What is the Gamma of a portfolio of 100 long puts and how can we Delta-Gamma hedge with the following parameterization? ୮ 1 1 = e Soo√T √√2π (So = 100, K = 110, T = 2,σ = 0.10, r = 0.05)

There is one period. Assume a representative agent with utility function U(ct) = αc_t − βc^2_tassume the following: α = 100, β = 1, and δ = 0.97. Consumption at t = 0 is C0 = 24. At t = 1 one of two states θ1 and θ2 eventuate with probability π1 = 0.5, and π2 = 0.5,respectively. There are two complex securities s^1 and s^2.s^1 has a payoff of 23 in θ1 and 27 in θ2.s^2 has a payoff of 20 in θ1 and 32 in θ2.What is the stochastic discount factor mt+1? hint: Recall mt+1 =δU′(ct+1)/U′(ct)

4. Suppose that there are 2 assets with ri 012 = 0.005. = 0.20, 01 = = 0.40, 2 = 0.10, 02 = 0.25 and (a) If ro = 0.02, what are the market portfolio return and variance? What are the corre- sponding weights (i.e. how much to invest in asset 1, asset 2, and the risk-free asset to get the market portfolio)? Answer. (b) If ro 0.05, what are the market portfolio return and variance? What are the corre- sponding weights? Answer.

Use the following CAPM equation for a portfolio to answer the questions that follow:E(RP) = RF + βP (RM – RF) = 1 + 0.8 (5 – 1) = 4.2% a) Is the portfolio defensive or aggressive. Why? b) If the actual portfolio return is 6%, what is the portfolio’s alpha?

Based on the following probability distribution, what is the security’s expected return? State         Probability                       r     1                 0.2                        –5.0% 2                 0.4                        10.0 3                 0.4                        30.0

Hi there, I am trying to figure out how to calculate 5 data points so I can plot the feasible set of portfolios for A, B and C on a graph. I am unsure of the process to calculate the data points.   ρ23 is +1.0 Thank you

Consider a single-index model economy. The index portfolio M has E(RM ) = 6%, σM = 18%.An individual asset i has an estimate of βi = 1.1 and σ2ei = 0.0225 using the single index modelRi = αi + βiRM + ei. The forecast of asset i’s return is E(ri) = 12%. rf = 4%. a) According to asset i’s return forecast, calculate αi.  (b) Calculate the optimal weight of combining asset i and the index portfolio M .  (c) Calculate the Sharpe ratio of the index portfolio M and the portfolio optimally combiningasset i and the index portfolio M .

The possible returns of a security I and research returns under three possible states are as follows. Probability % market % security 0.2 15 10 0.5 13 16 0.3 25 30 The risk free rate is 9%, determine the required rate of return of security I and sate whether it is correctly valued. 12-marks] 20 Ri= E(R) XW

The Black-Scholes equation for the value of an option V(S,t), where S is the value of the underlying asset and t is time, can be transformed into the diffusion equation for a new variable u(x,7) using S = Ee™, t = T − 27/0², k = 2r/o² and - (k-1) x (k+1)² + T V=Eu(x,T) exp 2 4 where r is the risk-free interest rate, σ is the volatility, E is the strike price, and T is the expiry date. The diffusion equation has the solution u(x,T): = 1 TT or L 2√Tπ 1 uo (s) exp 4T : (x - s)²) ds. Determine, showing all steps, the function uo(s) for a European asset- or-nothing call with the pay-off A(S) = SH(S – E). -

Assume that security returns are generated by the single-index model,Ri = αi + βiRM + eiwhere Ri is the excess return for security i and RM is the market’s excess return. The risk-free rate is 2%. Suppose also that there are three securities A, B, and C, characterized by the following data:  Security βi E(Ri) σ(ei) A 0.7 7 % 20 % B 0.9 9   6   C 1.1 11   15     a. If σM = 16%, calculate the variance of returns of securities A, B, and C.   b. Now assume that there are an infinite number of assets with return characteristics identical to those of A, B, and C, respectively. What will be the mean and variance of excess returns for securities A, B, and C? (Enter the variance answers as a percent squared and mean as a percentage. Do not round intermediate calculations. Round your answers to the nearest whole number.)