what is the heston model 
the heston model named after steve heston is a type of stochastic volatility model used to price european options understanding the heston modelthe heston model developed by associate finance professor steven heston in 1993 is an option pricing model that can be used for pricing options on various securities it is comparable to the more popular black scholes option pricing model 1overall option pricing models are used by advanced investors to estimate and gauge the price of a particular option trading on an underlying security in the financial marketplace options just like their underlying security will have prices that change throughout the trading day option pricing models seek to analyze and integrate the variables that cause fluctuation of option prices in order to identify the best option price for investment as a stochastic volatility model the heston model uses statistical methods to calculate and forecast option pricing with the assumption that volatility is arbitrary the assumption that volatility is arbitrary rather than constant is the key factor that makes stochastic volatility models unique other types of stochastic volatility models include the sabr model the chen model and the garch model key differencesthe heston model has characteristics that distinguish it from other stochastic volatility models namely the heston model is also a type of volatility smile model smile refers to the volatility smile a graphical representation of several options with identical expiration dates that show increasing volatility as the options become more in the money itm or out of the money otm the smile model s name derives from the concave shape of the graph which resembles a smile heston model methodologythe heston model is a closed form solution for pricing options that seeks to overcome some of the shortcomings presented in the black scholes option pricing model the heston model is a tool for advanced investors d s t r s t d t v t s t d w 1 t d v t k v t d t v t d w 2 t where s t asset price at time t r risk free interest rate theoretical rate on an asset carrying no risk v t volatility standard deviation of the asset price volatility of the v t long term price variance k rate of reversion to d t indefinitely small positive time increment w 1 t brownian motion of the asset price w 2 t brownian motion of the asset s price variance begin aligned ds t rs tdt sqrt v t s tdw 1t dv t k theta v t dt sigma sqrt v t dw 2t textbf where s t text asset price at time t r text risk free interest rate theoretical rate on an text asset carrying no risk sqrt v t text volatility standard deviation of the asset price sigma text volatility of the sqrt v t theta text long term price variance k text rate of reversion to theta dt text indefinitely small positive time increment w 1t text brownian motion of the asset price w 2t text brownian motion of the asset s price variance end aligned dst rst dt vt st dw1t dvt k vt dt vt dw2t where st asset price at time tr risk free interest rate theoretical rate on anasset carrying no riskvt volatility standard deviation of the asset price volatility of the vt long term price variancek rate of reversion to dt indefinitely small positive time incrementw1t brownian motion of the asset pricew2t brownian motion of the asset s price variance heston model vs black scholesthe black scholes model for option pricing was introduced in the 1970s and served as one of the first models for helping investors derive a price associated with an option on a security in general it helped to promote option investing as it created a model for analyzing the price of options on various securities both the black scholes and heston model are based on underlying calculations that can be coded and programmed through advanced excel or other quantitative systems the black scholes call option formula is calculated by multiplying the stock price by the cumulative standard normal probability distribution function thereafter the net present value npv of the strike price multiplied by the cumulative standard normal distribution is subtracted from the resulting value of the previous calculation in mathematical notation conversely the value of a put option could be calculated using the formula in both formulas s is the stock price k is the strike price r is the risk free interest rate and t is the time to maturity the formula for d1 is the formula for d2 is special considerationsthe heston model is noteworthy because it seeks to provide for one of the main limitations of the black scholes model which holds volatility constant the use of stochastic variables in the heston model provides for the notion that volatility is not constant but arbitrary both the basic black scholes model and the heston model still only provide option pricing estimates for a european option which is an option that can only be exercised on its expiration date various research and models have been studied for pricing american options through both black scholes and the heston model these variations provide estimates for options that can be exercised on any date leading up to the expiration date as is the case for american options 