تحلیل مقایسه‌ای کارایی مدل‌های بلک-شولز و انتشار پرش در مدل سازی قیمت مسکن: مراکز استان‌های ایران

Purpose: During the last two decades, housing price fluctuations in some countries including Iran have been a main challenge of the housing market and the country's economy. In one period, there was a significant increase in housing prices and, in another period, it decreased or stabilized. Relatively high and widespread, it governs the price of housing, as a result of which significant developments have occurred in the housing sector and in the entire economy. In new theories, housing prices can fluctuate over time, and housing price fluctuations can be divided into two important categories. First, minor fluctuations result from market structure based on fundamentals. The housing market is based on the housing supply and demand conditions and the endogenous factors of the housing sector. Hence, the gradual and slow changes in the housing price over time are caused by the basic and underlying factors of the housing market and through changes in the total cost. Housing production changes housing prices. Second, housing cyclical shocks or impulses, are the exogenous factors that create cyclical shocks in the housing sector, and the monetary policy's effect on asset prices, including real estate and housing, is determined. The capital market, household asset portfolio composition and macroeconomic variables are among them.
Methodology: We assume thatis the probability space,  is a filter created by Brownian  and Poisson process  with  is intensity. We also assume that Brownian process, Poisson process  and price jump  are independent of one another.  housing prices are based on time . In the Black-Scholes model (BSM), housing prices at time t are modeled by the following geometric Brownian process:
where  is the average and  standard deviation of housing prices. In the jump diffusion model (JDM), housing prices are calculated by the following equation:
where  is the expected growth rate,  is the turbulence of the Brownian process, and  is the housing price at time t and before the jump.
Results and discussion: In this research, using GEM algorithm, the five parameters of jump diffusion model were estimated and then two parameters of Black-Scholes model were estimated using the maximum likelihood method. Next, the simulation of the future housing price was done based on the Monte-Carlo method. The simulation was done in 100,000 repetitions, and then the best model was selected. The housing price was simulated based on the real price, so that the price at time t could be calculated with its next monthly price, i.e. t+1. This method was repeated until the last data. In this research, many models were simulated with random numbers generated for housing prices to get the best model with the least error. In three cases of 6 months, 12 months and 24 months, housing prices were simulated and predicted. One way to calculate the accuracy of the model was based on the confidence interval with the assumption of normal approximation. One way to check the stability of the obtained coefficients of the models was to repeat the simulation with different random numbers and calculate the average performance of each model. In this research, in order to avoid bringing a large number of estimated models, 25 models with the best performance and the least error, and among these 25 models, the best models were identified.
The results of the models show that, in most of the provincial centers of Iran, the jump diffusion model yields better results than the Black-Scholes model. Also, in some provincial centers, the 6-month performance is better, and, in some others, 12-month or 24-month performance is better. On the other hand, some provincial centers perform better in 6 months, 12 months and 24 months. The results of the average jump frequency in the centers of the provinces of Iran in the housing market show that, for most of the provinces, the average jump frequency is a high number, which indicates high fluctuations and the high impact of internal and external shocks in the Iranian housing market.
Conclusions and policy implications: Accurate modeling of the pricing of various assets, including the housing market, as well as its fluctuations, has always been one of the concerns of researchers and policymakers. Therefore, this research aimed at the comparative analysis of housing prices using Black-Scholes asset pricing models and jump diffusion in the provincial centers of Iran. This study used the monthly housing price data in the provincial centers of Iran for a period from March 2009 to March 2023. In addition, through the GEM algorithm, the jump diffusion model and the maximum likelihood method, the Black-Scholes model was fulfilled, and then the future housing prices in the centers of the provinces of Iran were simulated by the Monte Carlo method. The research results show that, in most provinces of Iran, the jump diffusion model has better and more accurate results than the Black-Scholes model in 6, 12 and 24 months of performance. It is worth mentioning that, in some provincial centers, the results of the Black-Scholes model were better than the jump diffusion model. According to the results of the average jump frequency, it is clear that the highest and lowest average jump frequencies belong to Khorasan Razavi and Kohgiluyeh-Boyer Ahmad Provinces with values of 0.58 and 0.09, respectively.