An Introduction to Mathematical Statistics and Its Applications (6th Edition)
6th Edition
ISBN: 9780134114217
Author: Richard J. Larsen, Morris L. Marx
Publisher: PEARSON
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Chapter 3.8, Problem 3Q
(a)
To determine
To find: The
(b)
To determine
To find: The pdf of X+Y and check that itbelongs to the same family of pdfs as of X and Y.
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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.)
Chapter 3 Solutions
An Introduction to Mathematical Statistics and Its Applications (6th Edition)
Ch. 3.2 - An investment analyst has tracked a certain...Ch. 3.2 - In a nuclear reactor, the fission process is...Ch. 3.2 - In 2009 a donor who insisted on anonymity gave...Ch. 3.2 - An entrepreneur owns six corporations, each with...Ch. 3.2 - The probability is 0.10 that ball bearings in a...Ch. 3.2 - Suppose that since the early 1950s some...Ch. 3.2 - Doomsday Airlines (Come Take the Flight of Your...Ch. 3.2 - Two lighting systems are being proposed for an...Ch. 3.2 - The great English diarist Samuel Pepys asked his...Ch. 3.2 - The gunner on a small assault boat fires six...
Ch. 3.2 - If a family has four children, is it more likely...Ch. 3.2 - Experience has shown that only 13 of all patients...Ch. 3.2 - Transportation to school for a rural countys...Ch. 3.2 - The captain of a Navy gunboat orders a volley of...Ch. 3.2 - A computer has generated seven random numbers over...Ch. 3.2 - Listed in the following table is the length...Ch. 3.2 - Redo Example 3.2.4 assuming n=12 and p=0.3.Ch. 3.2 - Prob. 18QCh. 3.2 - Prob. 19QCh. 3.2 - A corporate board contains twelve members. The...Ch. 3.2 - One of the popular tourist attractions in Alaska...Ch. 3.2 - A city has 4050 children under the age of ten,...Ch. 3.2 - Country A inadvertently launches ten guided...Ch. 3.2 - Anne is studying for a history exam covering the...Ch. 3.2 - Each year a college awards five merit-based...Ch. 3.2 - Keno is a casino game in which the player has a...Ch. 3.2 - A display case contains thirty-five gems, of which...Ch. 3.2 - Consider an urn with r red balls and w white...Ch. 3.2 - Prob. 29QCh. 3.2 - Prob. 30QCh. 3.2 - Prob. 31QCh. 3.2 - Prob. 32QCh. 3.2 - Prob. 33QCh. 3.2 - Some nomadic tribes, when faced with a...Ch. 3.2 - Suppose a population contains n1 objects of one...Ch. 3.2 - Prob. 36QCh. 3.3 - Prob. 1QCh. 3.3 - Repeat Question 3.3.1 for the case where the two...Ch. 3.3 - Suppose a fair die is tossed three times. Let X be...Ch. 3.3 - Suppose a fair die is tossed three times. Let X be...Ch. 3.3 - A fair coin is tossed three times. Let X be the...Ch. 3.3 - Suppose die one has spots 1, 2, 2, 3, 3, 4 and die...Ch. 3.3 - Suppose a particle moves along the x-axis...Ch. 3.3 - How would the pdf asked for in Question 3.3.7 be...Ch. 3.3 - Suppose that five people, including you and a...Ch. 3.3 - Prob. 10QCh. 3.3 - Prob. 11QCh. 3.3 - Prob. 12QCh. 3.3 - A fair die is rolled four times. Let the random...Ch. 3.3 - At the points x=0,1,...,6, the cdf for the...Ch. 3.3 - Find