1 / 29

Boundary Conditions

Boundary Conditions. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A. Boundary Conditions. Attempt to define and categorise BCs in financial PDEs Mathematical and financial motivations Unifying framework (Fichera function)

halona
Download Presentation

Boundary Conditions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Boundary Conditions TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAA

  2. Boundary Conditions • Attempt to define and categorise BCs in financial PDEs • Mathematical and financial motivations • Unifying framework (Fichera function) • One-factor and n-factor examples

  3. Background • ‘Fuzzy’ area in finance • Boundary conditions motivated by financial reasoning • BCs may (or may not) be mathematically correct • A number of popular choices are in use • We justify them

  4. Challenges • Truncating a semi-infinite domain to a finite domain • Imposing BCs on near-field and far-field boundaries • Boundaries where no BC are needed (allowed) • Dirichlet, Neumann, linearity …

  5. Techniques • Using Fichera function to determine which boundaries need BCs • Determine the kinds of BCs to apply • Discretising BCs (for use in FDM) • Special cases and ‘nasties’

  6. The Fichera Method • Allows us to determine where to place BCs • Apply to both elliptic and parabolic PDEs • We concentrate on elliptic PDE • Of direct relevance to computational finance • New development, not widely known

  7. Elliptic PDE (1/2) • Its quadratic form is non-negative (positive semi-definite) • This means that the second-order terms can degenerate at certain points • Use the Oleinik/Radkevic theory • The application of the Fichera function

  8. Domain of interest Unit inward normal Region and Boundary

  9. Elliptic PDE

  10. Remarks • Called an equation with non-negative characteristic form • Distinguish between characteristic and non-characteristic boundaries • Applicable to elliptic, parabolic and 1st-order hyperbolic PDEs • Applicable when the quadratic form is positive-definite as well • Subsumes Friedrichs’ theory in hyperbolic case?

  11. Boundary Types

  12. Choices

  13. Example: Hyperbolic PDE (1/2)

  14. Example: Hyperbolic PDE (2/2) y 1 x L

  15. Example: Hyperbolic PDE y x

  16. Example: CIR Model • Discussed in FDM book, page 281 • What happens on r = 0? • We discuss the application of the Fichera method • Reproduce well-known results by different means

  17. CIR PDE

  18. Convertible Bonds • Two-factor model (S, r) • Use Ito to find the PDE

  19. Two-factor PDE (1/2)

  20. Two-factor PDE (2/2) V S

  21. Asian Options • Two-factor model (S, A) • Diffusion term missing in the A direction • Determine the well-posedness of problem • Write PDE in (x,y) form

  22. PDE for Asian

  23. PDE Formulation I (1/2)

  24. PDE Formulation I (2/2) y x

  25. Special Case

  26. Example: Skew PDE • Pure diffusion degenerate PDE • Used in conjunction with SABR model • Critical value of beta • (thanks to Alan Lewis)

  27. PDE y S

  28. Fichera Function

  29. Boundaries

More Related