Pricing Floating-Rate Notes Across Intermediate Tree Steps

Summary

This note asks how to value a floating-rate note in a short-rate tree when the tree contains time steps between an IBOR fixing date and the coupon payment date. It proposes valuing backward from each fixing-date node by considering reachable payment-date nodes, discounting their values using transition probabilities, and projecting the coupon at all those payment nodes from the rate fixed at the earlier node. The key distinction is between the fixing rate known at the start of the accrual period and later short rates along the tree.

The document presents this as a question rather than a demonstrated implementation. It gives no worked numerical example, model specification, or references confirming the proposed treatment. The idea is relevant when tree dates include intermediate steps, including for bonds with embedded American-style options, but details such as coupon conventions, model calibration, and the appropriate discounting mechanics remain unresolved in the source.

Key ideas

• A floating coupon is associated with its fixing date even when payment occurs later.
• The proposed method uses the fixing-date node's rate to project cashflows at reachable payment-date nodes.
• Expected values are discounted backward using transition probabilities across the tree.
• The note raises an implementation question and does not establish the approach with a worked example.

Tags

# Tree Pricing FRN Implementation


# Tree Pricing FRN Implementation












When pricing a bond via a short rate model on a tree, it seems natural to include intermediate time steps in addition to those corresponding to cashflow dates (i.e. for bonds with American style embedded options).

One thing which isn't clear to me is how to handle the case of floating coupons when there are multiple time steps between an IBOR fixing date and a cashflow date. In particular which IBOR fixing should be assumed to drive the cashflows at the pay date nodes?

It seems like the correct approach would be to:

- start with a tree composed of nodes indexed by time and rate - node[t_i, r_j], say

- take for example two times t0, t1 where t0 corresponds to a fixing date and t1 a payment date and assume there exist some intermediate time steps between t0 and t1 on the tree

- compute the expected value at each node[t0, r_j] by averaging/discounting back across all node[t1, r_k] such that the transition probability (t0, r_j)->(t1, r_k) is non-zero, using the fixing rate determined at node[t0, r_j] to project cashflows at all node[t1, r_k]

For example:

Is my intuition on this approach correct? All of the reference implementations I have found simply assume that all node time steps fall exactly on cashflow dates an so this implementation detail is not quite explicitly handled.

Any insights/references would be much appreciated.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.