Discrete Mathematics With Applications
5th Edition
ISBN: 9780357035283
Author: EPP
Publisher: Cengage
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Chapter 7.4, Problem 36ES
To determine
To prove:
There are as many functions from
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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?
The Course Name Real Analysis please Solve questions by Real Analysis
Chapter 7 Solutions
Discrete Mathematics With Applications
Ch. 7.1 - Given a function f from a set X to a set Y, f(x)...Ch. 7.1 - Given a function f from a set X to a set Y, if...Ch. 7.1 - Prob. 3TYCh. 7.1 - Given a function f then a set X to a set Y, if...Ch. 7.1 - Prob. 5TYCh. 7.1 - Prob. 6TYCh. 7.1 - Prob. 7TYCh. 7.1 - Prob. 8TYCh. 7.1 - Prob. 9TYCh. 7.1 - Prob. 1ES
Ch. 7.1 - Let X={1,3,5} and Y={a,b,c,d}. Define g:XY by the...Ch. 7.1 - Indicate whether the statement in parts (a)-(d)...Ch. 7.1 - a. Find all function from X={a,b}toY={u,v} . b....Ch. 7.1 - Let Iz be the identity function defined on the set...Ch. 7.1 - Find function defined on the sdet of nonnegative...Ch. 7.1 - Let A={1,2,3,4,5} , and define a function F:P(A)Z...Ch. 7.1 - Let Js={0,1,2,3,4} , and define a function F:JsJs...Ch. 7.1 - Define a function S:Z+Z+ as follows: For each...Ch. 7.1 - Prob. 10ESCh. 7.1 - Define F:ZZZZ as follows: For every ordered pair...Ch. 7.1 - Let JS={0,1,2,3,4} ,and define G:JsJsJsJs as...Ch. 7.1 - Let Js={0,1,2,3,4} , and define functions f:JsJs...Ch. 7.1 - Define functions H and K from R to R by the...Ch. 7.1 - Prob. 15ESCh. 7.1 - Let F and G be functions from the set of all real...Ch. 7.1 - Prob. 17ESCh. 7.1 - Find exact values for each of the following...Ch. 7.1 - Prob. 19ESCh. 7.1 - Prob. 20ESCh. 7.1 - If b is any positive real number with b1 and x is...Ch. 7.1 - Prob. 22ESCh. 7.1 - Prob. 23ESCh. 7.1 - If b and y are positivereal numbers such that...Ch. 7.1 - Let A={2,3,5} and B={x,y}. Let p1 and p2 be the...Ch. 7.1 - Observe that mod and div can be defined as...Ch. 7.1 - Let S be the set of all strings of as and bs....Ch. 7.1 - Consider the coding and decoding functions E and D...Ch. 7.1 - Consider the Hamming distance function defined in...Ch. 7.1 - Draw arrow diagram for the Boolean functions...Ch. 7.1 - Fill in the following table to show the values of...Ch. 7.1 - Cosider the three-place Boolean function f defined...Ch. 7.1 - Student A tries to define a function g:QZ by the...Ch. 7.1 - Student C tries to define a function h:QQ by the...Ch. 7.1 - Let U={1,2,3,4} . Student A tries to define a...Ch. 7.1 - Prob. 36ESCh. 7.1 - On certain computers the integer data type goed...Ch. 7.1 - Prob. 38ESCh. 7.1 - Prob. 39ESCh. 7.1 - Prob. 40ESCh. 7.1 - Prob. 41ESCh. 7.1 - In 41-49 let X and Y be sets, let A and B be any...Ch. 7.1 - Prob. 43ESCh. 7.1 - Prob. 44ESCh. 7.1 - Prob. 45ESCh. 7.1 - Prob. 46ESCh. 7.1 - Prob. 47ESCh. 7.1 - Prob. 48ESCh. 7.1 - Prob. 49ESCh. 7.1 - Prob. 50ESCh. 7.1 - Each of exercises 51-53 refers to the Euler phi...Ch. 7.1 - Prob. 52ESCh. 7.1 - Each of exercises 51-53 refers to the Euler phi...Ch. 7.2 - If F is a function from a set X to a set Y, then F...Ch. 7.2 - If F is a function from a set X to a set Y, then F...Ch. 7.2 - Prob. 3TYCh. 7.2 - Prob. 4TYCh. 7.2 - Prob. 5TYCh. 7.2 - Prob. 6TYCh. 7.2 - Prob. 7TYCh. 7.2 - Given a function F:XY , to prove that F is not one...Ch. 7.2 - Prob. 9TYCh. 7.2 - Prob. 10TYCh. 7.2 - Prob. 11TYCh. 7.2 - The definition of onr-to-one is stated in two...Ch. 7.2 - Fill in each blank with the word most or least. a....Ch. 7.2 - When asked to state the definition of one-to-one,...Ch. 7.2 - Let f:XY be a function. True or false? A...Ch. 7.2 - All but two of the following statements are...Ch. 7.2 - Let X={1,5,9} and Y={3,4,7} . a. Define f:XY by...Ch. 7.2 - Let X={a,b,c,d} and Y={e,f,g} . Define functions F...Ch. 7.2 - Let X={a,b,c} and Y={d,e,f,g} . Define functions H...Ch. 7.2 - Let X={1,2,3},Y={1,2,3,4} , and Z= {1,2} Define a...Ch. 7.2 - a. Define f:ZZ by the rule f(n)=2n, for every...Ch. 7.2 - Define F:ZZZZ as follows. For every ordered pair...Ch. 7.2 - a. Define F:ZZ by the rule F(n)=23n for each...Ch. 7.2 - a. Define H:RR by the rule H(x)=x2 , for each real...Ch. 7.2 - Explain the mistake in the following “proof.”