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Multi-period BTM

Multi-period BTM. Dr. DAI Min. Outline:. Multi-period BTM Vanilla options: European and American Barrier options Strongly-path dependent options: Asian, lookback Single-state variable BTM for some strongly-path dependent options Forward shooting grid method.

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Multi-period BTM

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  1. Multi-period BTM Dr. DAI Min

  2. Outline: • Multi-period BTM • Vanilla options: European and American • Barrier options • Strongly-path dependent options: Asian, lookback • Single-state variable BTM for some strongly-path dependent options • Forward shooting grid method

  3. Two-period BTM for vanilla options • Single-period BTM • A local algorithm • Two-period BTM • Assumption: two-period tree

  4. S0u4 S0u3 S0u2 S0u2 S0u S0u S0 S0 S0 S0d S0d S0d2 S0d2 S0d3 S0d4 Multi-period BTM for vanilla options

  5. A concise expression of BTM

  6. With continuous dividend yield q

  7. Multi-period BTM for American options • At any nodes of the tree, check if early exercise is optimal.

  8. Barrier options • Barrier options are triggered by the action of the underlying hitting a prescribed value at some time before expiry. • Consider an up-out call: if the underlying price remains below the barrier level until maturity, the contract has a usual call payoff at expiry (red line). However, if the underlying reaches the level before expiry, then the option becomes worthless because it has ``knocked out'‘ (blue line).

  9. Multi-period BTM for barrier options • Barrier options are weakly path-dependent

  10. Strongly path-dependent options (I) • Strong path-dependent options: • Asian options • Lookback options • Type of payoffs: • Fixed strike • Floating strike

  11. Strongly path-dependent options (II) • Type of Asian payoffs • Arithmetic average • Geometric average

  12. Strongly path-dependent options (III) • Type of lookback payoffs • Maximum • Minimum

  13. BTM for Asian arithmetic options (I)

  14. BTM for Asian arithmetic options (II) • A concise expression

  15. BTM for lookback maximum

  16. Single-state variable BTM for strongly path-dependent options (I)

  17. Single-state variable BTM for strongly path-dependent options (II)

  18. Single-state variable BTM for floating strike lookback put A single-state variable tree:

  19. Single-state variable BTM for floating strike Asian arithmetic • The single-state variable tree does not recombine! • The amount of computation grows exponentially with the number of time steps.

  20. Single-state variable BTM for American-style floating strike path-dependent options

  21. Single-state variable BTM for fixed strike path-dependent options • For European case, a single-state variable BTM is available by a different transformation (Roger and Shi (1995)); • For American case, not available.

  22. Forward shooting grid method (FSGM) • Drawback of BTM • For some path-dependent options, the number of tree nodes may grow exponentially with the number of time steps • FSGM • A local binomial model • Implementation is confined to a lattice, instead of a tree. • The number of nodes of the lattice does not grow exponentially. • Single-state variable case • General case • Interpolation is needed in some scenarios.

  23. Single-state variable FSGM for floating strike Asian arithmetic options

  24. Two-state variable FSGM for arithmetic Asian options (I)

  25. Two-state variable FSGM for arithmetic Asian options (II)

  26. Two-state variable FSGM for lookback options • A similar procedure; • No interpolation is required.

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