Why Quantitative Finance Uses Math Suited to Its Models

# Why doesn’t quantitative finance use the kind of advanced math seen in physics


# Why doesn’t quantitative finance use the kind of advanced math seen in physics












If you look at even the most sophisticated quant finance models - such as the black scholes model - you quickly realize that they don't really use higher mathematics. I'm not saying it was trivial to develop this model, of course, and even applying it in the real world may take years of experience. But the underlying math is basically extremely simple. There is a normal distribution and a bit of math with exponential and logarithmic functions - all things that a high school student who is good at math can understand. But if you look at other fields where math gets applied, such as physics or engineering, you learn things like tensor calculus and path integrals in complex vector spaces already in the first semesters. And in state of the art research today, mathematical techniques are used that probably not even Einstein would have understood.

That's why I've been wondering why there are no such sophisticated models in quant finance. Do you think it's because it simply doesn't work/not required at all (after all, the world of finance was invented by humans and not by things like matter or the universe). Or are the people who could apply such sophisticated math are simply not interested in finance but stay at university research?

## Answer by Frido (score 9)

https://quant.stackexchange.com/a/83758

If I wanted to be snarky I'd say that your question indicates you've not been around long enough in quantitative finance and/or physics.

First of all, you don't need advanced maths to make huge progress even in physics. A case in point is special relativity. How advanced was the maths Einstein used to derive the Lorentz transformations really? Not much more advanced than high school maths. Quantum mechanics is similar: with basic knowledge of complex numbers and linear algebra, and some intuitive understanding of Hilbert spaces, you can do quantum mechanics (I'm not talking about quantum field theory). So it's not the advanced maths that matter and drive innovation, brilliant/good idea(s) do.

A wonderful idea in finance is in fact the Black-Scholes-Merton hedging argument. I hope at some point you'll appreciate how brilliant it was of them to combine the concepts of no-arbitrage and hedging/replication to arrive at the BS PDE.

Now let's discuss advanced maths. I suppose you're thinking about methods from for instance differential geometry: in fact these are applied/used in financial mathematics as well. For example heat kernel expansion techniques are used to arrive at tractable analytical approximations for implied volatility.

Yes, maybe the theory of principal/fibre bundles and other concepts are not used but why use these techniques if they're not applicable (yet)? Good research should not be about finding a problem for a solution. It's the other way around.

And to be honest, stochastic calculus (which includes such things as Malliavin calculus), which is the main tool in derivatives pricing, is quite advanced in my opinion. I'm quite sure many a physics graduate will struggle with it if exposed to it for the first time.

Btw, I have a (theoretical) physics background myself and ended up in quant finance, so I have a bit of experience in both areas in regards to the maths involved.

## Answer by D Stanley (score 1)

https://quant.stackexchange.com/a/83764

A non-scientific answer is that many aspects of physics obey laws that are unbreakable, thus you can use more sophisticated methods for better precision.

Finance relies on human behavior which is by no means deterministic, therefore higher math does not necessarily grant you any better answer since there is too much uncertainty.

For example, I can use very sophisticated math to determine the course of a baseball I throw to include gravity, air resistance, spin, wind, etc. There's no way to precisely measure what will happen once it contacts a bat that someone is swinging. I can model it in terms of probability distributions, but the extra precision obtained by better math after that point is useless.

I could use a third-order differential equation to try and predict the future price of a stock, but that precision is useless when that price is determined not by fundamental laws but by human decisions to buy or sell it.

## Answer by Rylan (score 0)

https://quant.stackexchange.com/a/83767

Truthfully I am not very knowledgeable about physics, but hopefully I can shed some light on why quant math might be "disappointingly simple" for some.

I disagree with the assessment that the underlying math in Black-Scholes is the computation of the expectation (which is how I'm interpreting your comment "a normal distribution and a bit of math with exponential and logarithmic functions". To me, the math is the hedging argument, and why that argument allows us to compute value of the hedging portfolio as a relatively straightforward expectation. Whether a high-school student should be able to understand that, I'm not going to comment, but I certainly interview a number of masters students from highly-ranked schools who miss that point.

Even if Black-Scholes is easy, there's much more complicated models in terms of

- The dynamic of the underlying, and/or

- The payoff.

For 1, it's worth remembering that if you are a desk quant, your job is to help build quantiative tools for the traders. In a lot of cases (though not all), a trader will prefer a simpler, more "robust" model like Black-Scholes to a more complicated but more "fragile" model even if it captures the dynamics of the underlying better.

For 2, you will likely hear less about these because vanilla puts and calls are much more commonly traded than any other options. I work in commodities, so I can't speak for other fields, but we have gas storage contracts which can be quite interesting to value. But they likely won't ever come up for people who aren't specifically interested in commodity derivative pricing.

Lastly, it's always worth remembering the incentives not to publish. If I found a grand, unifying theory of all financial markets that could be used to price any security or predict any stock, I would never tell anyone and just use it to become unfathomably rich. (Maybe others are less greedy than me...)