Explanation Given: R 1 = R 2 = 1 Ω , C = 1 F, L = 1H . The electrical circuit is shown below. Calculation: The three equations for the unknown currents are: I 2 R 2 − q C = 0 , I 1 = d q d t , − I 2 R 2 + I 3 R 1 + L d I 3 d t = 0 Substitute given values in above equations. I 2 = q ( 1 ) I 1 = d q d t = d I 2 d t ( 2 ) − I 2 + I 3 + d I 3 d t = 0 ( 3 ) By Kirchoff Current Law, I 1 + I 2 + I 3 = 0 . I 3 = − I 1 − I 2 ( 4 ) Substitute Equation ( 1 ) and Equation ( 2 ) in Equation ( 4 ) . I 3 = − d I 2 d t − q ( 5 ) Differentiate Equation ( 5 ) . d I 3 d t = − d 2 I 2 d t 2 − d q d t = q − I 3 = q + d I 2 d t + q = 2 q + d I 2 d t d I 3 d t = − d 2 I 2 d t 2 − d q d t I 2 − I 3 = − d 2 I 2 d t 2 − d q d t q + d I 2 d t + q = − d 2 I 2 d t 2 − d q d t d 2 I 2 d t 2 + d I 2 d t + d q d t + 2 q = 0 d 2 q d t 2 + 2 d q d t + 2 q = 0 ( 6 ) The auxiliary equation of Equation ( 6 ) is, r 2 + 2 r + 2 = 0 ( 7 ) Roots of Equation ( 7 ) are r = − 1 ± i . So, the general solution of Equation ( 6 ) is, q ( t ) = A 1 e − t cos ( t ) + B 1 e − t sin ( t ) Here, A 1 and B 1 are arbitrary constants

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 5.RP, Problem 17RP

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The general solution I1(t), I2(t) and I3(t) for the systems of differential equations.

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Chapter 5.RP, Problem 16RP

Chapter 5.RP, Problem 18RP

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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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Fundamentals of Differential Equations and Boundary Value Problems

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The general solution I 1 ( t ) , I 2 ( t ) and I 3 ( t ) for the systems of differential equations.

Solution Summary: The author explains the three equations for the unknown currents: lI_2R

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