Question

Consider two put options with the same expiration but different strikes, E1 and E2 (with E1 < E2). Formulate the conditions on their prices P1 and P2 so that the PNL graphs of the two puts (both long) intersect. Justify your answer by drawing all possible types of the graphs’ arrangement. Hint: there are three arrangements (not counting the borderline cases).

Answer 1

Let C(E) is defined and continuous on the interval [E_1, E_2] and differentiable on [E_1, E_2] then there is at least one number Ein the interval that is E_1 < E < E_2 such that c^prime (E) = C (E_2) - c(E_1)/E_2 - E_1 lessthanorequalto (b) Let E_2 = E_1 + delta E_1 - (4) From eq (3) partial differential c/ partial differential E = c (E_2) - c(E_1)/E_2 - E_1 - (3) putting value of E_2 from eqn (4) to eqn (3) doublerightarrow so partial differential c/ partial differential E = c(E_1 + delta E_1) + c(E_1)/E_1 + delta E_1 - E_1 doublerightarrow partial differential c/partial differential E = partial differential c(E_1)/partial differential E_1 Taking the limit lim partial differential c(E_1)/ partial differential E_1 = partial differential c/ partial differential E doublerightarrow partial differential c/ partial differential E = 0 doublerightarrow or -1 lessthanorequalto partial differential c/ partial differential E_1 lessthanorequalto 0