QuestionQuestion

Transcribed TextTranscribed Text

4. Suppose the functions 4,0 : [0,T X R R and ยง : R R are given and we are trying to find a function F : [0,T] X R R which satisfies the following boundary value problem: (*) a) Show that if F(t,x) solves the above problem, then the function G(t,x) e-r(T-t) x) solves the problem: (#) b) Show that if G(t,x) solves prolblem (#), then the function F(t,x) = er(T=t)G(t,r) solves problem (*) . Note. a) shows that the probabilistic method of solving (*) (given by Feynman-Kai Theorem) will work for the equation (#). When (x) = |X - Kl+, the problem (#) is identical with the Black-Scholes-Merton equation for the price of a European call option. 5. Consider the following process Y(t)=&"W(+) Use Ito's Lemma to show that Y(0)=0 = = 6. Consider the standard stock price process dS(t) =pS(t)dt+oS(t)dW(t) = (**) Let c(t,I be a function with continuous partial derivatives and suppose that we want to investigate the process Z(t) = c(t,S(t)). Ito's lemma states that Z(t) is an Ito process, i.e., it has the dynamic: Z(0) = c(0,5 S0) ( dZ(t) = a(t,S(t))dt +b(t,S(t))dW(t) = Using stochastic calculus laws (dt * dt = dW * dt = dt * dW = 0, dW * dW = dt) and Ito's Lemma, find the coefficients a(t,r) and b(t,x). Note: This calculation is a step in derivation of Black-Scholes-Merton equation where o(t, S(t)) is interpreted as the price of a call option.

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:
    Solution.pdf.

    $68.00
    for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Mathematical Finance Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Decision:
    Upload a file
    Continue without uploading

    SUBMIT YOUR HOMEWORK
    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats