I want to calculate option prices based on a realistic distribution of the underlying. The underlying is a liquid index such as Eurostoxx50. I think of two aproaches, both of them incorporate assumption of Markov Chains/independet returns:
1) calcualate option prices by monte carlo simulation with historical returns 2) get option prices by montecarlo simulation, simulating returns with a random number generator based on a stochastic variance distribution, such as generalized hyperbolic distribution, that is fitted to historical index returns.
For both approaches I see the problem of how to deal with the vola of the underlying. I assume that in reality vola realisiations are not totally independent from past values. That means that extreme vola changes will occur with both simulation methods more often than in reality. Using the overall historical vola as an estimator for
What is your opinion? Assuming vola being totally random and applying a distribution parameter fit made by using the whole avaiable index history or rather fitting time windows of the index?
Fot the latter I see the problem of how to estimate the whole set of parameters (up to 4) when using only a small dataset (I think of using something like the historical one month vola, but one month of data isn't enough at all to fit a hyperbolic model nor to apply the historical returns itself to simulation). Therefore I would need to scale the simulated returns by local historical vola. Since none of the distribution parameters are directly related to the vola/standard deviation, I would use the following scaling method for simmulated returns:
return scaled = simulated return * (sigma local historic vola) / (sigma calculated from fitted distribution parameters)
Would that be ok or a "no go"?
Furthermore, I didn't find any ready programmed estimators in Python or Scilab, that can fit any of the hyperbolic distributions or Heston like models to emprical returns. Aren't there any? And, I coudln't find any function, that calculates random numbers from the generalized hyperbolic distribution. Isn't there any? If not, how can it be derived from the densitiy function, using Python?
This is a huge bulk of questions, I just emerged myself into those things and need some "hints" into which direction to go. THanks a lot in advance.