Applied Statistics in Business and Economics
5th Edition
ISBN: 9780077837303
Author: David Doane, Lori Seward Senior Instructor of Operations Management
Publisher: McGraw-Hill Education
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Chapter 7, Problem 14ERQ
a.
To determine
Calculate the
b.
To determine
Calculate the probability that the probability that the speed of a randomly chosen vehicle is between 65 mph and 75 mph.
c.
To determine
Calculate the probability that the probability that the speed of a randomly chosen vehicle is less than 70 mph.
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Students have asked these similar questions
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.)
Problem 4. Margrabe formula and the Greeks (20 pts)
In the homework, we determined the Margrabe formula for the price of an option allowing you to
swap an x-stock for a y-stock at time T. For stocks with initial values xo, yo, common volatility
σ and correlation p, the formula was given by
Fo=yo (d+)-x0Þ(d_),
where
In (±²
Ꭲ
d+
õ√T
and
σ = σ√√√2(1 - p).
дго
(a) We want to determine a "Greek" for ỡ on the option: find a formula for
θα
(b) Is
дго
θα
positive or negative?
(c) We consider a situation in which the correlation p between the two stocks increases: what
can you say about the price Fo?
(d) Assume that yo< xo and p = 1. What is the price of the option?
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
Chapter 7 Solutions
Applied Statistics in Business and Economics
Ch. 7.1 - Flight 202 is departing Los Angeles. Is each...Ch. 7.1 - It is Saturday morning at Starbucks. Is each...Ch. 7.1 - Which of the following could not be probability...Ch. 7.1 - For a continuous PDF, why cant we sum the...Ch. 7.2 - Find the mean and standard deviation for each...Ch. 7.2 - Find each uniform continuous probability and...Ch. 7.2 - For a continuous uniform distribution, why is P(25...Ch. 7.2 - Assume the weight of a randomly chosen American...Ch. 7.3 - If all normal distributions have the same shape,...Ch. 7.3 - (a) At what x value does f (x) reach a maximum for...
Ch. 7.3 - State the Empirical Rule for a normal distribution...Ch. 7.3 - Discuss why you would or would not expect each of...Ch. 7.4 - Find the standard normal area for each of the...Ch. 7.4 - Find the standard normal area for each of the...Ch. 7.4 - Find the standard normal area for each of the...Ch. 7.4 - Find the standard normal area for each of the...Ch. 7.4 - Find the standard normal area for each of the...Ch. 7.4 - Bobs exam score was 2.17 standard deviations above...Ch. 7.4 - Joans finishing time for the Bolder Boulder 10K...Ch. 7.4 - Find the associated z-score for each of the...Ch. 7.4 - Prob. 21SECh. 7.4 - Find the associated z-score or scores that...Ch. 7.4 - Prob. 23SECh. 7.4 - High school students across the nation compete in...Ch. 7.4 - Prob. 25SECh. 7.4 - Daily output of Marathons Garyville, Lousiana,...Ch. 7.4 - Assume that the number of calories in a McDonalds...Ch. 7.4 - The weight of a miniature Tootsie Roll is normally...Ch. 7.4 - The pediatrics unit at Carver Hospital has 24...Ch. 7.4 - The cabin of a business jet has a cabin height 5...Ch. 7.4 - On January 1, 2011, a new standard for baseball...Ch. 7.4 - Last years freshman class at Big State University...Ch. 7.4 - The time required to verify and fill a common...Ch. 7.4 - The time required to cook a pizza at a...Ch. 7.4 - The weight of a McDonalds cheeseburger is normally...Ch. 7.4 - The weight of a small Starbucks coffee is a...Ch. 7.4 - The weights of newborn babies in Foxboro Hospital...Ch. 7.4 - The credit scores of 35-year-olds applying for a...Ch. 7.4 - The number of patients needing a bed at any point...Ch. 7.4 - A ski resort pays its part-time seasonal employees...Ch. 7.4 - The average cost of an IRS Form 1040 tax filing at...Ch. 7.4 - Use Excel to find each probability. a. P(X 110)...Ch. 7.4 - Use Excel to find each probability. a. P(80 X 10)...Ch. 7.4 - The weight of a small Starbucks coffee is a normal...Ch. 7.4 - A study found that the mean waiting time to see a...Ch. 7.4 - High-strength concrete is supposed to have a...Ch. 7.5 - The default rate on government-guaranteed student...Ch. 7.5 - In a certain store, there is a .03 probability...Ch. 7.5 - The probability is .90 that a vending machine in...Ch. 7.5 - When confronted with an in-flight medical...Ch. 7.5 - On average, 28 patients per hour arrive in the...Ch. 7.5 - For a large Internet service provider (ISP), web...Ch. 7.6 - In Santa Theresa, false alarms are received at the...Ch. 7.6 - Between 11 p.m. and midnight on Thursday night,...Ch. 7.6 - A passenger metal detector at Chicagos