ABSTRACT
FIGURE 6.3: Implied volatility for the SABR model τ = 1Y. α = 0.2, ρ = −0.7, ν22 τ = 0.5 and β = 1.
be automatically calibrated by construction if the short-term local volatility σloc(f) is
σloc(f) = C(f)amin(f) (6.52)
with
amin(f)2 = α2 + 2ρανq + ν2q2
q = ∫ f f0
df ′
C(f ′)
By short-term, we mean a maturity date less than 1 year in practice. Solving (6.52) according to q, we obtain
νq = −ραν + √ α2(−1 + ρ2) + σloc(f)
C(f)2
and if we derive under f , we have ( ψ(f) ≡ σloc(f)C(f) )
dψ√ ψ2 − α2(1− ρ2) =
ν
σloc(f) df
