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Parameter estimation in merton jump diffusion model

Özdemir, Tuğcan Adem
Over the years, jump diffusion models become more and more important. They areused for many purposes in several branches such as economics, biology, chemistry,physics, and social sciences. The reason for prevalent usage of these jump modelsis that they capture stochastic movements and they are sensitive to jump points. It ispossible to measure sudden decreases/increases caused by some reasons such as wars,natural disasters, market crashes or some dramatic news, by jump diffusion models.Recently, US Dollar to Turkish Lira exchange rate has showed dramatic increases/de-creases. It is very difficult to model this exchange rate data with classical modelingmethods. In this thesis, we try to model this data with Merton model which is amongthe well-known jump diffusion models. To obtain true parameter estimation algo-rithm, we simulate a data by using Merton structure. The values of parameters arefound with Maximum Likelihood Estimation (MLE). The initial parameter values insimulated data and the estimated parameter values are compared to control the pa-rameter estimation is true or not. Also, the values of Euler-Maruyama numerical ap-proximation method and analytical solution values are checked whether convergence is good or not. After the true parameter estimation algorithm is found, US Dollarto Turkish Lira exchange rate data is used. This data is between date of 01.02.2019and21.06.2019. By using this data, the parameter estimation is made and predictionis made for between date of23.06.2019and02.07.2019for both Merton Jump Dif-fusion model and Black-Scholes model. Finally, the fitting and forecasting accuracyperformances of them are compared.