By applying the following transformation on the dependent variable in the Black–Scholes equation 

while the auxiliary conditions are transformed to become 

Consider the following diffusion equation defined in a semi-infinite domain 

with initial condition: v(x, 0) = f (x) and boundary condition: v(0,t) = g(t), the solution to the diffusion equation is given by (Kevorkian, 1990)

Using the above form of solution, show that the price of the European downand-out call option is given by 

Assuming B

The last term represents the additional option premium due to the rebate payment.

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where a = 11/12-½ 2, B = totype diffusion equation c = eay+Bt₁ - r, show that (4.1.3a) is reduced to the pro- δω δτ W, ²² 2 ay ²'