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AnnaBegins
Topic Author
Posts: 58
Joined: January 9th, 2004, 8:27 am

### Local volatility

Can anyone explain what the term 'local volatility' means?Also, how (if at all) is it related to the implied volatility surface?

player
Posts: 2290
Joined: August 5th, 2002, 10:00 am

### Local volatility

SPAAGG
Posts: 785
Joined: March 21st, 2003, 1:31 pm

### Local volatility

check paper by Dupire, 1995, at paribas. He explains very well the link

Sizous
Posts: 5
Joined: April 21st, 2004, 4:14 pm

### Local volatility

Try look ivolatility.com

DrEvil
Posts: 36
Joined: April 22nd, 2004, 10:17 pm

### Local volatility

The 'local volatility' at price S and time t is a function v(S,t) such when the process dS/S = r dt + v(S,t) dZ is simulated, all option prices match those seen in the market. You can get this function either with the Dupire formula (partial derivatives of the implied vol surface) or by calibration (more reliable).

AnnaBegins
Topic Author
Posts: 58
Joined: January 9th, 2004, 8:27 am

### Local volatility

Thanks DrEvil.Ive actually implemented a version of Dupires formula (taken from Rebonato's book, Volatility and Correlation) which takes the implied volatility surface and returns a local vol surface.Its not working though.... If i put in a flat implied surface of 20% vols my model is returning an (almost) flat surface of around 4%. This should return a flat 20% surface.... or I am going crazy?---------------------------------------Its Friday and Im in love!

DrEvil
Posts: 36
Joined: April 22nd, 2004, 10:17 pm

### Local volatility

Is that 4% vol or 4% variance (vol^2)? It seems awfully coincidental that (20%)^2 = 4%...

AnnaBegins
Topic Author
Posts: 58
Joined: January 9th, 2004, 8:27 am

### Local volatility

Ah ha!Goin to go and count my 10 brain cells.Cheers.

asd
Posts: 663
Joined: August 15th, 2002, 9:50 pm

### Local volatility

QuoteOriginally posted by: DrEvilThe 'local volatility' at price S and time t is a function v(S,t) such when the process dS/S = r dt + v(S,t) dZ is simulated, all option prices match those seen in the market. I have a confusion to what is referred in Dr.Wilmott's text - chapter on implied vol in 'Quantitative Finance' book.It mentions that implied vol. is an estimate of future realized volatility at a point. (If I have understood properly! ) My confusion is, should not the market option prices reflect the realized local volatility v(S,t) + uncertainity component in estimation of future volatility ? Could someone please help?Thanks,Asd
Last edited by asd on April 23rd, 2004, 10:00 pm, edited 1 time in total.

DrEvil
Posts: 36
Joined: April 22nd, 2004, 10:17 pm

### Local volatility

QuoteOriginally posted by: asdQuoteOriginally posted by: DrEvilThe 'local volatility' at price S and time t is a function v(S,t) such when the process dS/S = r dt + v(S,t) dZ is simulated, all option prices match those seen in the market. I have a confusion to what is referred in Dr.Wilmott's text - chapter on implied vol in 'Quantitative Finance' book.It mentions that implied vol. is an estimate of future realized volatility at a point. (If I have understood properly! ) My confusion is, should not the market option prices reflect the realized local volatility v(S,t) + uncertainity component in estimation of future volatility ? They do; local volatility is for an instantaneous moment in time conditional on a price, and implied volatility is for a finite term of time and price (say now to expiration and strike to infinity). The relationship between IV and LV is like that between a sport interest rate and a forward (instantaneous) interest rate.

AnnaBegins
Topic Author
Posts: 58
Joined: January 9th, 2004, 8:27 am

### Local volatility

Can I just confirm.....Right now, we have an implied volatility surface. Using Dupires formula we can get the local vol surface.If we then simulate a stochastic process for equities, all option prices that are calculated from the monte carlo simulation should equal the market prices.Would this work for a more complicated equity process that dS/S = r dt + v(t,S) dZ ?-------------------------In August and Everything After, I am after Everything.