In this problem, we will derive the pricing kernel in a direct fashion. From linear pricing, we know that for any asset f0=E[ftt], where t is the pricing kernel (or stochastic discounter). In general, it can also be shown that fss=E[fttFs], where Ft represents the information flow up to time t. Using this property, we will derive an SDE for t. (a) Consider the linear pricing formula for the bond and stock (GBM) but over an instantaneous dt. Use this to derive an SDE for t. (Hint: Assume that d=adt+bdz, then try to express a and b with t.) (b) Solve the SDE. Relate your answer to the discounting and Girsanov change of measure that appears in risk-neutral pricing. (Hint: Your t process is of the GBM form with the initial condition 0=1. Use the change of measure argument with market price of risk.) In this problem, we will derive the pricing kernel in a direct fashion. From linear pricing, we know that for any asset f0=E[ftt], where t is the pricing kernel (or stochastic discounter). In general, it can also be shown that fss=E[fttFs], where Ft represents the information flow up to time t. Using this property, we will derive an SDE for t. (a) Consider the linear pricing formula for the bond and stock (GBM) but over an instantaneous dt. Use this to derive an SDE for t. (Hint: Assume that d=adt+bdz, then try to express a and b with t.) (b) Solve the SDE. Relate your answer to the discounting and Girsanov change of measure that appears in risk-neutral pricing. (Hint: Your t process is of the GBM form with the initial condition 0=1. Use the change of measure argument with market price of risk.)