Is it fair to say the phi processes we encounter in this subject (discrete or continuous) are only previsible because we have made an assumption about the volatility structure of our stock price processes in advance? So that without the initial volatility assumption (binomial tree shape or assumption for sigma) we would not have previsible phi processes? Or to put it another way, previsibilty of phi only exists because we know what the volatility of our stock price process is going to be over the next time period.
Not quite so! The assumption around the volatility can very well be relaxed, and we can have stochastic volatility as well. The important point to note is that the ratio of the volatilities of the two processes is constant, i.e the first order derivative of the option price w.r.t. the underlying (delta). It is here that we assume that delta is instantenously non-changing. Indeed, if gamma is also low, changes delta over a longer time interval too can be trivial. Note that the volatility of option is f(s)/s*vol, vol being volatility of stock proce process. The previsioble process is nothing but the ration of these two volatilities. Hope this helps.
Not quite so! The assumption around the volatility can very well be relaxed, and we can have stochastic volatility as well. The important point to note is that the ratio of the volatilities of the two processes is constant, i.e the first order derivative of the option price w.r.t. the underlying (delta). It is here that we assume that delta is instantenously non-changing. Indeed, if gamma is also low, changes delta over a longer time interval too can be trivial. Note that the volatility of option is f(s)/s*vol, vol being volatility of stock proce process. The previsioble process is nothing but the ration of these two volatilities. Hope this helps.
