October 3, 2025
We discuss the Simple Universal Model for market prices and cash flows, trading strategies, and the values and amounts involved. Every company trading in the market needs to describe instument contracts. They also need models to value, hedge, and manage their risk.
Inspired by Lexifi, we propose a mathematical theory that can be practially implemented on a computer. A key simplification is to seperate the notion of an increasing sequence of times to be used in the contact specifications and events such as cash flows or trading strategies. We identify primitive operations that apply only to increasing time series and a operations on them. These can be used to convert contract specifications into a stream of cash flows or trades.
It is important to include knobs for all possible actions an intrument owner can take. For example, an American option holder can exercise any time prior to expiration. A realistic model should not assume the holder will exercise optimally. In the mortgage-backed secuities market a number of individual mortages are bundled together and the principal and interest paid by individual mortages are passed through to MBS holders. It is well known that most individual mortage holders do not exercise their right to pay off their mortgage in an optimal fashion.1 Any model that assumes this will be unable to fit observed market prices. See (Kalotay, Yang, and Fabozzi 2004) for a prepayment model that takes this into account.
A fundamental problem when implementing date and time operations is how to convert two calendar dates to a time duration and a date plus a time duration back to a calendar date. Let t(d) be the time associated with date d. On Unix systems it is common to use the number of non-leap seconds since Unix epoch midnight 1 January, 1970 UTC. Most modern systems allow resolution to microseconds. The inverse function d(t) converts time to dates. Complications arise when considering durations such as weeks and months. For example, 31 Jan plus one month cannot be 31 Feb. Should it be 29 Feb on leap-years and 28 Feb otherwise? Such issues will not be considered here. We assume any such resolution satisfed d + (d' - d) = d' ant d(t + t') = d(t) + (t' - t).
It is also important to distinguish between prices and cash flows. Every actual trade has a price: the quotient of the amount the seller exchanges with the buyer. If a buyer receives 2 shares of Ford stock for $22 the price is $11 per share of Ford stock. The price 11 and number of shares 2 will be entered into books and records. A model of an instrument specifies possible future prices and cash flows. Models typically do not take into account the bid/ask spread, how that widens as the amounts get larger, much less how price is affected by the counterparties involved.
Owning an instrument can entail cash flows. Bonds have coupons, stocks have dividends, futures have daily margin adjustments. The price of a futures is always zero and futures quotes are a naturally occuring martingale in efficient markets.
A European call option is specified by a triple (X, k, d)\in I\times\boldsymbol{{R}}\times d(T) where X is the call underlying, k is the strike2, and d is the expiration date. On the expiration date the option issuer pays the option holder \max\{X_t - k, 0\} per unit held where X_t is the price of the underlying at time t(d). Prices are X_t\colon\Omega\to\boldsymbol{{R}}3 that are measurable with respect to information available at time t. We can also write this as X\colon\T\times\Omega\to\boldsymbol{{R}} where X(t,\omega) = X_t(\omega), t\in T, \omega\in\Omega.
An ordered time series is a non-decreasing sequence of times u_0 \le u_1 \le \cdots where the u_j belong to a totally ordered set T. Let U for the set \{u_j\}. We introduce the notation \text{?}U = \text{true} if U is not empty and \text{?}U = \text{false} otherwise, \text{*}U = \min U, and \text{+}U = U\setminus\text{*}U. In particular \text{*}U = u_0, \text{*}\text{+}U = u_1, etc., so U is an iterator.
If R\subset T\times T is a relation on T define URt for t\in T by u\in URt if and only if uRt and u\in U. For example, U < t is a stream that terminates at the first u_j \ge t.
Recall the disjoint union of two sets A\sqcup B = (\{0\}\times A)\cup(\{1\}\times B). An element of the disjoint union for the form (i,c) satisfies i = 0 and c\in A or i = 1 and c\in B. The order (i,c)\le(i',c') if and only if c < c' or c = c' and i < i' is a total order
The disjoint union of sets A_i indexed by i\in I is \sqcup_{i\in I} A_i = \cup_{i\in I} (\{i\}\times A_i)
The Simple Universal Model posits a mathematical model of the market that is arbitrage free. Possible trading times are denoted T. The set of all instruments is I. The sample space \Omega is the set of everything that can happen. Information available at time t\in T corresponds to a partition \mathcal{A}_t of \Omega. The prices and cash flows associated with each instrument are \mathcal{A}_t-measurable functions X_t,C_t\colon\mathcal{A}_t\to\boldsymbol{{R}}^I.
The SUM shows every arbitrage-free model has the form X_t D_t = M_t - \sum_{s < t} C_s Where D_t is a positive measure on \mathcal{A}_t and M_t is a \boldsymbol{{R}}^I-valued martingale measure on \mathcal{A}_t. We say (M_t)_{t\in T} is a martingale measure if M_t = M_u|_{\mathcal{A}_t} for t < u where the vertical bar and subscript algebra indicate restriction of M_u on \mathcal{A}_u to \mathcal{A}_t. This requires \mathcal{A}_u to be a refinement of \mathcal{A}_t if u < t.
The SUM can be used for any collection of instruments: bonds, stocks, futures, options, convertible bonds, FX spot/options/futures, commodities, etc.
We also need a mechanism to convert between calendar dates and times and assume t(d) is the time correpondng to calendar date d. This will not be considered in what follows.