Weak-Star Closure in the Fundamental Theorem of Asset Pricing

# No free Lunch and weak-star topology


# No free Lunch and weak-star topology












The no free lunch is stated as follows

What is the significance of the weak-star topology here .Also as far as I understand the weak-star topology is defined on the dual of a Banach space.So what is the space under consideration here

## Answer by Kevin (score 5, accepted)

https://quant.stackexchange.com/a/66550

The context of weak$^*$ topologies and no free lunch is often the proof of the first fundamental theorem of asset pricing. All the ideas below are from Delbaen and Schachermayer (1994).

### Notation

Suppose the price process is a semimartingale $S$. Let $K_0$ represent the space of all claims generated by admissible trading strategies (self-financing and zero initial cost). One can show that $K_0$ is a convex cone in $L^0$.

Let $C_0=K_0-L_+^0$. Thus, $C_0$ contains worse trading strategies than $K_0$ (the elements of $C_0$ are dominated by elements of $K_0$). Because $L_+^0$ contains $\{0\}$, $C_0$ also contains the claims from $K_0$. We restrict ourselves to bounded claims by setting $C=C_0\cap L^\infty$.

Summary: $C$ contains all bounded payoffs that can be replicated when trading $S$ (or worse claims).

Note: $L_+^0$ and $C$ are convex sets.

For more fun, let's introduce more notation:

- The set $\bar{C}$ is the closure of $C$ with respect to the norm topology of $L^\infty$,

- The set $\bar{C}^*$ is the weak$^*$ closure of $C$,

- The set $\tilde{C}$ contains all limits of weak$^*$ converging sequences of elements of $C$.

### What do we actually want to do? Standard Separation Argument

How does one prove the implication no arbitrage $\Rightarrow$ EMM exists in simple discrete settings? We find a pricing kernel (= EMM) that aligns with the no-arbitrage consideration that the elements of $L_+^0$ have positive prices and that the payoffs in $C$ have negative prices:

Suppose $L_+^0$ and $C$ only intersect in $0$ (this is the no-arbitrage condition). Then, there should exist a random variable $M\in L^1$ that separates the two subspaces, i.e., $\mathbb{E}^\mathbb{P}[MX]>0$ for $X\in L_+^0$ and $\mathbb{E}^\mathbb{P}[MX]<0$ for $X\in C$. Any such random variable $M$ is a SDF and defines an EMM via $\text{d}\mathbb{Q}=M\text{d}\mathbb{P}$

This works nicely in a discrete setting. However, in this more general setting, one needs a stronger version of a separation theorem that ensures that $M$ is integrable and strictly positive. So, all what follows does not really provide any further economic insights, it's just pure maths technicalities.

### No Arbitrage Definitions

Let's define different ways of prohibiting people getting rich:

- No Free Lunch (NFL) means $\bar{C}^*\cap L^\infty_+=\{0\}$

- No Free Lunch with Bounded Risk (NFLBR) means $\tilde{C}\cap L^\infty_+=\{0\}$

- No Free Lunch with Vanishing Risk (NFLVR) means $\bar{C}\cap L^\infty_+=\{0\}$

- No-Arbitrage (NA) means $C\cap L^\infty_+=\{0\}$

The definitions get increasingly more restrictive: $$NFL \Rightarrow NFLBR \Rightarrow NFLVR \Rightarrow NA$$

Because $C\cap L^\infty_+=\{0\} \Leftrightarrow K_0\cap L^\infty_+=\{0\}$, no-arbitrage means that trading admissible strategies alone must not yield a positive profit. However, this condition is too restrictive for an EMM to exist.

NFLVR generalises (it more restrictive than) NA. There is a sequence of elements in $C$, call it $(f_n)$ which converges almost surely to some $f_0\in L_+^0$. We thus have a sequence of attainable claims which converge to a positive payoff. ``In economic terms this amounts to almost the same thing as (NA), as the risk of the trading strategies becomes arbitrarily small.''

The difference between NFLVR and NFLBR is that NFLVR requires the risk of trading strategies to uniformly converge to zero, whereas NFLBR requires the risk to be bounded and the negative bound to tend to zero in probability. NFLBR has also been used in other earlier papers, such as Delbaen (1992) and Schachermayer (1994).

### Back to Separation Argument

Delbaen and Schachermayer (1994, Section 4) prove that for uniform bounded price processes $S$, the following holds: $S$ satisfies NFLVR $\Rightarrow$ $C$ is weak$^*$ closed in $L^\infty$. They can then apply the Kreps-Yan separation theorem which ensures that $M$ is integrable and strictly positive. This separation theorem extends the Hahn–Banach theorem and is due to Schachermayer (1994). However, this theorem precisely requires weak$^*$ closedness.

To summarise, the recipe is: assume NFLVR to get weak$^*$ closedness which in turn gives separation between $C$ and $L^0_+$ by a positive integrable SDF which defines the EMM. Thus, the weak$^*$ requirement, just like NFLVR condition, are just the result of what the maths requires. Economically, the concepts are close to the usual no-arbitrage condition.

### Final Result

The first FTAP in all its beauty:

- Let $S$ be a bounded, real-valued semimartingale. Then, $S$ satisfies NFLVR iff there exists an EMM for $S$.

- Let $S$ be a locally bounded, real-valued semimartingale. Then, $S$ satisfies NFLVR iff there exists an ELMM for $S$.

- Let $S$ be an arbitrary real-valued semimartingale. Then, $S$ satisfies NFLVR iff there exists an ESMM for $S$.