The risk-neutral process for the Constant Elasticity of Variance (CEV) model is:

dS=rSdt+S^ dZ

Assume you have a call option (V) that is a function of the spot (stock price S) and time (t), Ito tells us how the derivative evolves:

dV=V/S dS+V/t dt+1/2(^2 V)/(S^2 ) (dS)^2

Using the rules of stochastic calculus (i.e. dtdt=0,dtdZ=0,(dZ)^2=dt) substitute for dS and (dS)^2 and show how you get the following expression (hint: see Extra Notes 2, under Syllabus link) (5 points out of 15 points total):

dV=(rS V/S+V/t+1/2 ^2 S^2(^2 V)/(S^2 ))dt+S^V/S dZ

Then, using a replicating portfolio with =V-S (see class notes, page 76), derive the corresponding partial differential equation (PDE) which should be (5 points out of 15 points total):

V/t+1/2 ^2 S^2(^2 V)/(S^2 )+rS V/S-rV=0

Using a differencing scheme, replace the corresponding derivatives with appropriate approximations to write down the discretized equations for the explicit finite difference method and the implicit finite difference method (see the article Introduction to the Numerical Solution of Partial Differential Equations in Finance, Monk (2007), this was one of the readings for the final). Be sure to explain what forward, backward, and central difference are, and which is used for the time derivative for each of the explicit and implicit methods (3 points out of 15 points total).

Finally, how would you determine the values for ,and in using the CEV model to value derivatives? Write down the mathematical expression including your decision variables