What is Asymptotic Analysis and How is it Used in Financial Modelling?
Asymptotic analysis is about exploiting a large or small parameter in a problem to find simple(r) equations or even solutions. You may have a complicated integral that is much nicer if you approximate it. Or a partial differential equation that can be solved if you can throw away some of the less important terms. Sometimes these are called approximate solutions. But the word 'approximate does not carry the same technical requirements as 'asymptotic.
The SABR model is a famous model for a forward rate and its volatility that exploits low volatility of volatility in order for closed-form solutions for option prices to be found. Without that parameter being small we would have to solve the problem numerically.
Asymptotic analysis is about exploiting a large or small parameter to find simple(r) solutions/expressions. Outside finance asymptotic analysis is extremely common, and useful. For example, almost all problems in fluid mechanics use it to make problems more tractable. In fluid mechanics there is a very important non-dimensional parameter called the Reynolds number. This quantity is given by
where p is the density of the fluid, U is a typical velocity in the flow, L is a typical lengthscale, and x is the fluid's viscosity. This parameter appears in the Navier-Stokes equation which, together with the Euler equation for conservation of mass, governs the flow of fluids. And this means the flow of air around an aircraft, and the flow of glass. These equations are generally difficult to solve. In university lectures they are solved in special cases, perhaps special geometries. In real life during the design of aircraft they are solved numerically. But these equations can often be simplified, essentially approximated, and therefore made easier to solve, in special 'regimes.' The two distinct regimes are those of high Reynolds number and low Reynolds number. When Re is large we have fast flows, which are essentially inviscid to leading order. Assuming that Re " 1 means that the Navier-Stokes equation dramatically simplifies, and can often be solved analytically. On the other hand if we have a problem where Re <§; 1 then we have slow viscous flow. Now the Navier-Stokes equation simplifies again, but in a completely different way. Terms that were retained in the high Reynolds number case are thrown away as being unimportant, and previously ignored terms become crucial.
Remember we are looking at what happens when a parameter gets small, well, let's denote it by e. (Equivalently we also do asymptotic analysis for large parameters, but then we can just define the large parameter to be 1/e.) In asymptotic analysis we use the following symbols a lot: 0(), o()and ~. These are defined as follows:
In finance there have been several examples of asymptotic analysis.
Transaction costs are usually a small percentage of a trade. There are several models for the impact that these costs have on option prices and in some cases these problems can be simplified by performing an asymptotic analysis as this cost parameter tends to zero. These costs models are invariably non linear.
This model for forward rates and their volatility is a two-factor model. It would normally have to be solved numerically but as long as the volatility of volatility parameter is small then closed-form asymptotic solutions can be found. Since the model requires small volatility of volatility it is best for interest rate derivatives.
Fast drift and high volatility in stochastic volatility models
These are a bit more complicated, singular perturbation problems. Now the parameter is large, representing both fast reversion of volatility to its mean and large volatility of volatility. This model is more suited to the more dramatic equity markets which exhibit this behaviour.
References and Further Reading
Hagan, P, Kumar, D, Lesniewski, A & Woodward, D 2002 Managing smile risk. Wilmott magazine, September
Rasmussen, H & Wilmott, P 2002 Asymptotic analysis of stochastic volatility models. In New Directions in Mathematical Finance (eds Wilmott, P & Rasmussen, H). John Wiley & Sons Ltd
Whalley, AE & Wilmott, P 1997 An asymptotic analysis of an optimal hedging model for option pricing with transaction costs. Mathematical Finance 7 307-324