Homework Answers
Yes, V also satisfies Black Scholes PDE.
V is a portfolio of securities whose prices satisfy Black Scholes PDE separately.
The Black Scholes PDE is nothing but a second order differential equation with respect to the price of the underlying S and a first order equation with time t.
And we all know that:
- If a, b, c...are solutions of a second order PDE then any linear combination of a, b, c will also be the solution of the same PDE.
- If a, b, c are solutions of a first order PDE then any linear combination of a, b, c will also be the solution of the same PDE.
Hence, as long as the price of the securities P1, P2, ...Pn are function of underlying price S and time t, any portfolio made by liner combination of these securities will also be function of (S, t) and hence will satisfy the Black Scholes PDE.
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Suppose that P, P2, . ., Pn are prices of derivative securities and that each price...
X Text Question 4.3 Question Help Suppose a nonlinear price discriminating monopoly, can set three prices,...
X Text Question 4.3 Question Help Suppose a nonlinear price discriminating monopoly, can set three prices, depending on the quantity a consumer purchases. The firm's profit is T=P1 (Q1) +P2 (Q2-Q1) +P3 (Q3 - Q2) – mQ3, where p, is the high price charged on the first Q, units (first block), P, is a lower price charged on the next Q, -Q, units, P3 is the lowest price charged on the Q3-Q, remaining units, Q, is the total number of...
(a) Prove that if A1, A2, . . . , are mutually exclusive, then P(An) →...
(a) Prove that if A1, A2, . . . , are mutually exclusive, then P(An) → 0 as n → oo. (Recall that whenever Σοοι pn is finite and all the pn's are nonnegative, then Pn-+0 as n o.) (b) Suppose 1 flip a fair coin forever. Let An be the event that the rnth flip is a head. Since the coin is fair, P(An)-.Notice that P(An) 0 asnoo. How, then, can the previous problem still be true? Each An...
2. Suppose a monopoly firm is allowed to price discriminate in 3 markets where the prices for the good in each mark...
2. Suppose a monopoly firm is allowed to price discriminate in 3 markets where the prices for the good in each market are given by: P1 = 63 - 401 P2 = 105-502 P3 = 75 - 6Q3 The cost of the output is (Q) = 20 + 15Q+Q? where: Q = Q1 + Q2 + Q3 a) Give the profit function for the firm. b) Find the FOC's and find the p*'s and Qo's that maximize profit c) Find...
2. Suppose a monopoly firm is allowed to price discriminate in 3 markets where the prices...
2. Suppose a monopoly firm is allowed to price discriminate in 3 markets where the prices for the good in each market are given by: P1 = 63 - 401 P2 = 105 - 502 P3 = 75 - 603 where: Q = Q1 + Q2 + Q3 The cost of the output is (Q) = 20 + 15Q+Q2 a) Give the profit function for the firm. b) Find the FOC's and find the p*'s and Q*'s that maximize profit....
Q1. Consider these four points: P [,,5| , P2 = 2], P3 = [H]. Plot these...
Q1. Consider these four points: P [,,5| , P2 = 2], P3 = [H]. Plot these three points. (a) Find the Manhattan distance between Pi and P2 (b) Find the Manhattan distance between P1 and P3. (e) Find the Manhattan distance between P2 and P3. Q2. Consider the same points in Q1 and find the Euclidean distances between the points specified in parts (a), (b), and (e). In other words, you will be doing the above question again but now...
1. Suppose that a quantiies q and q2, respectively, and that it sells them at prices...
1. Suppose that a quantiies q and q2, respectively, and that it sells them at prices p1 and p2, respectively Suppose that the company's production costs are sporting goods company manufactures basketballs and soccer balls in given by C 2q 2q 10. (a) Find the maximum profit that the company can make assuming that prices are fixed the price p (b) Find the rate of change of the maximum profit you found in part (a) increases. Is it wise for...
Two firms are price-competing as in the standard Bertrand model. Each faces the market demand function D(p)=50-p. Firm 1 has constant marginal cost c1=10 and firm 2 has c2=20. As usual, if one of the...
Two firms are price-competing as in the standard Bertrand model. Each faces the market demand function D(p)=50-p. Firm 1 has constant marginal cost c1=10 and firm 2 has c2=20. As usual, if one of the firms has the lower price, they capture the entire market, and when they both charge exactly the same price they share the demand equally. 1. Suppose A1=A2={0.00, 0.01, 0.02,...,100.00}. That is, instead of any real number, we force prices to be listed in whole cents....
of 3. Suppose there is a firm that produces nonnegative output quantities q = 101 02...
of 3. Suppose there is a firm that produces nonnegative output quantities q = 101 02 " en n different goods. It does so using the very same output it produces as inputs, x = 2 22 ... 2n). For each output i (i = 1,...,n), let the following price vector represent the price of ouput i, p = P1 P2 ... Pn]. Let n = 5: a. derive the firm's revenue; b. derive the firm's cost; and, c. derive...
