Financial contracts as mathematical objects

Definition A state space $\Omega$ is the set of outcomes of a model. Each state $\omega \in \Omega$ fixes everything the contracts under study depend on, such as a terminal stock price, a whole price path, or whether a borrower defaulted and how much was recovered. In this unit the state space is finite,

and in a one-period model the true state is unknown at time $0$ and revealed at the single later date $T$. A quantity that depends on the state is a function on $\Omega$, and once the states are listed in a fixed order it is a vector in $\mathbb{R}^m$ with one entry per state. The order is only a labeling. Listing the states differently permutes every such vector in the same way and changes no conclusion.

Definition Fix a currency, a finite state space $\Omega$ and payment dates $0 \le t_1 < t_2 < \cdots < t_n$ in years. A financial contract specifies, for every date $t_k$ and every state $\omega$, the amount $C_{t_k}(\omega)$ that the holder receives at $t_k$ in state $\omega$, in that currency. Its cash-flow vector in state $\omega$ is

and the whole contract is the map $\omega \mapsto C(\omega)$, an $m \times n$ array with one row per state and one column per date. Each entry must be known from the information available at its own date, so a payment settled at $t_k$ cannot depend on an event observed only after $t_k$, although the contract may add a separate adjustment at a later date. In particular a payment due at time $0$ is known today and is the same in every state. Two contracts are the same mathematical object exactly when their arrays agree entry by entry with the same dates, currency and conventions. Moving a payment date, changing the currency or adding an obligation such as posting collateral gives a different object, while relabeling the states gives the same one.

Definition Cash flows are recorded from the holder’s point of view. A receipt is a positive entry and a payment is a negative entry, each in a stated currency at a stated date. For a contract between two parties with no payments to anyone else, the counterparty’s array is $-C(\omega)$, so the two arrays add to zero entry by entry. The sign gives the direction of a payment only. Whether a trade is profitable depends on a valuation of all its entries, not on the sign of any one of them. Entries at the same date in the same currency add directly, and entries at different dates or in different currencies are combined only after a valuation rule or an exchange rate says how.

Definition When a contract makes a single payment at maturity $T$ and that payment depends on the state only through an observed quantity, such as the terminal stock price $S_T$, the payoff function is the rule that maps the quantity to the payment. A European call with strike $K$ and maturity $T$ gives its holder the right, exercisable only at $T$, to pay $K$ for one share. Exercising is worth $S_T - K$ at $T$, which is positive only when $S_T > K$, so a holder who exercises exactly when $S_T > K$ receives

where $(x)^{+} = \max(x, 0)$ is the positive part. A cash-settled call pays the same amount directly. As a function of the state, $X(\omega) = (S_T(\omega) - K)^{+}$, and on a finite state space the payoff is the vector of these values. The payoff is an amount paid at $T$. It does not include the premium paid at time $0$, and it involves no probabilities.

Definition Consider a one-period model with finite state space $\Omega = \{\omega_1, \ldots, \omega_m\}$ in which every payoff is paid at $T$ in one currency. A valuation operator $\Pi$ assigns to each payoff vector $X \in \mathbb{R}^m$ a value at time $0$ in the same currency. The state price $\pi_j$ is the time-$0$ price of the unit claim on $\omega_j$, the contract that pays 1 at $T$ in state $\omega_j$ and 0 in every other state, also called an Arrow-Debreu security. Given state prices, the operator is

It is linear, because the sum distributes over $aX + bY$, so for all payoffs $X$ and $Y$ and real numbers $a$ and $b$

Conversely, every linear map from $\mathbb{R}^m$ to $\mathbb{R}$ has this form with $\pi_j = \Pi(e_j)$, where $e_j$ is the payoff vector of the unit claim on $\omega_j$. The operator is called positive if $X \ge 0$ in every state implies $\Pi(X) \ge 0$, and strictly positive if moreover $\Pi(X) > 0$ whenever $X \ge 0$ and $X \ne 0$. The state-price operator is positive exactly when every $\pi_j \ge 0$, and strictly positive exactly when every $\pi_j > 0$. The sum of the state prices is $\Pi(\mathbf{1})$, the value of one sure unit at $T$, where $\mathbf{1}$ is the vector of ones. Using one strictly positive linear operator for every contract is an economic assumption. It is consistent with frictionless trading and no arbitrage, which the next lesson makes precise, while fees, bid-ask spreads or position limits can give one contract different purchase and sale values. The lesson does not say which states or state prices describe a particular market.

Definition The payoff of a contract paying at $T$ is the vector $X = (X(\omega_1), \ldots, X(\omega_m))$ of amounts received at $T$, one for each state. Its present value is the single number $\Pi(X)$ at time $0$, which depends on the valuation rule as well as on the contract. The profit of a trade is a third quantity. It combines payments made at different dates, so it needs a convention that moves them to one date. Buying the payoff for $V_0$ at time $0$ with money borrowed by selling $V_0/\Pi(\mathbf{1})$ sure units at $T$, which requires $\Pi(\mathbf{1}) > 0$, gives the time-$T$ profit

in each state. A payoff of 20 is therefore not a profit of 20, and a present value is not an amount that any state pays.