(a) Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

Question

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.)

Answer 1

Question

Chapter 6.5, Problem 79ES

To determine

(a)

Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

To determine

(b)

Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

To determine

(c)

Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

To determine

(d)

Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

To determine

(e)

Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

To determine

(f)

Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.

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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.)

3. Consider the polynomial equation 6-iz+7z² - iz³ +z = 0 for which the roots are 3i, -2i, -i, and i. (a) Verify the relations between this roots and the coefficients of the polynomial. (b) Find the annulus region in which the roots lie.

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…

Answer 2

Question

Chapter 6.5, Problem 79ES

To determine

(a)

Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

To determine

(b)

Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

To determine

(c)

Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

To determine

(d)

Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

To determine

(e)

Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

To determine

(f)

Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.

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Answer 3

Question

Chapter 6.5, Problem 79ES

To determine

(a)

Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

To determine

(b)

Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

To determine

(c)

Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

To determine

(d)

Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

To determine

(e)

Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

To determine

(f)

Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.

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Answer 4

Question

Chapter 6.5, Problem 79ES

To determine

(a)

Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

To determine

(b)

Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

To determine

(c)

Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

To determine

(d)

Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

To determine

(e)

Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

To determine

(f)

Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.

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Answer 5

Chapter 6.5, Problem 79ES

To determine

(a)

Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

To determine

(b)

Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

To determine

(c)

Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

To determine

(d)

Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

To determine

(e)

Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

To determine

(f)

Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.

Answer 6

Chapter 6.5, Problem 79ES

To determine

(a)

Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

Answer 7

Chapter 6.5, Problem 79ES

To determine

(a)

Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

Answer 8

To determine

(b)

Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

Answer 9

To determine

(b)

Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

Answer 10

To determine

(c)

Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

Answer 11

To determine

(c)

Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

Answer 12

To determine

(d)

Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

Answer 13

To determine

(d)

Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

Answer 14

To determine

(e)

Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

Answer 15

To determine

(e)

Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

Answer 16

To determine

(f)

Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.

Answer 17

To determine

(f)

Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.

Answer 18

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Ch. 6.1 - Complete the crossword puzzle shown on the...Ch. 6.1 - Prob. 2EE Ch. 6.1 - Write 357 as a traditional Chinese numeral.Ch. 6.1 - Prob. 4EE Ch. 6.1 - Prob. 5EE Ch. 6.1 - Prob. 6EE Ch. 6.1 - Prob. 1ES Ch. 6.1 - Prob. 2ES Ch. 6.1 - Prob. 3ES Ch. 6.1 - Prob. 4ES

(a) Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

Solution Summary: The author analyzes whether the statement 'There are infinitely many twin primes' is a theorem or conjecture.

(a)

Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

(b)

Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

(c)

Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

(d)

Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

(e)

Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

(f)

Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.

Chapter 6 Solutions

(a) Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

Solution Summary: The author analyzes whether the statement 'There are infinitely many twin primes' is a theorem or conjecture.

(a) Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

(b) Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

(c) Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

(d) Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

(e) Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

(f) Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.

Chapter 6 Solutions

(a)

Whether the statement'There are infinitely many twin primes' is a theorem or a conjecture.

(b)

Whether the statement'There are infinitely many primes of the formn2+1' is a theorem or a conjecture.

(c)

Whether the statement'There is always a prime between n and 2n forn≥2' is a theorem or a conjecture.

(d)

Whether the statement'There is always a prime betweenn2and(n+1)2' is a theorem or a conjecture.

(e)

Whether the statement'Every odd number greater than 5 can be written as the sum of threeprimes'is a theorem or a conjecture.

(f)

Whether the statement'Every positive even number can be written as the difference of twoprimes'is a theorem or a conjecture.