If s follows the geometric Brownian motion process ds - S dt+oS dz what is the...
If s follows the geometric Brownian motion process ds - S dt+oS dz what is the process followed by (a) y = 2S, (b) y S 5.
1. Consider a stock and assume it follows a geometric Brownian motion dS = µdt+σdz. Consider now a function G = G(S, t). i) Use Itˆo’s lemma to find the stochastic process dG followed by G. ii) Use Itˆo’s lemma to find the stochastic process followed by G(S) = ln S.
Assume an asset price St follows the geometric Brownian motion, dSt = u Stdt+oS+dZt, So =s > 0 where u and o are constants, r is the risk-free rate, and Zt is the Brownian motion. 1. Using the Ito's Lemma find to the stochastic differential equation satisfied by the process X+ = St. 2. Compute E[Xt] and Var[Xt]. 3. Using the Ito's Lemma find the stochastic differential equation satisfied by the process Y1 = Sert'. 4. Compute E[Y] and Var[Y].
2. Solving the generalized geometric Brownian motion equation. Let S(t) be a positive stochastic process that satisfies the generalized geometric Brownian motion SDE dS(t) = u(t)S(t) dt + o(t)S(t) dW(t), where u(t) and o(t) are processes adapted to the filtration Ft, t > 0. (a) Use Itô’s lemma to compute d log S(t). Simplify so that you have a formula for d log S(t) that does not involve S(t). (b) Integrate the formula you obtained in (a), and then exponentiate...
PROBLEM 2. [5 points] Let X(t) be a Brownian motion. Assume that the stock price follows the stochastic differential equation dS ơSdX+1Sdt. what stochastic differential equation does the stochastic process (a) Y 25, (b) Y = S (c) Y-es, (d) YeT-/S follow? In each cases express the coefficients of dX and dt in terms of Y rather than S. Use Ito's lemma
PROBLEM 2. [5 points] Let X(t) be a Brownian motion. Assume that the stock price follows the stochastic...
use the chain rule to find dz/ds and dz/dt. z=arcsin(x-y), x=s^2+t^2, y=2-6st. dz/ds=? dz/dt=?
Let S(t), t >=0 be a Geometric Brownian motion process with drift mu = 0.1 and volatility sigma = 0.2. Find P(S(2) >S(1) > S(0))
A geometric Brownian motion has parameters mu= 0.2 and sigma=2. What is the probability that the process is above 4 at time t=4, given that it starts at 3. Round answer to 3 decimals.
3. Brounian motion f(O)eR+ is a special case of a Gaussian process with mean zero and covariance C(s, t) = min(s, t) (a) What is the distribution of f(1), the Brownian motion at time t = 4? (Hint: it may be useful to function recall that for any random variable X, var(X)-(x, X) (b) Fix tE R. What is the distribution of f(t)? (c) What is the distribution of f(4)-f(2)? (Hint: it may be useful to utilize var(X-Y) = var(X)...
4. A variable, x, starts at 10 and follows a generalized Wiener process dx a dt+b dz where a 3, b = 2, and dz is a Wiener process. What is the mean value of the variable after two years? What is its standard deviation ofx after two years?
4. A variable, x, starts at 10 and follows a generalized Wiener process dx a dt+b dz where a 3, b = 2, and dz is a Wiener process. What is...