1. Convergence Analysis of BTM
Dr. DAI Min

2. Why convergence analysis is important?
To better understand the BTM and the continuous-time model
American options
Lookback options
To develop new schemes
FSGM
Finite difference scheme

3. Consistency of BTM and PDE model for vanilla options (I)

4. Consistency of BTM and PDE model for vanilla options (II)
Consistency of assumptions
Continuous-time assumption:
BTM assumption:
Connection:

5. Consistency of BTM and PDE model for vanilla options (III)

6. Consistency of BTM and PDE model for American options
From BTM to continuous-time model

7. Continuous-time model for American options
PDE model (variational inequality):
Optimal stopping problem:

8. Derivation of PDE model for American options
Delta-hedging

9. Consistency of BTM and PDE model for barrier options
V=V(S,t): up-out option price before the barrier is hit
PDE model:
BTM:

10. PDE models for Asian options (I)

11. PDE models for Asian options (II)
Derivation (arithmetic Asian)
Ito lemma:
Arithmetic average:
Delta hedging

12. Consistency of BTM and PDE model for Asian options
Geometric Asian
Arithmetic Asian

13. PDE model for lookback options
Pricing model (lookback maximum)
V=V(S,A,t)
For S<A,
Neumann boundary condition at S=A,

14. Consistency of BTM and PDE model for lookback options
Case i)
consistent with the PDE
Case ii)
consistent with the Neumann boundary condition

15. Convergence analysis Consistency does not mean convergence
Convergence proof (not required)

16. Convergence analysis (continued)
The analysis works for all (linear) BTMs for European-style products.

17. Convergence analysis of BTM for American options
The BTM for American options is nonlinear.
A criterion:
Usually we only check i) and ii), which hold for BTM for American options.

18. FSGM BTM FSGM: a modified BTM confined to a lattice
option pricing is a backward procedure;
the evolution of the underlying price is forward
FSGM: a modified BTM confined to a lattice
at any lattice point, we generate a single-period (forward) binomial tree through which a backward pricing applies
interpolation is likely needed

19. Convergence analysis of FSGM (involving interpolation)
Does the interpolation affect the convergence?
Consistency:

20. Convergence analysis of FSGM (continued)

21. Convergence analysis of FSGM (continued)
Consistency:
Monotonicity:
holds for the nearest point or linear interpolation
no longer true for quadratic interpolation
Conclusion:
The FSGM with linear interpolation is recommended
Quadratic interpolation can improve the order of consistency, but damage the monotonicity. No convergence is guaranteed in this case!

22. Convergence analysis of FSGM (continued)