Transcript FE5106 Advanced Numerical Methods

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 stronglypath dependent options
• Forward shooting grid method
Two-period BTM for vanilla options
• Single-period BTM
• A local algorithm
• Two-period BTM
– Assumption: two-period tree
Multi-period BTM for vanilla options
S0u
S0
S0d
S0u2
S0
S0d2
u3
S0u4
S0u
S0u2
S0d
S0
S0d3
S0d2
S0
S0d4
A concise expression of BTM
With continuous dividend yield q
Multi-period BTM for American options
• At any nodes of the tree, check if early exercise is optimal.
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).
Multi-period BTM for barrier options
• Barrier options are weakly
path-dependent
Strongly path-dependent options (I)
• Strong path-dependent options:
– Asian options
– Lookback options
• Type of payoffs:
– Fixed strike
– Floating strike
Strongly path-dependent options (II)
• Type of Asian payoffs
– Arithmetic average
– Geometric average
Strongly path-dependent options (III)
• Type of lookback payoffs
– Maximum
– Minimum
BTM for Asian arithmetic options (I)
BTM for Asian arithmetic options (II)
• A concise expression
BTM for lookback maximum
Single-state variable BTM for strongly pathdependent options (I)
Single-state variable BTM for strongly pathdependent options (II)
Single-state variable BTM for floating strike
lookback put
A single-state variable tree:
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.
Single-state variable BTM for American-style
floating strike path-dependent options
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.
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.
Single-state variable FSGM for floating
strike Asian arithmetic options
Two-state variable FSGM for arithmetic
Asian options (I)
Two-state variable FSGM for arithmetic
Asian options (II)
Two-state variable FSGM for lookback options
• A similar procedure;
• No interpolation is required.