Abstract
In this study, we implement the GARCH models of Duan andWei (1999), and Heston and Nandi
(2000) to price options on illiquid foreign currencies. In contrast with the common practice of
combining both historical spot transaction data with options data from the market to calibrate
GARCH models, we estimate the GARCH parameters purely from returns on historical FX spot
rate transactions. This choice is motivated by the absence of data on options in the case of
illiquid market. We use daily spot foreign exchange rates from Reuters (Eikon Refinitiv) for the
period 2016 to 2021 to produce spot volatilities from GARCH. Using both GARCH models, we
obtain option prices, which are then converted into GARCH implied volatilities. Our objective
is to determine the size of an error between the historical return-based model price and the
market price. This should indicate to us the extent to which we may rely on the historical only
method, when market options data is not available.
For liquid pairs such as EURUSD and USDZAR, both models give a minimum error of 0.01%
at least for short maturity. For illiquid pairs such as EURZAR, we observe larger errors ranging
between 0.03% up to 2.61%, even for small maturities. We note that the reference price in
case of illiquid are synthetic and not the market price. We also find that the GARCH method
has out-performed the triangle rule.
Keywords: GARCH, stylised facts, volatility smile and surface.