THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MATH4210 Financial Mathematics 2020-2021 T1 Assignment 2 Due date: 16 October 2020 11:59 p.m. Please submit this assignment on blackboard. If you have any questions regarding this assignment, please email your TA Wong Wing Hong ([email protected]). 1. Suppose the continuous compounding interest rate is r and the price of a stock is S(t) at time t. If it pays dividend d×S(tD) at time tD, where 0 < tD < T and 0 < d < 1, show that its forward price F (0, T ) satisfies F (0, T ) = 1 1 + d S(0)erT under no arbitrage opportunity assumption. 2. (Put-Call Parity Relation with Dividend) Assume that the value of the dividends of the stock paid during [t, T ] is a deterministic constant D at time tD ∈ (t, T ]. Let S(t) be the stock price, r be the continuous compounding interest rate, CE(t,K) and PE(t,K) be the prices of Eu- ropean call and put option at time t with strike K and maturity T respectively. Show that CE(t,K) PE(t,K) = S(t) Ke r(T t) De r(tD t) for all
t < T . 3. Assume that the value of the dividends of the stock paid during [t, T ] is a deterministic constant D at time tD ∈ (t, T ]. Let S(t) be the stock price, r be the continuous compounding interest rate, CA(t,K) and PA(t,K) be the prices of American call and put option at time t with strike K and maturity T respectively. Show that CA(t,K) PA(t,K) < S(t) Ke r(T t) for all t < T . 4. Suppose the continuous compounding interest rate is r, two European call options has same strike K and different maturity T1 < T2. Suppose 2the underlying asset pays a deterministic dividend D at tD ∈ (T1, T2]. Prove CE(t, T1) < CE(t, T2) + (De r(tD t) K(e r(T1 t) e r(T2 t)))+ for all t < T1. 5. Suppose that we have the following 3 European call with the same maturity T in the financial market: Type Strike Price Price at time 0 Call 90 15 Call 100 12 Call 110 5 Suppose that the continuous compounding interest rate is r = 0 in the market and the maturity time is T = 1. Can you construct an arbitrage portfolio with the above options