Abstract
Many risk-neutral pricing problems proposed in the finance literature do not admit closed-form expressions and have to be dealt with by solving the corresponding partial integro-differential equation. Often, these PIDEs have singular diffusion matrices and coefficients that are not Lipschitz-continuous up to the boundary. In addition, in general, boundary conditions are not specified. In this paper, we prove existence and uniqueness of (continuous) viscosity solutions for linear PIDEs with all the above features, under a Lyapunov-type condition. Our results apply to European and Asian option pricing, in jump-diffusion stochastic volatility and path-dependent volatility models. We verify our Lyapunov-type condition in several examples, including the arithmetic Asian option in the Heston model.
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Notes
The Heston model is considered in Example 3.6 in Pascucci [28], but Assumption 3.3 is not satisfied as (3.9) does not hold.
Admissibility restrictions on the parameters required for the existence of an affine process are discussed, e.g., in Duffie et al. [14].
This condition ensures that the volatility process V is always positive if V 0 is positive; see Heston [20].
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Costantini, C., Papi, M. & D’Ippoliti, F. Singular risk-neutral valuation equations. Finance Stoch 16, 249–274 (2012). https://doi.org/10.1007/s00780-011-0166-8
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DOI: https://doi.org/10.1007/s00780-011-0166-8
Keywords
- Degenerate integro-differential equations
- Viscosity solutions
- Asian options
- Stochastic volatility
- Jump-diffusion