We consider a 4-dimensional stock price model given (under P) by dẴ₁ = µ· Xt dt + йt · ΣdŴt where (W) is an n-dimensional Brownian motion, π = (0.02, 0.01, -0.02, 0.05), 0.2 0 0 0 0.3 0.4 0 0 Σ= -0.1 -4a За 0 0.2 0.4 -0.1 0.2) and a E R. We assume that ☑0 = (1, 1, 1, 1) and that the interest rate on the market is r = 0.02. (a) Give a condition on a that would make stock #3 be the one with largest volatility. (b) Find the diversification coefficient for this portfolio as a function of a. (c) Determine the maximum diversification coefficient d that you could reach by varying the value of a? 2

Question 1. Your manager asks you to explain why the Black-Scholes model may be inappro- priate for pricing options in practice. Give one reason that would substantiate this claim? Question 2. We consider stock #1 and stock #2 in the model of Problem 2. Your manager asks you to pick only one of them to invest in based on the model provided. Which one do you choose and why ? Question 3. Let (St) to be an asset modeled by the Black-Scholes SDE. Let Ft be the price at time t of a European put with maturity T and strike price K. Then, the discounted option price process (ert Ft) t20 is a martingale. True or False? (Explain your answer.) Question 4. You are considering pricing an American put option using a Black-Scholes model for the underlying stock. An explicit formula for the price doesn't exist. In just a few words (no more than 2 sentences), explain how you would proceed to price it. Question 5. We model a short rate with a Ho-Lee model drt = ln(1+t) dt +2dWt. Then the interest rate…

In this problem, we consider a Brownian motion (W+) t≥0. We consider a stock model (St)t>0 given (under the measure P) by d.St 0.03 St dt + 0.2 St dwt, with So 2. We assume that the interest rate is r = 0.06. The purpose of this problem is to price an option on this stock (which we name cubic put). This option is European-type, with maturity 3 months (i.e. T = 0.25 years), and payoff given by F = (8-5)+ (a) Write the Stochastic Differential Equation satisfied by (St) under the risk-neutral measure Q. (You don't need to prove it, simply give the answer.) (b) Give the price of a regular European put on (St) with maturity 3 months and strike K = 2. (c) Let X = S. Find the Stochastic Differential Equation satisfied by the process (Xt) under the measure Q. (d) Find an explicit expression for X₁ = S3 under measure Q. (e) Using the results above, find the price of the cubic put option mentioned above. (f) Is the price in (e) the same as in question (b)? (Explain why.)

The managing director of a consulting group has the accompanying monthly data on total overhead costs and professional labor hours to bill to clients. Complete parts a through c.         Question content area bottom Part 1 a. Develop a simple linear regression model between billable hours and overhead costs.   Overhead Costsequals=212495.2212495.2plus+left parenthesis 42.4857 right parenthesis42.485742.4857times×Billable Hours ​(Round the constant to one decimal place as needed. Round the coefficient to four decimal places as needed. Do not include the​ $ symbol in your​ answers.) Part 2 b. Interpret the coefficients of your regression model.​ Specifically, what does the fixed component of the model mean to the consulting​ firm?   Interpret the fixed​ term, b 0b0​, if appropriate. Choose the correct answer below.     A. The value of b 0b0 is the predicted billable hours for an overhead cost of 0 dollars.   B. It is not appropriate to interpret b 0b0​, because its value…

Using the accompanying Home Market Value data and associated regression​ line, Market ValueMarket Valueequals=​$28,416+​$37.066×Square ​Feet, compute the errors associated with each observation using the formula e Subscript ieiequals=Upper Y Subscript iYiminus−ModifyingAbove Upper Y with caret Subscript iYi and construct a frequency distribution and histogram.   LOADING... Click the icon to view the Home Market Value data.       Question content area bottom Part 1 Construct a frequency distribution of the​ errors, e Subscript iei.   ​(Type whole​ numbers.) Error Frequency minus−15 comma 00015,000less than< e Subscript iei less than or equals≤minus−10 comma 00010,000 0   minus−10 comma 00010,000less than< e Subscript iei less than or equals≤minus−50005000 5   minus−50005000less than< e Subscript iei less than or equals≤0 21   0less than< e Subscript iei less than or equals≤50005000 9…

The managing director of a consulting group has the accompanying monthly data on total overhead costs and professional labor hours to bill to clients. Complete parts a through c   Overhead Costs    Billable Hours345000    3000385000    4000410000    5000462000    6000530000    7000545000    8000

Using the accompanying Home Market Value data and associated regression​ line, Market ValueMarket Valueequals=​$28,416plus+​$37.066×Square ​Feet, compute the errors associated with each observation using the formula e Subscript ieiequals=Upper Y Subscript iYiminus−ModifyingAbove Upper Y with caret Subscript iYi and construct a frequency distribution and histogram. Square Feet    Market Value1813    911001916    1043001842    934001814    909001836    1020002030    1085001731    877001852    960001793    893001665    884001852    1009001619    967001690    876002370    1139002373    1131001666    875002122    1161001619    946001729    863001667    871001522    833001484    798001589    814001600    871001484    825001483    787001522    877001703    942001485    820001468    881001519    882001518    885001483    765001522    844001668    909001587    810001782    912001483    812001519    1007001522    872001684    966001581    86200

a. Find the value of A.b. Find pX(x) and py(y).c. Find pX|y(x|y) and py|X(y|x)d. Are x and y independent? Why or why not?

The PDF of an amplitude X of a Gaussian signal x(t) is given by: