A large assortment of price and interest rate theories creates model risk and returns constants, deterministic functions of time or price σ(t), σ(t, P) or stochastic σ. Changing the probability measure of the underlying price to an equivalent martingale measure, they avoid arbitrage and "ignore" price drift that is relevant for a trader.

Autoregressive models approximate conditional heteroscedasticity of time series. (Heteroscedasticity refers to varying levels of correlation between observed data.) They simulate effects such as volatility clusters, time intervals aggregating large and small price fluctuations and provide new definitions.

Second, there is a tendency to measure trend and volatility using price increments sometimes adjusted by time increments. It is strange that such polar properties are measured by the same quantities. Ironically, a price change cannot separate deterministic trend and random volatility contributions. We need further assumptions. For instance, whatever is responsible for the trend and volatility for one increment remains constant for others. Mapping notions expressed by the same words is especially confusing when economic data are treated by a mixture of methods.

The lesson here is that while mainstream market analysis offers many options for measuring trend and volatility, none of them is perfect and, perhaps, many are worse than random. Indeed, the best resolution may be to avoid terms such as "trend" and "volatility" altogether, admitting that we cannot define them well.

**Alternative analysis**

MPS does not try to decipher the market. It instead attempts to learn from whatever the market is teaching. Under equal conditions, MPS depends on costs. In futures, commissions and fees do not depend on price; they change based on the number of traded contracts. We can vary these costs to get a new MPS. In addition, futures prices are discrete. They rise and fall by whole numbers of minimal price fluctuations, specified by a contract’s specifications.

The first task of the MPS framework is to build the MPS for a chain of ticks and costs. This can be done manually or by developing an algorithm for a given set of price data.

The second task is to aggregate the MPS’s transactions into trades. A trade is a complete round trip in the market.

We describe MPS transactions using a unique system for trade identification. For instance, the strategy (1_{1}, 0_{2}, -2_{3}, 1_{4}) is buy one contract at *t*_{1}, do nothing at *t*_{2}, sell two contracts reversing long to short position at *t*_{3}, exit buying one contract at *t*_{4}. Three transactions (from four actions) are combined into two trades (1_{1}, -1_{3}) and (-1_{3}, 1_{4}). Their durations are *t*_{3} to *t*_{1} and *t*_{4} to *t*_{3}.

The third task is statistical analysis of the optimal trades. Notice how we distinguish transactions and trades. The former are individual buy and sell actions and the latter are ordered sets of offsetting transactions. In all cases, the net trade action, algebraic sum, is equal to zero. MPS transactions can be combined into trades with zero net action each because the MPS is free to add the last mark-to-market transaction.

This leads us to the optimal trading element. This is a collective name for properties associated with an optimal trade returned by an MPS. This measure is an alternative, or complement, to traditional trend and volatility analysis.