Multi-period BTM Dr. DAI Min. Outline: Multi-period BTM –Vanilla options: European and American...
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Transcript of Multi-period BTM Dr. DAI Min. Outline: Multi-period BTM –Vanilla options: European and American...
![Page 1: Multi-period BTM Dr. DAI Min. Outline: Multi-period BTM –Vanilla options: European and American –Barrier options –Strongly-path dependent options: Asian,](https://reader035.fdocuments.in/reader035/viewer/2022081501/56649c9e5503460f9495e3e7/html5/thumbnails/1.jpg)
Multi-period BTM
Dr. DAI Min
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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
![Page 3: Multi-period BTM Dr. DAI Min. Outline: Multi-period BTM –Vanilla options: European and American –Barrier options –Strongly-path dependent options: Asian,](https://reader035.fdocuments.in/reader035/viewer/2022081501/56649c9e5503460f9495e3e7/html5/thumbnails/3.jpg)
Two-period BTM for vanilla options
• Single-period BTM
• A local algorithm
• Two-period BTM– Assumption: two-period tree
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Multi-period BTM for vanilla options
S0
S0u
S0d S0 S0
S0u2
S0d2
S0u2
S0u3 S0u4
S0d2
S0u
S0d
S0d4
S0d3
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A concise expression of BTM
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With continuous dividend yield q
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Multi-period BTM for American options
• At any nodes of the tree, check if early exercise is optimal.
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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).
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Multi-period BTM for barrier options
• Barrier options are weakly path-dependent
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Strongly path-dependent options (I)
• Strong path-dependent options: – Asian options– Lookback options
• Type of payoffs:– Fixed strike
– Floating strike
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Strongly path-dependent options (II)
• Type of Asian payoffs– Arithmetic average
– Geometric average
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Strongly path-dependent options (III)
• Type of lookback payoffs– Maximum
– Minimum
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BTM for Asian arithmetic options (I)
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BTM for Asian arithmetic options (II)
• A concise expression
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BTM for lookback maximum
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Single-state variable BTM for strongly path-dependent options (I)
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Single-state variable BTM for strongly path-dependent options (II)
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Single-state variable BTM for floating strike lookback put
A single-state variable tree:
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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.
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Single-state variable BTM for American-style floating strike path-dependent options
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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.
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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.
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Single-state variable FSGM for floating strike Asian arithmetic options
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Two-state variable FSGM for arithmetic Asian options (I)
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Two-state variable FSGM for arithmetic Asian options (II)
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Two-state variable FSGM for lookback options• A similar procedure;• No interpolation is required.