7Multi-period BTM for American options At any nodes of the tree, check if early exercise is optimal.
8Barrier optionsBarrier 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).
9Multi-period BTM for barrier options Barrier options are weakly path-dependent
16Single-state variable BTM for strongly path-dependent options (I)
17Single-state variable BTM for strongly path-dependent options (II)
18Single-state variable BTM for floating strike lookback put A single-state variable tree:
19Single-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.
20Single-state variable BTM for American-style floating strike path-dependent options
21Single-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.
22Forward shooting grid method (FSGM) Drawback of BTMFor some path-dependent options, the number of tree nodes may grow exponentially with the number of time stepsFSGMA local binomial modelImplementation is confined to a lattice, instead of a tree.The number of nodes of the lattice does not grow exponentially.Single-state variable caseGeneral caseInterpolation is needed in some scenarios.
23Single-state variable FSGM for floating strike Asian arithmetic options
24Two-state variable FSGM for arithmetic Asian options (I)
25Two-state variable FSGM for arithmetic Asian options (II)
26Two-state variable FSGM for lookback options A similar procedure;No interpolation is required.