Exam 2

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  1. (25 points) Consider the following two-period binomial tree for a stock price:
    MATH

    Furthermore, assume that the interest rate on a money market per period is 10%. At all nodes, the probability p of going up is 0.8.

    1. Determine the risk-neutral probabilities q, at time 0, and in the two possible states at time 1; note that q is different at each of these three nodes.

    2. Consider a derivative X which pays off (at time $T=2)$ $50$ if the first movement is up (i.e., if $S_{1}=120),$ and an additional $50$ if the second movement is up as well (if $S_{2}=150)$; otherwise the payoff is zero. Set this up as a tree for $X$. Also determine the value of the derivative at time 1 and 0.

  2. (25 points) Consider the process
    MATH
    where $a$, $b$, and $\sigma >0$ are constants, and $W_{t}$ is a standard Brownian motion.

    1. Use Ito's lemma to obtain the differential equation for
      MATH

    2. What is the solution for $X_{t}?$

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