2 Suppose a stock price is log-normal with volatility o. Consider a derivative with maturity Tand payoff f(s(T)) = s³(T).(a) What is its value at time 0?(b) What is the Delta of the option considered in this Problem?(Hint: your task is to evaluate e e-rTERN (s). Recall that under the risk-neutral probabilitydistribution, sã is lognormal, and therefore s is also log-normal. Use the fact that if Z isGaussian with mean m and standard deviation s then E[e²] = em+¾s².

The present value of the anticipated payoff at maturity determines the derivative's value at time…

Answered: 2 Suppose a stock price is log-normal…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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7. (a) If land that can be farmed L is eroding continuously at a rate of 4.5% per year because of changing climatic conditions, how much of this land will be left after 25 years? (b) A piece of art currently worth $2500 is appreciating in value 'V' according to the formula: 2500et0.5 However, assuming continuous discounting, the present value of V is given by: P = Vert = How long should the piece of art by kept to maximise its present value under continuous compounding with: (i) r = 0.05 (ii) r = 0.09
For the above model, determine the following: a. Mean Excess Loss Function b. Limited Expected Value Function c. The 75th Percentile
3
You are given a two dice with 4 sides each, with equal probability of landong on all sides. Dice one has values 1 - 4 and the second has values 5-8. The low-valued die (dice one) is rolled first. If you get a 1 or 2, then you roll the high-valued die (dice 2). If you get 3 or 4, you roll the low-valued die. X= value of the 1st roll Y = value of the 2nd roll Z = X + Y %3D Find H(X), H(Y), H(Z) Find H(Y|X) Find H(X,Y), H(XIҮ) Find I (X;Y)
A radio manufacturer believes that the length of life of WPPX model radio is normal with µ=12 years and σ=2.5 years. (a) What percent of the radios will function for more than 16 years? (b) Suppose the company decides to replace 1% of the radios. Find the length of guarantee period.i.e. find X. (c) What percent of the radios that will fail to satisfy the guarantee period of 10 years? , i.e. less than 10 years? (d) If 10% of the radios will function for more than X years, find X.
Suppose that the value of an index of the stock market increases on average about 0.02% per day (calculated with continuous discounting) and a volatility (i.e., standard deviation) of 1% per day. Assuming that the returns are Normally distributed, what would be the 5% Daily Expected Shortfall (ES) expressed as a percent return? (Note: Enter your answer rounded to the nearest 2 decimal places. (For example, -1.2345% should be entered as -1.23%).
Suppose that the borrowing rate that your client faces is 9%. Assume that the equity market index has an expected return of 13% and standard deviation of 25%, that rf = 5%. Your fund manages a risky portfolio, with the following details: E(rp) = 11%, p = 15%. What is the largest percentage fee that a client currently lending (y 1)? (Negative values should be indicated by a minus sign. Do not round intermediate calculations. Round your answers to 1 decimal place.) y 1
Let Y be the present value random variable of a 10-year term annuity-due of 1 issued at an individual (35). Calculate its actuarial present value if (a) Mortality follows De Moivre's Law with w = 100. (b) 8 = 0.05.
The "kernel trick" is a quick way to integrate when you can recognize a distribution in some equation g(x) which you want to integrate. It takes advantage of the fact that the pdf f(x) of a proper probability distribution must integrate to 1 over the support. Thus if we can manipulate an equation g(x) into the pdf of a known distribution and some multiplying constant c in other words, g(x) = c · f(x) --- then we know that C • --- Saex 9(x)dx = c Smex f(x)dx = c ·1 = c Steps: 1. manipulate equation so that you can recognize the kernel of a distribution (the terms involving x); 2. use the distribution to figure out the normalizing constant; 3. multiply and divide by the distribution's normalizing constant, and rearrange so that inside the integral is the pdf of the new distribution, and outside are the constant terms 4. integrate over the support. The following questions will be much easier if you use the kernel trick, so this question is intended to give you basic practice. QUESTION:…
2) An insurance policy on a ground-up loss process X has a uniform distribution between 0 and 3000, with a deductible of 100, coinsurance of 50% (a = 0.5), and a maximum policy payment per loss of 2500. Find the expected insurer payment per loss, E(Y), and the expected insurer payment per payment, E (YP).
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5. Suppose that the random variable X follows an exponential distribution with probability density function S(z) = de-A";0 0). Define a new random variable Y = x. (a) Show that the cumulative density function of Y is given by Fy(y) = {-exp(-Ay"; and hence, or otherwise, find the probability density function of Y. (b) Find an expression for the maximum likelihood estimator of the parameter A, using a sample Y1, 92, , Yn, from the distribution of Y. v20

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