The Fundamental Theorems

Abstract

This chapter presents the three fundamental theorems of asset pricing. These theorems are the basis for pricing and hedging derivatives, characterizing price bubbles, and understanding the risk return relations among assets including the notion of systematic risk, idiosyncratic risk, portfolio optimization, and equilibrium pricing.

Appendix

This appendix proves that the original definition of NFLVR in Delbaen and Schachermayer [44, Proposition 3.6], is equivalent to Definition 27 of NFLVR given in the text. And, it uses Theorem 12 to give another characterization of NUPBR.

Definition 32 (No Free Lunch with Vanishing Risk (D&S))

A free lunch with vanishing risk (D&S) is: (i) a simple arbitrage opportunity (NA) , or (ii) a sequence of zero initial investment admissible s.f.t.s.’s $(\alpha _{0},\alpha )_{n}\in \mathcal {A}(0)$ with value processes $X_{t}^{n}$, lower admissibility bounds c _n ≤ 0, and a $\mathcal {F}_{T}$-measurable random variable χ ≥ 0 with $\mathbb {P}(\chi >0)>0$ such that c _n → 0 and $X_{T}^{n}\rightarrow \chi$ in probability. We denote condition (ii) as D&S(ii).

The market satisfies D&S if there exist no D&S trading strategies.

As seen, a market satisfies D&S if there are no simple arbitrage opportunities and no approximate arbitrage opportunities $(\alpha _{0},\alpha )_{n}\in \mathcal {A}(0)$, that in the limit, become simple arbitrage opportunities. Both conditions (i) and (ii) are needed in this definition. Indeed, condition (ii) does not imply condition (i). To see why, consider a (simple) arbitrage opportunity $(\alpha _{0},\alpha )\in \mathcal {A}(0)$ with admissibility bound c and value process X _t that satisfies X ₀ = 0, X _T ≥ 0, and $\mathbb {P}(X_{T}>0)>0$. Define the (constant) sequence of admissible s.f.t.s.’s using the simple arbitrage opportunity, i.e. (α ₀, α)_n = (α ₀, α) whose time T value process $X_{T}^{n}=X_{T}$ approaches (equals) χ = X _T where χ ≥ 0 with $\mathbb {P}(\chi >0)>0$. This constant sequence of admissible s.f.t.s.’s satisfies all of the properties of condition (ii) except that the sequence’s admissibility bounds c _n = c do not converge to 0.

We now show that Definition 27 of NFLVR is equivalent to D&S.

Theorem 24 (Equivalence of NFLVR Definitions)

NFLVR is equivalent to D&S

Proof

(Step 1) Show a FLVR implies a D&S.

Let $(\alpha _{0},\alpha )_{i}\in \mathcal {A}(x)$ be a FLVR, i.e. $(\alpha _{0},\alpha )_{i}\in \mathcal {A}(x)$ with initial value x ≥ 0, value processes $X_{t}^{i}$, lower admissibility bounds 0 ≥ c _i where ∃c ≤ 0 with c _i ≥ c for all i, and a $\mathcal {F}_{T}$-measurable random variable χ ≥ x with $\mathbb {P}(\chi >x)>0$ such that $X_{T}^{i}\rightarrow \chi$ in probability.

Now, we have 0 ≥ c _i ≥ c for all i.

Thus, there exists a subsequence n such that c _i → c ^∗∈ [0, c] with c ≤ 0. This implies 0 ≥ c ^∗.

Consider $(\alpha _{0},\alpha )_{n}\in \mathcal {A}(x)$. Define $(\tilde {\alpha }_{0},\tilde {\alpha })_{n}=(\alpha _{0}-c^{*},\alpha )_{n}$.

This trading strategy has initial wealth $\tilde {x}=x-c^{*}\geq 0$.

The value process is $\tilde {X}_{t}=X_{t}-c^{*}$.

It is self-financing because this trading strategy just adds − c ^∗≥ 0 additional dollars to the mma in the original s.f.t.s. for all t and holds this new position for all t.

The admissibility bounds are $\tilde {c}_{n}=min\{c_{n}-c^{*},0\}$ for all n with $\tilde {c}_{n}=min\{c_{n}-c^{*},0\}\rightarrow 0$.

Finally, we have that

$\tilde {X}_{T}^{n}\rightarrow \tilde {\chi }=\chi -c^{*}$ in probability and

$\tilde {\chi }=\chi -c^{*}\geq x-c^{*}=\tilde {x}$ with $\mathbb {P}(\tilde {\chi }=\chi -c^{*}>x-c^{*}=\tilde {x})>0$.

Hence, this is a D&S.

(Step 2) Show a D&S implies a FLVR.

(Case a) Let $(\alpha _{0},\alpha )_{n}\in \mathcal {A}(x)$ be a simple arbitrage opportunity with admissibility bound c ≤ 0 and value process X _t that satisfies X ₀ = 0, X _T ≥ 0, and $\mathbb {P}(X_{T}>0)>0$. Then, define the (constant) sequence of admissible s.f.t.s.’s using the simple arbitrage opportunity, i.e. (α ₀, α)_n = (α ₀, α) whose time T value process $X_{T}^{n}=X_{T}$ approaches (equals) χ = X _T where χ ≥ 0 with $\mathbb {P}(\chi >0)>0$ with lower admissibility bounds c _n = c ≥ c for all n. This is a FLVR.

(Case b) Let $(\alpha _{0},\alpha )_{n}\in \mathcal {A}(0)$ with value processes $X_{t}^{n}$, lower admissibility bounds c _n ≤ 0, and a $\mathcal {F}_{T}$-measurable random variable χ ≥ 0 with $\mathbb {P}(\chi >0)>0$ such that c _n → 0 and $X_{T}^{n}\rightarrow \chi$ in probability.

Note that since c _n → 0, there exists a N and c < 0 such that for all n ≥ N, c _n ≥ c.

Consider the sequence of admissible s.f.t.s.’s $(\alpha _{0},\alpha )_{n=N}^{\infty }\in \mathcal {A}(0)$. This is a FLVR. This completes the proof.

We can restate this theorem as NFLV R⇔NA + DS(ii). From Theorem 12, we have NFLV R⇔NA + NUBPR. Consequently, NUBPR⇔DS(ii). This completes the characterization.