Consider a one-period market model with initial stock price So, initial bond price Ao, bond return...

  

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Consider a one-period market model with initial stock price So, initial bond price Ao, bond return r. The terminal stock price ST is a random variable with values in (0, +∞) with expected value > 0 and variance o² > 0. Because of this assumption, the market model is not necessarily binomial but it covers the one-period binomial model as a special case. In this market, consider a portfolio (x, y) that consists of a shares of stock and y bonds. a. (5) What are the expected value and the standard deviation of the value of (x, y) at terminal time T? The expected value of the return of a portfolio is called its expected return. The stan- dard deviation of this return is called the risk of the portfolio. b. (5) What are the expected return and the risk of (x, y)? c. (5) Consider all portfolios (x, y) with a fixed initial value Vo> 0. Calculate the expected returns and risks of these portfolios as functions of a (but not y). Does the expected return increase with a? When? How about the risk? Now, suppose that ST takes values in the set {u₁ So, ...,uK So}, where u₁, ..., uk are distinct and strictly positive factors and K≥ 2 is an integer. For each k = {1,..., K}, let Pk P {ST = uk So}, = which is assumed to be strictly positive. Note that k-1pk = 1. K d. (5) Show that K K μ = So Epiuk, σ' = 5% Σpu k=1 k=1 Pkuk Pe(uk-ue) lzk e. (5) Take K = 2 with u₁ = u, u2 = d, p1 = P, P2 = 1 - p. Show that μ = [pu + (1-P)d] So, ²p(1-p) (u - d)² S². Consider a one-period market model with initial stock price So, initial bond price Ao, bond return r. The terminal stock price ST is a random variable with values in (0, +∞) with expected value > 0 and variance o² > 0. Because of this assumption, the market model is not necessarily binomial but it covers the one-period binomial model as a special case. In this market, consider a portfolio (x, y) that consists of a shares of stock and y bonds. a. (5) What are the expected value and the standard deviation of the value of (x, y) at terminal time T? The expected value of the return of a portfolio is called its expected return. The stan- dard deviation of this return is called the risk of the portfolio. b. (5) What are the expected return and the risk of (x, y)? c. (5) Consider all portfolios (x, y) with a fixed initial value Vo> 0. Calculate the expected returns and risks of these portfolios as functions of a (but not y). Does the expected return increase with a? When? How about the risk? Now, suppose that ST takes values in the set {u₁ So, ...,uK So}, where u₁, ..., uk are distinct and strictly positive factors and K≥ 2 is an integer. For each k = {1,..., K}, let Pk P {ST = uk So}, = which is assumed to be strictly positive. Note that k-1pk = 1. K d. (5) Show that K K μ = So Epiuk, σ' = 5% Σpu k=1 k=1 Pkuk Pe(uk-ue) lzk e. (5) Take K = 2 with u₁ = u, u2 = d, p1 = P, P2 = 1 - p. Show that μ = [pu + (1-P)d] So, ²p(1-p) (u - d)² S².

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SOLUTION a To find the expected value and standard deviation of the portfolio x y at terminal time T we need to consider the expected value and varian... View the full answer