# Lecture 3(2): Convergence Analysis of BTM

```Convergence Analysis of BTM
Dr. DAI Min
Why convergence analysis is important?
• To better understand the BTM and the continuoustime model
– American options
– Lookback options
• To develop new schemes
– FSGM
– Finite difference scheme
Consistency of BTM and PDE model for vanilla
options (I)
• PDE model:
• BTM:
• Consistency:
Consistency of BTM and PDE model for vanilla
options (II)
• Consistency of assumptions
– Continuous-time assumption:
– BTM assumption:
– Connection:
Consistency of BTM and PDE model for vanilla
options (III)
Consistency of BTM and PDE model for
American options
• From BTM to continuous-time model
Continuous-time model for American options
• PDE model (variational inequality):
• Optimal stopping problem:
Derivation of PDE model for American options
• Delta-hedging
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:
PDE models for Asian options (I)
PDE models for Asian options (II)
• Derivation (arithmetic Asian)
– Ito lemma:
– Arithmetic average:
– Delta hedging
Consistency of BTM and PDE model for Asian
options
• Geometric Asian
• Arithmetic Asian
PDE model for lookback options
• Pricing model (lookback maximum)
– V=V(S,A,t)
– For S<A,
– Neumann boundary condition at S=A,
Consistency of BTM and PDE model for lookback
options
• BTM:
• Case i)
consistent with the PDE
• Case ii)
consistent with the Neumann boundary condition
Convergence analysis
• Consistency does not mean convergence
• Convergence proof (not required)
Convergence analysis (continued)
• The analysis works for all (linear) BTMs for Europeanstyle products.
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.
FSGM
• BTM
– 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
Convergence analysis of FSGM (involving
interpolation)
• Does the interpolation affect the convergence?
• Consistency:
Convergence analysis of FSGM (continued)
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!
Convergence analysis of FSGM (continued)
```