Question

No arbitrage pricing Consider the following two-period setting. There are 3 aseta A, B, C with respectivo payoffs EA, 1B, re while there are three possible future states w (1, 2, 3). Cuent asset prices are respectively A).Qac) 11/22 1 21/6 2a 3 1/3 2 21 Price in S 15 21 Construct risk-free bond in this cconomy and compute its price, following which compute risk free rate in this economy

Answer 1

Q - 4

Please observe the payoff table. A risk free situation will be when returns from the portfolio is the same in either of the three states. Let's say we create a portfolio comprising of N_A, N_B and N_C numbers of three assets A, B & C respectively.

Payoff from portfolio in state 1 = P₁ = (N_A, N_B, N_C ) . (1, 2, 1) = N_A+2N_B + N_C

Payoff from portfolio in state 2 = P₂ = (N_A, N_B, N_C ) . (2, 1, 3) = 2N_A+ N_B + 3N_C

Payoff from portfolio in state 3 = P₃ = (N_A, N_B, N_C ) . (2, 2, 1) = 2N_A+ 2N_B + N_C

Let's put P₂ = P₃

2N_A+ N_B + 3N_C = 2N_A+ 2N_B + N_C

Hence, N_B = 2N_C

Let's put P₁ = P₃

N_A+2N_B + N_C = 2N_A+ 2N_B + N_C

Hence, N_A = 0

So, our portfolio is 0 nos. of A, 2 nos. of B and 1 no. of C

State 1: Payoff from the portfolio, P₁ = (0, 2, 1) . (1, 2, 1) = 5

State 2: Payoff from the portfolio, P₂ = (0, 2, 1) . (2, 1, 3) = 5

State 3: Payoff from the portfolio, P₃ = (0, 2, 1) . (2, 2, 1) = 5

Thus this portfolio pays off 5 irrespective of the state. Thus this portfolio is a risk free portfolio.

Hence, the risk free bond is: 2 units of B + 1 unit of C

Price today = Further investment required to create this portfolio = 2 x q(x_B) + 1 x q(x_C) = 2 x 2 + 1 x 1 = 5

Risk free rate = (Future payoff - Initial investment) / Initial investment = (5 - 5) / 5 = 0%