the pdf for the infinite-valued discrete...Ch. 3.3 - Recall the game of Fantasy Five from Example...Ch. 3.4 - Suppose fY(y)=4y3,0y1. Find P(0Y12).Ch. 3.4 - For the random variable Y with pdf...Ch. 3.4 - Let fY(y)=23y2,1y1. Find P(|Y12|14). Draw a graph...Ch. 3.4 - For persons infected with a certain form of...Ch. 3.4 - For a high-risk driver, the time in days between...Ch. 3.4 - Let n be a positive integer. Show that...Ch. 3.4 - Find the cdf for the random variable Y given in...Ch. 3.4 - If Y is an exponential random variable,...Ch. 3.4 - If the pdf for Y is fY(y)={0,|y|11|y|,|y|1 find...Ch. 3.4 - Prob. 10QCh. 3.4 - Prob. 11QCh. 3.4 - Prob. 12QCh. 3.4 - Prob. 13QCh. 3.4 - Prob. 14QCh. 3.4 - The logistic curve F(y)=11+ey,y, can represent a...Ch. 3.4 - Prob. 16QCh. 3.4 - Prob. 17QCh. 3.4 - Let Y be a random variable denoting the age at...Ch. 3.5 - Recall the game of Keno described in Question...Ch. 3.5 - The roulette wheels in Monte Carlo typically have...Ch. 3.5 - The pdf describing the daily profit, X, earned by...Ch. 3.5 - In the game of redball, two drawings are made...Ch. 3.5 - Suppose a life insurance company sells a $50,000,...Ch. 3.5 - A manufacturer has one hundred memory chips in...Ch. 3.5 - Records show that 642 new students have just...Ch. 3.5 - Prob. 8QCh. 3.5 - Recall Question 3.4.4, where the length of time Y...Ch. 3.5 - Let the random variable Y have the uniform...Ch. 3.5 - Show that the expected value associated with the...Ch. 3.5 - Show that fY(y)=1y2,y1 is a valid pdf but that Y...Ch. 3.5 - Based on recent experience, ten-year-old passenger...Ch. 3.5 - Prob. 14QCh. 3.5 - A city has 74,806 registered automobiles. Each is...Ch. 3.5 - Regulators have found that twenty-three of the...Ch. 3.5 - An urn contains four chips numbered 1 through 4....Ch. 3.5 - A fair coin is tossed three times. Let the random...Ch. 3.5 - Prob. 19QCh. 3.5 - For the St. Petersburg problem (Example 3.5.5),...Ch. 3.5 - Prob. 21QCh. 3.5 - Prob. 22QCh. 3.5 - Suppose that two evenly matched teams are playing...Ch. 3.5 - An urn contains one white chip and one black chip....Ch. 3.5 - Prob. 25QCh. 3.5 - Prob. 26QCh. 3.5 - Find the median for each of the following pdfs:...Ch. 3.5 - Suppose X is a binomial random variable with n=10...Ch. 3.5 - A typical days production of a certain electronic...Ch. 3.5 - Let Y have probability density function...Ch. 3.5 - Prob. 31QCh. 3.5 - A box is to be constructed so that its height is...Ch. 3.5 - Prob. 33QCh. 3.5 - If Y has probability density function fY(y)=2y,0y1...Ch. 3.5 - Prob. 35QCh. 3.5 - Prob. 36QCh. 3.6 - Find Var(X) for the urn problem of Example 3.6.1...Ch. 3.6 - Find the variance of Y if...Ch. 3.6 - Ten equally qualified applicants, six men and four...Ch. 3.6 - A certain hospitalization policy pays a cash...Ch. 3.6 - Use Theorem 3.6.1 to find the variance of the...Ch. 3.6 - If fY(y)=2yk2,0yk for what value of k does...Ch. 3.6 - Calculate the standard deviation, , for the random...Ch. 3.6 - Consider the pdf defined by fY(y)=2y3,y1 Show that...Ch. 3.6 - Frankie and Johnny play the following