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Convergence Analysis of BTM

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).

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Convergence Analysis of BTM

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  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) • PDE model: • BTM: • Consistency:

  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 • BTM: • 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 • 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)

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