...Ch. 7.2 - In each of 15-18 a function f is defined on a set...Ch. 7.2 - Prob. 16ESCh. 7.2 - Prob. 17ESCh. 7.2 - Prob. 18ESCh. 7.2 - Referring to Example 7.2.3, assume that records...Ch. 7.2 - Define Floor: RZ by the formula Floor (x)=x , for...Ch. 7.2 - Prob. 21ESCh. 7.2 - Let S be the set of all strings of 0’s and 1’s,...Ch. 7.2 - Define F:P({a,b,c})Z as follaws: For every A in...Ch. 7.2 - Les S be the set of all strings of a’s and b’s,...Ch. 7.2 - Let S be the et of all strings is a’s and b’s, and...Ch. 7.2 - Prob. 26ESCh. 7.2 - Let D be the set of all set of all finite subsets...Ch. 7.2 - Prob. 28ESCh. 7.2 - Define H:RRRR as follows: H(x,y)=(x+1,2y) for...Ch. 7.2 - Define J=QQR by the rule J(r,s)=r+2s for each...Ch. 7.2 - Prob. 31ESCh. 7.2 - a. Is log827=log23? Why or why not? b. Is...Ch. 7.2 - Prob. 33ESCh. 7.2 - The properties of logarithm established in 33-35...Ch. 7.2 - Prob. 35ESCh. 7.2 - Prob. 36ESCh. 7.2 - Prob. 37ESCh. 7.2 - Prob. 38ESCh. 7.2 - Prob. 39ESCh. 7.2 - Suppose F:XY is one—to—one. a. Prove that for...Ch. 7.2 - Suppose F:XY is into. Prove that for every subset...Ch. 7.2 - Prob. 42ESCh. 7.2 - Prob. 43ESCh. 7.2 - In 44-55 indicate which of the function in the...Ch. 7.2 - In 44-55 indicate which of the function in the...Ch. 7.2 - Prob. 46ESCh. 7.2 - Prob. 47ESCh. 7.2 - Prob. 48ESCh. 7.2 - Prob. 49ESCh. 7.2 - Prob. 50ESCh. 7.2 - Prob. 51ESCh. 7.2 - Prob. 52ESCh. 7.2 - Prob. 53ESCh. 7.2 - Prob. 54ESCh. 7.2 - Prob. 55ESCh. 7.2 - Prob. 56ESCh. 7.2 - Write a computer algorithm to check whether a...Ch. 7.2 - Write a computer algorithm to check whether a...Ch. 7.3 - If f is a function from X to Y’,g is a function...Ch. 7.3 - Prob. 2TYCh. 7.3 - If f is a one-to=-one correspondence from X to Y....Ch. 7.3 - Prob. 4TYCh. 7.3 - Prob. 5TYCh. 7.3 - Prob. 1ESCh. 7.3 - In each of 1 and 2, functions f and g are defined...Ch. 7.3 - In 3 and 4, functions F and G are defined by...Ch. 7.3 - In 3 and 4, functions F and G are defined by...Ch. 7.3 - Define f:RR by the rule f(x)=x for every real...Ch. 7.3 - Define F:ZZ and G:ZZ . By the rules F(a)=7a and...Ch. 7.3 - Define L:ZZ and M:ZZ by the rules L(a)=a2 and...Ch. 7.3 - Let S be the set of all strings in a’s and b’s and...Ch. 7.3 - Define F:RR and G:RZ by the following formulas:...Ch. 7.3 - Prob. 10ESCh. 7.3 - Define F:RR and G:RR by the rules F(n)=3x and...Ch. 7.3 - The functions of each pair in 12—14 are inverse to...Ch. 7.3 - G:R+R+ and G1:RR+ are defined by G(x)=x2andG1(x)=x...Ch. 7.3 - H and H-1 are both defined from R={1} to R-{1} by...Ch. 7.3 - Explain how it follows from the definition of...Ch. 7.3 - Prove Theorem 7.3.1(b): If f is any function from...Ch. 7.3 - Prove Theorem 7.3.2(b): If f:XY is a one-to-one...Ch. 7.3 - Prob. 18ESCh. 7.3 - If + f:XY and g:YZ are functions and gf is...Ch. 7.3 - If f:XY and g:YZ are function and gf is onto, must...Ch. 7.3 - Prob. 21ESCh. 7.3 - If f:XY and g:YZ are functions and gf is onto,...Ch. 7.3 - Prob. 23ESCh. 7.3 - Prob. 24ESCh. 7.3 - Prob. 25ESCh. 7.3 - In 26 and 27 find (gf)1,g1,f1, and f1g1 , and...Ch. 7.3 - In 26 and 27 find (gf)1,g1,f1 , and f1g1 by the...Ch. 7.3 - Prob. 28ESCh. 7.3 - Suppose f:XY and g:YZ are both one-to-one and...Ch. 7.3 - Prob. 30ESCh. 7.4 - A set is finite