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  3. let btt greaterthanorequalto 0 be brownian motion let xt ...

Question: let btt greaterthanorequalto 0 be brownian motion let xt ...

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7. Let Bi 20 be Brownian motion, let t To exp(Aut o Bt) be the stock price model ttt (where o 0), and let D e rtXt be the discounted stock price. (a) [4] Show that if Au 3r then (D) is a martingale. Hint: Dont forget the previous question.] (b) (6) Show that if Au r- then (r )S log(K/Xo) log(K/Xo) Elle max 0, As-K) o GP rS where (u Juoo t2/2 dv is the cdf of a standard normal distribution. Hint: Write the expectation as an integral with respect to the density function for Bs. Then, break up the integral into the part where Xs K 20 and the part where Xs K <0.] [Note: This is the famous Black-Scholes formula. You do not need to memorise it.]
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