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Killtech

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- TL;DR Summary
- Kochen-Specker type of contextuality in classical probability theory and finance. How does it differ from QT?

I am trying to figure out what the issue is that supposedly prevents QT being modeled via classical probabilities and in terms of untypical behavior Kochen-Specker theorem maybe one of the interesting features to look at.

As like all other theorems of this type it uses additional assumptions about the random variables needed to prove its statement. In this case it's that that non-contextual random variables are not able to describe QT. The premise about the type of random variables needs to be assumed because it does not hold for all. Other hand it is a well motivated premise for how one expects measurements to behave normally.

Anyhow, it rises the question what about the random variables in classical probability theory that do not meet Kochen Specker requirements, i.e. are contextual. A good example that produces such random variables is a Markov decision process and the type of variables in question can be described as predicted outcomes of a possible action before it was taken (in the link denoted as ##R_a(s,s')## with ##a## representing the action, ##s## the state at time ##t## before the action, ##s'## the state after the action). Now obviously the realization of the result is only possible after the action was taken, not before and is also conditional on the action being indeed taken. For example the predictions on the economic impact of the Brexit can only realize and take a concrete value once Brexit is actually done. This creates a contextuality of two random variables in a situation where they depend on actions that are mutually exclusive. For example staying in the EU and Brexit are incompatible actions and as such it is not possible to know the economic impact of both - the system only determines the variables of the action that realized while the other variable become obsolete representing a reality that did not come to pass. So contextuality means here that we will never be able to tell/measure with certainty that Brexit was economically the better/worse choice but we will be able to tell/measure the consequences of Brexit exactly (discounting the possibility of MWI).

In finance one will encounter a lot of random variables of this type. For example the market value ##MV_t## of a security at time ##t## is normally a theoretical expectation value - a prediction of what the security would trade for on a market. It's by no means the real value you will get. You can only know its real worth, when you actually find a buyer so that real cash flows and changes hands. Thus there is an inherent uncertainty ##\Delta MV_t## that is a result of the bid-ask spread and trades are a type of measurement which make prices realize with very real money.

An American option is a security that has an additional feature that it can be exercised at any time yielding a cash flow depending on some underlying item value. On the other hand, selling the option will make you lose it thus forfeiting the ability to exercise it. Selling and exercising are mutually exclusive. Let's denote ##CF_t## the value of the cash flow if the option is exercised at time ##t##. On top of the uncertainity of the underlying an OTC uncollaterized option has a chance that your counterpart goes default and is unable to pay up, so ##CF_t## has it's very own stochastics. ##MV_t## and ##CF_t## exhibit a strong contextually where it is impossible for both to realize simultaneously and doing so would also break risk neutral measure. - i.e. you cannot cash in twice (legally) as that means you cheated someone and your expected profit would be better then the risk free rate.

A Markov decision process can be used to model the time evolution of those random variables, for example to find a good exercise strategy or simply to have a model to calculate an expectation of ##MV_t##.

Checking Kochen-Specker for these random variables immediately points to the problem that a value function like ##v(MV_t)## only exist in rare cases where trades are made but not otherwise. So it doesn't apply here.

The interpretation of the classical case never caused a head ache, though asking questions about how such random variables describe properties of the market system, well, that is actually not so easy. Not even clear to me in how much something like the theoretical market value is entirely real to begin with. The problem is that ##E(MV_t)##, ##\Delta MV_t## and ##Volatility(MV_t)## are all important properties of the market watched at all times, more so then something like ##v(MV_t)##, however the latter stands for trades that drive the time evolution of the market. On the other hand ##MV_t## depends on the pricing method chosen so in practice it depends on the observer and with Elon Musk we know that ##MV_t## can also depend on the decision to tweet (thus he can have a better ##MV_t## model then we do). Thus it's actually a really tough one to answer.

Well, the people doing these kind of calculations are incidentally called Quants ;). But on a serious note, how does that differ from measurements in QT?

As like all other theorems of this type it uses additional assumptions about the random variables needed to prove its statement. In this case it's that that non-contextual random variables are not able to describe QT. The premise about the type of random variables needs to be assumed because it does not hold for all. Other hand it is a well motivated premise for how one expects measurements to behave normally.

Anyhow, it rises the question what about the random variables in classical probability theory that do not meet Kochen Specker requirements, i.e. are contextual. A good example that produces such random variables is a Markov decision process and the type of variables in question can be described as predicted outcomes of a possible action before it was taken (in the link denoted as ##R_a(s,s')## with ##a## representing the action, ##s## the state at time ##t## before the action, ##s'## the state after the action). Now obviously the realization of the result is only possible after the action was taken, not before and is also conditional on the action being indeed taken. For example the predictions on the economic impact of the Brexit can only realize and take a concrete value once Brexit is actually done. This creates a contextuality of two random variables in a situation where they depend on actions that are mutually exclusive. For example staying in the EU and Brexit are incompatible actions and as such it is not possible to know the economic impact of both - the system only determines the variables of the action that realized while the other variable become obsolete representing a reality that did not come to pass. So contextuality means here that we will never be able to tell/measure with certainty that Brexit was economically the better/worse choice but we will be able to tell/measure the consequences of Brexit exactly (discounting the possibility of MWI).

In finance one will encounter a lot of random variables of this type. For example the market value ##MV_t## of a security at time ##t## is normally a theoretical expectation value - a prediction of what the security would trade for on a market. It's by no means the real value you will get. You can only know its real worth, when you actually find a buyer so that real cash flows and changes hands. Thus there is an inherent uncertainty ##\Delta MV_t## that is a result of the bid-ask spread and trades are a type of measurement which make prices realize with very real money.

An American option is a security that has an additional feature that it can be exercised at any time yielding a cash flow depending on some underlying item value. On the other hand, selling the option will make you lose it thus forfeiting the ability to exercise it. Selling and exercising are mutually exclusive. Let's denote ##CF_t## the value of the cash flow if the option is exercised at time ##t##. On top of the uncertainity of the underlying an OTC uncollaterized option has a chance that your counterpart goes default and is unable to pay up, so ##CF_t## has it's very own stochastics. ##MV_t## and ##CF_t## exhibit a strong contextually where it is impossible for both to realize simultaneously and doing so would also break risk neutral measure. - i.e. you cannot cash in twice (legally) as that means you cheated someone and your expected profit would be better then the risk free rate.

A Markov decision process can be used to model the time evolution of those random variables, for example to find a good exercise strategy or simply to have a model to calculate an expectation of ##MV_t##.

Checking Kochen-Specker for these random variables immediately points to the problem that a value function like ##v(MV_t)## only exist in rare cases where trades are made but not otherwise. So it doesn't apply here.

The interpretation of the classical case never caused a head ache, though asking questions about how such random variables describe properties of the market system, well, that is actually not so easy. Not even clear to me in how much something like the theoretical market value is entirely real to begin with. The problem is that ##E(MV_t)##, ##\Delta MV_t## and ##Volatility(MV_t)## are all important properties of the market watched at all times, more so then something like ##v(MV_t)##, however the latter stands for trades that drive the time evolution of the market. On the other hand ##MV_t## depends on the pricing method chosen so in practice it depends on the observer and with Elon Musk we know that ##MV_t## can also depend on the decision to tweet (thus he can have a better ##MV_t## model then we do). Thus it's actually a really tough one to answer.

Well, the people doing these kind of calculations are incidentally called Quants ;). But on a serious note, how does that differ from measurements in QT?