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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)

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Boundary Conditions

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  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

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