game....Ch. 3.6 - Let Y be a random variable whose pdf is given by...Ch. 3.6 - Suppose that Y is an exponential random variable,...Ch. 3.6 - Suppose that Y is an exponential random variable...Ch. 3.6 - Let X be a random variable with finite mean ....Ch. 3.6 - Suppose the charge for repairing an automobile...Ch. 3.6 - If Y denotes a temperature recorded in degrees...Ch. 3.6 - Prob. 16QCh. 3.6 - Suppose U is a uniform random variable over [0,1]....Ch. 3.6 - Recovering small quantities of calcium in the...Ch. 3.6 - Let Y be a uniform random variable defined over...Ch. 3.6 - Find the coefficient of skewness for an...Ch. 3.6 - Calculate the coefficient of kurtosis for a...Ch. 3.6 - Suppose that W is a random variable for which...Ch. 3.6 - If Y=aX+b,a0, show that Y has the same...Ch. 3.6 - Let Y be the random variable of Question 3.4.6,...Ch. 3.6 - Prob. 25QCh. 3.7 - Prob. 1QCh. 3.7 - Prob. 2QCh. 3.7 - Prob. 3QCh. 3.7 - Find c if fX,Y(x,y)=cxy for X and Y defined over...Ch. 3.7 - Prob. 5QCh. 3.7 - Four cards are drawn from a standard poker deck....Ch. 3.7 - An advisor looks over the schedules of his fifty...Ch. 3.7 - Consider the experiment of tossing a fair coin...Ch. 3.7 - Suppose that two fair dice are tossed one time....Ch. 3.7 - Let X be the time in days between a car accident...Ch. 3.7 - Let X and Y have the joint pdf...Ch. 3.7 - A point is chosen at random from the interior of a...Ch. 3.7 - Find P(X2Y) if fX,Y(x,y)=x+y for X and Y each...Ch. 3.7 - Prob. 14QCh. 3.7 - A point is chosen at random from the interior of a...Ch. 3.7 - Prob. 16QCh. 3.7 - Find the marginal pdfs of X and Y for the joint...Ch. 3.7 - Prob. 18QCh. 3.7 - For each of the following joint pdfs, find fX(x)...Ch. 3.7 - For each of the following joint pdfs, find fX(x)...Ch. 3.7 - Prob. 21QCh. 3.7 - Prob. 22QCh. 3.7 - Prob. 23QCh. 3.7 - Prob. 24QCh. 3.7 - Consider the experiment of simultaneously tossing...Ch. 3.7 - Prob. 26QCh. 3.7 - For each of the following joint pdfs, find...Ch. 3.7 - Prob. 28QCh. 3.7 - Prob. 29QCh. 3.7 - Prob. 30QCh. 3.7 - Given that FX,Y(x,y)=k(4x2y2+5xy4),0x1,0y1, find...Ch. 3.7 - Prob. 32QCh. 3.7 - Prob. 33QCh. 3.7 - Prob. 34QCh. 3.7 - Prob. 35QCh. 3.7 - Prob. 36QCh. 3.7 - Prob. 37QCh. 3.7 - Prob. 38QCh. 3.7 - Prob. 39QCh. 3.7 - Suppose that each of two urns has four chips,...Ch. 3.7 - Let X and Y be random variables with joint pdf...Ch. 3.7 - Are the random variables X and Y independent if...Ch. 3.7 - Prob. 43QCh. 3.7 - Find the joint cdf of the independent random...Ch. 3.7 - Prob. 45QCh. 3.7 - Prob. 46QCh. 3.7 - Prob. 47QCh. 3.7 - Prob. 48QCh. 3.7 - Prob. 49QCh. 3.7 - Prob. 50QCh. 3.7 - Suppose that X1,X2,X3, and X4 are independent...Ch. 3.7 - Prob. 52QCh. 3.8 - Prob. 1QCh. 3.8 - Prob. 2QCh. 3.8 - Prob. 3QCh. 3.8 - Prob. 4QCh. 3.8 - Prob. 5QCh. 3.8 - Prob. 6QCh. 3.8 - Prob. 7QCh. 3.8 - Prob. 8QCh. 3.8 - Prob. 9QCh. 3.8 - Prob. 10QCh. 3.8 - Prob. 11QCh. 3.8 - Prob. 12QCh. 3.8 - Prob. 13QCh. 3.9 - Prob. 1QCh. 3.9 - Prob. 2QCh. 3.9 - Suppose that fX,Y(x,y)=23(x+2y),0x1,0y1 [recall...Ch. 3.9 - Marksmanship competition at a