# Payoff of a Butterfly spread under risk neutral measure is always positive for any t<T

In a situation where $$K_3-K_2=K_2-K_1=h>0$$ and $$K_1\le S_t\le K_3$$ where $$S_T=S_t.e^{[(r-\sigma^2/2)(T-t)+\sigma(W_T-W_t)]}$$ (i.e. Stock process follows GBM under the risk neutral measure).

I know the value of the call under the risk neutral measure is: $$f(S_t)= e^{-r(T-t)}*E((S_T-K_1)^+-2(S_T-K_2)^++(S_T-K_3)^+|\mathcal{F_t})$$ How do we know that the value of the payoff of the butterfly spread using calls is positive for any t<T.

• It is based on the convexity of the call option payoff. May 11 at 14:09
• Can you elaborate or prove it? May 11 at 14:10
• What do you mean with "payoff ... for any t<T"? The payoff happens exactly at $T$, no? Do you mean value instead of payoff, by any chance? May 11 at 17:58
• Vandalizing questions is not allowed, edit rolled back May 12 at 5:19

Note that \begin{align*} K_2 = \frac{K_1+K_3}{2}. \end{align*} Then \begin{align*} &\ \max(S_T-K_1, \, 0) + \max(S_T-K_3, \, 0) \\ =&\ \max\big(S_T-K_1 + \max(S_T-K_3, \, 0), \, \max(S_T-K_3, \, 0)\big)\\ =&\ \max\big(\max(S_T-K_3 + S_T-K_1, \, S_T-K_1), \, \max(S_T-K_3, \, 0)\big)\\ =&\ \max\big(2S_T-(K_1+K_3), S_T-K_1, S_T-K_3, 0\big)\\ \ge&\ \max\big(2S_T-(K_1+K_3), 0\big)\\ =&\ 2\max(S_T-K_2, \, 0). \end{align*}
• Also how is this valid for every $$S_t$$ since $$S_T$$ is also variate on $$r, t and \sigma$$ May 11 at 14:59