Consider 3 assets. Asset 1 is a risk-free bond which has normalized price of unity at time t=0. Assume the risk-free rate of interest is 5% per year. The price of Asset 2 is $50 at time t=0. In addition, it is estimated that, if the economy is in an “upturn”, the price of Asset 2 would be $55, while it would be $45 if the economy is in a “downturn”. The price of Asset 3 is unknown at time t=0 and it is $3 if the economy is in an “upturn”, while it is $2 if the economy is in a “downturn”. Use the non-arbitrage theorem to derive the equations that the price of Asset 3 at time t=0 has to satisfy in order to be an arbitrage-free price. (Note: you do not have to solve the equations.)   

Let's replicate the Asset 3 by a combination of Asset 1 and Asset 2.

Let's say Asset 3 can be fully replicated by N1 units of Asset 1 and N2 units of asset 2.

hence, Asset 3 = N1 x Asset 1 + N2 x Asset 2

If we say the price of asset 3 at t = 0 is P then

At t = 0,

Price of Asset 3 = N1 x Price of asset 1 + N2 x Price of asset 2

Hence, P = N1 x 1 + N2 x 50

Or, P = N1 + 50N2

If the economy is in an upturn:

Price of Asset 3 = N1 x Price of asset 1 + N2 x Price of asset 2

hence, 3 = N1 x 1 + N2 x 55

hence, N1 + 55N2 = 3

If the economy is in a downturn:

Price of Asset 3 = N1 x Price of asset 1 + N2 x Price of asset 2

hence, 2 = N1 x 1 + N2 x 45

hence, N1 + 45N2 = 2

Hence,  the equations that the price of Asset 3 at time t = 0, denoted by P,  has to satisfy in order to be an arbitrage-free price are:

P = N1 + 50N2

such that,

N1 + 55N2 = 3 and

N1 + 45N2 = 2