Black–Scholes PDE

The question is can we derive an equation for v(t, x)? The answer is yes, and the equation is a Partial Differential Equation (PDE): an equation connecting the partial derivatives of v in t and x, hence the name. This equation is of interest because if we can solve it, then to decide Vt we only need to plug in St for x. Of course we can decide Vt by taking Expectation via the Independence Lemma, which leads to the Black-Scholes formula. Numerically, this would lead to the pricing by simulation method: we simulate the paths of St and summing over the paths as way to approximate the expectation. The pricing of Vt by by figuring out v(t, x) would like to the numerical solution of PDE approach. This provides us with an alternative (and sometimes possibly more powerful) approach to the simulation method described above.