FIN 538 Stochastic Foundation of Finance Problem Set 2
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Stochastic Foundation of Finance (FIN 538)
Problem Set 2
Problem 1 Consider the random walk approximation t of the Brownian motion in LN1 with 2 time steps, that is
0 = 0
1 = X1
2 = X1 + X2
Denote an upward move of the random walk by + and a downward move by -, each of whcih happens with probability . Here X1 and X2 have identical and independent distribution as follows: Xi = ∆ for an + move, and Xi = -∆ for an - move, i = 1, 2. Similar to Example 1 of LN1, consider the following sample space
Ω = (++, +-, -+, --}
1. Describe i , i = 0, 1, 2 as maps from Ω into the set of real numbers R.
2. Find the σ-algebra generated by i for each i = 0, 1, 2.
3. What is the smallest σ-algebra F that makes i , i = 0, 1, 2 a random map?
4. What is the probability measure P defined on F that tells us how likely an event in F happens? Describe P as a map from F into [0, 1].
Problem 2 Modify the Excel file ”NormalApprox100.xlsx” posted on Canvas to plot the distri- bution of 1 where t is the random walk that approximates the Brownian motion on the interval t e [0, 1], with increments dt = 0.001. Calculate the mean and variance of 1 .
Problem 3 Let Bt be a Brownian motion.
1. Calculate 匝[Bt Bs] for t > s where Bt .
2. Calculate Var[Bt + Bs] for t > s where Bt .
3. Calculate 匝[(Bs - Bt )4], s > t > 0.
2022-09-18