Chapter 7 No arbitrage and pricing theory
7.1 Fundamental Theorem of Asset Pricing
7.1.1 Framework
This section owes a lot to Schachermayer (n.d.). Let \(X = L^p(\Omega,\mathscr F,\mathbb P)\), where \(1 ≤ p≤\infty\) and \((\Omega,\mathscr F, (\mathscr F_t)_{0≤t≤T} ,\mathbb P)\) is the underlying filtered probability space.
In an ideal perfect market, a set of final wealth can be achieved by actively trading In the context of a stock price process \(S = (S_t)_{0≤t≤T}\)
APT is about extrapolating the price of some assets to large variety of different others. How can this be done?
A key ingredient is first to understand the power of dynamic trading. Imagine a situation where \(S_t\) is non random. You can implement a trading strategy defined by a function \(f\) which consists in holding \(h(S_t)\) stock at date \(t\). Final wealth will be \(X_T-X_0=\int_0^T f(s)ds\) i.e. the integral on the trajectory of prices. Obviously, this generates a final value which is non path dependent: \(X_T-X_0=F(S_T)-F(S_0)\) where \(F\) is the primitive function of \(f\).
Stochastic calculus allows to generalize such intuition to general case where the price process is a, let say, a diffusion. In that case, \[\begin{equation} X_T = X_0 +\int_0^TH_tdS_t \tag{7.1} \end{equation}\]
where \(H\) is an admissible trading strategy. This notion
APT is very much related to the idea that a lot of payoffs can be obtained from an existing price “dynamic replication” which is not so intuitive. The precise admissibility conditions which should be imposed on the stochastic integral (7.1), in order to make sense both mathematically as well as economically, are a subtle issue. Much of the early literature on the Fundamental Theorem of Asset Pricing struggled exactly with this question.
Let M be the set of claims that can be attained by a dynamic trading self financing strategy. It seems natural that the price of those claims shall be the initial amount of money needed, i.e. \(x\). If we restrict ourselves to linear pricing, we obtain that the pricing functional shall be represented (in Riesz terms) by an element \(g\in X^*\) i.e. an element of\(L^q\) where \(q\) and \(p\) are conjugate: \[\pi(X)=\mathbb E(gX)\]
Suppose \(\pi(1)\) is finite and positive and define the measure \(\frac{d\mathbb Q}{d\mathbb P}= g/g(1)\). One can write the pricing functional: \[\pi(X)=\pi(1)\times\mathbb E^ \mathbb Q(X)\]
7.1.2 Theorem
Let Let \(p_0\) the set of observed prices. In particular, \(\forall i, p_0(S^i)=S_0^i.\). Let \(M_0 = \{m \in M : \pi(m) = 0\}\)
Definition 7.1 (No Arbitrage) Let \(X^*_+\) the strictly positive orthant of X. \(m\in X^*_+\) iff:
- \(\mathbb m\geq 0 \mathbb P-a.s.\)
- \(\mathbb P [m>0]>0\)
An arbitrage is an element which would belong both to \(M_0\) and \(X^*_+\). So the No arbitrage condition is equivalent to \[M_0 \cap X^*_+=\varnothing\].
The fact that \(M_0\) and \(X^*_+\) do not intersect might allow to separate them under some extra conditions. Those conditions are some conditions that ensure some version of the Hahn Banach theorem to hold. Unfortunately, those conditions involve some closeness and boundedness properties which none sets enjoy. This explains why the corresponding literature is so involved mathematically. See Delbaen and Schachermayer (2006) for a review.
In reality, the No Arbitrage condition is not strong enough and one has to impose stronger condition such as “No Free Lunch” or “No Free Lunch with Vanishing Risk”. The No Free lunch condition of Harrison and Kreps (1979) is defined in the following manner.
Definition 7.2 (No Free Lunch) Let \(A=\overline{M_0-X_+}\). A Free Lunch is a claim belonging both to \(A\) and \(X_+\).
