MTH3251 HOMEWORK 9
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MTH3251
HOMEWORK 9
Consider the two-step Binomial model, which is specified by the following. The price of one share of stock st at time u is given by s0 = 1, s1 = ∈ 1 s0 , s2 = ∈2 s1 with random variables ∈ 1 , ∈2 taking two values d and ., and having the distribution P (∈1 = d, ∈2 = d) = 1/4, P (∈1 = d, ∈2 = .) = 1/8, P (∈1 = ., ∈2 = d) = 1/8, P (∈1 = ., ∈2 = .) = 1/2. The savings account is specified by 8t = 8t , u = 0, 1, 2. These parameters satisfy d < 8 < ..
(a) Give the marginal distributions of ∈1 and ∈2 (P (∈t = d), P (∈t = .), u = 1, 2) and state
whether they are independent under the original (real world) probability measure P.
(b) Show that the discounted stock price process st/8t , u = 0, 1, 2, is not a martingale
under P.
(c) Describe the EMM in this model, and give the derivative Λ = do/dP .
(Λ(w) = , where w runs over all possibilities (d, d), (d, .), (.d), (., .).) Show that the discounted stock price process st/8t , u = 0, 1, 2, is a martingale under
.
(d) Consider the following strategy: start with portfolio at time u = 0, a1 = 1, b1 = − 1. (Borrow $1 to buy 1 share), and at time u = 1 take a2 = 1, b2 = − 1 (do nothing, no rebalancing). Show that this is a self-financing portfolio (see eqn (8) p. 63 Notes). Show that the value process is given by V0 = 0, V1 = s1 − 8 , V2 = s2 − 82 . Which contract x does this portfolio replicate? Is this strategy an arbitrage strategy?
(e) Start with portfolio at time u = 0, a1 = 1, b1 = − 1. (Borrow $1 to buy 1 share). At
time u = 1 let a2 = a2 (s1 ) = 0 if s1 = . and a2 = a2 (s1 ) = 2 if s1 = d, in other words a2 = I(∈1 = .) + 2I(∈1 = d). Find b2 = b2 (s1 ), so that the self-financing condition holds. Give the value of the portfolio at times u = 1 and u = 2. Which contract x does this portfolio replicate?
2022-10-11