We show existence and uniqueness of a strong solution to a linear non-uniformly parabolic equation, which gives the fair price of a normalized European call option. We then provide a direct link between local and implied volatilities in the form of a quasilinear degenerate parabolic partial differential equation. We also establish closed-form asymptotic formulae for the implied volatility near expiry as well as for deep in and out of the money options, using a generalized comparison principle on bounded domains.
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