THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MATH4210 (2016/17 Term 1) Financial Mathematics Assignment 3 solution Note: If you have any questions about the solution or assignment score, please let me know by sending an Email to kckchan@math.cuhk.edu.hk 1. ([2;p55]) Draw the expiry payoff diagrams for each of the following portfolios: (a) Short one share, long two calls with exercise price E (this combination is called a straddle); (b) Long one call and one put, both with exercise price E (this is also a straddle: why?); (c) Long one call and two puts, all with exercise price E (a strip); (d) Long one put and two calls, all with exercise price E (a strap); (e) Long one call with exercise price E1 and one put with exercise E2 . Compare the three cases E1 > E2 (known as a strangle), E1 = E2 , E1 < E2 . (f) As (e) but also short one call and one put with exercise price E (when E1 < E < E2 , this is called a butterfly spread). Solution: ( −S (a) Λ = S − 2E S≤E S>E (b) Λ = |E − S| 1 ( 2(E − S) S ≤ E (c) Λ = S−E S>E ( E−S S≤E (d) Λ = 2(S − E) S > E 2   E2 − S S ≤ E2 (e) (i) If E1 > E2 , Λ = 0 E2 < S < E1   S − E1 S > E1 (ii) If E1 = E2 = E, Λ = |E − S| 3   E2 − S S ≤ E1 (iii) If E1 > E2 , Λ = E2 − E1 E2 < S < E1   S − E1 S > E2 (f) We have Λ = max(S − E1 , 0) + max(E2 − S, 0) − max(S − E, 0) − max(E − S, 0) Here we have 9 cases: (a) E1 > E2 > E (b) E1 > E2 = E (c) E1 > E > E2 (d) E2 > E1 > E (e) E2 > E1 = E 4 (f) E2 > E > E1 (g) E > E2 > E1 (h) E > E2 = E1 (i) E > E1 > E2 By explicitly writing down the equation, we can obtain the 9 graphs. Remark: When you are asked to draw a diagram, please don’t just show me the line without indicating any information. You have to do more than stating the equation of the graph. Please state additional information, such as x-intercept, y-intercept and slope, to let people know what you are drawing. This time marks are given to you, but please make sure you draw a graph with sufficient informati