The Finite Difference Method in Computational Finance Theory, Applications and Computation Boundary Conditions TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A AAA A

2 The Finite Difference Method in Computational Finance Theory, Applications and Computation 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 The Finite Difference Method in Computational Finance Theory, Applications and Computation 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 The Finite Difference Method in Computational Finance Theory, Applications and Computation 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 The Finite Difference Method in Computational Finance Theory, Applications and Computation 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 Finite Difference Method in Computational Finance Theory, Applications and Computation 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 The Finite Difference Method in Computational Finance Theory, Applications and Computation 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 The Finite Difference Method in Computational Finance Theory, Applications and Computation Domain of interest Region and Boundary Unit inward normal

9 The Finite Difference Method in Computational Finance Theory, Applications and Computation Elliptic PDE

10 The Finite Difference Method in Computational Finance Theory, Applications and Computation Remarks Called an equation with non-negative characteristic form Distinguish between characteristic and non- characteristic boundaries Applicable to elliptic, parabolic and 1 st -order hyperbolic PDEs Applicable when the quadratic form is positive- definite as well Subsumes Friedrichs’ theory in hyperbolic case?

11 The Finite Difference Method in Computational Finance Theory, Applications and Computation Boundary Types

12 The Finite Difference Method in Computational Finance Theory, Applications and Computation Choices

13 The Finite Difference Method in Computational Finance Theory, Applications and Computation Example: Hyperbolic PDE (1/2)

14 The Finite Difference Method in Computational Finance Theory, Applications and Computation Example: Hyperbolic PDE (2/2) y x 1 L

15 The Finite Difference Method in Computational Finance Theory, Applications and Computation Example: Hyperbolic PDE y x

16 The Finite Difference Method in Computational Finance Theory, Applications and Computation 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 The Finite Difference Method in Computational Finance Theory, Applications and Computation CIR PDE

18 The Finite Difference Method in Computational Finance Theory, Applications and Computation Convertible Bonds Two-factor model (S, r) Use Ito to find the PDE

19 The Finite Difference Method in Computational Finance Theory, Applications and Computation Two-factor PDE (1/2)

20 The Finite Difference Method in Computational Finance Theory, Applications and Computation Two-factor PDE (2/2) S V

21 The Finite Difference Method in Computational Finance Theory, Applications and Computation 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 The Finite Difference Method in Computational Finance Theory, Applications and Computation PDE for Asian

23 The Finite Difference Method in Computational Finance Theory, Applications and Computation PDE Formulation I (1/2)

24 The Finite Difference Method in Computational Finance Theory, Applications and Computation PDE Formulation I (2/2) x y

25 The Finite Difference Method in Computational Finance Theory, Applications and Computation Special Case

26 The Finite Difference Method in Computational Finance Theory, Applications and Computation Example: Skew PDE Pure diffusion degenerate PDE Used in conjunction with SABR model Critical value of beta (thanks to Alan Lewis)

27 The Finite Difference Method in Computational Finance Theory, Applications and Computation PDE S y

28 The Finite Difference Method in Computational Finance Theory, Applications and Computation Fichera Function

29 The Finite Difference Method in Computational Finance Theory, Applications and Computation Boundaries