Implied Volatility in Models Without Moment Explosions

In this chapter, sharp asymptotic formulas for the implied volatility are established under the assumption that the moments of all positive (or all negative) orders of the asset price are finite. Such models are called asset price models without moment explosions. The chapter also discusses an asymptotic formula conjectured by V.V. Piterbarg. In an unpublished working paper, he formulated a formula that might replace R. Lee’s moment formula for large strikes in the environment of asset price models without moment explosions. It is shown in Chap. 11 that Piterbarg’s formula is valid in a slightly modified form, and that the original conjecture holds under very mild restrictions. Chapter 11 also provides numerous formulas describing the asymptotic behavior of the implied volatility in special models without moment explosions. The list of such models includes the displaced diffusion model, the constant elasticity of variance model, the finite moment log-stable model of P. Carr and L. Wu, and SV1 and SV2 models developed by L.C.G. Rogers and L.A.M. Veraart.