Analytic solutions and complete markets for the Heston model with stochastic volatility
| dc.contributor.author | Alziary, Benedicte | |
| dc.contributor.author | Takac, Peter | |
| dc.date.accessioned | 2022-03-09T17:50:55Z | |
| dc.date.available | 2022-03-09T17:50:55Z | |
| dc.date.issued | 2018-10-11 | |
| dc.description.abstract | We study the Heston model for pricing European options on stocks with stochastic volatility. This is a Black-Scholes-type equation whose spatial domain for the logarithmic stock price x ∈ ℝ and the variance v ∈ (0, ∞) is the half-plane ℍ = ℝ x (0, ∞). The volatility is then given by √v. The diffusion equation for the price of the European call option p = p(x, v, t) at time t ≤ T is parabolic and degenerates at the boundary ∂ℍ = ℝ x {0} as v → 0+. The goal is to hedge with this option against volatility fluctuations, i.e., the function v ↦ p(x, v, t): (0, ∞) → ℝ and its (local) inverse are of particular interest. We prove that ∂p/∂v (x, v, t) ≠ 0 holds almost everywhere in ℍ x (-∞, T) by establishing the analyticity of p in both, space (x, v) and time t variables. To this end, we are able to show that the Black-Scholes-type operator, which appears in the diffusion equation, generates a holomorphic C0-semigroup in a suitable weighted L2-space over ℍ. We show that the C0-semigroup solution can be extended to a holomorphic function in a complex domain in ℂ2 x ℂ, by establishing some new a priori weighted L2-estimates over certain complex "shifts" of ℍ for the unique holomorphic extension. These estimates depend only on the weighted L2-norm of the terminal data over ℍ (at t = T). | |
| dc.description.department | Mathematics | |
| dc.format | Text | |
| dc.format.extent | 54 pages | |
| dc.format.medium | 1 file (.pdf) | |
| dc.identifier.citation | Alziary, B., & Takác, P. (2018). Analytic solutions and complete markets for the Heston model with stochastic volatility. <i>Electronic Journal of Differential Equations, 2018</i>(168), pp. 1-54. | |
| dc.identifier.issn | 1072-6691 | |
| dc.identifier.uri | https://hdl.handle.net/10877/15464 | |
| dc.language.iso | en | |
| dc.publisher | Texas State University, Department of Mathematics | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.source | Electronic Journal of Differential Equations, 2018, San Marcos, Texas: Texas State University and University of North Texas. | |
| dc.subject | Heston model | |
| dc.subject | Stochastic volatility | |
| dc.subject | Black-Scholes equation | |
| dc.subject | European call option | |
| dc.subject | Degenerate parabolic equation | |
| dc.subject | Terminal value problem | |
| dc.subject | Holomorphic extension | |
| dc.subject | Analytic solution | |
| dc.title | Analytic solutions and complete markets for the Heston model with stochastic volatility | |
| dc.type | Article |