Friday, May 27, 2016

calibration - How to calibrate a volatility surface using SVI


I've read the following paper by Gatheral and Jacquier and have several question regarding the calibration of a volatility surface in a arbitrage free way and some theoretical aspects. Let me first introduce some notation. They define the log strike as


k:=logKF


where F denotes the forward. Moreover, σBS(k,t) denotes the implied Black-Scholes volatility with strike k and maturity t and



w(k,t):=σ2BS(k,t)t


the so called total implied variance. With θt:=σ2BS(0,t)t we denote the at the money implied total variance. What follows they present different parametrization of a single slice in the surface, i.e. not depending on t. My first question:


1. Question: Why do they authors use the total implied variance instead of the directly observable σBS(k,t) for the parametrization? Is there any advantage / meaning of that? Naturally, I would fit a model to the implied volatility


There are three parametrizations of a single surface slide:



  • raw SVI: For a parameter set ξR:={a,b,ρ,m,σ} the raw parametrization is given by: w(k,ξR):=a+b(ρ(km)+(km)2+σ2)

  • natural SVI: For a parameter set ξN:={Δ,μ,ρ,ω,ζ} the natural parametrization is given by: w(k,ξN):=Δ+ω2(1+ζρ(kμ)+(ζ(kμ)+ρ)2+(1ρ2))

  • SVI Jump Wings (SVI_JW): For a given time to expiry t>0 and a parameter set ξJ:={vt,ψt,pt,ct,~vt} the SVI-JW parametrization is given in raw SVI parameters: vt=a+b(ρm+m2+σ2)tψt=b2wt(mm2+σ2+ρ)pt=bwt(1ρ)ct=bwt(1+ρ)~vt=1t(a+bσ1ρ2)
    where wt:=vtt.


2. Question: Why is it an advantage of having a dependency on time to expiration t in the SVI-JW parametrization? As far as I see, you still fit the model to a given slice in all of the above parametrization, that is: You fix time to expiry and fit the model to the observed quotes. So that you could also introduce a time to expiry parameter in the raw/natural SVI.



The authors introduce now a new parametrization for a complete surface, the SSVI.



  • SSVI: For a smooth function ϕ (with some additional properties) the SSVI parameterization is given by: w(k,θt):=θt2(1+ρϕ(θt)k+(ϕ(θt)k+ρ)2+(1ρ2))
    a common choice is ϕ(θ)=ηθγ(1+θ)1γ


They are translations how to convert one parametrization to another.


3. Question: Is it correct that the SSVI tries to fit a whole surface not just a single slice at once?


My last question is more about the actual calibration. For the raw and natural parametrization you would try to find optimal parameters so that ni=1(w(ki,ξR)w(ki)market)2

is minimized, where w(ki)market are observed market quotes (calculated from σBS) for strike k1,,kn for a fixed time to expiry t.


Now for the SSVI, if its really about fitting the whole surface, what function are you minimizing?


ti(ni=1(w(ki,θti)w(ki,ti)market)2)

where you also sum over the maturities?


4. Question: How does the minimization function for the SSVI look like? It seems that the authors are using still for a fixed time to expiry ti a slice parametrization and then compare it with previous / next slice, run additional calibration if needed to avoid calendar spread arbitrage. See page 21 "An example SVI calibration recipe".





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