2. Suppose two firms are competing in prices (Bertrand) in an industry where demand is P-200-8Q....
2. Suppose two firms are competing in prices (Bertrand) in an industry where demand is P-200-8Q. Assume neither firm faces any fixed costs. (a) If both firms have MC-120, what is the equilibrium price and profits for each firm? (b) Suppose one firm has MC-150 and one has MC-0. How much profit does each firm make? (c) Suppose one firm has MC-120 and one has MC-100. Approximately how much profit does each firm make?
1. Let demand be P(Q) = 6 - 2. What is the price elasticity of demand...
1. Let demand be P(Q) = 6 - 2. What is the price elasticity of demand at Q = 4? a. E = C. b. E= E = -4 d. E= -2 2. Suppose we have 3 types of households each with private demand for a public good (like flood protection) of P1(Q) = 5, P2(Q) = 10 - Q, and P3(Q) = 20 – 2Q. What is the social demand curve for the range Q < 10? a. Ps(0=...
X Text Question 4.3 Question Help Suppose a nonlinear price discriminating monopoly, can set three prices, depending on the quantity a consumer purchases. The firm's profit is T=P1 (Q1) +P2 (Q2-Q1) +P3 (Q3 - Q2) – mQ3, where p, is the high price charged on the first Q, units (first block), P, is a lower price charged on the next Q, -Q, units, P3 is the lowest price charged on the Q3-Q, remaining units, Q, is the total number of...
(a) Prove that if A1, A2, . . . , are mutually exclusive, then P(An) → 0 as n → oo. (Recall that whenever Σοοι pn is finite and all the pn's are nonnegative, then Pn-+0 as n o.) (b) Suppose 1 flip a fair coin forever. Let An be the event that the rnth flip is a head. Since the coin is fair, P(An)-.Notice that P(An) 0 asnoo. How, then, can the previous problem still be true? Each An...
2. Suppose a monopoly firm is allowed to price discriminate in 3 markets where the prices for the good in each market are given by: P1 = 63 - 401 P2 = 105-502 P3 = 75 - 6Q3 The cost of the output is (Q) = 20 + 15Q+Q? where: Q = Q1 + Q2 + Q3 a) Give the profit function for the firm. b) Find the FOC's and find the p*'s and Qo's that maximize profit c) Find...
2. Suppose a monopoly firm is allowed to price discriminate in 3 markets where the prices for the good in each market are given by: P1 = 63 - 401 P2 = 105 - 502 P3 = 75 - 603 where: Q = Q1 + Q2 + Q3 The cost of the output is (Q) = 20 + 15Q+Q2 a) Give the profit function for the firm. b) Find the FOC's and find the p*'s and Q*'s that maximize profit....
Q1. Consider these four points: P [,,5| , P2 = 2], P3 = [H]. Plot these three points. (a) Find the Manhattan distance between Pi and P2 (b) Find the Manhattan distance between P1 and P3. (e) Find the Manhattan distance between P2 and P3. Q2. Consider the same points in Q1 and find the Euclidean distances between the points specified in parts (a), (b), and (e). In other words, you will be doing the above question again but now...
1. Suppose that a quantiies q and q2, respectively, and that it sells them at prices p1 and p2, respectively Suppose that the company's production costs are sporting goods company manufactures basketballs and soccer balls in given by C 2q 2q 10. (a) Find the maximum profit that the company can make assuming that prices are fixed the price p (b) Find the rate of change of the maximum profit you found in part (a) increases. Is it wise for...
of 3. Suppose there is a firm that produces nonnegative output quantities q = 101 02 " en n different goods. It does so using the very same output it produces as inputs, x = 2 22 ... 2n). For each output i (i = 1,...,n), let the following price vector represent the price of ouput i, p = P1 P2 ... Pn]. Let n = 5: a. derive the firm's revenue; b. derive the firm's cost; and, c. derive...
2. Suppose two firms are competing in prices (Bertrand) in an industry where demand is P-200-8Q. Assume neither firm faces any fixed costs. (a) If both firms have MC-120, what is the equilibrium price and profits for each firm? (b) Suppose one firm has MC-150 and one has MC-0. How much profit does each firm make? (c) Suppose one firm has MC-120 and one has MC-100. Approximately how much profit does each firm make?
1. Let demand be P(Q) = 6 - 2. What is the price elasticity of demand at Q = 4? a. E = C. b. E= E = -4 d. E= -2 2. Suppose we have 3 types of households each with private demand for a public good (like flood protection) of P1(Q) = 5, P2(Q) = 10 - Q, and P3(Q) = 20 – 2Q. What is the social demand curve for the range Q < 10? a. Ps(0=...
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