9. The risk-neutral process for the Constant Elasticity of Variance (CEV) model is: dSrSdt + Sad...

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9. The risk-neutral process for the Constant Elasticity of Variance (CEV) model is: dSrSdt + Sad Z 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 = -dS + dt +- at 10V 2 as (ds) Using the rules of stochastic calculus (i.e. dt. dt = 0, dt dZ = 0, (dZ) = dt) substitute for dS and (dS)2 and, a. Show how you get the following expression: av av 1 aV dV = (rS + as at 2 as) dt + Sa av dt +SadZ as Then, using a replicating portfolio with = V AS, b. Derive the corresponding partial differential equation (PDE) which should be: at av 1 + 2 av as +rs rV = 0 c. 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 (i.e., replace the partial derivatives with approximations to the derivatives). d. Explain what forward, backward, and central difference are, and which is used for the time derivative for each of the explicit and implicit methods. 9. The risk-neutral process for the Constant Elasticity of Variance (CEV) model is: dSrSdt + Sad Z 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 = -dS + dt +- at 10V 2 as (ds) Using the rules of stochastic calculus (i.e. dt. dt = 0, dt dZ = 0, (dZ) = dt) substitute for dS and (dS)2 and, a. Show how you get the following expression: av av 1 aV dV = (rS + as at 2 as) dt + Sa av dt +SadZ as Then, using a replicating portfolio with = V AS, b. Derive the corresponding partial differential equation (PDE) which should be: at av 1 + 2 av as +rs rV = 0 c. 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 (i.e., replace the partial derivatives with approximations to the derivatives). d. Explain what forward, backward, and central difference are, and which is used for the time derivative for each of the explicit and implicit methods.