Why, in a option pricing model, one need of stochastic process describing the dynamics of the underlying?And then, is it true that a models for asset return may be good in representing time series but not good in option pricing model?

The stochastic process tells us what distribution the asset has in our model.(Solving the SDE in the BS world for instance shows us that the asset is lognormally distributed)Therefore we are able to calculate the expectation of the future payoff under the risk neutral measure.

Sure, but one couldn't calculate the distribution just from time series?Anyway, what about the second question?

I'm not really familiar with econometric techniques, but I suppose the anlaytical tractability in a stochastic process model is apealing, and allows you to bring all sorts of results to bear on the problem.Also, if you have an actual time series, you don't have the risk neutral distribution, you have the real world distribution, so the discounted expected payoff wrt to this distribution is not the option price.

As I see it, a time series would be an instance, whereas an assumed type of stochastic process is a template. If you want to do anything using the time series that goes beyond just description ('The returns on that date looked like this'), you need to assume an underlying model for that as well; if you don't do that, you are purely calculating a product value based on a single historical instance. Time series models (ARMA, GARCH..) imply stochastic generating processes so the two approaches ultimately resolve to the same thing. Ito processes are popular given the Black-Scholes theory was built on top of them.Also be careful to distinguish between expectation pricing (discount the 'actual' process payoff to derive product value) with risk-neutral pricing, which alone is arbitrage free. The initial pages of Baxter and Rennie give a good description of this.

QuoteOriginally posted by: marcsterAs I see it, a time series would be an instance, whereas an assumed type of stochastic process is a template. Yes, anyway every time you want utilize such template you need historical data in order to calibrate it...Further, when youQuoteAlso, if you have an actual time series, you don't have the risk neutral distribution, you have the real world distribution, so the discounted expected payoff wrt to this distribution is not the option price.A way to calibrating a model is just estimates paramter from time series so you obtain, as you're saying, real world distribution...So: Do i need always risk neutral distribution? Can't I pricing option using real world distribution?

Parameters for price processes are often estimated from the market....So for instance, there may be a deep liquid market for oil futures, and you have a stochastic model for the oil price, you would fit your parameters to the model so that the model gave the same price for the forward contract as that obtained in the market. You could then use this process to price more complex products....The parameters you obtain in this way will be risk neutral, not historical.Also you cant price an option using the real world distribution... this is not an arbitrage free price as Marcster noted.In the simplest example, the Feynman Kac theorem allows us to write the solution to the BS PDE as the discounted risk neutral expectation of the future payoff (Not the real world expectation)

Mmm...However i still have some doubts about my first question...And what about my second question:"is it true that a models for asset return may be good in representing time series but not good in option pricing model?"

What kind of models are you thinking of ?

QuoteOriginally posted by: hoareMmm...However i still have some doubts about my first question...And what about my second question:"is it true that a models for asset return may be good in representing time series but not good in option pricing model?"May be. I think "tractability" is a key determinant in a model's usage, for example, GBM.

QuoteOriginally posted by: hoareWhy, in a option pricing model, one need of stochastic process describing the dynamics of the underlying?An option is a derivative, so it derives its value from some underlying asset. The assets that are modeled usually follow a stochastic process, e.g. Brownian Motion or Geometric Brownian Motion. These processes are used because they actually model the true dynamics of the asset. These processes are also continuous almost everywhere but nowhere differentiable (which is a property that we would want). If this were not true, then we can just apply the first-order conditions to find the optimal response. QuoteAnd then, is it true that a models for asset return may be good in representing time series but not good in option pricing model?For this issue, if I remember correctly, you can refer to an asset pricing text that talks about risk-neutral probabilities. The answer to why asset returns are not necessary in options pricing model should be somewhere there.