**I**.

a) Technically derivation looks good and the essence of the concept can be summarized as following. Observations over options data lead us to the fact that volatility coefficient of the underlying of the BS formula do not correspond to volatility coefficient of the underlying security. The volatility of the underlying that corresponds to BS formula is called implied volatility, IV.

My point is that Market prices options not fully based on no arbitrage principle. In particular, chances to make or lose money also effect on derivatives pricing. Such chances do not represented by BS pricing concept.

b) Local volatility equation derived with the help of Dupire equation is aimed to get explicit formula for implied volatility. We cannot mix two different probabilistic representations of the BSE solution C ( t , S ; T , K ) in two coordinate space ( t , S ) and ( T , K ). First representation is BS formula for BSE solution and other is probabilistic representation of the Dupire equation solution. Thus there are three different issues: stock GBM , BS implied volatility, local volatility corresponding Dupire equation. We used stock process to get BSE in which option price is defined by the risk neutral process having the same volatility as stock process. Hence, dynamics of the call option on [ 0 , T ] is defined by [$]C ( t , S ( t ) ; T , K ; \sigma_{S} )[$]. Now we replace risk neutral underlying process in definition [$] C ( t , x ; T , K ; \sigma_{iv} ) [$] by the implied volatility process. There is a formula that represents implied volatility by using local volatility

[$]\sigma^2 ( T , S ( T )) = F ( T , K ; C )[$]

On the other hand implied volatility is a function of time on [ 0 , T ] and Dupire representation of the implied volatility is given only for the final moment T. Hence the attempt to present implied volatility by using local volatility does not look a complete representation.

**II**. Other problem is derivation of the Dupire equation derivation. For simplicity putting [$]r = \mu = 0[$] and following Dupire’s original derivation we begin with BSE solution

[$]C ( t , x ; T , K ) = E max ( S ( T ; t , x ) – K , 0 ) [$]

Differentiating twice with respect to K leads to the formula

[$]\frac{\partial^{2} C ( t , x ; T , K ) }{\partial K^{2} } \,=\, p ( t , x ; T , K )[$]

Here p is transition density of S ( T ; t , x ). Next general SDE

[$]dy ( t ) = b ( t , y ) dw ( t ) [$] (1)

with some diffusion coefficient b ( y , t ) and assuming that drift is zero is introduced. The forward Kolmogorov eq for density f ( t , x ; T , y ) of the process y ( t )

[$] \frac{\partial^{2} b^2 ( T , y )f ( t , x ; T , y ) }{\partial y^{2} } \,=\, \frac {\partial f ( t , x ; T , y )}{\partial T}[$] (2)

Next he puts y = K and is making changes

[$]f ( t , x ; T , K ) = p ( t , x ; T , K ) = \frac{\partial^{2} C ( t , x ; T , K ) }{\partial K^{2} }[$]

Here we should make a comment. We should note that p ( t , x ; T , K ) is the known density of the stock process which itself satisfies equation (2). If densities corresponding two SDEs are equal , ie f = p that implies that equations and their solutions should be identical, ie dS ( t ) = dy ( t ) and therefore

[$]b ( t , y ) = \sigma ( t ) y[$]

In such case derivation of the formula for volatility smile does not look make sense. Of course I could be wrong but I could not see my error.

Next, we can make transformations similar to Dupire but they did not lead us to [$]\sigma ( T , K ) [$] which presents volatility smile.

From my point of view the error in this derivation is based on the fact that assuming that densities p and f are equal the fact that Kolmogorov equations for f and p should be equal too.