Bachelier model call option pricing formula with leverage and spread

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).

• Habe you already tried anything so far? If so, please let is know where you’re stuck. Nov 26, 2020 at 21:04

\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:
$$max(x \times a,y \times a) = a \; max(x,y) \ \ if \ \ (a \geq 0)$$