Results 1 to 2 of 2

Thread: Derivatives and Securities

  1. #1
    Member
    Joined
    Aug 2007
    Posts
    239

    Derivatives and Securities

    Consider a derivative that pays off $\displaystyle S_{T}^{n} $ at time $\displaystyle T $, where $\displaystyle S_T $ is the stock price at that time. When the stock price follows geometric Brownian motion, it can be shown that is price at time $\displaystyle t $ ($\displaystyle t \leq T $) has the form $\displaystyle h(t,T)S^{n} $ where $\displaystyle S $ is the stock price at time $\displaystyle t $ and $\displaystyle h $ is a function only of $\displaystyle t $ and $\displaystyle T $.

    (a) Derive and ordinary differential equation satisfied by $\displaystyle h(t,T) $. I am guessing that I should use Black-Scholes here?

    (b) What is the boundary condition for the differential equation for $\displaystyle h(t,T) $?

    (c) Show that $\displaystyle h(t,T) = e^{[0.5 \sigma^{2} n(n-1) +r(n-1)](T-t)} $ where $\displaystyle r $ is the risk free interest rate and $\displaystyle \sigma $ is the stock price volatility. This need part (a) right?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    GAMMA Mathematics
    colby2152's Avatar
    Joined
    Nov 2007
    From
    Alexandria, VA
    Posts
    1,172
    Thanks
    1
    Awards
    1
    Quote Originally Posted by shilz222 View Post
    Consider a derivative that pays off $\displaystyle S_{T}^{n} $ at time $\displaystyle T $, where $\displaystyle S_T $ is the stock price at that time. When the stock price follows geometric Brownian motion, it can be shown that is price at time $\displaystyle t $ ($\displaystyle t \leq T $) has the form $\displaystyle h(t,T)S^{n} $ where $\displaystyle S $ is the stock price at time $\displaystyle t $ and $\displaystyle h $ is a function only of $\displaystyle t $ and $\displaystyle T $.

    (a) Derive and ordinary differential equation satisfied by $\displaystyle h(t,T) $. I am guessing that I should use Black-Scholes here?

    (b) What is the boundary condition for the differential equation for $\displaystyle h(t,T) $?

    (c) Show that $\displaystyle h(t,T) = e^{[0.5 \sigma^{2} n(n-1) +r(n-1)](T-t)} $ where $\displaystyle r $ is the risk free interest rate and $\displaystyle \sigma $ is the stock price volatility. This need part (a) right?
    Which Black-Scholes equation would you be using here? The stochastic difeq that is used for geometric Brownian motion is: $\displaystyle \frac{dS(t)}{S(t)}=\alpha dt+\sigma dZ(t)$ where $\displaystyle \alpha$ is the drift factor and $\displaystyle \sigma$ is the volatility or variance factor. That exponential function is derived from the differential equation.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Derivatives and Anti-Derivatives
    Posted in the Calculus Forum
    Replies: 7
    Last Post: Feb 6th 2011, 06:21 AM
  2. Replies: 1
    Last Post: Jul 19th 2010, 04:09 PM
  3. Two derivatives
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Apr 12th 2010, 04:14 AM
  4. Replies: 4
    Last Post: Feb 10th 2009, 09:54 PM
  5. Trig derivatives/anti-derivatives
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Feb 10th 2009, 01:34 PM

Search Tags


/mathhelpforum @mathhelpforum