Local Volatility Calibration

Stylz · October 7th, 2005, 1:08 am

Two local volatility questions:1. In practice is it common to use the analytic form for local volatility or to construct the function by calibration?2. When calibrating, is there any agreement over the main types of functional forms that perform well in this application?I am mainly speaking about listeds and OTC equity index volatility.Rgds

PaperCut · October 7th, 2005, 1:17 am

Rebonato's book seems to indicate we're better off using the "analytic" approach (Dupire formula) as opposed to a tree approach. I've been working on this recently for the SPX European suface, and one thing I've been trying is using a SABR vol surface from which to calculate the (various) finite differences. This is somewhat contradictory, I think, to Hagan's concept for SABR, but I thought I might give it a try for smoothing purposes. So far my pricing is crap, but then again I haven't been able to give it full attention recently. I'll let you know if I get it right.

Stylz · October 10th, 2005, 3:33 pm

Thx Paper for your reply.Any other suggestions for functional form?Rgds

ldrage · October 11th, 2005, 5:27 am

I'm not answering exactly the question you asked but the following paper may be of interest. It gives a detailed demonstration of how to calibrate a local vol model to observed European option vols.Andersen, L. B.G. and R. Brotherton-Ratcliffe (1998), The equity option volatilitysmile: an implicit finite-difference approach, Journal of Computational Finance, 1,5-37

PaperCut · October 11th, 2005, 12:37 pm

Firms where I've worked have used simple polynomials to fit implieds at a given slice of tau. This gives you the smoothness you need.

Rufus · November 21st, 2007, 12:15 am

QuoteOriginally posted by: PaperCutFirms where I've worked have used simple polynomials to fit implieds at a given slice of tau. This gives you the smoothness you need.Does anyone have this paper in pdf?

Rez · November 21st, 2007, 9:33 am

QuoteOriginally posted by: StylzTwo local volatility questions:1. In practice is it common to use the analytic form for local volatility or to construct the function by calibration?2. When calibrating, is there any agreement over the main types of functional forms that perform well in this application?I am mainly speaking about listeds and OTC equity index volatility.RgdsHi Stylz,The attached examples might help.Cheers, Kyriakos Edt: haha.. just noticed that the mess was from 05 :--)

amit7ul · November 27th, 2007, 4:35 am

how is local vol fitted using dupire(or some other smooth function) for a given expiry used to evolve spot?impled vols are not of much help beyond calibration, i am searching for some good references.

thecount · January 9th, 2008, 9:36 am

PaperCut -- or anyone else -- , do you mean that the interpolation along the strike axis for a given tau is performed by polynomials, and that the interpolation along tau is purely linear -- since we only need the first derivative in that direction? Do you know of anyone who replaces the spline interpolation by the stochastic volatility inspired (SVI) parameterization of Gatheral? And do you happen to know any reference on functional forms -- no matter how many parameters -- which yield a reasonable fit both in the strike and maturity direciton at the same time? Thanks in advance

devito · January 9th, 2008, 3:57 pm

Hey Papercut, I have been trading equity and equity index derivatives in a very illiuid market. I have been using slope and convexity to fit or construct a vol curve. Additionally, just using these two parameters does not really fit the curve. Can you please recommend me additional polynomials to fit the curve. Rez for some reason cannot open that zip fileThanks, D

quant99trader · January 10th, 2008, 1:35 am

I have fitted Gatheral's SVI parameterization to SPX options. It fits every time slice very well, especially when using a vega weighed least squares objective function for your calibration. You need to be careful about parameter restrictions though - it is not guarenteed to be arbitrage free (in the strike direction). Gatheral gives a parameter restriction that supposedly guartees no arb - Hoever, I have never seen a proof of this result and have tried to replicate it with no luck. Anyone have a proof? Calendar spread arbitrage can be detected more easily - you just ned to make sure the total implied variance (implied voil squared * time) is increasing in time. QuoteOriginally posted by: thecountPaperCut -- or anyone else -- , do you mean that the interpolation along the strike axis for a given tau is performed by polynomials, and that the interpolation along tau is purely linear -- since we only need the first derivative in that direction? Do you know of anyone who replaces the spline interpolation by the stochastic volatility inspired (SVI) parameterization of Gatheral? And do you happen to know any reference on functional forms -- no matter how many parameters -- which yield a reasonable fit both in the strike and maturity direciton at the same time? Thanks in advance

thecount · January 25th, 2008, 10:35 am

Thanks quant99trader. Now I am having trouble with the conceptual interpretation of the local volatility formula (for instance page 13 of "The Volatility Surface" by Gatheral). I am examining the Quantlib implementation and I don't know if it is confusing me, if it is wrong, or that I just didn't understand the formula correctly. Let's assume that I have a volatility surface at time t = 0, for K (strikes) and T (times to maturity) for call options w(K,T), where w(K,T) = impliedVol^2(K,T)*T. Now I calculate the local volatility at t = 0 for an option which expires at t = T and strike = k. My question is related to the value of w and its derivatives in the local vol formulaat t = 0what is localVol(S(0),0) ??? what is w and its derivatives for this calculation in the local vol formula?My interpretation is w = w(k,T)For the time derivative, I would calculate w_- = w(k,T-dt) and compute it as dw/dT = (w-w_-)/dtif t = t1what is localVol(S(t1),t1) ??? what is w and its derivatives for this?I think it is w = w(k,T-t1) and I would compute the time derivative in a similar way.Well, the answer in quantlib ist = 0 w = w(S(0),0)t = 1 w = w(S(t1),t1)I am pretty sure I am wrong but I am mentally blocked, so I post this running the risk of making a fool of myself.Thanks in advance

thecount · January 25th, 2008, 12:57 pm

I think I finally figured out I was wrong and that the implementation in quantlib is right, but feel free to comment on it if you're of a different opinion. Let's say that the fact of recovering the functional form of the local vol from the implied volatility surface Impl(K,T), and then replacing the strike (K) and time to maturity (T) by spot (S) and time (t) confused me.

jfuqua · January 25th, 2008, 5:44 pm

The Madan paper can be found at [no I did not put it there so don't send the lawyers----I just found it]:http://www.nuclearphynance.com/User%20F ... ration.pdf ==========="Calibratiing and Pricing with Embedded Local Volatility Models"Opps. Put this in the wrong Forum. Still may be of interest.

thecount · January 28th, 2008, 8:34 am

I have an existential question related to the absence of arbitrage on the implied volatility surface. More specifically I mean avoiding the calendar spreads, by which the implied volatility surface must be upward sloped as a function of time to maturity ( impVol(T1)^2*T1 < impVol(T2)^2*T2 for T2 > T1 with T1 and T2 time to maturity). Apparently, it must not be a very interesting arbitrage possibility, since this week, after all this financial turmoil, I am only seeing downward sloped volatility surfaces. Needless to say, if I use local volatility to model this, I end up with negative squared local volatilities and the whole thing fails. Do you see any chance to model this still within local volatility? You may contend local vol is useless, but I am afraid any stochastic volatility model would be useless as well, since they all yield an arbitrage-free surface. Any suggestions here? Thank you