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Let \begin{align*} C(S, K, t) = SN(d_1) - e^{-rt}KN(d_2) \end{align*} denote the Black-Scholes call option price with initial asset value $S$, strike $K$, and maturity $t$. Note that \begin{align*} \frac{\partial C}{\partial S} = N(d_1). \end{align*} For the above barrier option, note that \begin{align*} E_0 &= V_0 N(d_1)-e^{-rt}KN(d_2) -\bigg[V_0 ...


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This is an interesting and not so easy question. Here's my 2 cents: First, you should distinguish between mathematical models for the dynamics of an underlying asset (Black-Scholes, Merton, Heston etc.) and numerical methods designed to calculate financial instruments' prices under given modelling assumptions (lattices, Fourier inversion techniques etc.). ...



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