Prices of Barrier and First-Touch Digital Options in Lévy-Driven Models, Near Barrier

Abstract

We calculate the leading term of asymptotics of the prices of barrier options and firsttouch digitals near the barrier for wide classes of Lévy processes with exponential jump densities, including the Variance Gamma model, the KoBoL (a.k.a. CGMY) model and Normal Inverse Gaussian processes. In the case of processes of infinite activity and finite variation, with the drift pointing from the barrier, we prove that the price is discontinuous at the boundary. This observation can serve as the basis for a simple robust test of the type of processes observed in real financial markets. In many cases, we calculate the second term of asymptotics as well. By comparing the exact asymptotic results for prices with those of Carr's randomization approximation, we conclude that the latter is very accurate near the barrier. We illustrate this by including numerical results for several types of Lévy processes commonly used in option pricing.

Efficient Laplace Inversion, Wiener-Hopf Factorization and Pricing Lookbacks

“…q under the integral sign in (B.13). See [5] for proofs of similar estimates. Consider first the case µ > 0 and ν ∈ [0+, 1).…”

“…Further significant improvement can be obtained if the characteristic exponent admits analytic continuation into the complex plane with two cuts i(−∞, λ − ] and i[λ + , +∞) [5,20].…”

“…We regard the VG model as a process of order ν = 0+. Modifications for other sRLPE are also straightforward; for further examples of sRLPEs, see [5,20].…”

Static and Semi-Static Hedging as Contrarian or Conformist Bets

“…In all popular models used in finance, the prices of vanilla options are infinitely smooth before the maturity date and up to the boundary but prices of barrier options in Lévy models are not smooth at the boundary, the exceptions being double jump diffusion model, hyper-exponential jump diffusion model, and other models with rational characteristic functions. For wide classes of purely jump models, it is proved in [21,9] that the price of an "out" barrier option near the barrier behaves as c(T )|x − h| κ , where κ ∈ [0, 1) is independent of time to maturity T , c(T ) > 0, and |x − h| is the log-distance from the barrier. For finite variation processes with the drift pointing from the boundary, κ = 0, and the limit of the price at the barrier is positive.…”

Pricing of Discretely Sampled Asian Options Under Levy Processes

“…These properties are valid for wide classes of processes used in the theoretical and empirical studies of financial markets. See M. Boyarchenko et al [11], Levendorskiȋ [43].…”