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Published byAmelia Chappel Modified about 1 year ago

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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

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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 S0S0 S0uS0u S0dS0d S0S0 S0S0 S0u2S0u2 S0d2S0d2 S0u2S0u2 S0u3S0u3 S0u4S0u4 S0d2S0d2 S0uS0u S0dS0d S0d4S0d4 S0d3S0d3

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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.

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