| In this thesis we firstly express a direct link between local and implied volatilities in the form of a quasilinear degenerate parabolic partial differential equation, following Berestycki, Busca and Florent's work in [1]. Then, we derive the asymptotic form of pricing formula with constant coefficients. Next, we introduce the parametrix method. Using the parametrix method leads us to the lower and upper bounds of the fundamental solution of the parabolic partial differential solution with variable coefficients. Using the expansion of the transition probability of 1-dimensional Brownian Motion with drift, and applying Girsanov's theorem, we could get the leading term in the fundamental solution. Finally, we give the asymptotic formula of the implied volatility near expiry. |
