Arbitrage and Pricing: The Arbitrage Pricing Theory (APT)
Apart from mean-variance preference analysis and CAPM pricing, there is another pricing theory. Arbitrage Opportunity In the very beginning of the previous lecture note, we defined a simple hypothetical world of financial assets, in which we have
⎡v1 ⎤ v=⎢ ⎥ ⎢ ⎥ ⎢vn ⎥ ⎣ ⎦
is a vector of prices of individualfinancial assets, and these assets provide payoff of
⎛ Y11 … Y1n ⎞ ⎜ ⎟ Y =⎜ ⎟ ⎜Y Ysn ⎟ ⎝ s1 ⎠
Thus the return of those financial assets could be defined as
⎛ Z11 … Z1n ⎞ ⎜ ⎟ Z = Y ⋅v = ⎜ ⎟ ⎜Z Z sn ⎟ ⎝ s1 ⎠
We defined the weight of individual assets in a portfolio as
$ invested in i-th asset . total $ invested
We define ηi as dollar amount invested in asset i . Consider thefollowing exposure: We invest the following
⎛η1 ⎞ ⎜ ⎟ η =⎜ ⎟ ⎜η ⎟ ⎝ n⎠
The total cost of investment
We know that the payoff of η at state s is defined as
η1Z s1 + η2 Z s 2 + ... + ηn Z sn
Thus we have a payoff vector of
Z snη n
Weak and Strong Arbitrage Define η exposure as an arbitrage if it fulfills either 1) the weak arbitrage condition, i.e.
1Τη ≤ 0
totalinvestment cost is either equal or less than 0; And
Zη > 0 at least one state provides positive return,
2) the strong arbitrage condition, i.e.
1Τη < 0
total investment is strictly negative
Zη ≥ 0
return of the portfolio may be equal to zero.
State Prices & Non-Arbitrage Condition We can use State Prices to identify whether or not a return matrix Z contains arbitrageopportunities. (State Price, however, doesn’t tell where, within Z, the arbitrage opportunity might be.) Definition: We start by assigning a price to each state of occurrence
⎛P ⎞ 1 ⎜ ⎟ Ps = ⎜ ⎟ ⎜P ⎟ ⎝ s⎠
State price, by definition, recovers property price from state contingent payoffs so that
vi = ∑ PYsi s
(in another word, the price for i -th asset is to be the same as the sum of payoff in eachstate adjusted by the state prices), and
vi = ∑ PYsi ⇒ s
or in matrix format:
vi Y = ∑ Ps si ⇒ 1 = ∑ Ps Z si vi vi
v = Y ΤP ⇒ 1 = Z ΤP
Now there is a very important theorem:
There exists a positive state price vector, supporting a return matrix Z, if and only if Z contains no arbitrage opportunity.
Proof: Suppose η = (η1 ,....,ηn ) is an arbitrage. Then by definition of arbitrageabove, we have the following relationship 1Τη ≤ 0 (the cost of portfolio η is at most zero, possibly negative) and Zη ≥ 0 (the payoff of η is at least zero, possibly positive and is strictly positive in at least one state.). Now if P is positive, as assumed by the theorem, then P Τ ( Zη ) > 0 . We also assume that Z Τ P = 1 . Thus P Τ ( Zη ) = ( P Τ Z )η = 1Τη > 0 . It contradicts with the conditionabove under arbitrage situation. It implies that if P is positive, then it must be a non-arbitrage situation. -- end of proof Example: A simple example we have a payoff matrix
⎛ 3,1 ⎞ ⎜ ⎟ Z = ⎜ 2, 2 ⎟ ⎜1,3 ⎟ ⎝ ⎠
Is there an arbitrage opportunity?
⎛P ⎞ 1 ⎛ 3, 2,1⎞ ⎜ ⎟ ⎛1⎞ Z P=⎜ ⎟ ⋅ ⎜ P2 ⎟ = ⎜ ⎟ , ⎝1, 2,3 ⎠ ⎜ ⎟ ⎝1⎠ ⎝ P3 ⎠
we should solve for P , and there is no arbitrage,
P >0 1
ifone solution with P2 > 0 exists.
P3 > 0
Risk neutral Pricing We know we can write the asset price v as the discounted expected future payoffs.
E ( y) (1 + r )
In CAPM, we make risk adjustment to the discount rate In Risk Neutral Pricing, we make risk adjustment to the expected future payoff, more precisely to the probabilities of states.
There is a second property of state prices,which is: If a risk free asset is traded with return R then the sum of the state prices P must always 1 1 equal to . (i.e. P Τ1 = ) R R Proof If portfolio w = ( w1 ,..., wn ) is risk free, then by the fact that w is a proper portfolio
1Τ w = 1 (it simply states wi must add up to 1), and by the fact w is risk free,
Zw = R1 . Thus P Τ ( Zw) = ( P Τ Z ) w = 1Τ w = 1 (by Z Τ P = 1 , the same as...