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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**

S0u S0d S0u2 S0d2 S0u3 S0u4 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.

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