Explanation Given information: The gambler is betting on 6 when a die is thrown, if the die shows up 6 he wins $ 1 otherwise he loses $ 1 each time. The given P k is the probability that the gambler is eventually ruined if he has k dollars. Proof: The probability that gambler wins = 1 6 And, the probability that gambler loses = 5 6 P k is the probability that the gambler is eventually ruined if he has k dollars. Similarly, P k − 1 will be the probability that the gambler is eventually ruined if he has k − 1 dollars

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Author: EPP

Publisher: Cengage

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Chapter 9.9, Problem 33ES

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The equation Pk−1=16Pk+56Pk−2 using the definition of conditional probability.

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Chapter 9.9, Problem 32ES

Chapter 9.9, Problem 34ES

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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.)

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?

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Chapter 9 Solutions

Discrete Mathematics With Applications

Ch. 9.1 - A sample space of a random process or experiment...Ch. 9.1 - An event in a sample space is .Ch. 9.1 - To compute the probability of an event using the...Ch. 9.1 - Prob. 4TY Ch. 9.1 - Toss two coins 30 times and make a table showing...Ch. 9.1 - In the example of tossing two quarters, what is...Ch. 9.1 - In 3-6 use the sample space given in Example...Ch. 9.1 - In 3-6 use the sample space given in Example...Ch. 9.1 - In 3-6 use the sample space given in Example...Ch. 9.1 - In 3-6 use the sample space given in Example...

