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

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.