April 25, 2024
Every trader knows about slippage. The price quoted by an exchange is not necessarily the price at which a trade is executed. Market orders get matched against limit orders like blocks dropping in tetris. Limit order quantities get eaten up until the market order is filled. If the market order is larger then the amount of the total of the level 1 limit orders it spills over to the next best offers by the liquidity provider. It is also possible other market orders or limit orders are placed in the meantime.
If you have spent time on a trading floor you have probably heard a discussion between a trader and a back office person about how their positions were marked-to-market. Traders care about this very much because that is used to determine their profit and loss, but they sometimes don’t understand exactly how this is calculated.
This document provides a precise description of how P&L is calculated.
The atoms of finance are positions: a legal entity owns some amount of an instrument. Positions change via transactions: amounts and instruments are exchanged at some time between two entities. A portfolio is a collection of positions belonging to an entity. A market is a collection of portfolios.
We can express this using math by a position is an amount, instrument, and entity: \pi = (a, i, e) and a transaction is the exchange of positions at some time t: \chi = (t, \pi, \pi'). The market {\{\ldots,(a, i, e),\ldots,(a',i',e'),\ldots\}} becomes {\{\ldots,(a', i', e),\ldots,(a,i,e'),\ldots\}} after the transaction.
If \mathcal{M} = \{\pi_j\} is a market define the net amount of instrument i held by entity e by N(i, e) = \sum_{\pi_j = (a_j, i_j, e_j)}\{a_j\mid i_j = i, e_j = e\}.
The market price of a transaction \chi = (t,\pi,\pi') is X^{i'}_{i} = a'/a. To calculate P&L in native currency i_0 it is necessary to specify “prices” X^{i_0}_{i} for each instrument i. The position (a,i,e) can, in principal, be exchanged for (aX^{i_0}_i, i_0, e_0) with some imaginary entity e_0. The mark-to-market in currency i_0 for entity e is N(e) = \sum_{\pi_j = (a_j, i_j, e_j)}\{a_jX^{i_0}_{i_j}\mid e_j = e\}. The profit-and-loss over an interval of time is the difference of the mark-to-mark at the end and beginning of the interval.
It is not controversial to assume X^{i_0}_{i_0} = 1 but the other “prices” can be contentious. If i is exchange traded and liquid then its closing price can be used. If i is rarely traded then its price is problematic to determine. Firms often have pricing models for illiquid instruments that everyone agrees to use in order to prevent heated arguments.
If you work in a back office you have to deal with fiddly, but important, details. It is not as simple as specifying a position as an amount, instrument, and entity. The amount can be precisely modeled as an integer. Every instrument has a lowest level of divisibility and trades in an integer multiple of that.
Instruments have ISINs that are supposed to uniquely identify them. If you can afford to pay for them.
There are LEIs to identify legal entities, but legal entities have internal subdivisions. The back office has to keep track of the ever changing portfolio managers.
We have to give names to mathematical concepts, but I always think in terms of the mathematical concepts. The names are just shorthand. The fundamental problem of risk management is assessing the risk of future P&L, where P&L is a stochastic process defined by a model.
In the one-period case the model is a set I of instruments, initial prices x \in \RR^I, a set \Omega of possible outcomes over the period, a function X\colon\Omega \to \RR^I of final prices given \omega\in\Omega occurred, and a probability measure P on Omega. P&L is defined by specifying a portfolio \xi\in\RR^I. To be completely rigorous, \xi is in the dual space of \RR^I with dual pairing xi^* x = \langle x, xi\rangle = sum_{i\in I} xi_i x_i. In this case the stochastic process for the P&L is the number xi^* x at time 0 and the random variable xi^*X\colon\Omega \to \RR at time 1. What is the “risk” of the one period model P&L?