Butterfly spread model price

Question

Consider a butterfly spread with strikes $K_1, K_2, K_3$. My professor wrote the model price, $V$, was equal to the following: $$V = exp(-rT) * P(K_1<S_T<K_3) * (1/2) \Delta K$$

where $\Delta K = K_2-K_1 = K_3 - K_2$. I asked after class why this was true. He said it was obvious and that its just the probability times the area of the spread or something like that. I understand that he is discounting the expected payoff of the option. The option only has value when $S_T$ is between $K_1$ and $K_3$, but why multiply by $0.5\Delta k$. What am I not seeing? Can someone provide a rigorous proof with more steps?

Answer 1

Under the risk neutral measure, the expected present value of the butterfly payoff is: $$V_0 = e^{-rT} * \int_{S_T=K_1}^{K_3}P(T,S_T)f_{S_T}dS_T$$

And if we assume that $f_{S_T}$ is constant from $K_1$ to $K_3$, then:

$$V_0 = e^{-rT} * \dfrac{1}{\Delta K} \int_{S_T=K_1}^{K_3}P(T,S_T)dS_T = e^{-rT} *\dfrac{\delta^2}{\Delta K} $$