Modeling exchange rate behavior in Iran using random differential equations: Merton model and NGARCH Approach

The main objective of this paper is to model the exchange rate behavior in Iran using random differential equations. In order to model the behavior of this market, three random differential equations have been used which include the Black-Scholes model, the Merton model, and the geometric Brownian motion with the non-linear GARCH. Also, in order to estimate the coefficients of equations, the maximum likelihood approach has been used and the drift and propagation parameters are calculated monthly and yearly for the period 2007-2017. According to the research findings, the probability of the exchange rate jump in the market is 0.87. The average rate of jump is 0.10 and the jump variance is 0.03. This confirms that the efficient-market hypothesis(EMH) does not exist in the Iranian foreign exchange market. In this paper, the non-linear GARCH model (NGARCH) based on Merton's model has been used to investigate the impact of good and bad news and positive and negative shocks. According to the results of the research, the estimated coefficient in the foreign exchange market is positive. That is, the exchange rate is most affected by bad news, negative shocks, and systematic risks. The numerical value of the coefficient in the currency market is 2.9, indicating that it is least affected by good news. according to the maximum likelihood function, in the Iranian currency market, the Merton model has more explanatory power than the nonlinear GARCH model and the Black-Scholes model.