I need to prove that vertical spread is bounded, by using no arbitrage condition.
0 > (C(T,K1 )- C(T,K2))/(K1- K2 ) >-e^(-r*T )
I have documented my solution below. Can you kindly review it and comment on my approach.
In order to prove vertical spread condition, I am using following inequality, which is lower bound on option value:
C ≥ S(0) − Ke^(−rT)
Vertical spread is created by going long (1) call at strike (K1) and going short (-1) call at strike (K2), where K2 ≥ K1.
C(T,K1) ≥ S(0) – K1e^(−rT) and C(T,K2) ≥ S(0) – K2e^(−rT) , when I calculate difference in call price I get:
C(T,K1) - C(T,K2) ≥ K2e^(−rT) - K1e^(−rT) --(1)
C(T,K1) - C(T,K2) ≥ e^(−rT)* (K2 - K1), and as K2 ≥ K1 , I get:
C(T,K1) - C(T,K2) ≥ 0 --(2)
If I rearrange following inequality K2 ≥ K1 , I get:
K1 - K2 ≤ 0 -- (3)
If I divide (2) by (3), I get:
(C(T,K_1 )- C(T,K_2))/(K1- K2 ) <0 --(4)
As dividing positive numerator with negative numerator gives a negative number. You may notice that I have removed equals to symbol from the inequality. Reason being, when K1= K2 , then the numerator is equal to zero and so is the denominator, leading to undefined state of solution. (4) satisfies the upper bound of inequalities. By rearranging inequality (1), I get:
C(T,K1 )-C(T,K2 )≥ -e^(-rT ) (K1- K2)
Dividing above inequality by (3.3), I get:
(C(T,K1 )- C(T,K2))/(K1- K2 ) >-e^((-r)T ) --(5)
Combining inequality (4) and (5), gives me required vertical spread condition
0 > (C(T,K1 )- C(T,K2))/(K1- K2 ) >-e^((-rT )