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 strongly-path 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

S0

S0u

S0d S0 S0

S0u2

S0d2

S0u2

S0u3 S0u4

S0d2

S0u

S0d

S0d4

S0d3

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 path-dependent options (I)

Single-state variable BTM for strongly path-dependent 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.