ABSTRACT
FIGURE 6.4: Implied volatility for the SABR model τ = 5Y. α = 0.2, ρ = −0.7, ν22 τ = 2.5 and β = 1.
Solving this ODE, we obtain that C(f) is fixed to
C(f) = σloc(f)
α √
1− ρ2 cosh ( ν ∫ f f0
) If ν = 0, we have C(f) = σloc(f) and we reproduce the Dupire formula. Using the BBF formula (5.39), we have
C(f) = fσBS(f)
( 1− f ln( ff0 )
) α √
1− ρ2 cosh ( ν
) The short term smile is automatically calibrated when using this function in the case of the λ-SABR model.
