# Measure theory in quantitative finance

When I read up on stochastic modeling, the use of "measure" comes up a lot. So far I just read the word "measure" as "probabilities" or "distribution" and was able to get away with it when trying to understand informally the concept and results of various models (Black, Dupire, Hull White..), at a basic level.

I must admit that I was not able to follow the rigorous proofs of every step in their derivation.

Can anyone tell me what is the significance of measure theory in stochastic modeling?

I do understand that the definition of a probability measure is more general and rigorous than just saying "probabilities". But apart from a rigorous definition, was there some useful results from measure theory that are often used in stochastic modeling?

Thanks very much and pardon my ignorance.

• The Borel-Cantelli Lemma, the Kolmogorov Zero One Law, the various forms of stochastic convergence cannot really be formulated or proven except with Measure Theory. Commented Aug 2, 2018 at 20:20
• Measure theory does not just "add rigour", it provides new results and tools for dealing with infinite sets and infinite sequences, etc. in aprobaility context. This was is a great advance in Probability Theory for which we can thank academicain A. N. Kolmogorov. Commented Aug 2, 2018 at 20:25
• OTOH if you just trade options all day long you probably don't need it Commented Aug 3, 2018 at 16:56

Measure theory helps us overcome some of the drawbacks of constructing measures (measure of probability when ranged at $$[0,1]$$). Classic probability theory is effective for probability models whose sample space $$\Omega$$ is a finite or countable set ($$P: 2^{\Omega} \to [0,1]$$). But for uncountable sets, such as $$\mathbb{R}$$ (which is uncountable) the construction falls short. Measure theoretic probability, thanks to H.Lebesgue and C. Carathéodory enables measure construction on uncountable sets. Definition of a measure $$\mu$$, requires firstly the definition of the outer measure $$\mu ^*.$$ An outer measure is a mapping $$\mu^*:2^{\Omega} \to [0,+ \infty)$$:

1) $$\mu^*(\emptyset)=0$$

2) if $$A \subset B$$, then, $$\mu^*(A) \leq \mu^*(B)$$

3) if $$\{A_i\} \in 2^{\Omega}$$, a sequence of events, then $$\mu^*(\bigcup A_i) \le \sum_i \mu^*(A_i)$$

We would like to define a subset $$\mathcal{F} \subset 2^{\Omega},$$ such that 3) becomes an equality (This property is called countable additivity). Using Carathéodory's condition we are able to construct such a set $$\mathcal{F}$$, which is called a $$\sigma-$$algebra. A measure is $$\mu$$ is just the outer measure restricted in $$\mathcal{F}$$, $$\mu=\mu^{*} \mid_\mathcal{F}$$.The next step is to define the smallest $$\sigma-$$algebra, which is the intersection of all $$\sigma-$$algebras. This set is the Borel algebra. We continue on this manner.

Measure theory provides more robust methods for integration in measureable functions, almost everywhere convergence, Fatou's Lemma, Borel Cantelli Lemma and so on.

Scholes and Merton 1973 originally derived the Black-Scholes equation with Partial Differential Equations -- they assume continuous-time uninterrupted trading and nonarbitrage, and find a PDE that can be solved with some hardcore (but engineering-type, i.e. not dizzyingly abstract, just really really hard to do) maths.

Later Black-Scholes was reformulated in terms of stochastic differential equations. So instead of assuming continuous trading you assume that stock (log-)prices behave like Brownian motion $$d\log(S) = \mu + \sigma dW_t$$ (which is more or less the same because Brownian motion is the continuous-time limit of all random walks) with drift. But to specify a drift parameter is to make an assumption about the future behavior of stocks, which of course doesn't work.

But why doesn't it work? Arbitrage, of course. And arbitrage was built in the Scholes-Merton PDE approach. What do we do to reflect this in the world of stochastic processes? We change the measure.

Everything in "naïve probability theory" is implicitly defined in terms of an "ambient" measure $$\mathbb P$$ that in finance is called the "historical" or "physical" probability measure. It's effectively the probability that holds in the real world. But as it turns out (Girsanov's theorem), there is a probability measure $$\mathbb Q$$, the "risk-free measure" where you can change the drift of your process from its value under the physical measure to something else. With some additional assumptions and math that defines a portfolio and arbitrage opportunities, you get that you should be using the risk-free interest rate as the drift under $$\mathbb Q$$. Then you solve the equation without making assumptions about the physical drift.

There's actually trading lore of Black-Scholes-like models in use by traders well before 1973, but these all used Brownian motion with drift -- and who can really claim that the stock market moves together with risk-free rates? It turns out that thanks to the Girsanov theorem and portfolio processes we can do the whole math as if we lived in the risk-free world.

At risk of stating the obvious, common sense is by any measure - pun intended - more important than knowing the details of measure theory if you want to trade options. Quite a few results that have been derived using measure theory were actually known to traders already by just applying common sense and no arbitrage principles. I am not downplaying the importance of measure theory here, but imho there is a tendency to over-engineer financial engineering these days. It is and remains an applied science, it's not pure mathematics or theoretical physics.