Convergence Analysis of BTM Dr. DAI Min

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

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

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 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 European-style 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 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

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)