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Math Help - Derivatives and Securities

  1. #1
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    Derivatives and Securities

    Consider a derivative that pays off  S_{T}^{n} at time  T , where  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  t (  t \leq T ) has the form  h(t,T)S^{n} where  S is the stock price at time  t and  h is a function only of  t and  T .

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

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

    (c) Show that  h(t,T) = e^{[0.5 \sigma^{2} n(n-1) +r(n-1)](T-t)} where  r is the risk free interest rate and  \sigma is the stock price volatility. This need part (a) right?
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  2. #2
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    Quote Originally Posted by shilz222 View Post
    Consider a derivative that pays off  S_{T}^{n} at time  T , where  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  t (  t \leq T ) has the form  h(t,T)S^{n} where  S is the stock price at time  t and  h is a function only of  t and  T .

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

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

    (c) Show that  h(t,T) = e^{[0.5 \sigma^{2} n(n-1) +r(n-1)](T-t)} where  r is the risk free interest rate and  \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: \frac{dS(t)}{S(t)}=\alpha dt+\sigma dZ(t) where \alpha is the drift factor and \sigma is the volatility or variance factor. That exponential function is derived from the differential equation.
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