October 8th, 2017, 1:17 pm

Here is the essence of my comment. The change in the value of the Black Scholes hedged portfolio at [$] t \in [ 0 , T ) [$]

[$] d \Pi ( t ) = d C ( t , S ( t ) ) - C ^{\prime}_S ( t , S ( t ) ) d S ( t ) [$]

does not have risky term dw ( t ). This fact lead them to the statement that

[$] d \Pi ( t ) = r \Pi ( t ) dt [$]

from which Black Scholes equation is followed.

My point is to calculate rate of return of the BS portfolio. Then we note that

[$]\frac{d\,\Pi \,( t )} {\Pi \,( t ) } = \frac { C^{\prime}_{t} ( t , x ) - C ^{\prime \prime}_{S S} ( t , x )) \, \sigma^{2} S ^2 ( t )}{ C ( t , x )) - C _{S}^{\prime} ( t , x )\, x ) } | \, _ { x = S ( t ) } \, dt [$] (1)

The BS logic of statement that if the change in the value of the portfolio does not hold 'dw' then it is equal to risk free rate r is correct for a single point of time. The formal statement that is correct on arbitrary interval is that [ 0 , T ] is that

[$] \frac{d\,\Pi ( t ) } {\Pi ( t ) } = r [$] (2)

[font=Times New Roman, serif]and it is wrong on the [ 0 , T ].If (2) is true then BSE is takes place on [ 0 , T ]. [/font] We can not say that rate of return of the portfolio (2) is nonrandom on [ 0 , T ] as far as it depends on random process S ( t ). It is incorrect to write that rate of return of the hedged portfolio which is a random function is equal to nonrandom constant r.

Details is in

https://www.slideshare.net/list2do/remark-on-no-arbitrage-black-scholes-options-pricing