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## Finite Difference Schemes

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**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 Crank-Nicolson scheme: second order accuracy**Explicit scheme**European put option: Lattice:**Explicit scheme (continued)**Monotone scheme**Explicit scheme for a transformed equation**Transformed Black-Scholes equation:**Why use implicit scheme?**• Explicit scheme is conditionally convergent**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.**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**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**Examples**• European call options • CIR model for zero coupon bond**CIR models (continued)**• Method 1: confined to [0,M] • Method 2: a transformation