Financial markets are important since they allow fort wo main activities – two faces of the same coin. These are betting and hedging/insuring.

The question we have to ask ourselves is whether it is always possible to perfectly hedge everything. Can we hedge specific future cash-flows?

We define cash-flow at t=1 as contingent claim X which is any vector in ℝ^{K} such that

This contingent claim can be perfectly hedged (replicated) when there exists a trading strategy θ such that

V_{θ}(1)(w_{k}) = X(w^{k}), k=1,…,K

This equation shows that at t= 1 we will have the same value of assets invested in the contingent claim or the or the replicating strategy. With a little bit of algebra this condition leads to the following system:

If this system has solutions, we say that the market is complete.

Thus a market is complete if any contingent claim X can be perfectly hedged i.e. for any vector X in ℝ^{K} there exists a strategy θ such that

Aθ = X

Equivalently a one-period market is complete if and only if there are as many linearly independent securities as states and which is equivalent to saying that

Rank(A) = K

This leads us to the second fundamental theorem of finance

**2 ^{nd} Fundamental Theorem of Finance**

The following statements are equivalent

- The market is both arbitrage-free and complete
- There exists one and only one state-price vector
- There exists one and only one risk-neutral probability.

__Pricing by No-arbitrage__

In order to price securities, we start again in a one period market with N risky securities. This is called the pre-existing market. We assume that the pre-existing market is arbitrage free. Then, an additional security is added and it is our goal to find the right price for this security.

The payoff of this new security at t= 1 is contingent claim X such that

What is the rational price of this contingent claim X today i.e. S_{x}(0)=?

There are two cases which have to be differentiated.

**Case 1: The payoff promised by the new security can be perfectly hedged i.e. the new security is redundant and the market is complete.**

There exists some trading strategy θ_{x} that employs the pre-existing securities such that

Aθ_{x} = X

Then the following three propositions are then equivalent

- No arbitrage holds also in the extended market
- S
_{x}(0) = Vθ_{x}(0) for any strategy θ_{x}such that Aθ_{x}= X. This says that the price of the contingent claim X is equal to the price of the strategy replicating this contingent claim. If the two prices differed there would be a violation of the law of one price and thus arbitrage opportunities would exist. - S
_{x}(0) = . This says that the price of the contingent claim is equal to the discounted expected cash flow tomorrow (discounted by the risk-neutral probability Q)

**Case 2: The new payoff cannot be perfectly hedged i.e the new security is non-redundant. **

Given a contingent claim X, we say that a trading strategy θ in the initial market super- replicates X if V_{θ}(1)(w_{k})=> X(w_{k}) for k = 1,…,K . The value process of the strategy at t = 1 is equal to or larger than the value of the contingent claim in every possible scenario tomorrow. This leads to the following result.

If the new security is non-redundant, these three conditions are equivalent:

- No-arbitrage holds in the extended market.
- MAXδ – Vθ(0) < Sx(0) < MINδ Vθ(0). The maximum is obtained from all market strategies at t=0 that super-replicate -X and the minimum from all strategies at t= 1 that super-replicate X
- INF(1/(1+r)EQ(X) < Sx(0) < SUPQ 1/(1+r)EQ(X). Supremum and infimum are obtained from calculations over risk-neutral probabilities related to the initial market.

This case 2 got pretty complicated but we can conclude.

- The t= 0 prices of the new securities in a no-arbitrage case have to be greater than the maximum amount obtainable selling at t= 0 a portfolio of securities that entails a t = 1 liability at most equal to the payoff of the new security.
- The t= 0 prices of the new security must be lower than the minimum cost incurred to super-replicate at t = 0 the t = 1 payoff of the new security.

Are your heads exploding as well?

Let´s see where this brings us.

Best,

Nils