2. Solving the generalized geometric Brownian motion equation. Let S(t) be a positive stochastic process that...
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...
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.
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. 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.
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].
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.
3. Some computations related to a stock S(t) following the Merton-Black-Scholes Model. (a) Let S(t) = S(0) exp((u - 02/2)t +oW(t)), where W(t) is a standard Brownian motion. Compute that u is the expected annual return rate, i.e., E[S(T)] = S(O)eMT, where T > 0. Is o2 the variance of S(T)/S(O)? (b) Let X be the continuously compounded annual rate of return between 0 and T, i.e., S(T) = S(0) exp(XT). Compute E(X) and Var(X) (find the distribution of X...
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))
5. Let (S2,F,P) be a probability space and let {W(t),t 2 0) be Brownian mo- tion with respect to the filtration Ft, t 2 0. By considering the geometric Brownian motion where Q R, σ > 0, S(0) > 0. Show that for any Borel-measurable function f(y), and for any 0 〈 8くthe function 2 2 g(x) =| f(y) da 0 satisfies Ef(S(t))F (s)-g(S(s)), and hence S(t) has a Markov property. We may write qlx as q We may write...
Let aſt),o(t), and r(t) be non-random adapted processes. For a stock process that follows: dS(t) = a(t) S(t)dt + o(t)S(t)dW(t) and a discount factor given by: D(t) = e-Soudu derive the formula at any time 0 <t< T for a forward agreement to purchase the stock at time T for price K.