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Mathematical Foundations for Finance Exercise Sheet 1 by qfc86623


   ETH Z¨rich HS 2009
   D-MATH                                                                                         Coordinator
   Prof. E.W. Farkas, Prof. M. Schweizer                                                 Christoph Czichowsky

                 Mathematical Foundations for Finance
                                           Exercise Sheet 1
Please hand in until Friday, 25.9.2009, 13:00, either in the lecture or the assistant’s box in HG F 27.

   Exercise 1-1
   Let (S 0 , S 1 ) be a multiplicative model as in the lecture. Suppose that we only consider trading
   strategies ϕ = (ϕ0 , ϑ) with strictly positive (discounted) value process, i.e., Vk (ϕ) > 0 È-a.s.
   for all k = 0, 1, . . . , T and that we change the parameterization of our trading strategies from
   number of shares to fractions of wealth π = (π 0 , π 1 ) and set V (ϕ) = V (π).
     a) State a similar formula to (2.1) (in the lecture notes) in terms of fractions of wealth.
     b) State a similar formula to the self-financing condition (2.7) (in the lecture notes) in terms
        of fractions of wealth.
     c) Combine a) and b) to get a formula for the (discounted) value process of a self-financing
        trading strategy in terms of fractions of wealth only involving V0 (π), π 1 and S.

   Exercise 1-2

   Consider the probability space (Ω, F, È) with
         Ω = {0, 1}T = ω = (x1 , . . . , xT ) | xn ∈ {0, 1} for all n = 1, . . . , T ,     and   F = 2Ω .
   Fix p ∈ (0, 1) and consider i.i.d. random variables Xn : Ω → {0, 1} such that Xn (ω) = xn and
   È[Xn = 1] = p for n = 1, . . . , T .
                                      We want to use the random variables Xn to model the stock
   price S in the Cox-Ross-Rubinstein binomial model (CRR) on (Ω, F, È) with u > d > −1.
      a) Introduce random variables Sn which have the same distribution as the stock price at
         time n and only depend on u, d and Xk for k = 1, . . . , n.
     b) Write down the distribution of Sn .

   Exercise 1-3

   Let (Ω, F, È) be a probability space and X : Ω → R a random variable with X ≥ 0 È-a.s. Prove
   that [X] = 0 implies that X = 0 È-a.s.
                      Organization of the exercise classes
Coordinators: Christoph Czichowsky (first two weeks) and Nicoletta Gabrielli.
Organization: The new exercise sheets will be available on Fridays on
During each exercise class, the assistants will discuss the current exercises as well as the solu-
tions of the exercises of the previous week with the students. It is expected that the students
prepared the current exercise sheet before coming to the exercise class. Students should hand
in the solved exercise sheet until the Friday following the exercise class at 13:00 either in the
lecture or the assistant’s box in HG F 27. Students will receive the corrected exercises back
during the next exercise class on Tuesday.
Question times (Pr¨senz): Monday and Thursday, 12:00 to 13:00 in HG G32.6.
Requirements for the certificate (Testat): 2/3 of all exercises reasonably tackled.
Exercise classes:
          Assistant                Time            Room         Students (preliminary)
          Nicoletta Gabrielli      Tuesday 9-10    HG D 3.2     A-G
          Christoph Czichowsky     Tuesday 9-10    HG D 7.1     H-R
          Roman Muraviev           Tuesday 9-10    HG D 7.2     S-Z

             Exercise sheets and further information are also available on:

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