Boundary Conditions

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Boundary Conditions
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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

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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

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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

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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

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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

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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

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Domain of interest
Unit inward normal
Region and Boundary
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Elliptic PDE
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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?

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Boundary Types
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Choices
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Example Hyperbolic PDE (1/2)
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Example Hyperbolic PDE (2/2)
y
1
x
L
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Example Hyperbolic PDE
y
x
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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

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CIR PDE
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Convertible Bonds

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

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Two-factor PDE (1/2)
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Two-factor PDE (2/2)
V
S
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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

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PDE for Asian
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PDE Formulation I (1/2)
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PDE Formulation I (2/2)
y
x
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Special Case
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Example Skew PDE

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

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PDE
y
S
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Fichera Function
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Boundaries