The famous Black-Scholes framework is usually derived using a hedging approach where a self-financing portfolio is constructed and the resulting stochastic differential equation is being solved under some conditions.
The self-financing portfolio is basically a dynamic trading strategy where according to the actual price development and the strike of the replicated option parts of the underlying are being bought or sold. For example when you want to replicate a call you would have to buy when the price rises and sell when it develops in the direction of the strike.
My question: I want to build a framework where you could match different dynamic trading strategies with derivatives. For example I would like to find the characteristics of a moving average approach in terms of a derivative, e.g. the P/L-diagram of this would in my opinion look like an covered call writing combination.
How would you do that?
First of all I guess in this example one could take the price process (e.g., a geometric Brownian motion) and e.g. convolute that with a rectangular function to model the moving average. After that one would have to find the resulting distribution density function...
But all of that is just speculation... (also quite literally ;-)
Has any one of you some ideas on how to do that - or are there even ready-to-use approaches (on the web, in articles, books, etc...)