Suppose the value of an option is given by $v(s_0)$ where $s_0$ is the current price of the underlying asset and $v:\mathbb{R}_+\to\mathbb{R}_+$.
It seems that the literature is mostly focused on getting an estimate of $v(s_0)$. My question is whether there is any benefit from estimating the whole function $v$ instead of just a estimate at one point??? Is there any benefit in terms of hedging, etc???
Thank you.
Some numerical methods, e.g. finite difference schemes, enable you to compute the entire function $s \mapsto v(s)$ at once. This can be useful as no additional pass is required to compute the delta and the gamma.
Ciao,
Of course it would be amazing to know the future (i.e. the whole trajectory of the price process) and it would be really usefull for hedging (it makes the hedging itself a trivial problem). However I think you have to think about that everything in the future depends on stochastic variables and that the only place where hedging makes sense is the present (you always work to be covered right now...at every time!).
In general you can work on the future in a very simple way. Infact at each time $t$ you can just use the initial point $s_t$ instead of $s_0$ in your model.
What changes is that $s_t$ is a stochastic variable rather than a real value like $s_0$. In any case, depending on your model, $s_t$ will depend on $s_0$ (see for example Black Scholes model where $s_t$ depends linearly from $s_0$).
The big problem at this point is that you can not do calibration in $t$ since by definition you need the market data (ok..maybe you can do something with futures value) and this situation stop you from having a proper model for anything.
I hope I was helpful!
Ciao ciao,
AM
