Black-Scholes

Still WIP

Model:

\[ \frac{dS}{S}(t) = r(t) dt + \sigma(t) dW_{t} \]

After isolation and application:

\[ \int dS = \int S(t)(r(t) dt + \sigma(t) dW_{t}) \]

After resolution:

\[ S(t) - S(0) = \int S(t)(r(t) dt + \sigma(t) dW_{t}) \]

and final re-write:

\[ S(t) = S(0) + \int S(t)(r(t) dt + \sigma(t)dW_{t}) \]

and then linearity of the integral:

\[ S(t) = S(0) + \int S(t)r(t) dt + \int S(t)\sigma(t) dW_{t} \]

Now Stratonovich translation:

\[ S(t) = S(0) + \int S(t)r(t) dt + \int S(t)\sigma(t) \circ dW_{t} + \int \frac{1}{2}S(t)\sigma(t)\frac{d}{dS}(S(t)\sigma(t)) dt \]

THIS IS THE POINT WHERE WE NEED TO FIGURE OUT THE STRAT EXPANSION IN DETAIL... FOR THE SEMANTICS.. IF IT WORKS LIKE TAYLOR THEN THE STRAT INTEGRAL IS THE USUAL???

Q: the strat integral is not a new dimension? then what is it wrt to the register?

THEN CONSTRUCT CIRCUIT AND DO STRATONOVICH expansion

Supposedly we can just implement the above circuit and using the expansion technique compute the BS price!! We need to test this soooooooon......