1. Consider a stock and assume it follows a geometric Brownian motion dS = µdt+σdz. Consider...
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.
Homework Answers
1) Following equations can be written as:
a) d(G) =[{(dG/dS)*dS} + {(dG/dt)*dt}]
b) dG =(µ-σ^2/2)dx + σdy
Add Answer to:
1. Consider a stock and assume it follows a geometric Brownian
motion dS = µdt+σdz. Consider...
PROBLEM 2. [5 points] Let X(t) be a Brownian motion. Assume that the stock price follows the stoc...
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...
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.
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...
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...
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...
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...
3. Assume a stock price follows a lognormal Brownian motion, i.e. its price satisfies So eo)...
3. Assume a stock price follows a lognormal Brownian motion, i.e. its price satisfies So eo) where W(t) is standard Brownian motion, So-$265, ơ 18%. Calculate the probability that in 20 days 22), or less, the stock price reaches the value H 275, i.e. calculate Prob. max St > 275
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) &g...
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))
Consider a market with X W, where W is a Brownian motion process. Define the trading...
Consider a market with X W, where W is a Brownian motion process. Define the trading strategy (Vo. ) by vo 1, ,2W Find Gr(p) Select one: WA-T 2WT W2+T
Consider a market with X W, where W is a Brownian motion process. Define the trading strategy (Vo. ) by vo 1, ,2W Find Gr(p) Select one: WA-T 2WT W2+T
0.2. The time-t price of a stock is s(t). You are given: (1) A stock's price...
0.2. The time-t price of a stock is s(t). You are given: (1) A stock's price follows geometric Brownian motion with a = 0.01 and (ii) S(O)= 40. Calculate Pr(40<S(5) < 50).
You own one share of a stock. The price is 10. The stock price changes according to a geometric Brownian motion process, with time measured in months and coefficients µ=.1 σ= 2. Calculate the probabil...
You own one share of a stock. The price is 10. The stock price changes according to a geometric Brownian motion process, with time measured in months and coefficients µ=.1 σ= 2. Calculate the probability that: a. The stock price is at least 20 in one year. b. The stock reaches the price of 20 before it drops to 5. c. You will be able to sell the stock at 20 in a year if the price at the end...
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...
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.
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].
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...
3. Assume a stock price follows a lognormal Brownian motion, i.e. its price satisfies So eo) where W(t) is standard Brownian motion, So-$265, ơ 18%. Calculate the probability that in 20 days 22), or less, the stock price reaches the value H 275, i.e. calculate Prob. max St > 275
Consider a market with X W, where W is a Brownian motion process. Define the trading strategy (Vo. ) by vo 1, ,2W Find Gr(p) Select one: WA-T 2WT W2+T
Consider a market with X W, where W is a Brownian motion process. Define the trading strategy (Vo. ) by vo 1, ,2W Find Gr(p) Select one: WA-T 2WT W2+T
0.2. The time-t price of a stock is s(t). You are given: (1) A stock's price follows geometric Brownian motion with a = 0.01 and (ii) S(O)= 40. Calculate Pr(40<S(5) < 50).
Most questions answered within 3 hours.
-
What is the expression for the uncertainty in the total energy
Etot of equation?
Assume an...
asked 13 minutes ago -
A sledge loaded with bricks has a total mass of 18.5 kg and is
pulled at...
asked 5 minutes ago -
Write a command line to view the long listing of all files,
including hidden files, with...
asked 10 minutes ago -
Predict the reactions, if any, when the
following hydrogen compounds are added to water, and
explain...
asked 10 minutes ago -
In response to the increasing weight of airline passengers, the
Federal Aviation Administration told airlines to...
asked 13 minutes ago -
Events A and B are independent. Suppose event A occurs with
probability 0.63 and event B...
asked 23 minutes ago -
Rubies are essentially alumina Al2O3 with a contaminant of
chromium in the form of Cr2O3. A...
asked 31 minutes ago -
Write a full program that will accept
exactly three command line arguments
that can be interpreted as...
asked 33 minutes ago -
1.histones are proteins that make up nucleosomes. they bind to
the backbone of DNA in chromosomes....
asked 42 minutes ago -
What is the relationship between a variables value and their
probabilities as it is summarized by...
asked 51 minutes ago -
Balance Sheet
DR
CR
Cash
10000
AR
7000
Supplies
500
Equipment
5,000
Accum depreciation
...
asked 49 minutes ago -
Country
Continent
GDP (millions of US$)
Afghanistan
Asia
 
asked 51 minutes ago