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Dzertyx
Topic Author
Posts: 2
Joined: November 22nd, 2011, 9:59 am

### Volatility surface interpolation

Hello, I was used to interpolate the volatility surface (in the strikes dimension) using a linear interpolation on the volatility. It seems that some people prefer to use a linear interpolation on the variance and using a logarithmic moneyness.What is the reason behind that ? The interpolated surface is better generated this way ?Is there any conventions to interpolate the surface in the maturity dimension ? Why ?Thanks.

pimpel
Posts: 363
Joined: May 12th, 2006, 5:26 pm
Location: Warsaw

### Volatility surface interpolation

In the maturity dimension usually linear interpolation of the full variance is used. Additionally weights are used to give more/less importance to special economic dates (e.g. announcment of unemployment rates, CPI etc.), weekends or holidays.Regarding strikes - I don't think you are right. There are many other methods used, just dig the forum. Depending on the market, I guess that that would be something like SVI, SABR, vanna-volga, spline.

gergely
Posts: 16
Joined: February 19th, 2004, 8:10 am

### Volatility surface interpolation

To prevent calendar-spread arbitrage, the total variance should be strictly increasing. So fitting a linear function is a crude way of preventing calendar spread arbitrage. However, if your surface is more complex you should use monotone splines.Why people prefer to use moneyness or log-moneyness is described here:www.fea.com/resources/pdf/risk_desk_imp ... dfQuoteThe basic assumption of this approach is that, while the implied volatility as a function of strike does not adequately capture volatility market movements, the implied volatility as a function of "moneyness" parameter does. For example, if yesterday an option with strike K was in the money today it might be out of the money due to a movement in the market forward curve. Hence yesterday's associated implied volatility is not today's correct implied volatility. In essence, due to a move in the market forward curve we should move along the smile.Also, the function you use to model the IVS depends on what you want to do with the implied volatility surface and how liquid your data is.For example, Figlewski (2009) points out that cubic splines do not work if you want to extract the risk-neutral probability distribution from option prices. He uses a fourth order spline with one knot:http://papers.ssrn.com/sol3/papers.cfm? ... 492However, if you want to incorporate some info from other maturities and do not need an arbitrage-free surface, thin-plate splines work well:http://papers.ssrn.com/sol3/papers.cfm? ... 911147Note that if you fit a nonlinear basis you have to apply some transformation to the strike prices. This is another reason why people use moneyness. Also, none of the above methods gives you an arbitrage-free IVS. If you need an arbitrage-free surface, see this recent thread:http://wilmott.com/messageview.cfm?cati ... d=81646Two good reviews of modeling the IVS are here:SHORT ONE:http://papers.ssrn.com/sol3/papers.cfm? ... 922441LONG ONE (very comprehensive) http://papers.ssrn.com/sol3/papers.cfm? ... 567Finally, if your goal is to model the local volatility surface, you might be better off by fitting a parametric or nonparametric local volatility surface by minimizing the nonlinear loss function (instead of fitting an IVS and using Dupire's equation).http://www.iijournals.com/doi/abs/10.39 ... 0310011069
Last edited by gergely on November 24th, 2011, 11:00 pm, edited 1 time in total.

Dzertyx
Topic Author
Posts: 2
Joined: November 22nd, 2011, 9:59 am

### Volatility surface interpolation

Thanks a lot for your replies

MCarreira
Posts: 1724
Joined: July 14th, 2002, 3:00 am

### Volatility surface interpolation

And log-moneyness is closely related to delta; hence it's useful in FX (quoted on delta).

LH1984
Posts: 12
Joined: November 4th, 2014, 7:37 am

### Volatility surface interpolation

Hi Pimpel, Could you elaborate a bit of of interpolating across maturity dimension? What is T? For example, if I have two impvol, A and B for month and 3 months receptively, do I just linear interpret the two month impvol as (A^2 + B^2)/2 ?Thanks in advance.

Jhammer
Posts: 37
Joined: February 24th, 2014, 8:05 pm

### Volatility surface interpolation

Here is the simple solution for linear interpolation of total vol in your example:$$\sigma^{*^2} \times 2 = \frac{A^2 + 3B^2}{2} \quad \rightarrow \quad \sigma^{*} = \sqrt{\frac{A^2 + 3B^2}{4}}$$

LH1984
Posts: 12
Joined: November 4th, 2014, 7:37 am

### Volatility surface interpolation

Thanks for you explanation Jhammer.One stupid question, what is the rational of having T here? It looks like the interpolation give higher weight on vol or variance with longer maturity.But the Vol is annualised vol, no? Is there any website/resource for this? Thanks

pimpel
Posts: 363
Joined: May 12th, 2006, 5:26 pm
Location: Warsaw

### Volatility surface interpolation

You can find it for example in chapter 50. in Paul Wilmott on Quantitative Finance. If you don't have, sufficient is to have a look on this paper where you will also find practical approach related to time weighting, where different weights are assigned to weekends, holidays and special days where important information is published.

Jhammer
Posts: 37
Joined: February 24th, 2014, 8:05 pm

### Volatility surface interpolation

QuoteOriginally posted by: LH1984Thanks for you explanation Jhammer.One stupid question, what is the rational of having T here? It looks like the interpolation give higher weight on vol or variance with longer maturity.But the Vol is annualised vol, no? Is there any website/resource for this? ThanksVolatilities are quoted on an annual term. Interpolation is performed on the total variance $\sigma^2 T$ to guarantee longer dated options with equal strikes ($r=q=0$) have higher prices.