A volatility focus : the appropriateness of black-Scholes modelling in hedging ZAR volatility

Abstract

The Black-Scholes Model infers preconditions for its application and this paper refers to them as the Black-Scholes assumptions. The model makes the following assumptions, volatility and interest rates are constant, the underlying instrument does not pay dividends, the options can only be exercised at expiry, there are no transaction costs in buying or selling of the underlying instrument and that the price follows random walk theory (Black & Scholes 1973a). Implied volatility is an important variable as it forms the volatility smile and terms structure which together derive the implied volatility surface. The shape of the volatility surface enables the paper to test certain aspect of the Black-Scholes Model assumptions. The study focuses on USDZAR currency pair and the literature review section of the paper investigates the history of the South African Rand from 1961 to 2005. The value of an option premium can be better understood with the aid of two option dynamics i.e. intrinsic value and time value. Intrinsic value is the amount of money that is realized by exercising the option, when the price of a call or put option is greater than its intrinsic value, it is said that the option has time value. The moneyness of an option is a crucial concept that describes the potential gain or loss emanating from exercising an option, an option can be in-the-money, out-the-money and at-the-money (Chisholm, 2010) and (Hull, 2015). The study finds that the original Black-Scholes Model is not an appropriate valuation model for hedging against USDZAR volatility purely based on its assumptions and its inability to incorporate domestic and foreign interest rates. The Black-Scholes Model assumes an important foundation in the understanding of option valuation. Since its inception, there have been extensive expansions of the Original Black-Scholes Model but the most significant one (for this study) is the Garman Kohlhagen Model which was specifically derived to deal with two interest rates (foreign and domestic). The paper suggest that the Garman Kohlhagen Model is a more appropriate alternative to the Original Black Scholes Model, it is worth noting that this model has similar limitation by assuming that volatility remains constant during the life of the option. It is more appropriate (not perfect) in the sense that it alleviates the Black-Scholes Model assumption that risk-free rates of the domestic currency are the same as the risk-free rate of the foreign currency, this is not true (Garman & Kohlhagen 1983).