I need to prove that the butterfly condition is always positive under no arbitrage theorem. We are constructing a long butterfly using European call options
C(T,K+∆K) - 2C(K) + C(T,K-∆K) > 0 where ∆K < K
I have managed to prove for greater than or equal to zero using the following steps:
Lower bound of a European call option of a non-divided paying stock is as follows:
C(T,K) >= S(0) - Ke^-rT
K = strike price, T = time to maturity, r = interest rate, S(0) = stock price at time=0
Hence for options in the butterfly this evaluates to
C(T,K+∆K) >= S(0) - (K+∆K)e^-rT --- (Eq:1)
C(T,K) >= S(0) - (K)e^-rT --- (Eq:2)
C(T,K-∆K) >= S(0) - (K-∆K)e^-rT --- (Eq:3)
Doing (Eq:1) - 2*(Eq:2) + (Eq:3)
, I get the following
C(T,K+∆K) - 2C(T,K) + C(T,K-∆K) >= 0
However, how do i go further and prove that the above inequality is not equal to zero under no arbitrage.