if, and only if,________Ch. 7.4 - Prob. 2TYCh. 7.4 - The reflexive property of cardinality says that...Ch. 7.4 - The symmetric property of cardinality says that...Ch. 7.4 - The transitive property of cardinality say that...Ch. 7.4 - Prob. 6TYCh. 7.4 - Prob. 7TYCh. 7.4 - Prob. 8TYCh. 7.4 - Prob. 9TYCh. 7.4 - Prob. 1ESCh. 7.4 - Show that “there are as many squares as there are...Ch. 7.4 - Let 3Z={nZn=3k,forsomeintegerk} . Prove that Z and...Ch. 7.4 - Let O be the set of all odd integers. Prove that O...Ch. 7.4 - Let 25Z be the set of all integers that are...Ch. 7.4 - Prob. 6ESCh. 7.4 - Prob. 7ESCh. 7.4 - Use the result of exercise 3 to prove that 3Z is...Ch. 7.4 - Show that the set of all nonnegative integers is...Ch. 7.4 - In 10-14 s denotes the sets of real numbers...Ch. 7.4 - Prob. 11ESCh. 7.4 - In 10-14 S denotes the set of real numbers...Ch. 7.4 - Prob. 13ESCh. 7.4 - Prob. 14ESCh. 7.4 - Show that the set of all bit string (string of 0’s...Ch. 7.4 - Prob. 16ESCh. 7.4 - Prob. 17ESCh. 7.4 - Must the average of two irrational numbers always...Ch. 7.4 - Prob. 19ESCh. 7.4 - Give two examples of functions from Z to Z that...Ch. 7.4 - Give two examples of function from Z to Z that are...Ch. 7.4 - Define a function g:Z+Z+Z+ by the formula...Ch. 7.4 - âa. Explain how to use the following diagram to...Ch. 7.4 - Prob. 24ESCh. 7.4 - Prob. 25ESCh. 7.4 - Prove that any infinite set contain a countable...Ch. 7.4 - Prove that if A is any countably infinite set, B...Ch. 7.4 - Prove that a disjoint union of any finite set and...Ch. 7.4 - Prove that a union of any two countably infinite...Ch. 7.4 - Prob. 30ESCh. 7.4 - Use the results of exercise 28 and 29 to prove...Ch. 7.4 - Prove that ZZ , the Cartesian product of the set...Ch. 7.4 - Prob. 33ESCh. 7.4 - Let P(s) be the set of all subsets of set S, and...Ch. 7.4 - Prob. 35ESCh. 7.4 - Prob. 36ESCh. 7.4 - Prove that if A and B are any countably infinite...Ch. 7.4 - Prob. 38ES
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- We consider a 4-dimensional stock price model given (under P) by dẴ₁ = µ· Xt dt + йt · ΣdŴt where (W) is an n-dimensional Brownian motion, π = (0.02, 0.01, -0.02, 0.05), 0.2 0 0 0 0.3 0.4 0 0 Σ= -0.1 -4a За 0 0.2 0.4 -0.1 0.2) and a E R. We assume that ☑0 = (1, 1, 1, 1) and that the interest rate on the market is r = 0.02. (a) Give a condition on a that would make stock #3 be the one with largest volatility. (b) Find the diversification coefficient for this portfolio as a function of a. (c) Determine the maximum diversification coefficient d that you could reach by varying the value of a? 2arrow_forwardQuestion 1. Your manager asks you to explain why the Black-Scholes model may be inappro- priate for pricing options in practice. Give one reason that would substantiate this claim? Question 2. We consider stock #1 and stock #2 in the model of Problem 2. Your manager asks you to pick only one of them to invest in based on the model provided. Which one do you choose and why ? Question 3. Let (St) to be an asset modeled by the Black-Scholes SDE. Let Ft be the price at time t of a European put with maturity T and strike price K. Then, the discounted option price process (ert Ft) t20 is a martingale. True or False? (Explain your answer.) Question 4. You are considering pricing an American put option using a Black-Scholes model for the underlying stock. An explicit formula for the price doesn't exist. In just a few words (no more than 2 sentences), explain how you would proceed to price it. Question 5. We model a short rate with a Ho-Lee model drt = ln(1+t) dt +2dWt. Then the interest rate…arrow_forwardIn 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.)arrow_forward
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