218 Frequently Asked Questions In Quantitative Finance
We say that f( )∼g( )as → 0
if lim
→ 0
f( )
g( )
= 1.
In finance there have been several examples of asymp-
totic analysis.
Transactions costs: 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 per-
forming an asymptotic analysis as this cost parameter
tends to zero. These costs models are invariably non
linear.
SABR: This model for forward rates and their volatil-
ity 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 asymp-
totic 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 prob-
lems. Now the parameter is large, representing both
fast reversion of volatility to its mean and large volatil-
ity 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 Man-
aging smile risk.Wilmottmagazine, September