Midway...Ch. 7.6 - The Johnson family uses a propane gas grill for...Ch. 7.6 - At a certain Noodles Company restaurant,...Ch. 7.6 - The time it takes a ski patroller to respond to an...Ch. 7.6 - Between 11 p.m. and midnight on Thursday night,...Ch. 7.6 - The shoplifting sensor at a certain Best Buy exit...Ch. 7.6 - Between 2 a.m. and 4 a.m. at an all-night pizza...Ch. 7.6 - The mean life of a certain computer hard disk in...Ch. 7.7 - Suppose that the distribution of order sizes (in...Ch. 7.7 - Suppose that the distribution of oil prices (/bbl)...Ch. 7 - (a) Why does a point have zero probability in a...Ch. 7 - Define (a) parameter, (b) PDF, and (c) CDF.Ch. 7 - For the uniform distribution: (a) tell how many...Ch. 7 - For the normal distribution: (a) tell how many...Ch. 7 - Prob. 5CRCh. 7 - (a) What is the transformation to standardize a...Ch. 7 - (a) Explain the difference between Appendix C-1...Ch. 7 - Write an example of each of the four normal...Ch. 7 - Prob. 9CRCh. 7 - For the exponential distribution: (a) tell how...Ch. 7 - Prob. 11CRCh. 7 - For the triangular distribution: (a) tell how many...Ch. 7 - Which of the following is a continuous random...Ch. 7 - Which of the following could be probability...Ch. 7 - Applicants for a night caretaker position arc...Ch. 7 - Passengers using New Yorks MetroCard system must...Ch. 7 - Discuss why you would or would not expect each of...Ch. 7 - Why might the following not be normally...Ch. 7 - Scores on a certain accounting exam were normally...Ch. 7 - Chlorine concentration in a municipal water supply...Ch. 7 - The weekly demand for Baked Lays potato chips at a...Ch. 7 - The weekly demand for Papa Chubbys pizzas on a...Ch. 7 - The amounts spent by customers at a Noodles ...Ch. 7 - The length of a Colorado brook trout is normally...Ch. 7 - The caffeine content of a cup of home-brewed...Ch. 7 - The fracture strength of a certain type of...Ch. 7 - Tire pressure monitoring systems (TPMS) warn the...Ch. 7 - In a certain microwave oven on the high power...Ch. 7 - Procyon Manufacturing produces tennis balls. Their...Ch. 7 - Shower temperature at the Oxnard Health Club...Ch. 7 - Tests show that, on average, the Li-ion Hitachi...Ch. 7 - The time it takes to give a man a shampoo and...Ch. 7 - The length of a time-out during a televised...Ch. 7 - If the weight (in grams) of cereal in a box of...Ch. 7 - Demand for residential electricity at 6:00 p.m. on...Ch. 7 - Jims systolic blood pressure is a random variable...Ch. 7 - A statistics exam was given. Calculate the...Ch. 7 - Are the following statements true or false?...Ch. 7 - John can take either of two routes (A or B) to LAX...Ch. 7 - The amount of fill in a half-liter (500 ml) soft...Ch. 7 - The length of a certain kind of Colorado brook...Ch. 7 - Times for a surgical procedure are normally...Ch. 7 - Prob. 95CECh. 7 - Among live deliveries, the probability of a twin...Ch. 7 - Nationwide, the probability that a rental car is...Ch. 7 - The probability of being in a car accident when...Ch. 7 - A multiple-choice exam has 100 questions. Each...Ch. 7 - The probability that a certain kind of flower seed...Ch. 7 - Prob. 101CECh. 7 - Prob. 102CECh. 7 - Prob. 103CECh. 7 - Automobile warranty claims for engine mount...Ch. 7 - Prob. 105CECh. 7 - Prob. 106CECh. 7 - The price (dollars per 1,000 board feet) of...Ch. 7 - Prob. 108CECh. 7 - Prob. 109CECh. 7 - (a) Write an Excel formula to generate a random...Ch. 7 - (a) Write an Excel formula to generate a random...Ch. 7 - On a police sergeants examination, the historical...Ch. 7 - Which type of probability (empirical, classical,...Ch. 7 - For the following contingency table, find (a) P(H ...Ch. 7 - If P(A) = .30, P(B) = .70, and P(A B) = .25, are...Ch. 7 - Prob. 4ERQCh. 7 - Which statement is true? Why not the others? a....Ch. 7 - Prob. 6ERQCh. 7 - Assuming independent arrivals with a mean of 2.5...Ch. 7 - If a random experiment whose success probability...Ch. 7 - In a random experiment with 50 independent trials...Ch. 7 - Prob. 10ERQCh. 7 - Prob. 11ERQCh. 7 - Prob. 12ERQCh. 7 - Prob. 13ERQCh. 7 - Prob. 14ERQCh. 7 - Prob. 15ERQCh. 7 - Prob. 16ERQCh. 7 - If arrivals follow a Poisson distribution with...Ch. 7 - In the previous problem, find (a) the 95th...Ch. 7 - Which statement is correct concerning the normal...Ch. 7 - Prob. 20ERQ
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- 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…arrow_forwardIn 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.)arrow_forwardThe 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…arrow_forward
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