certain level...Ch. 3.9 - Suppose that Xi is a random variable for which...Ch. 3.9 - Prob. 6QCh. 3.9 - Prob. 7QCh. 3.9 - Suppose two fair dice are tossed. Find the...Ch. 3.9 - Prob. 9QCh. 3.9 - Suppose that X and Y are both uniformly...Ch. 3.9 - Prob. 11QCh. 3.9 - Prob. 12QCh. 3.9 - Prob. 13QCh. 3.9 - Prob. 14QCh. 3.9 - Prob. 15QCh. 3.9 - Let X and Y be random variables with...Ch. 3.9 - Suppose that fX,Y(x,y)=2e(x+y),0x,0y. Find...Ch. 3.9 - Prob. 18QCh. 3.9 - Prob. 19QCh. 3.9 - Let X be a binomial random variable based on n...Ch. 3.9 - Prob. 21QCh. 3.9 - Prob. 22QCh. 3.9 - Prob. 23QCh. 3.9 - A gambler plays n hands of poker. If he wins the...Ch. 3.10 - Suppose the length of time, in minutes, that you...Ch. 3.10 - A random sample of size n=6 is taken from the pdf...Ch. 3.10 - What is the probability that the larger of two...Ch. 3.10 - Prob. 4QCh. 3.10 - Prob. 5QCh. 3.10 - Let Y1,Y2,...,Yn be a random sample from the...Ch. 3.10 - Calculate P(0.6Y40.7) if a random sample of size 6...Ch. 3.10 - A random sample of size n=5 is drawn from the pdf...Ch. 3.10 - Prob. 9QCh. 3.10 - Suppose that n observations are chosen at random...Ch. 3.10 - In a certain large metropolitan area, the...Ch. 3.10 - Consider a system containing n components, where...Ch. 3.10 - Prob. 13QCh. 3.10 - Prob. 14QCh. 3.10 - Prob. 15QCh. 3.10 - Suppose a device has three independent components,...Ch. 3.11 - Prob. 1QCh. 3.11 - Suppose a die is rolled six times. Let X be the...Ch. 3.11 - Prob. 3QCh. 3.11 - Five cards are dealt from a standard poker deck....Ch. 3.11 - Given that two discrete random variables X and Y...Ch. 3.11 - Prob. 6QCh. 3.11 - Suppose X, Y, and Z have a trivariate distribution...Ch. 3.11 - Prob. 8QCh. 3.11 - Let X and Y be independent Poisson random...Ch. 3.11 - Prob. 10QCh. 3.11 - Prob. 11QCh. 3.11 - Prob. 12QCh. 3.11 - Prob. 13QCh. 3.11 - Prob. 14QCh. 3.11 - Prob. 15QCh. 3.11 - Prob. 16QCh. 3.11 - Prob. 17QCh. 3.11 - Prob. 18QCh. 3.11 - Prob. 19QCh. 3.11 - Prob. 20QCh. 3.11 - For continuous random variables X and Y, prove...Ch. 3.12 - Let X be a random variable with pdf pX(k)=1/n, for...Ch. 3.12 - Two chips are drawn at random and without...Ch. 3.12 - Prob. 3QCh. 3.12 - Find the moment-generating function for the...Ch. 3.12 - Which pdfs would have the following...Ch. 3.12 - Prob. 6QCh. 3.12 - The random variable X has a Poisson distribution...Ch. 3.12 - Prob. 8QCh. 3.12 - Prob. 9QCh. 3.12 - Find E(Y4) if Y is an exponential random variable...Ch. 3.12 - Prob. 11QCh. 3.12 - Prob. 12QCh. 3.12 - Prob. 13QCh. 3.12 - Prob. 14QCh. 3.12 - Prob. 15QCh. 3.12 - Find the variance of Y if MY(t)=e2t/(1t2).Ch. 3.12 - Prob. 17QCh. 3.12 - Let Y1,Y2, and Y3 be independent random variables,...Ch. 3.12 - Use Theorems 3.12.2 and 3.12.3 to determine which...Ch. 3.12 - Calculate P(X2) if MX(t)=(14+34et)5.Ch. 3.12 - Suppose that Y1,Y2,...,Yn is a random sample of...Ch. 3.12 - Suppose the moment-generating function for a...Ch. 3.12 - Suppose that X is a Poisson random variable, where...Ch. 3.12 - Prob. 24Q
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