Contingent Claims and Market Completeness

\[ \def\BB#1{{\mathbb{#1}}} \def\BF#1{{\mathbf{#1}}} \def\E{{\mathbb{E}}} \def\R{{\mathbb{R}}} \]

yedlu, Winter 2024

Contingent Claim Markets

Settings

We have:

Notation	Description
\(S\)	Set of states
\(x \in \BB{R}^S\)	Payoff vector for every possible states

Contingent Claim

We call a security contingent claim if it pays $1 in one particular future state \(s^{*} \in S\) and zero in all other states.

In matrix notation, this is a standard-unit vector in \(\BB{R}^S\). By definition, contingent claims are mutually independent and span the whole payoff space.

We use the notation \(pc(s^*)\) as the current price of the contingent claim for state \(s^*\).

Complete Market through Contingent Claims

In a complete market, there exists a contingent claim for every possible state, either directly or through synthesizing.

Market completeness is an intriguing concept through this view: we are able to hedge any type of risk through contingent claims under market completeness.

Contingent Claim Pricing

Happy Meal Theorem

In a complete market, we can price an asset as:

\[\begin{aligned} p(x) &= \sum_{s \in S} pc(s) \cdot x(s) \end{aligned}\]

Intuition: the price of a payoff stream (which can be represent by linear combinations of contingent claims) should be the linear combination of contingent claim prices.

Note

We will later discuss the assumption needed for this theorem to hold (namely law of one prices and/or arbitrage-free conditions).

Links to Consumption Pricing

We will derive and show how to link contingent claim pricing to the consumption pricing model.

Let \(\pi(s)\) be the probability that state \(s \in S\) occurs. Then for the pricing equation:

\[\begin{aligned} p(x) &= \sum_{s \in S} \pi(s) \frac{pc(s)}{\pi(s)} x(s) \end{aligned}\]

We define the state-dependent tochastic discount factor (SDF) in this model as:

\[\begin{aligned} m(s) := \frac{pc(s)}{\pi(s)} \end{aligned}\]

In this way, the contingent claim pricing equation becomes identical to the consumption pricing model:

\[\begin{aligned} p(x) &= \sum_{s \in S} \pi(s) \frac{pc(s)}{\pi(s)} x(s) \\ &= \sum_{s \in S} \pi(s) m(s) x(s) \\ &= \E[mx] \end{aligned}\]

In a complete market, if SDF exists, then it is given by the set of contingent claim prices scaled by probability measures.

What’s different now is that we do not assume investor-specific preferences (including utility functions and discount factors).

Risk-Neutral Probabilities

Risk-Free Rate

We have previously derived the risk-free rate using SDFs in consumption pricing. Now we try to derive it using contingent claim pricing. First, we write down the expected SDF under contingent claim pricing:

\[\begin{aligned} \E[m] &= \sum_{s \in S} \pi(s) m(s) \\ &= \sum_{s \in S} \pi(s) \frac{pc(s)}{\pi(s)} \\ &= \sum_{s \in S} pc(s) \end{aligned}\]

Then, using the finding from consumption pricing on risk-free rate, we have:
\[\begin{aligned} R^f &= \frac{1}{\E[m]} \\ &= \frac{1}{\sum_{s \in S} pc(s)} \end{aligned}\]

Intuition

We know from the definition of risk-free asset that its payoff vector would look like:
\[\begin{aligned} \mathcal{A} &= (1, ..., 1)^{'} \in \R^n \end{aligned}\] given state set \(S = (s_1, ..., s_n)\).

If we were to assume complete markets, then the risk-free asset with the said payoff would have price:
\[\begin{aligned} p(\mathcal{A}) &= \sum_{i = 1}^{n} pc(s_i) \end{aligned}\] The return of investing this asset would be \(1/\sum_{i = 1}^{n} pc(s_i) = R^f\).

Definition: Risk-Neutral Probabilities

We define:
\[\begin{aligned} \pi^* (s) &:= R^f m(s) \pi(s) \\ &= R^f pc(s) \\ &= \frac{pc(s)}{\sum_{s \in S} pc(s)} \end{aligned}\] as the risk-neutral probability of state \(s \in S\).

Risk-Neutral Pricing

Under risk-neutral probabilities, we can derive further the contingent claim pricing model:
\[\begin{aligned} p(x) &= \sum_{s \in S} pc(s) x(s) \\ &= \sum_{s \in S} \frac{\pi^* (s)}{R^f} x(s) \\ &= \frac{\E^*[x]}{R^f} \end{aligned}\]

Where \(\E^*\) is the expected payoff under risk-neutral probabilities.

Linking Two Probabilities

Although investors might have disagreements over the real probability, they must all agree on one set of risk-neutral probabilities. Rationale would be:

• all investors observe the same asset price;
• all investors infer and agree the same contingent claim prices

Then, by the definition of risk-neutral probabilities:

\[\begin{aligned} \pi^* (s) &= \frac{pc(s)}{\sum_{s \in S} pc(s)} \end{aligned}\]

all investors must all agree on the same \(\pi^* (s)\).

Linking Two Pricing Models

Comparing the consumption pricing model:
\[\begin{aligned} p &= \E[mx] \\ &= \sum_s \pi(s) m(s) x(s) \\ &= \frac{\E[x]}{R^f} + \sigma(m,x) \end{aligned}\]

With the contingent claim pricing model:
\[\begin{aligned} p &= \sum_s pc(s) x(s) \\ &= \frac{\E^* [x]}{R^f} \end{aligned}\]

Note

We can see that in addition to making risk adjustments to real-world probabilities, we can also discount the payoff directly as if we are in risk-neutral world. This makes sense intuitively if we consider:

• the marginal investor in the economy can shift their risk preferences (hence hypothetically they can be risk-neutral)
• the asset price has to be unanimously agreed upon (to reach an equilibrium) regardless of the marginal investor’s risk preference
• thus we can calculate directly the asset pricing as if everyone’s in the risk-neutral world