What do improvements in the modelling of stochastic price processes imply?

[GeekTheMartian]

Stochastic processes describe the probabilistic evolution of the value of a variable trough time. The main application of describing stochastic behavior is for the purpose of derivative pricing. In general, the price of any derivative is a function of the stochastic variables underlying the derivative and time. The price of the derivative is then obtained using no arbitrage conditions. Using Ito's lemma we can additionally determine the pricing process of a derivative, which is usually strong related to the stochastic process of the underlying variable..

Given this result, a different stochastic description would imply a different derivative price. In case we improve the models for our stochastic processes of the underlying variable, i.e. X, in the sense that it approximates more to the reality (actual stock price behavior), we get closer to the true value of that derivative.

However, what would such an action imply? Moreover, what are the applications of models that describe the stochastic process of the underlying variables better?

My suggestion is that it can be useful for 'speculative' investors who believe that the price of a stock or derivative will converge to the derived theoretical price. This off course does not have to be case, the value of a stock will not always converge to it's theoretical value.

--> So in short, do you agree with this and what are your beliefs on the role of modelling stochastic processes? Do you know a good resource which discusses this issue?

 

[financeguy]

Can you give an example of what you are talking about?
I think there may be a couple things you're missing.
1) The drift of an asset may be incorporated into a model simply because there are costs of carry associated with holding it or selling it. Just because you don't believe the forward doesn't mean you'll make any money if the asset in fact doesn't follow the forward.
2) Depending on what type of derivative it is, you might find the range of possible prices to be smaller than you think because at the end of the day no matter what model you use it must be able to be calibrated to vanilla surfaces.