Reference no: EM131237883
Q1. Do the following:
a. Show that for positive values of monthly mean return not greater than µ = .01 (1 percent), the approximation T · µ given by (1.21) under-estimates the exact result of equation (1.20) by less than 1 percent. An acceptable solution will be to write simple R code to show the result for a range of positive monthly mean returns µ less than or equal to 1 percent. A desirable analytic solution is to show the result for a 1 percent monthly mean return, and prove that the difference between the exact and approximate solution increases with increasing µ and correspondingly decreases with decreasing µ.
b. Show that for a windfall constant monthly mean return of 2% the exact formula gives 26.8% for annual mean return, whereas the approximate gives 24%, an under estimate of nearly 3%.
Q2. Derive the expression (1.22) for the variance of multi-period arithmetic returns under the assumption that the returns are independent with constant mean µ and constant variance σ2. Hint: Make use of the fact that the expectation of a product of independent random variables is the product of the expectations of the individual random variables.
Q3. Use the fact that the log is a concave function to prove the inequality (1.47). Hint: Use the fact that for any set of values x1, x2,··· , xn and any set of non-negative constants a1,a2,··· ,an such that i=1∑nai = 1, a concave function g(x) has the property that i=1∑nai g(xi) ≤ g (i=1∑nai xi) with equality if and only if the xi are all equal.
Q4. (Advanced) Prove the approximate formula (1.48) relating the geometric mean to the true mean and volatility.
Q5. Show that if f(r) is any probability density function that is symmetric about a location parameter µ and that has a finite third moment E(r3), and hence finite third central moment E(r-µ)3, then the coefficient of skewness is zero.
Q6. Show that if f(r) is a normal density with mean µ and variance σ2 the ∫(r -µ)4f(r)dr = 3·σ4, and so the (excess) kurtosis is zero.
Q7. Derive the equations (1.57) and (1.58) for the mean and variance of a two component normal mixture distribution.
Q8. Show that a kernel density estimator ˆf(r) given by (1.64) is a bona fide probability density, i.e., fˆ(r) is non-negative and integrates to one.
Attachment:- Assignment.rar
|
What was major influence in the formation of solar system
: What is a solar nebula, and what is its relationship to the condensation theory? What was the major influence in the formation of the solar system? What is astrometry? Name two indirect methods by which other planetary systems can be detected?
|
|
Explain what is meant by crashophobia
: What volatility smile is likely to be observed for 6-month options when the volatility is uncertain and positively correlated to the stock price?
|
|
Conducting a case analysis
: In 100 words, what are the key areas you must consider when conducting a case analysis? Are there some you consider more important than others?
|
|
Why is pluto not considered a major planet
: Which planets have large, distinguishing "spots," and what are these spots called? What are dwarf planets? Why is Pluto not considered a major planet, and what other peculiarity distinguishes it from the major planets?
|
|
Derive the equations for the mean and variance
: Show that if f(r) is any probability density function that is symmetric about a location parameter µ and that has a finite third moment E(r3), and hence finite third central moment E(r-µ)3, then the coefficient of skewness is zero. Derive the equati..
|
|
Pattern of implied volatilities for options on exchange rate
: Suppose that a central bank's policy is to allow an exchange rate to fluctuate between 0.97 and 1.03. What pattern of implied volatilities for options on the exchange rate would you expect to see?
|
|
What is the sum of the squared deviations
: In a multiple choice exam consisting of 48 questions, with four possible answers for each question, (a) what is the probability of getting more than 15 correct just by guessing? (b) what is the probability of getting at least 15 correct just by gu..
|
|
Which planet''s axis of rotation is a peculiarity and why
: Are there "canals" on Mars? Explain. What are the major distinctions between the Jovian planets and the terrestrial planets? What is the Cassini Division? Which planet's axis of rotation is a peculiarity, and why?
|
|
Why can''t we see the surface of venus
: What proofs are there that the Earth is rotating and that it is revolving around the Sun? What are the terrestrial planets, and why are they called this? Why can't we see the surface of Venus?
|