Exchange rate movements in the Indian rupee (and many other emerging market currencies) are characterised by long periods of placidity punctuated by abrupt and sharp changes. Many, but by no means all, of these sharp changes are currency depreciations. This paper shows that econometric models of changing volatility like Generalised AutoRegressive Conditional Heteroscedasticity (GARCH) with non normal residuals which perform quite well in other financial markets fail quite miserably in the case of the INR-USD process because they do not allow for such jumps in the exchange rate. The empirical results very convincingly demonstrate the need to model the exchange rate process as a mixed jump-diffusion (or normal mixture) process. Equally importantly, the empirical results provide strong evidence that the jump probabilities are not constant over time. From a statistical point of view, changes in the jump probabilities induce large shifts in the kurtosis of the process. The failure of GARCH processes arises because they allow for changes in volatility but not for changes in kurtosis. The time varying mixture models are able to accommodate regime shifts by allowing both volatility and kurtosis (not to mention skewness) to change. This also shows that the periods of calm in the exchange rate are extremely deceptive; in these periods, the variance of rate changes is quite low, but the kurtosis is so high (in the triple digit range) that the probability of large rate changes is non trivial. The empirical results also show that the Black-Scholes-Garman-Kohlhagen model for valuation of currency options is quite inappropriate for valuing rupee-dollar options and that the Merton jump-diffusion model is the model of choice for this purpose.