Financial Modeling: A Warning About Models in Practice

We are now entering Part 5 of Wilmott’s book: Advanced Topics. Everything so far was classical foundation. Lognormal random walks, Black-Scholes, delta hedging, portfolio theory. Well-established stuff. From here on out, we go beyond the standard model and into territories where things get interesting, controversial, and sometimes dangerous.

But before diving into the advanced material, Chapter 45 is a pause. A warning label. Wilmott steps back to talk about how modeling actually works in practice, and it is not pretty.

Part 5: What Is Coming

All the Black-Scholes assumptions are wrong to some degree. The rest of the book is about relaxing those assumptions and building better models. Some ideas fit neatly into the existing framework. Others take us far from anything conventional.

One key thing to watch: whether a model is linear or non-linear. If it is linear, the value of a portfolio is the sum of values of individual parts. If it is non-linear, it is not. Non-linear models capture important features that linear models miss entirely. This will come up again and again in the chapters ahead.

The upcoming chapters cover discrete hedging, transaction costs, stochastic volatility, uncertain parameters, jump diffusion, crash modeling, speculating with options, static hedging, feedback effects in illiquid markets, utility theory, advanced American options, dividend modeling, and more. It is a big menu.

The Four Rules of Practical Modeling

Wilmott has seen how modeling works in the real world, and he distills it into four honest (and slightly depressing) rules.

Rule 1: The Simpler the Model, the More Popular

This one Wilmott actually agrees with. Simplicity is a virtue. A simple, transparent model like Black-Scholes can be more robust in practice than a sophisticated model with many parameters. When something goes wrong, you want to understand why. Simple models let you do that.

Einstein said: make things as simple as possible, but no simpler. Wilmott adds: if you must complicate things, do so one step at a time.

Rule 2: Closed-Form Solutions Beat Numerical Solutions

This one bothers Wilmott. Everyone has massive computing power now. There is no reason to be afraid of numerical methods. But people are. They love formulas they can type into Excel. They distrust anything that requires actual number crunching.

The popularity of the binomial method proves this point. It is the simplest possible numerical method, and yet people use it precisely because it is the most accessible. Not because it is the best.

Rule 3: Never Test Your Model with Real Data

This one shocks Wilmott, who analyzes data every day. But the logic in practice is clear. All financial models are inaccurate. Everyone knows this. So what is the upside of proving it? If you test your model and show it is wrong, you have not gained anything useful. You have just made yourself look bad.

The result: very few people on the sell side of investment banks bother with actual testing. Wilmott suspects that hedge funds (the buy side) are somewhat more sensible about this.

Rule 4: People Do Not Like Models That Are Different

Keynes said it best: “It is better to fail conventionally than to succeed unconventionally.” Most models in quant finance are of the Brownian motion / diffusion / parabolic PDE type. They all live in the same mathematical world. Nobody feels threatened by them.

If you build a model that lives in a different mathematical world, even if it works better, people will not use it. It is too unfamiliar. Too risky for careers. If quantitative finance ever had scope for creativity, human nature squashes it.

The Find-and-Replace School of Modeling

This is Wilmott at his most sarcastic, and he earns every bit of it.

In 1973, Black, Scholes, and Merton published their options pricing work. They started with a stochastic differential equation for a stock price $S$ and derived an equation for pricing options. Wilmott gives them seven out of ten for accuracy. Great work.

In 1977, Vasicek applied the same ideas to interest rates. He used a stochastic differential equation for the short rate $r$. Same mathematical framework, same type of equations. Five out of ten for accuracy. Interest rate modeling is harder.

Then in the following decade, people applied the same ideas to credit risk. They modeled the hazard rate $p$ (instantaneous risk of default) using a stochastic differential equation. And the resulting models were not just similar to the interest rate models. They were exactly the same. The only difference was that everywhere there was an $r$ it was replaced by a $p$.

Wilmott calls this the Find-and-Replace School of Mathematical Modeling. Open your old interest rate paper in a word processor, hit Edit then Replace, swap $r$ for $p$, and you have a brand new credit risk paper. Double your CV length overnight.

The Modeling Escalation

Wilmott describes the lifecycle of a quant model with biting accuracy.

Stage 1: Determinism. You start by pretending everything is predictable. The stock is $100 today and will be $137 in two months. You can answer questions. But two months later, the stock is not $137. Your model is not just wrong, it is demonstrably wrong. And the eleventh commandment is “Don’t Get Caught.”

Stage 2: Stochasticity. You introduce randomness. Model the stock as a random walk. This is the Black-Scholes world. It works well for years.

Stage 2 Prime: Iterate. Eventually your model gives prices that differ from other banks. Solution: make something else random. Volatility is the natural choice. Now you have stochastic volatility. Extra variable, extra parameters, more degrees of freedom to match any price you want. Bonus: since volatility is unobservable, nobody can prove your model is wrong without a lot of statistical work.

This buys you a year or two. Then iterate again. Add a third stochastic variable. Credit risk, maybe. More variables, more parameters. You can match anything.

Wilmott notes that three is currently the optimal number of stochastic variables. Not for any scientific reason. Three makes you look “sophisticated.” Four means you have lost the plot.

Stage 3: Jumps. When three diffusive variables are not enough, add Poisson jumps. Let one of your variables gap. You now have a jump-diffusion process. Back on top.

Stage 3 Prime: Iterate. One jumping variable not enough? Add jumps to volatility too. Now you have stochastic asset and stochastic volatility, both with jumps. So many parameters. Surely that will do.

Wilmott asks: “I’m not sure, did that convey the right amount of sarcasm?”

The Real Problem

The issue is not that these models are mathematically wrong. Each extension genuinely captures something that simpler models miss. The problem is the motivation. People do not add complexity because they have analyzed data and found a real-world effect they need to capture. They add complexity because they need to match the prices their competitors are quoting. The goal is not accuracy. It is career safety.

This chapter is a warning to anyone entering the field: understand why you are using a model. Is it because it captures reality well? Or because everyone else uses it? Those are very different reasons, and they lead to very different outcomes.

Now that Wilmott has properly warned us, the rest of Part 5 dives into the actual advanced models. Starting with everything that is wrong with Black-Scholes.

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