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unsmt


Total Posts: 196
Joined: Jul 2014
 
Posted: 2015-04-01 00:02
It is probably initial step in understanding local probability. One can ignore for a while direct connection of the problem to finance. Consider a parabolic PDE with constant coefficients which is known as BSE with correspondent initial condition at T representing call option payoff. We wish to estimate sigma based on observation of the solution of the BSE. It is clear that any method represent estimate that does not equal to unknown sigma. My feeling is that an estimate of the unknown constant sigma must depend on parameters of the BSE problem, ie T, t, K.

unsmt


Total Posts: 196
Joined: Jul 2014
 
Posted: 2015-04-02 12:56
Other fact that is related to LV is following. Bruno Dupire showed that if the function C ( t , S ; T , K ) is the solution parabolic PDE ( BSE ) in the area ( 0 , T ) x ( 0 , + infinity ) with boundary condition at T which corresponds to call option payoff { ie ( t , S ) are variables and ( T, K ) are fixed parameters }. Next this function as a function of the variables T , K satisfies other equation with a nonlinear diffusion coefficient. For this new equation T , K are variables taking values in area T > t and K > 0. Here t , S are fixed parameters.
This result justifies the observation that values of volatility does not a constant but also depend on T - t and K, ie we should consider C ( * ) not as a function on t , S but as a function on T , K. Of course it makes sense if we could not comprehend the difference between constant sigma in BSE and its estimate based on observation BSE solution. On the other hand its only phenomena that explain the 'paradox'. But it does not make any specification of the BSE approach in option pricing. Two approaches classic BSE and LV dealing with the same values of the function C ( * ) and therefore we do not get more precise pricing formula replacing traditional B-S approach by LV approach and therefore we do not present any progress in option pricing. Graphs of local volatility surface is just an applied mathematical exercise which actually does not have a connection to option pricing.
Of course companies can make money selling programs with LVs. But they and buyers probably do not comprehend that the LV model does not present a new pricing construction of the same BSE world. The option price is the same in two approaches to option pricing.
Some math details can be read on http://www.slideshare.net/list2do/supplement-to-local-voatility
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