Portfolio value on selling and buying calls Consider buying a call option with strike $K − δ$, selling two call options with strike $K$ and buying a call option with strike $K + δ$, where $K, δ > 0$ and $K > \delta$, all with maturity $T > 0$. Draw the terminal payoff function of this portfolio of options. Does this portfolio have a positive value at time zero?
So this is what I did:
$C(S, t) = e^{−r(T −t)}E^Q(1_{S_T >K− δ} + 1_{S_T >K + δ} - 2* 1_{S_T >K} | St = S)$
$C(S, t) = e^{−r(T −t)} Q(S_T >K− δ) + Q(S_T >K + δ) - 2Q (S_T >K) $
$Q(Log\frac{S_T}{S_0} >Log\frac{K− δ}{S_0})$ = $Q(\frac{Log\frac{S_T}{S_0}-(r-\frac{σ^2}2)t}{σt^{0.5}} >\frac{Log\frac{K− δ}{S_0}-(r-\frac{σ^2}2)t}{σt^{0.5}})$
$=Q(z >\frac{Log\frac{K− δ}{S_0}-(r-\frac{σ^2}2)t}{σt^{0.5}})$
=$Φ^c(\frac{Log\frac{K− δ}{S_0}-(r-\frac{σ^2}2)t}{σt^{0.5}}))$
So $C(S, t) = e^{−r(T −t)} Q(S_T >K− δ) + Q(S_T >K + δ) - 2Q (S_T >K) $
$= e^{−r(T −t)}[Φ^c(\frac{Log\frac{K− δ}{S_0}-(r-\frac{σ^2}2)t}{σt^{0.5}})+Φ^c(\frac{Log\frac{K+ δ}{S_0}-(r-\frac{σ^2}2)t}{σt^{0.5}})-2Φ^c(\frac{Log\frac{K}{S_0}-(r-\frac{σ^2}2)t}{σt^{0.5}})]$
I'm not sure what to do from here.

A call option with strike $K$ maturing at $t=T$ has terminal payoff $(S_T - K)^+$. This holds from the definition of a call option in a model-free manner, so you should not be applying the Black-Scholes equation here.
The case you have been given is known as a "butterfly spread". Here, you have terminal payoff equal to
$$\varphi (S_T) = (S_T - (K - \delta))^+ - 2(S_T - K)^+ +(S_T - (K+\delta))^+$$
For instance, if $K = 1$ and $\delta = 0.1$, our payoff looks like this:

One can check that for $\delta > 0$, $\varphi \geq 0$ and $\varphi \neq 0$. Thus, no-arbitrage pricing tells us that at $t = 0$, the portfolio has positive value.
