I give a little more explaination may be some body can help me with this
To compute option price from the Heston model, one needs the inputparameters that are not observable from market data. The input parametersfall into two group, the structure parameters that govern the diffusion process of the underlying asset (μ, κ, θ , η, and ρ), and the spot variance vt and risk premium λ. Because volatility is random and risk premium λ is unobservable, the exact maximum likelihood function for the
Heston stochastic volatility model cannot be traced, so the traditionalmaximum likelihood estimation cannot be applied to estimate Heston model directly. Of course, on can always use option panel data to back out structure parameters, as Bakshi, Cao and Chen (1997) and Nandi (1998) do. However, the option is priced under risk neutral probability,
it is not clear whether the option implied structure parameters truly reflect the original information contained in the underlying asset return distribution. In this study, we adopt a two-step procedure to estimate the stochastic diffusion process as well as the option risk premium jointly. In the first step, the simulation-based indirect inference method
is used to estimate the structure parameters that govern the underlying asset process (μ, κ, θ , η). In the second step, the remaining parameters (vt , λ, ρ) are estimated by a non-linear least square method. The indirect inference method is a simulation based, moment matching procedure.The method works in the following way: suppose the true data generating process is governed by a stochastic diffusion process, one can simulate discrete time observations from this process by the Euler approximation given any set of structural parameters. The simulated data is estimated using some discrete time model called auxiliary model. The market data is also estimated by the same auxiliary model. If the
moments from the simulated data match the moments from the market data, which means that the simulated data has the same property as the market data. Then the structural parameters that generate the simulated
data represent the true data generating process. In our study, we estimate
both the market data and simulated data with the GARCH (1,1)
model and search for the structural parameters that match the GARCH
parameters of the simulated data to the market data.
After κ, θ , η are estimated, we need to estimate the correlation coefficient
ρ, the spot variance vt , and the risk premium λ. This is done by a
non-linear least square method. On each day, there are many option
quotes with different times to maturity and different exercise prices.
Some former studies only use at-the-money options, but we use all
options available on the particular day to estimate required parameters
on that day since more information is included. Define:
εt = p − p(ρ, λ, vt)
as the error of between the market price and theoretical price computed
from Heston model, we want to search for a set of parameters (ρ, λ, vt)
that minimize the sum of square errors:
where I is the total number of options on the day. This is done on each
day.
