Question

4. (10 marks) (Black-Scholes) Fill in the details of L9.24. Specifically, a. (3 marks) Assume that C(S, t) satisfies the first PDE of L9.24 (Black-Scholes). Show that U(z,T) satisfies the second PDE of L9.24 (the heat/diffusion equation b. (5 marks) Using L9.23, solve for U Write your final answer using the standard normal CDF N Hint: Replace the lower limit of integration (the -o) with a correctly chosen quantity that lets you get rid of the "positive-part function" in the integrand, thus replacing + with (..). Then split the integral into two integrals. Introduce a new variable of integration x into each integral. In the first integral, let x := (く--σ2r)/(ovF); in the second integral, let x := (Ç-)/(oVF). Also have a look at Question 3 on tutorial week 9. c. (2 marks) Transform the solution U(z, T) back to find C(S, t).

Answer 1

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