Serving the Quantitative Finance Community

 
User avatar
Paul
Topic Author
Posts: 6604
Joined: July 20th, 2001, 3:28 pm

What is asymptotic analysis and how is it used in financial modeling?

March 1st, 2005, 3:48 pm

..such as SABR. Inspired by terrorbyte.P
 
User avatar
J
Posts: 1
Joined: November 1st, 2001, 12:53 am

What is asymptotic analysis and how is it used in financial modeling?

March 10th, 2005, 2:42 am

Very good question!!!!I am not an expert in asymptotic analysis. "Martingale" may be one of good candidates to give the best anwer.My understandiing is that asymptotical analysis is to analyze the limite behavior of a random experiment. Using MC simulation to comput the expectation of payoff is an example I can think.
 
User avatar
Cuchulainn
Posts: 20254
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

What is asymptotic analysis and how is it used in financial modeling?

March 11th, 2005, 8:17 am

QuoteOriginally posted by: JVery good question!!!!I am not an expert in asymptotic analysis. "Martingale" may be one of good candidates to give the best anwer.My understandiing is that asymptotical analysis is to analyze the limite behavior of a random experiment. Using MC simulation to comput the expectation of payoff is an example I can think.Under AA do you mean Singular Perturbation problem in which you have SDE with different time scales for example? For example, a PDE with very small VOL is effectively a 1st order deterministic equation, that kind of idea.Do you have specifics?
 
User avatar
Paul
Topic Author
Posts: 6604
Joined: July 20th, 2001, 3:28 pm

What is asymptotic analysis and how is it used in financial modeling?

March 11th, 2005, 11:49 am

Asymptotic analysis is about exploiting a large or small parameter to find simple(r) solutions/expressions. You may have a complicated integral that is much nicer if you approximate it. Or a pde that can be solved if you can throw away some of the terms. In finance there have been several examples:1. Transactions costs. You can find nonlinear pdes for option prices when there are costs of hedging. The pdes can often be simplified if you exploit the smallness of costs.2. SABR. Pat Hagan's model assumes small drift and vol of rates so you can do an expansion like Taylor series.3. Fast drift and high vol in stochastic vol models. These are a bit more complicated, singular perturbation problems. To be discussed another time perhaps!P
 
User avatar
Cuchulainn
Posts: 20254
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

What is asymptotic analysis and how is it used in financial modeling?

March 11th, 2005, 12:17 pm

QuoteOriginally posted by: PaulAsymptotic analysis is about exploiting a large or small parameter to find simple(r) solutions/expressions. You may have a complicated integral that is much nicer if you approximate it. Or a pde that can be solved if you can throw away some of the terms. In finance there have been several examples:1. Transactions costs. You can find nonlinear pdes for option prices when there are costs of hedging. The pdes can often be simplified if you exploit the smallness of costs.2. SABR. Pat Hagan's model assumes small drift and vol of rates so you can do an expansion like Taylor series.3. Fast drift and high vol in stochastic vol models. These are a bit more complicated, singular perturbation problems. To be discussed another time perhaps!PThis ie VERY interesting. Do you have any articles? I suppose this is well-known knowledge?BTW my research on FDM 1979 used SP expansion for BS-type equations with a small parameter in the Gamma derivative. Funny that things repeat themselves . In PDE small VOL is the most tricky in those days.
 
User avatar
Paul
Topic Author
Posts: 6604
Joined: July 20th, 2001, 3:28 pm

What is asymptotic analysis and how is it used in financial modeling?

March 11th, 2005, 12:30 pm

Small vol is tricky for the numerics but is quite straightforward for the asymptotics (there'll be a regime around ATMF where you'll have to zoom in).Hagan, P, Kumar, D, Lesniewski, A & Woodward, D 2002 Managing smile risk. Wilmott magazine September 2002Rasmussen, H & Wilmott, P 2002 Asymptotic analysis of stochastic volatility models. In "New Directions in Mathematical Finance," Ed. Wilmott, P & Rasmussen, H, John Wiley & SonsWhalley, AE & Wilmott, P 1997 An asymptotic analysis of an optimal hedging model for option pricing with transaction costs. Mathematical Finance 7 307-324P
 
User avatar
Cuchulainn
Posts: 20254
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

What is asymptotic analysis and how is it used in financial modeling?

March 11th, 2005, 12:39 pm

thxBTW my expontially fited scheme are specially made for small VOL (in CFD this is boundary layer behaviour).
 
User avatar
satishv
Posts: 0
Joined: February 11th, 2005, 10:51 am

What is asymptotic analysis and how is it used in financial modeling?

March 23rd, 2005, 1:20 am

QuoteOriginally posted by: PaulAsymptotic analysis is about exploiting a large or small parameter to find simple(r) solutions/expressions. You may have a complicated integral that is much nicer if you approximate it. Or a pde that can be solved if you can throw away some of the terms. -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------This is a pretty good explanation by paul . One very basic example which is used again and again in financial math is the calculation of present value based on continious compounding i.e PV = exp(-rt) i.e e raised to the power of -rt. Lets take a look at the definition of e the base of a natural logarithm at the following link:Here you will find an applet which will allow you to change the independent variable x and see the effecton the dependent variable yTry increasing the value of x and observe what happens to y for large values of x (>200)? expCan you envision a stage where y will stop increasing in a meaningfully significant way for a corresponding increase in x?Going through this thought process is one example of asymptotic analysis.btw, this is my first post here. so the curiosity about how it will be received is there. let me know. tx
Last edited by satishv on March 22nd, 2005, 11:00 pm, edited 1 time in total.
 
User avatar
seanster
Posts: 0
Joined: December 10th, 2006, 12:32 pm

What is asymptotic analysis and how is it used in financial modeling?

November 3rd, 2008, 1:17 pm

2 pieces of work come into mind:hagan with sabrroger lee with his work on moments of rr/flies