Ch. 9.1 - In 7-10, use the sample space given in Example...Ch. 9.1 - In 7-10, use the sample space given in Example...Ch. 9.1 - In 7-10, use the sample space given in Example...Ch. 9.1 - In 7-10, use the sample space given in Example...Ch. 9.1 - Suppose that a coin is tossed three times and the...Ch. 9.1 - Suppose that each child born is equally likely to...Ch. 9.1 - Suppose that on a true/false exam you have no idea...Ch. 9.1 - There people have been exposed to a certain...Ch. 9.1 - Prob. 15ES Ch. 9.1 - Two faces of a six-sided die are painted red, two...Ch. 9.1 - Prob. 17ES Ch. 9.1 - Prob. 18ES Ch. 9.1 - An urn contains two blue balls (denoted B1 and B2)...Ch. 9.1 - Relet to Example 9.1.3. Suppose you are appealing...Ch. 9.1 - Prob. 21ES Ch. 9.1 - a. How many positive three-digit integers are...Ch. 9.1 - Suppose A[1],A[2],A[3],....,A[n] is a...Ch. 9.1 - Suppose A[1],A[2],...,A[n] is a one-dimensional...Ch. 9.1 - Suppose A[1], A[2],…A[n] is a one-dimensional...Ch. 9.1 - Prob. 26ES Ch. 9.1 - What is the 62nd element in the one-dimensional...Ch. 9.1 - If the largest of 56 consecutive integers is 279,...Ch. 9.1 - Prob. 29ES Ch. 9.1 - How many even integers are between 1 and 1,.001?Ch. 9.1 - Prob. 31ES Ch. 9.1 - A certain non-leap year has 365 days, and January...Ch. 9.1 - Prove Theorem 9.1.1. (Let m be any integer and...Ch. 9.2 - The multiplication rule says that if an operation...Ch. 9.2 - A permutation of a set of elements is_________.Ch. 9.2 - Prob. 3TY Ch. 9.2 - Prob. 4TY Ch. 9.2 - Prob. 5TY Ch. 9.2 - Prob. 6TY Ch. 9.2 - Prob. 1ES Ch. 9.2 - Prob. 2ES Ch. 9.2 - Prob. 3ES Ch. 9.2 - Prob. 4ES Ch. 9.2 - Prob. 5ES Ch. 9.2 - One urn contains two black balls (Labeled B1 and...Ch. 9.2 - One urn contains one blue ball (labeled B1) and...Ch. 9.2 - A person buying a personal computer system is...Ch. 9.2 - Suppose there are three roads from city A to city...Ch. 9.2 - Suppose there are three routes from North Point to...Ch. 9.2 - (a)A bit string is a finite sequence of 0’s and...Ch. 9.2 - Hexadecimal numbers are made using the sixteen...Ch. 9.2 - A coin is tossed four times. Each time the result...Ch. 9.2 - Suppose that in a certain stale, all automobile...Ch. 9.2 - A combination lock requires three selections of...Ch. 9.2 - a. 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P(n+1,3)=n3n Ch. 9.2 - Prob. 41ES Ch. 9.2 - Prob. 42ES Ch. 9.2 - Prob. 43ES Ch. 9.2 - Prove Theorem 9.2.1 by mathematical induction.Ch. 9.2 - Prove Theorem 9.2.2 by mathematical induction.Ch. 9.2 - Prob. 46ES Ch. 9.2 - Prob. 47ES Ch. 9.3 - The addition rule says that if a finite set A...Ch. 9.3 - Prob. 2TY Ch. 9.3 - Prob. 3TY Ch. 9.3 - Prob. 4TY Ch. 9.3 - Prob. 5TY Ch. 9.3 - (a) How many bit string consist of from one...Ch. 9.3 - (a) How many string of hexadecimal digits consist...Ch. 9.3 - (a) How many integers from 1 through 999 do not...Ch. 9.3 - How many arrangements in a row of no more than...Ch. 9.3 - (a) How many five-digit integers (integers from...Ch. 9.3 - In a certain stale, all license plain consist of...Ch. 9.3 - At a certain company, passwords must be from...Ch. 9.3 - In a certain country license plates consist of...Ch. 9.3 - a. Consider the following algorithm segment: for...Ch. 9.3 - A calculator has an eight-digit display and a...Ch. 9.3 - a. 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A store sells 8 colors of balloons with at...Ch. 9.6 - A large pile of coins consists of penruey nickels,...Ch. 9.6 - Suppose the bakery in exercise 3 has at least...Ch. 9.6 - Suppose the camera shop in exercise 4 can obtain...Ch. 9.6 - Prob. 21ES Ch. 9.7 - If n and r are nonnegative integers with rn , then...Ch. 9.7 - Prob. 2TY Ch. 9.7 - Prob. 3TY Ch. 9.7 - Prob. 4TY Ch. 9.7 - Prob. 5TY Ch. 9.7 - Prob. 6TY Ch. 9.7 - Prob. 7TY Ch. 9.7 - Prob. 1ES Ch. 9.7 - Prob. 2ES Ch. 9.7 - Prob. 3ES Ch. 9.7 - Prob. 4ES Ch. 9.7 - Prob. 5ES Ch. 9.7 - Prob. 6ES Ch. 9.7 - Prob. 7ES Ch. 9.7 - Prob. 8ES Ch. 9.7 - Prob. 9ES Ch. 9.7 - (a) Use Pascal’s triangle given in Table 9.7.1 to...Ch. 9.7 - Prob. 11ES Ch. 9.7 - Use Pascal’s formula repeatedly to derive a...Ch. 9.7 - Use Pascal’s formula to prove by mathematical...Ch. 9.7 - Prob. 14ES Ch. 9.7 - Prove the following generalization of exercise 13:...Ch. 9.7 - Prob. 16ES Ch. 9.7 - Prove that for