no-arb-neural-vol-surfacesmicrostructuremartingalestat-arb<M>_texp(-1/2 x' Sigma^-1 x)argminN(mu, Sigma)constraint set Cconstraint set CKL(p||q)Var[X]pricingquadratic variationSigmamicrostructurevolatilityP(A | B)quadratic variationVegad log S_tlocal volKKTlocal volN(mu, Sigma)entropy rateDeltaimplied surfaceGammaH(X)sigma_t dW_tphi(x)S(f)sigma^2exp(-1/2 x' Sigma^-1 x)Q Lambda Q^TKKTlambda_iargminquadratic variationE[X]microstructureP(A | B)volatilitydS_tNyquistrank(K)S(f)local volDeltaOptionsLabnabla L<M>_tDeltaS(f)implied surfaceOptionsLabKL(p||q)KKTP(A | B)Q Lambda Q^Texp(-1/2 x' Sigma^-1 x)dP / dQDeltamax E[R] - lambda Var[R]Ito driftP(A | B)GammaIVno-arb-neural-vol-surfacesBS PDEBS PDEKKTIto driftlocal vollocal volN(mu, Sigma)IVno-arb-neural-vol-surfaceslocal volOptionsLabconstraint set CmartingaleQ Lambda Q^Tsigma^2entropy rateexp(-1/2 x' Sigma^-1 x)lambda_iGammaOptionsLab<M>_tKKTvolatilitymicrostructurerank(K)||Ax-b||_2constraint set Cphi(x)Ito driftentropy ratephi(x)sigma^2<M>_td log S_tmax E[R] - lambda Var[R]VegaDeltaNyquistquadratic variationBS PDEKKTQ Lambda Q^Tlambda_i<M>_tS(f)entropy rateN(mu, Sigma)volatility||Ax-b||_2argminnabla Ldual gapdS_tquadratic variationGammaE[X]VegaS(f)p(x | theta)sigma_t dW_tdual gapSigmaN(mu, Sigma)martingaleBS PDEp(x | theta)quadratic variationexp(-1/2 x' Sigma^-1 x)Nyquistimplied surfaceno-arb-neural-vol-surfacessigma_t dW_tentropy rateN(mu, Sigma)local volimplied surfacesigma^2local volE[X]<M>_tOptionsLabS(f)dual gapVega||Ax-b||_2cov(r_i,r_j)d log S_texp(-1/2 x' Sigma^-1 x)stat-arbstat-arbdual gapKKTexp(-1/2 x' Sigma^-1 x)local volE[X]entropy ratenabla LDeltaOptionsLabimplied surfaceargminS(f)Nyquistcov(r_i,r_j)exp(-1/2 x' Sigma^-1 x)I(X;Y)SigmaOptionsLabentropy ratemartingaleIVconstraint set CI(X;Y)max E[R] - lambda Var[R]local volFFTmartingalecov(r_i,r_j)lambda_idP / dQstat-arblocal vold log S_tcov(r_i,r_j)local vollocal volargminI(X;Y)pricingnabla Lrank(K)dual gappricingFFTmicrostructuresigma_t dW_tcov(r_i,r_j)P(A | B)exp(-1/2 x' Sigma^-1 x)VegavolatilitypricingBS PDEN(mu, Sigma)Var[X]macroOptionsLabVar[X]macrorank(K)max E[R] - lambda Var[R]nabla Llambda_iE[X]local volSigmamicrostructuresigma^2rank(K)local volcov(r_i,r_j)P(A | B)sigma^2local volmacroE[X]constraint sIVNyquistFFTrank(K)quadratic va||Ax-b||_2I(X;Y)<M>_tGammaIto driftsigma^2dual gapdS_tVar[X]dS_tQ Lambda Q^TKL(p||q)exp(-1/2 x' argminBS PDE<M>_tmacroQ Lambda Q^Tquadratic vastat-arbKKTIto driftquadratic vadP / dQstat-arbphi(x)lambda_irank(K)BS PDEIto driftvolatilityrank(K)FFTdS_tdual gapI(X;Y)nabla LpricingFFTFFTKL(p||q)martingaleS(f)H(X)quadratic variationSigmalambda_idual gapQ Lambda Q^TP(A | B)Sigmarank(K)||Ax-b||_2d log S_tBS PDEvolatilitysigma^2S(f)FFTphi(x)sigma_t dW_tsigma_t dW_tNyquistvolatilityKKTlambda_idS_tKL(p||q)cov(r_i,r_j)