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I understand that to describe financial data, we build stochastic models and calibrate their parameters to past data. When coming up with new algorithms, we rely on rigorous backtesting to convince ourselves that our future-oblivious algorithm would have worked well in the past. In either case, the past data is our best friend for understanding the future.

However this is paradoxical - eg, for investors, we clearly sound a warning that the past returns are not a guarantee for future performance. This is because our models are non-causal - they describe random data without providing explanations for their causes. For example, the most effective model or algorithm will never be able to predict correctly if the President will make a tweet that can impact the markets.

This brings to my question - what is the actual use of mathematical models in finance? Are they meant to predict the future? Or are they simply meant to provide a convincing explanation of past data? If the latter, why do people (like in the case of Renaissance) claim that mathematical modelling gives them an edge in financial markets?

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All models serve an important purpose: They are an appropriate tool to deal with the uncertainty. You need to base your decisions on something, you can't just guess an option price. Finance models are by no means perfect. But they are helpful.

For instance, a GARCH volatility model is fitted to past data to predict future values of volatility in the real world. This obviously then leads to important risk metrics (VAR and such). So, here you use a mathematical model to predict the future. The high autocorrelation of squared returns hints that this is reasonably possible.

Many option price models do not predict the future. Look at the Heston model. You calibrate it to observed liquid options to obtain the best possible parameter given the current market conditions. Then, you use those numbers to price more complex products for which there is no (liquid) market. Once the market conditions change, you re-calibrate your model. Here, you do not predict future stock movements. You price new options relative to existing products in a way which avoids arbitrage. Remember that the true drift of the asset under the $\mathbb{P}$ measure does not feature in option prices.

Many asset pricing models (C-CAPM, Long-Run Risk & co) try to fit expected future stock returns (under the real-world measure) into a theoretical concept and relate returns on stocks to deeper economic state variables (say consumption growth). This would be a mathematical framework which tries to explain why the assets move how they do.

But people always long for better models. I mentioned the GARCH models earlier. With the available of high-frequency data, researchers and practitioners use model-free elements like realised variance as an alternative. Similarly, newer machine-learning algorithms can be used to find patterns in historical stock returns and help to predict them. In 20 years, we will all be reading about some new interesting method. But all these models, algorithms and approaches have the same goal: Help investors to deal with the natural uncertainty about the future.

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The purpose of mathematical models in both economics and finance isn't to be exactly right, but to structure discussion and assist in decision making.

Some models such as the Black-Scholes-Merton model in option pricing is still widely used by practitioners, in spite of the fact that it is known to be flawed in all matters of ways. However, it has the advantage of simplicity and of presenting problems that are well defined, well known and with which you can try to cope informally. It has also gifted us with a way to think about option prices: implied volatility is computed by asking which volatility level you would need for BSM to match observed prices exactly. This gave us the volatility smile and the knowledge that the slope and level of that beast changes with maturity and across time. This gives you a hint as to how you can improve on BSM: you need to find a way to put more probability density than BSM does in the zones where IV is high.

Now, moving back to the issue of causality, it is not entirely fair. Take again the example of option pricing. Ideally, what you would like to do is to reconcile the cross-section of option prices with the time series of returns of the underlying -- as Bates (1996) pointed out. Although many people force fit their option pricing model directly under the Q measure, caring only about getting the cross-section part right, more recent papers have tried to reconcile both sources of data. The link between both the risk-neutral and the physical measure is the pricing kernel or the stochastic discount factor: this beast soaks up concerns about things like risk aversion as it would invariably emerge in a general equilibrium model. That actually is an economic explanation, something structural which hinges on investor preferences.

This is even more explicit if you look at the more interesting option pricing models: all of them showcase incomplete markets, which means there is an infinite number of ways to eliminate risk from the model. In the earlier literature, people would pick a price for risk by motivating their choice from a general equilibrium model. That is what Heston (1993) did, even if it is a footnote seemingly no one remembers and this is what Duan (1995) did in the option pricing context. When you use the Heston model, for example, you're actually relying on an economic theory behind -- i.e., you know for a fact that there exist some equilibrium that supports your choice of measure. A more recent exception would be Christoffersen, Elkami, Fenou and Heston (2010) that proposed to slightly generalize the work of Duan (1995), but without relying on an equilibrium model. They do not have a formal theory behind the quadratic kernel used in Christoffersen, Heston and Jacobs (2013), but there is an empirical motivation (matching ratios of density estimates) and an informal economic intuition (people value volatility directly).

Clearly, there is a lot more going on here that simply describing statistical patterns. These models can be used to "predict the future," although it depends on what you mean by prediction. They are designed to capture intricate behavior and changes in the higher moments of conditional distributions. If you want to think about the likelihood of something bad happening, they are well suited to do just that. They aren't peculiarly interesting model of the conditional mean -- that's what people usually try to model and predict, at least as far as the squared loss they use in estimation and evaluation suggests.

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  • $\begingroup$ This addresses the part of your question about the type of quant models ren tech builds.Note that in statistical arbitrage ( this is another word for what they do ) , you're not necessarily trying to build causal models. You're trying to predict behavior. This is because, if you can predict behavior, then you can have a way to predict markets. $\endgroup$ – mark leeds Mar 28 at 4:25

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