finite difference schemes
DESCRIPTION
Finite Difference Schemes. Dr. DAI Min. Type of finite difference scheme. Explicit scheme Advantage There is no need to solve a system of algebraic equations Easy for programming Disadvantage: conditionally convergent Implicit scheme Fully implicit scheme: first order accuracy - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/1.jpg)
Finite Difference Schemes
Dr. DAI Min
![Page 2: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/2.jpg)
Type of finite difference scheme
• Explicit scheme– Advantage
• There is no need to solve a system of algebraic equations• Easy for programming
– Disadvantage: conditionally convergent
• Implicit scheme– Fully implicit scheme: first order accuracy– Crank-Nicolson scheme: second order accuracy
![Page 3: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/3.jpg)
Explicit scheme• European put option:
• Lattice:
![Page 4: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/4.jpg)
Explicit scheme (continued)
![Page 5: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/5.jpg)
Explicit scheme (continued)
![Page 6: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/6.jpg)
Explicit scheme (continued)
![Page 7: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/7.jpg)
Explicit scheme (continued)
• Monotone scheme
![Page 8: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/8.jpg)
Explicit scheme for a transformed equation
• Transformed Black-Scholes equation:
![Page 9: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/9.jpg)
Explicit scheme for a transformed equation
![Page 10: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/10.jpg)
Explicit scheme for a transformed equation (continued)
![Page 11: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/11.jpg)
Explicit scheme for a transformed equation (continued)
![Page 12: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/12.jpg)
Equivalence of explicit scheme and BTM
![Page 13: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/13.jpg)
Equivalence of explicit scheme and BTM (continued)
![Page 14: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/14.jpg)
Why use implicit scheme?
• Explicit scheme is conditionally convergent
![Page 15: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/15.jpg)
Fully implicit scheme
![Page 16: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/16.jpg)
Fully implicit scheme (continued)
![Page 17: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/17.jpg)
Matrix form of an explicit scheme
![Page 18: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/18.jpg)
Monotonicity of the fully implicit scheme
![Page 19: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/19.jpg)
Second-order scheme: Crank-Nicolson scheme
![Page 20: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/20.jpg)
Crank-Nicolson scheme in matrix form
![Page 21: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/21.jpg)
Convergence of Crank-Nicolson scheme
• The C-N scheme is not monotone unless t/h2 is small enough. • Monotonicity is sufficient but not necessary• The unconditional convergence of the C-N scheme (for linear
equation) can be proved using another criterion (see Thomas (1995)).
• Due to lack of monotonicity, the C-N scheme is not as stable/robust as the fully implicit scheme when dealing with tough problems.
![Page 22: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/22.jpg)
Iterative methods for solving a linear system
![Page 23: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/23.jpg)
Linearization for nonlinear problems
![Page 24: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/24.jpg)
Newton iteration
![Page 25: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/25.jpg)
Handling non-smooth terminal conditions
• C-N scheme has a better accuracy but is unstable when the terminal condition is non-smooth.
• To cure the problem– Rannacher smoothing– Smoothing the terminal value condition
![Page 26: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/26.jpg)
Upwind (upstream) treatment
![Page 27: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/27.jpg)
An example for upwind scheme in finance
![Page 28: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/28.jpg)
Artificial boundary conditions
• Solution domain is often unbounded, but implicit schemes should be restricted to a bounded domain– Truncated domain– Change of variables
• Artificial boundary conditions should be given based on– Properties of solution, and/or– PDE with upwind scheme
![Page 29: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/29.jpg)
Examples
• European call options
• CIR model for zero coupon bond
![Page 30: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/30.jpg)
CIR models (continued)
• Method 1: confined to [0,M]
• Method 2: a transformation
![Page 31: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/31.jpg)
Test of convergence order
![Page 32: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/32.jpg)
Test of convergence order (alternative method)
![Page 33: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/33.jpg)
An example: given benchmark values
![Page 34: Finite Difference Schemes](https://reader031.vdocuments.us/reader031/viewer/2022020711/5681681b550346895ddda998/html5/thumbnails/34.jpg)
An example: no benchmark values