Heston model can be used to value options by modeling the underlying asset such as the stock of a company. The one major feature of the Heston model is that it inocrporates a stochastic volatility term.

\begin{eqnarray} dS_t &=& \mu S_tdt + \sqrt{V_t} S_t dW_t^1 \\ dV_t &=& \kappa(\theta-V_t) + \sigma \sqrt{V_t} dW_t^2 \end{eqnarray}

Here :

• $S_t$ is the asset's value at time $t$
• $\mu$ is the expected growth rate of the log normal stock value
• $V_t$ is the variance of the asset $S_t$
• $W_t^1$ is the stochastic process governing the $S_t$ process
• $\theta$ is the value of mean reversion for the variance $V_t$
• $\kappa$ is the strengh of mean reversion
• $\sigma$ is the volatility of volatility
• $W_t^2$ is the stochastic process governing the $V_t$ process
• The correlation between $W_t^1$ and $W_t^2$ is $\rho$

In contrast, the Black-Scholes-Merton process assumes that the volatility is constant.

Let us consider a European call option for AAPL with a strike price of \$130 maturing on 15th Jan, 2016. Let the spot price be \$127.62. The volatility of the underlying stock is know to be 20%, and has a dividend yield of 1.63%. We assume a short term risk free rate of 0.1%. Lets value this option as of 8th May, 2015.

Using the above inputs, we construct the European option as shown below.

In order to price the option using the Heston model, we first create the Heston process. In order to create the Heston process, we use the parameter values: mean reversion strength kappa = 0.1, the spot variance v0 = volatility*volatility = 0.04, the mean reversion variance theta=v0, volatility of volatility sigma = 0.1 and the correlation between the asset price and its variance is rho = -0.75.

On valuing the option using the Heston model, we get the net present value as:

Performing the same calculation using the Black-Scholes-Merton process, we get:

The difference in the price between the two models is: bs_price - h_price = 0.21840667525992163. This difference is due to the stochastic modeling of the volatility as a CIR-process.