Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: THEend8_
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?
