A Unified Market Model

USD/JPY = 123 means a USD can be exchanged for 123a JPY.

Price of Ford is 8 means a shares of can be exchanged for 8a dollars.

F/\$ = 8.

time

amount

instrument

entity

Time T.

Amount A = \mathbf{R}

Instruments I market instruments.

Entities E legal entities.

A holding is an element h = (t, a, i, e)\in T\times A\times I\times E. At time t entity e owns a units of instrument i.

A market is a (multi-)set of holdings.

Issuers can add holdings to a market.

A transaction is a function \xi

M\mapsto M + \xi.

Portfolios can be netted by measures on finite subsets of holdings.

Define a \mathbf{R}^{I\times E} measure on A\times I\times E by A(\Pi)(i, e) = \sum\{ \pi.a\mid \pi\in\Pi, \pi.i = i, \pi.e = e\}.

A market is a subset of \mathbf{R}\times I\times E.

Define A(\Pi) = \{(A(\Pi)(i,e),i,e)\mid i\in I, e\in E\}

An exchange is \xi = (\pi, \pi'), where e is the buyer and e' is the seller.

Define \xi\colon I\times E\to \mathbf{R} by \xi(j,f) = -a if j = i and f = e, \xi(j,f) = a' if j = i' and f = e, \xi(j,f) = a if j = i and f = e', \xi(j,f) = -a' if j = i' and f = e', and \Xi(j,f) = 0 otherwise.

Exchanges act on markets by \xi\Pi = \Pi + \xi = \{\pi + \xi\mid \pi\in\Pi\}.

Prices are a function X\colon I\times I\to\mathbf{R}. Amount a of instrument i can be exchanged for X(i,i')a of instrument i'.

A market model is a function from time to prices. X_t\colon\mathcal{A}_t\to R^{I\time I}.

A trading strategy is a finite set of increasing stopping times \tau_j and trades \Gamma_j\in \mathbf{R}^{I\times E}.

Trades accumulate to a position \Delta_t = \sum_{\tau_j \le t} \Gamma_j.