NEED HELP WITH 2B AND 2C PLEASE DO NOT COPY OTHER QUESTION'S ANSWERS THEY ARE INCORRECT

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Question 2 2. (Black-Scholes) Fill in the details of L9.24. Specifically, a. 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. Using L9.23, solve for U. Write your final answer using the standard normal cumulative distribution function N(u, o).. Hint: Replace the lower limit of integration (the -) 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 := (5 - 2 - )/(0 V ); in the second integral, let x := (5 z)/(o V ). Also have a look at Question 3 on the tutorial where the normal density function is discussed. c. Transform the solution U(z, 1 ) back to find C(S, t). L9.23 Intuition of the diffusion kernel 1 The (z 5)2 ] exp V27027 2027 S is sometimes called the diffusion kernel" or "fundamental solution or Greens function for the diffusion PDE. Intuition: op{- 17 { This is the Normal(5,02T) density. Regard the initial unit mass at as many many particles each following a Brownian motion. Alternative derivation: let 5 = 0. Note that if U(2, 1) solves the PDE, then so does U(cz, c-7). So look for a solution U(z,T) = otaul). Plug into PDE to produce an ODE for u, and solve. Question 2 2. (Black-Scholes) Fill in the details of L9.24. Specifically, a. 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. Using L9.23, solve for U. Write your final answer using the standard normal cumulative distribution function N(u, o).. Hint: Replace the lower limit of integration (the -) 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 := (5 - 2 - )/(0 V ); in the second integral, let x := (5 z)/(o V ). Also have a look at Question 3 on the tutorial where the normal density function is discussed. c. Transform the solution U(z, 1 ) back to find C(S, t). L9.23 Intuition of the diffusion kernel 1 The (z 5)2 ] exp V27027 2027 S is sometimes called the diffusion kernel" or "fundamental solution or Greens function for the diffusion PDE. Intuition: op{- 17 { This is the Normal(5,02T) density. Regard the initial unit mass at as many many particles each following a Brownian motion. Alternative derivation: let 5 = 0. Note that if U(2, 1) solves the PDE, then so does U(cz, c-7). So look for a solution U(z,T) = otaul). Plug into PDE to produce an ODE for u, and solve