Lecture Notes for Finance 1 (and More).
Lecture Notes for Finance 1 (and More).
Lecture Notes for Finance 1 (and More).
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4.2. THE SINGLE PERIOD MODEL 45<br />
Proof. Assume πc < ψ 1c1 + . . . + ψ ScS. Buy the security c <strong>and</strong> sell the<br />
portfolio θc. The price of θc is by assumption higher than πc, so we receive<br />
a positive cash-flow now. The cash-flow at time 1 is 0. Hence there is an<br />
arbitrage opportunity. If πc > ψ 1c1 + . . . + ψ ScS reverse the strategy. �<br />
Definition 22 The market is complete if <strong>for</strong> every y ∈ R S there exists a<br />
θ ∈ R N such that<br />
D ⊤ θ = y<br />
i.e. if the rows of D (the columns of D ⊤ ) span R S .<br />
Proposition 8 If the market is complete <strong>and</strong> arbitrage-free, there exists precisely<br />
one state-price vector ψ.<br />
The proof is exactly as in Chapter 3 <strong>and</strong> we are ready to price new<br />
securities in the financial market; also known as pricing of contingent claim. 2<br />
Here is how it is done in a one-period model: Construct a set of securities<br />
(the D-matrix,) <strong>and</strong> a set of prices. Make sure that (π, D) is arbitrage-free.<br />
Make sure that either<br />
(a) the model is complete, i.e. there are as many linearly independent<br />
securities as there are states<br />
or<br />
(b) the contingent claim we wish to price is redundant given (π, D).<br />
Clearly, (a) implies (b) but not vice versa. (a) is almost always what is<br />
done in practice. Given a contingent claim c = (c1, . . .,cS). Now compute<br />
the price of the contingent claim as<br />
π (c) = 1<br />
R0<br />
E q (c) ≡ 1<br />
R0<br />
S�<br />
i=1<br />
qici<br />
(4.1)<br />
where qi = ψ i<br />
d0 ≡ R0ψ i. The method in Equation (4.1) (<strong>and</strong> the generalizations<br />
of it we’ll meet in Chapters 5 <strong>and</strong> 6) is often called risk-neutral pricing.<br />
Arbitrage-free prices are calculated as discounted expected values (with some<br />
new or artificial probabilities, the q’s), ie. as if agents were risk-neutral. But<br />
the “as if” is important to note. No assumption of actual agent risk-neutrality<br />
is use to derive (4.1), just (slightly implicitly, but spelled out more in the<br />
next section) that they prefer more to less. As a catch-phrase: Risk-neutral<br />
2 A contingent claim just a r<strong>and</strong>om variable describing pay-offs; the pay-off is contingent<br />
on ω. The term (financial) derivative (asset, contract, or security) is largely synonymous,<br />
except that we are usually more specific about the pay-off being contingent on another<br />
financial asset such as a stock. We say option, we have even more specific pay-off structures<br />
in mind.<br />
complete market<br />
contingent claim<br />
risk-neutral<br />
pricing
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