When pricing options, one must always be careful to model the implied volatility of the instrument appropriately. This is not actually particularly easy, because implied volatility is, by definition, not observable in the market. It is essentially a value that, when you plug it into your pricing model, returns back the actual market value of the instrument you are trying to price. A badly devised or poorly understood volatility model is therefore costly to the user, because it will result in incorrect valuations of the instrument itself, and costly hedging against risks in the process.

Finding an “appropriate” way of accounting for implied volatility in the context of option pricing is a challenging business. The classic Black-Scholes option pricing formula is a truly brilliant construction, as is its simplified Black-1976 formulation for pricing options on forwards and futures. But these OG models make some quite strong assumptions about implied volatility. We are in no way disrespecting the genius of the men who devised that original model when we admit that those assumptions just do not correspond to market realities.

When Pretty Theory Meets Messy Reality

Chief among the assumptions used in the Black-Scholes (1973) and Black (1976) pricing models is the view that the implied volatility does not change based on the strike price of the option, or on the maturity of the option itself. While this was true when the model itself was original devised, it is assuredly not true today, and indeed has not been true since the late 1980s. Ever since then, we have seen that, in most asset classes, there exist very clear differences between implied volatilities at different strikes for the same underlying asset.

A number of different solutions have been proposed to deal with this problem. Because of my background in capital markets and finance, I happen to be pretty familiar with the approach proposed by Hagan, Lesniewski, Woodward, and Kumar in the classic 2002 paper, Managing Smile Risk. Everything that follows is a very simplified and, hopefully, easily understood examination of that alternative method.

Motivations and Key Questions

More than 10 years ago, I had the opportunity to offer my knowledge and experience to students in the Columbia University Graduate School of Arts and Sciences (GSAS) MA Mathematics of Finance programme. What you see in this post, and in all of the accompanying materials, is a direct evolution of the knowledge that I shared in that seminar.

I graduated from that course all the way back in 2007 – right before the worst financial crash of our lifetimes. (As anyone who knows me well can personally attest, my sense of timing is not, generally, particularly good.) Over the next 10 years, as I gained knowledge across a variety of asset classes, particularly within the fascinating and very complex world of interest rate derivatives, I came to understand the Stochastic Alpha Beta Rho Nu (SABR) model reasonably well – or so I thought.

For, as any teacher can tell you, it is one thing to learn something from reading a paper or a book. It is entirely another matter to explain that idea to someone else who hasn’t got a clue. My first Practitioner’s Seminar was, in my opinion, a moderate success – I always felt that I could have done better. My second go-around was better, but I still felt that I did not do a good enough job of actually explaining how the model itself is calibrated.

And that is the core issue. The SABR model, as explained in the Hagan et al paper, is another brilliant concept, but the paper is awfully short on details of how to calibrate and run the actual model itself. You kind of have to figure it out on your own.

That is where this post comes in. Everything that you see here is designed to help you understand and implement this model for yourself. Whether you are a casual reader, a student of mathematical finance, or an experienced practitioner, you will come away from this with the tools and techniques to implement the model yourself.

This post, then, and all of the accompanying material, is essentially a pet project designed to show you how this model really works, where and how it can be used, and why it is so powerful and effective that virtually every bank trading non-linear products uses it.

Explaining the Idea

The following long video explains the entire model and how it is used, without going into the deep technical details:

You can download the actual slides in PDF format right here. But, in order to do anything with this, you need to be able to see it and run it for yourself. And that is where the coding aspect of this model comes in.

The actual R code that I created can also be downloaded from my site – just click here to download a source-code tarball. This file can easily be installed in R for Windows, Mac OS X, or Linux using R Studio – see here for straightforward instructions. It is also easy to install and use if you are using the R program in Windows alone. The vignette included within this package is an HTML file that shows you actual code and results from the package itself, along with some very nice plots, so you can easily see how everything fits together.

Basics of the SABR Model

At its core, the SABR model simply says that we should relax some of the stronger assumptions behind the Black-76 option pricing model. It treats the implied volatility as a random variable in its own right, which follows a reasonably smooth and predictable process, and says that the forward rate and implied volatility are coupled together, or correlated. That model can then be put into a near-analytical form and used to derive market-equivalent volatilities by calibrating a set of four input parameters:

• Beta, which affects the distribution of forward rates and is provided BEFORE the calibration step;
• Rho, which defines the strength of the relationship between forward rates and volatilities;
• Nu, which defines the volatility of the implied volatility itself;
• Alpha, which defines the volatility of the forward rate;

While the mathematics seem forbidding when you see the equations in the slides, you will quickly realise that, if you dispose of all of the jargon and symbols, it really isn’t that hard to figure out what is going on.

The actual hard part involves calibrating the model. Without going deep into the specifics here, you need to use a non-linear optimisation procedure to test different values of the three calibration inputs above, until the difference between the model-computed values of implied volatility match those that market participants actually use, as closely as possible.

And, in fact, this isn’t so hard at all. You just need a decent non-linear optimisation programme and, possibly, a root-solver. All of this is implemented in my code, which you can examine and unpack entirely for free, at your own leisure.

Benefits and Costs

The upshot of using the SABR model is that it results in clean, market-consistent implied volatlities that, in turn, result in stable and appropriately priced hedges. And, because the equation that governs the relationship between lognormal volatility and the SABR core parameters is essentially analytical in nature, all you have to do is plug in a bunch of calibrated constants, get a number back, and then plug that number straight into your standard option-pricing formula, to get a fast and simple quote for your option price.

These features explain exactly why the SABR model is an industry-standard approach to calculating implied volatility across multiple asset classes. It is heavily used for pricing interest rate swaptions, where the at-the-money quotes for volatility are all-important, as well as for options on FX, equity, commodity, and other futures.

On a personal note, the bottom line is that, if you can walk into an interview with a Market Risk or Product Control or Valuations team at any large bank, and you can intelligently and articulately explain the SABR model properly, you will go a very long way toward making yourself look very, very good.

Limitations and Issues

That is not to say that the SABR model works well all the time and everywhere. There are known issues with using this model for pricing illiquid high-strike options in second-order applications, such as for pricing CMS products or step-up Bermudans. I go into these issues in my presentation and the video above, which should help you understand where and when and how to use the toolkit that I have provided appropriately.

Moreover, there are certain times when the calibration process itself can fail. I encourage you to download the R code, examine it for yourself, and test it out. The vignette in the package was written rather in the tradition of the best TV chefs, by essentially saying, “here’s something I prepared earlier”, which looks impressive – but, in reality, quite a lot of hard work and trial-and-error went into writing that code. Only by testing and trying it for yourself can you be sure that you really UNDERSTAND what is going on.

Conclusions

The SABR model is an industry-standard approach to pricing options that anyone looking to get into the technical side of banking should try to understand and unpack. The model itself is one of those odd things in life that look really complicated from a distance, but turn out to be actually very fast, simple, and easy to use when you just sit down and examine them closely. The model itself is not difficult at all to use, as long as you do it intelligently and with respect for the limitations of the framework and approach. Ultimately, understanding it well will be of great help to you if you ever find yourself in a job interview at a bank or hedge fund for a technical role. And for that reason alone, I recommend becoming familiar with it, even if you are not overly fond of mathematics in general.