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(k−m)+√(k−m)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(k−μ)+√(ζ(k−μ)+ρ)2+(1−ρ2))
- 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=b2√wt(−m√m2+σ2+ρ)pt=b√wt(1−ρ)ct=b√wt(1+ρ)~vt=1t(a+bσ√1−ρ2)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+(1−ρ2))a common choice is ϕ(θ)=ηθγ(1+θ)1−γ
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 n∑i=1(w(ki,ξR)−w(ki)market)2
Now for the SSVI, if its really about fitting the whole surface, what function are you minimizing?
∑ti(n∑i=1(w(ki,θti)−w(ki,ti)market)2)
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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