Question

II. Arbitrage Suppose the economy can be in one of the following three states: (i) Boom or “good” state, (ii) Neutral state, and (iii) Recession or “bad” state. Each state can occur with an equal probability. There are three securities available in the economy: A, B, C. The net payoffs of these securities are as follows: • Security A: at the end of the year, the security is expected to yield a net payoff of $30 in the good state, $10 in the neutral state, and -$10 in the bad state. • Security B: at the end of the year, the security is expected to yield a net payoff of -$10 in the good state, $10 in the neutral state, and $30 in the bad state. • Security C: at the end of the year, the security is expected to yield a net payoff of $35 in the good state, $30 in the neutral state, and $25 in the bad state. The current prices of these three securities, A, B, and C, are $10, $10, and $20, respectively.

1. Relative to the fundamental values, which of the three securities (A, B, and C) are fairly priced? Which securities are under-priced? Which securities are over-priced? For simplicity, please ignore the discount rate. (Hint: Use Net Payoffs to calculate the asset prices)

2. Construct an arbitrage portfolio using these securities, which yields a positive payoff in each state. Mention clearly which securities (and their quantities) you would long and/or short. Calculate the net payoffs of the arbitrage portfolio in the three states.

Answer 1

1. If Current Price is > Intrinsic price, security is over priced and vice versa. If both are same, it is fairly priced.

Security A
	Probability	Payoff	Value
	A	B	C = A * B
Boom	0.333	30	9.99
Neutral	0.333	10	3.33
Recession	0.333	-10	-3.33
Intrinsic Value			9.99
Current Price			10
Valuation			Fairly Priced

Security B
	Probability	Payoff	Value
	A	B	C = A * B
Boom	0.333	-10	-3.33
Neutral	0.333	10	3.33
Recession	0.333	30	9.99
Intrinsic Value			9.99
Current Price			10
Valuation			Fairly Priced

Security C
	Probability	Payoff	Value
	A	B	C = A * B
Boom	0.333	35	11.655
Neutral	0.333	30	9.99
Recession	0.333	25	8.325
Intrinsic Value			29.97
Current Price			10
Valuation			Under Priced

2. For Arbitrage strategy, we will short the security which have lower payoff than the security having maximum payoff. Quantity will be 1

We will go long the security which has maximum payoff. Quantity will be 2 invest the fund received from shorting 2 above securities

	Strategy
	A	B	C	Total
Boom	Short 1	Short 1	Long 2	16.65
Neutral	Short 1	Short 1	Long 2	13.32
Recession	Short 1	Long 2	Short 1	14.985

	Payoff
	A	B	C	Net Payoff
Boom	-9.99	3.33	23.31	16.65
Neutral	-3.33	-3.33	19.98	13.32
Recession	3.33	19.98	-8.325	14.985

As seen above, no investment is required to form above portfoilio since amount received from shorting 2 securities is investing in going long 1 securitie with 2 quantity. Profit from strategy is also shown above.