every integer n0 ,...Ch. 9.7 - Prob. 18ES Ch. 9.7 - Prob. 19ES Ch. 9.7 - Prob. 20ES Ch. 9.7 - Prob. 21ES Ch. 9.7 - Use the binomial theorem to expand the expressions...Ch. 9.7 - Use the binomial theorem to expand the expressions...Ch. 9.7 - Use the binomial theorem to expand the expressions...Ch. 9.7 - Prob. 25ES Ch. 9.7 - Prob. 26ES Ch. 9.7 - Prob. 27ES Ch. 9.7 - Prob. 28ES Ch. 9.7 - Prob. 29ES Ch. 9.7 - Prob. 30ES Ch. 9.7 - In 29-34, find the coefficient of the given term...Ch. 9.7 - In 29-34, find the coefficient of the given term...Ch. 9.7 - Prob. 33ES Ch. 9.7 - In 29-34, find the coefficient of the given term...Ch. 9.7 - Prob. 35ES Ch. 9.7 - For every integer n1 ,...Ch. 9.7 - Prob. 37ES Ch. 9.7 - Prob. 38ES Ch. 9.7 - Prob. 39ES Ch. 9.7 - Prob. 40ES Ch. 9.7 - Prob. 41ES Ch. 9.7 - Prob. 42ES Ch. 9.7 - Prob. 43ES Ch. 9.7 - Prob. 44ES Ch. 9.7 - Prob. 45ES Ch. 9.7 - Prob. 46ES Ch. 9.7 - Prob. 47ES Ch. 9.7 - Prob. 48ES Ch. 9.7 - Prob. 49ES Ch. 9.7 - Prob. 50ES Ch. 9.7 - Express each of the sums in 43—54 in closed form...Ch. 9.7 - Prob. 52ES Ch. 9.7 - Prob. 53ES Ch. 9.7 - Prob. 54ES Ch. 9.7 - Prob. 55ES Ch. 9.8 - If A is an event in a sample space S,P(A) can...Ch. 9.8 - Prob. 2TY Ch. 9.8 - Prob. 3TY Ch. 9.8 - Prob. 4TY Ch. 9.8 - Prob. 5TY Ch. 9.8 - Prob. 1ES Ch. 9.8 - Prob. 2ES Ch. 9.8 - Prob. 3ES Ch. 9.8 - Prob. 4ES Ch. 9.8 - Prob. 5ES Ch. 9.8 - Prob. 6ES Ch. 9.8 - Prob. 7ES Ch. 9.8 - Prob. 8ES Ch. 9.8 - Let A and B be events in a sample space S, and let...Ch. 9.8 - Prob. 10ES Ch. 9.8 - Prob. 11ES Ch. 9.8 - Prob. 12ES Ch. 9.8 - Prob. 13ES Ch. 9.8 - A lottery game offers $2 million to the grand...Ch. 9.8 - A company offers a raffle whose grand prize is a...Ch. 9.8 - An urn contains four balls numbered 2, 2, 5, and...Ch. 9.8 - Prob. 17ES Ch. 9.8 - An urn contains five balls numbered 1,2,2,8, and...Ch. 9.8 - Prob. 19ES Ch. 9.8 - Suppose a person offers to play a game with you....Ch. 9.8 - Prob. 21ES Ch. 9.8 - Prob. 22ES Ch. 9.8 - Prob. 23ES Ch. 9.9 - Prob. 1TY Ch. 9.9 - Prob. 2TY Ch. 9.9 - Prob. 3TY Ch. 9.9 - Prob. 4TY Ch. 9.9 - Prob. 1ES Ch. 9.9 - Prob. 2ES Ch. 9.9 - Prob. 3ES Ch. 9.9 - Prob. 4ES Ch. 9.9 - Suppose that A and B are events in a sample space...Ch. 9.9 - An urn contains 25 red balls and 15 blue balls....Ch. 9.9 - Prob. 7ES Ch. 9.9 - A pool of 10 semifinalists for a job consists of 7...Ch. 9.9 - Prob. 9ES Ch. 9.9 - Prob. 10ES Ch. 9.9 - One urn contains 12 blue balls and 7 white balls,...Ch. 9.9 - Redo exercise 11 assuming that the first urn...Ch. 9.9 - Prob. 13ES Ch. 9.9 - Prob. 14ES Ch. 9.9 - Prob. 15ES Ch. 9.9 - Three different supplier.-X, Y. and Z-provide...Ch. 9.9 - Prob. 17ES Ch. 9.9 - Prob. 18ES Ch. 9.9 - Prob. 19ES Ch. 9.9 - Prob. 20ES Ch. 9.9 - Prob. 21ES Ch. 9.9 - Prob. 22ES Ch. 9.9 - Prob. 23ES Ch. 9.9 - Prob. 24ES Ch. 9.9 - A coin is loaded so that the probability of heads...Ch. 9.9 - Describe a sample space and events A,B, and C,...Ch. 9.9 - Prob. 27ES Ch. 9.9 - Prob. 28ES Ch. 9.9 - Suppose that ten items are chosen at random from a...Ch. 9.9 - Suppose the probability of a false positive result...Ch. 9.9 - Prob. 31ES Ch. 9.9 - Prob. 32ES Ch. 9.9 - Prob. 33ES Ch. 9.9 - Prob. 34ES

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To prove: The equation P k − 1 = 1 6 P k + 5 6 P k − 2 using the definition of conditional probability.

Solution Summary: The author explains how the equation P_k-1=16pv — if a die is thrown, the gambler wins and loses

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To prove:

The equation Pk−1=16Pk+56Pk−2 using the definition of conditional probability.

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Chapter 9 Solutions

To prove: The equation P k − 1 = 1 6 P k + 5 6 P k − 2 using the definition of conditional probability.

Solution Summary: The author explains how the equation P_k-1=16p­v — if a die is thrown, the gambler wins and loses

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Videos

To prove: The equation Pk−1=16Pk+56Pk−2 using the definition of conditional probability.

Want to see the full answer?

Chapter 9 Solutions

Solution Summary: The author explains how the equation P_k-1=16pv — if a die is thrown, the gambler wins and loses

To prove:

The equation Pk−1=16Pk+56Pk−2 using the definition of conditional probability.