On solutions of a partial integro-differential equation in Bessel potential spaces with applications in option pricing m
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On solutions of a partial integro‑differential equation in Bessel potential spaces with applications in option pricing models José M. T. S. Cruz1 · Daniel Ševčovič2 Received: 5 September 2019 / Revised: 14 February 2020 © The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2020
Abstract In this paper we focus on qualitative properties of solutions to a nonlocal nonlinear partial integro-differential equation (PIDE). Using the theory of abstract semilinear parabolic equations we prove existence and uniqueness of a solution in the scale of Bessel potential spaces. Our aim is to generalize known existence results for a wide class of Lévy measures including with a strong singular kernel. As an application we consider a class of PIDEs arising in the financial mathematics. The classical linear Black–Scholes model relies on several restrictive assumptions such as liquidity and completeness of the market. Relaxing the complete market hypothesis and assuming a Lévy stochastic process dynamics for the underlying stock price process we obtain a model for pricing options by means of a PIDE. We investigate a model for pricing call and put options on underlying assets following a Lévy stochastic process with jumps. We prove existence and uniqueness of solutions to the penalized PIDE representing approximation of the linear complementarity problem arising in pricing American style of options under Lévy stochastic processes. We also present numerical results and comparison of option prices for various Lévy stochastic processes modelling underlying asset dynamics. Keywords Partial integro-differential equation · Sectorial operator · Analytic semigroup · Bessel potential space · Option pricing under Lévy stochastic process · Lévy measure Mathematics Subject Classification 45K05 · 35K58 · 34G20 · 91G20
* Daniel Ševčovič [email protected] 1
ISEG, University of Lisbon, Rua de Quelhas 6, 1200‑781 Lisbon, Portugal
2
Comenius University in Bratislava, Mlynská dolina 84248, Bratislava, Slovakia
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J. M. T. S. Cruz, D. Ševčovič
1 Introduction In this paper, we analyze solutions to the semilinear parabolic partial integro-differential equation (PIDE) of the form:
𝜕u 𝜎 2 𝜕2 u 𝜕u (𝜏, x) = (𝜏, x) + 𝜔 (𝜏, x) + g(𝜏, u(𝜏, x)) 𝜕𝜏 2 𝜕x2 𝜕x ] [ 𝜕u + u(𝜏, x + z) − u(𝜏, x) − (ez − 1) (𝜏, x) 𝜈(dz), ∫ℝ 𝜕x u(0, x) =u0 (x),
(1)
x ∈ ℝ, 𝜏 ∈ (0, T) , where g is Hölder continuous in the 𝜏 variable and it is Lipschitz continuous in the u variable. Here 𝜈 is a positive Radon measure on ℝ such that ∫ℝ min(z2 , 1)𝜈(dz) < ∞. Our purpose is to prove existence and uniqueness of a solution to (1) in the framework of Bessel potential spaces. These functional spaces represent a nested scale {X 𝛾 }𝛾≥0 of Banach spaces such that X 1 ≡ D(A) ↪ X 𝛾1 ↪ X 𝛾2 ↪ X 0 ≡ X,
for any 0 ≤ 𝛾2 ≤ 𝛾1 ≤ 1 where A is a sectorial operator in the Banach space X with a dense domain D(A) ⊂ X . For example, if A = −Δ is the Laplacian operator in ℝn with the domain D(A) ≡ W 2,p (ℝn ) ⊂ X ≡ Lp (ℝn ) then X 𝛾 is embedded in the
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