreign exchange rates, interest rates, etc., make it difficult to anticipate orguess the next price or exchange rate or interest rates. Hence hedging ones'self against thesemovements becomes a hurdle that is difficult to overcome. Coming to the fore of the derivativesmarkets made a relief to many traders, but still then, no one could be certain about the move ofthe market which he is trading in. Forecasting appeared as an educated guess as to whichdirection and by how much the market will move.This research report focusses on how to forecast the foreign exchange rates using theDollar/Rand as an example. I have gathered the historical daily data for the DoIIar/Rand spot rateswhich includes the mayhem period that happened in February 1996. The data was obtained fromone of the biggest banks of South Africa; it was drawn from the Reuters historical data giving theopen, high, low and close prices of the Dollar/Rand (USD/ZAR) spot rates. The data was thendownloaded and copied to the spreadsheet for the calculation of the historical volatilities fordifferent periods. To have a genuine comparison with the implied volatilities, a data of historicalimplied volatilities tor approximately the same period was gathered from the SAIMB (SouthAfrican International Money Brokers). The only snag with the data was that it only catered forspecific traded periods, like 1 month, 2 months, 3 months, 6 months, 9 months and 12 monthsonly. Most financial institutinns are using these implied volatilities for their pricing and end-of-dayor -month or -year revaluation. By the same token the data was downloaded to the spreadsheetfor further analysis and arrangement.Chapter 1 gives the purpose and the meaning of'forecasting, together with different methods thatthis process can be achieved. Views from Makridakis et al., (1983) are used to beautify the worldof forecasting and its importance. In Chapter 2 the concept of volatility and its causes, isdiscussed in detail. Besides the implied and historical volatility discussions, volatility 'smile'concept is discussed and expanded. Volatility slope trading strategies and constraints on the slopeof the volatility term structure are discussed in detail.Chapter 3 discusses different models used to calculate both the historical and the impliedvolatility. This includes models by Kawaller et al., (1994) and Figlewski et al., ( 1990). TheNewton-Raphson method is among of the methods that can be used to get a good estimate of theimplied volatility. For a lot accurate estimates the Method of Bisection can be used in place of theNewton-Raphson method. Mayhew (1995) even suggest a method, which involves the use ofmore weighting with higher vegas (Latane and Rendleman 1976) or weighting not by vegas butelasticity (Chiras and Manaster 1978).Chapter 4 dwells on different forecasting models for foreign exchange markets. This includesmodels by Engle (1993), who is one of the pioneers of the autoregression theory, He discusses theARCH, GARCH and EGARCH models; Heynen et al., (1994,1995) discusses the models for theterm structure of volatility implied by foreign exchange. In the 1995 article he dwells on thespecifications of the different autoregressive conditional heteroskedastic models. U.A. Muller etal., (1990,1993) discusses some of the models for the changing time scale for short-termforecasting in financial markets. This includes discussion of some statistical properties of FX ratestime-series. Xu and Taylor (1994) also discuss the term structure of volatility as implied, inparticular, by FX options. Regression is used in computation of implied volatilityChapter 5 dwells on the empirical evidence and the market practice. This includes the statisticalanalysis of the data; applying the scaling law; proprietary model which depicts the edge betweenthe historical volatility and implied volatility; empirical tests and the volatility forecast evaluationapplied to historical USD/ZAR daily data, using different models.In the statistical analysis, using U.A. Muller et al., (1993) theory, the scaling law, which involvesthe absolute price changes, which are directly related to the interval At, is discussed. Using myGSD/ZAR data Imanaged to calculate the parameters described by the scaling law, using At asone day since my data is a daily data Icould not calculate the activity model function, whichcalculates the intra-day and intra-hour trading using tick-by-tick data, because of the nature of mydata. Had it not been the case, f would have been able to calculate the intra-day and intra-hourvolatilities. These statistics would have been able to depict the daily volatility, more especially onvolatile days, like the day when the Rand took its first knock in February 1996.In the second section of the chapter the proprietary model is discussed, where an edge betweenthe actual volatility and implied volatility was identified. There is a positive correlation betweenthe actual and implied volatility although the latter is always higher than the former; hence traderscan play with this situation for arbitrage purposes. To get the estimates of historical volatility, I used the Well-known formula of using the log-relatives of the returns of any two consecutive days.Annnalised standard deviation of these log-relatives resulted into the required historical volatilityestimates. Moving averages were used to get estimates of different periods, as can be seen in thetext.The main theme of the research report is to expose forecasting models that can be used in foreignexchange currencies using DolIar/Rand as an example. Random walk model was used asbenchmark to other models like stochastic volatility, ARCH, GARCH( 1,1), and EGARCH (1,1).Due to the complexity of the specifications of these models, I used the SHAZAM 7.0 econometricprogram to generate the necessary parameters. Complex formulas of these models are given in theAppendices at the end of the report, together with the program itself.The significance of the forecasted volatility estimates was checked using the p-value correlationstatistic and the Akaike Information Criterion (AIC). The p-value gives us the significance of theparameters and the AlC gives us an indication of the goodness-of-fit of the model. The formulasused to calculate these statistics are given at the end of the report as part of the Appendices. Anaccount of where and how shese results can be of help in the practical situation is given under thesection of market practice. One of the areas worth mentioning is in risk management, whereestimates of the historical volatility can be used together with correlation in risk-metrics tocalculate VArt (value-at-risk). VAR is defined in simple terms as the 5thpercentile (quantile) ofthe distribution of value changes. The beau.y of working with the percentile rather than, say thevariance of a distribution, is that a percentile corresponds to both a magnitude e.g., dollar amountat risk, and exact probability e.g., the probability that the magnitude will not be exceeded. Thisroughly the gist of the research report.

[发布机构] University of the Witwatersrand