The economic interpretation of the “no free lunch” condition is a sharpening of the “no arbitrage condition”. A free lunch is a claim in \(X^*_+\) which can be approximated by claim of the form \(m_i-h_i\) i.e. claim which are the difference between null price claim and positive claim. This that agents are allowed to “throw away money”, i.e. to abandon a positive element.
A being convex can be separated it from any single non null element \(h\) of \(X^+\) by \(\pi_h\). So \(\pi_h(h)>0\) and \(\pi_h|_A\leq0\).
To be continued
7.1.3 Risk Neutral Probability
The pricing measure being positive it is sometimes called the “Risk Neutral Probability”. Since the price of asset S is \(S_0\), any Risk Neutral Probability shall ensure that:
\[ \forall t: \mathbb E ^\mathbb Q (S_t)=S_0B^{-1}(0,t) \]
In other terms, the expected value of the spot under the risk neutral probability must be equal to the compounding factor. The historical expected returns doesn’t have any impact.
7.1.4 Completeness
While the risk-neutral probability is not the historically estimated probability, it reflects to some degree the average expectations of the market. When the NFLVR condition is formulated, the existence of that probability is certain, but not its uniqueness. The fact that several risk-neutral probabilities may exist would imply that the same option can have several prices. In this case, we use the term “incomplete market”. But when only a single risk neutral probability exists, the market is complete.
In an incomplete market, there are several risk neutral probabilities, meaning that several prices are possible. Individual preferences, which had disappeared in the pricing process, now resurface. Perfect duplication via a self-financed strategy is no longer possible: option profiles are no longer “attainable”. At best, they can be neared — in which case the meaning of “near” becomes highly subjective. We can envisage two pricing techniques, which are generally hard to specify.
The first technique consists in selecting from the range of existing strategies the ones that replicate the option as closely as possible, as defined by the agent’s preferences. The second technique does not seek to accurately replicate the option but rather to find a strategy with a terminal value systematically greater than the option’s value. In this case, the option is said to be super-replicated. Unfortunately, we will see that prices obtained with this technique are generally disappointing because they coincide with trivial arbitrage bounds. In this way, the majority of super-replication models suggest that the best way to hedge a call option is actually to buy the underlying asset !
7.2 Ross APT
7.2.1 The model
The Arbitrage Pricing Theory developed by Ross (1976) is another interesting example of the consequences of the asymptotic absence of Free Lunch. The objective of Ross is to better understand the CAPM relationship and derive it from a different line of thoughts: “on empirical grounds the conclusions as well as the assumptions of the theory have also come under attack.”
What are the conditions for a return factor model to hold at equilibrium. Define: \[\pi=m+\beta ' f+\epsilon\] where \(<e>=D \;\mathrm {(diagonal), <f>=I,\mathbb E (f)= 0, \mathbb E (e)=0}\) The dimension \(p\)of \(f\) is supposed to be much smaller than the number of assets \(n\). Hence, \(f\) represents a limited number of common factors.
With \(n>>p\) there are plenty of portfolios \(\alpha\) such that \(\beta \alpha=0\) and \(\alpha_i\approx \frac{1}{n}\). The return of such portfolio yields: \(\pi(\alpha)=\alpha'm+\alpha\epsilon\). By the law of large number \(<\alpha\epsilon>\approx\frac{1}{n}\)
Hence in such model, the risk of such portfolio can be made arbitrarily small. This is not exactly an arbitrage but this could be called a “Free Lunch with Vanishing Risk”. If one prevents this situation to happen we must necessarily have \(m'\alpha=0.\) Differently stated:
\[\forall \alpha :\beta\alpha=0\Rightarrow m'\alpha=0\]
The so-called Farkas lemma implies that \(\alpha\) belongs to the span of \(\beta\) or equivalently: \[\exists \bar f \;|\; m=\beta' \bar f \]
which is equivalent to write a factor model where \(\mathbb E (f)= \bar f\) and \[\pi=\beta ' f+\epsilon\]
Nice extension of the CAPM. Only non diversifiable risks can generate premium