Electronic Companion A Two-Sided Laplace Inversion Algorithm with Computable Error Bounds and Its Applications in Financial Engineering

where the second equality holds because of the change of measure with dP̃ dP |Ft = e Xt−rt. Because St has a continuous distribution under both P and P̃ , then f1(k) and f2(k) are in C. Besides, P̃ ( St ≥ e−k ) and P ( St ≥ e−k ) are both increasing in k. Therefore, applying Lemma 3.1 yields that both f1(k) and f2(k) satisfy Assumption 1. Note that for any σ ∈ (−1,−σl − 1), ∫ +∞ −∞ ef2(k)dk =e −rt ∫ +∞ −∞ e−(σ+1)kP ( St ≥ e−k ) dk