Importance of Markov property

[Kamil]

Hi all,

What is the main application for Markov property in finance? I am learning that topic, got the definition, but why it is important and when.

Say I build a model for the stock price which is not Markov. What are my limitations? Can I still solve the PDE with finite difference?

I am mostly interested what is the connection of Markov property with finite difference.

thanks!

 

[DStahl]

I think (but don't quote me) that Feyman-Kac's theorem only works if the underlying diffusion processes are markov. That is, you cannot "convert" an expectation to a PDE without the process being markov. Intuitively this kind of makes sense...if the expectation depends on information not available at this time we cannot calculate the expectation, and we cannot price the security by this method.

Edit: Apparently solutions to SDEs are markov so you don't even have to worry about your model as long as your underlying processes are SDEs.

 

[dmv]

what markov property talks about? It talks about dependencies between successive values of random variable. If future value depends only on current value then you try to express this dependence in PDE/ODE form. Because any differential equation tells about link between current value and "next" value. If you dont have markov property you have to work with complex joint distributions. Btw markov property often is assumed to be held, because it makes model much more flexible and simple. Additionally, if you work in markov framework you can use markov-chains tools for analysis.

 

[DoubleTrouble]

Well, as DStahl mentioned in his Edit every solution to a SDE is markov. This is essential in the usual Martingale approach to option pricing since otherwise we would not be able to use the Feynman-Kac theorem even though the theorem it self doesn't require the Markov property.

If we use the delta heding approach to option pricing we don't need the Markov property. However in the delta-gamma hedging approach for stochastic volatility it's absolutely essential to have the Markov property to be able to derive anything useful at all.