6533b7d7fe1ef96bd126840a
RESEARCH PRODUCT
A Sequential Quadratic Programming Method for Volatility Estimation in Option Pricing
Bertram DüringAnsgar JüngelStefan Volkweinsubject
Hessian matrixMathematical optimizationLine searchComputer scienceMathematicsofComputing_NUMERICALANALYSISOptimal controlsymbols.namesakeValuation of optionsLagrange multipliersymbolsDescent directionVolatility (finance)Dupire equation parameter identification optimal control optimality conditions SQP method primal-dual active set strategySequential quadratic programmingdescription
Our goal is to identify the volatility function in Dupire's equation from given option prices. Following an optimal control approach in a Lagrangian framework, we propose a globalized sequential quadratic programming (SQP) algorithm with a modified Hessian - to ensure that every SQP step is a descent direction - and implement a line search strategy. In each level of the SQP method a linear-quadratic optimal control problem with box constraints is solved by a primal-dual active set strategy. This guarantees L^1 constraints for the volatility, in particular assuring its positivity. The proposed algorithm is founded on a thorough first- and second-order optimality analysis. We prove the existence of local optimal solutions and of a Lagrange multiplier associated with the inequality constraints. Furthermore, we prove a sufficient second-order optimality condition and present some numerical results underlining the good properties of the numerical scheme.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2006-03-29 |