the call option pricing formula for the plain/vanilla payoff ($S_T-K)^+$) has been resolved, under the Bachelier model here: Bachelier model call option pricing formula
But can anyone help me with with the generalized payoff (with a leverage and a spread): $(L*(S_T+a)-K)^+$ ?
For this pay-off, what would be the call option pricing formula?
Thanks in advance for the help, and sorry if this is an obvious question (i'm new in the field).
\begin{align} \left( L\times(S_T+\alpha)-K \right)^ + {} & = max \{ L\times(S_T+\alpha)-K,0\} \\ {}&= max \left \{ L \times\left( S_T+\alpha-\frac{K}{L}\right),0\right \} \\ {}&\stackrel{\dagger}{=}L \times max \left \{ S_T+\alpha-\frac{K}{L} ,0\right \} \\ {}& =L \times max \left \{ S_T- \left( \frac{K}{L} -\alpha\right),0\right \} \\ \end{align} By setting $K':= \frac{K}{L} -\alpha$ you can value the option as a vanilla call with strike $K'$ and scale the resulting price by $L$, accordingly. Note that in $(\dagger)$, we have used the property:
$$max(x \times a,y \times a) = a \; max(x,y) \ \ if \ \ (a \geq 0) $$
