Forward-Starting Swap Valuation and the Discounting Horizon

Summary

The document poses a valuation problem for a forward-starting interest-rate swap in a binomial short-rate lattice. The swap begins after the valuation date, receives floating payments, pays a fixed rate, and makes payments in arrears. The questioner reports a spreadsheet result they believe is incorrect and asks how to apply forward equations across the lattice.

The brief answer identifies a horizon issue: the lattice must extend through the final payment’s discounting date, which is later than the swap’s start and its final accrual period endpoint. The stated attempt uses too short a lattice, so the needed discounting information is unavailable. No complete valuation, corrected numerical price, or detailed lattice calculation is given, so the note is limited to identifying the missing horizon requirement. It illustrates that swap cash-flow dates and discounting dates must both be represented when valuing a forward-starting contract.

Key ideas

• A forward-starting swap’s value depends on discounting every future payment to the valuation date.
• Payments made in arrears can require lattice information beyond the swap’s start date and final accrual interval.
• The answer points to an incomplete short-rate lattice as the source of the attempted valuation problem.
• The document does not provide a full calculation or the corrected swap value.

Tags

# How to calculate a forward-starting swap with forward equations?


# How to calculate a forward-starting swap with forward equations?












I have been trying to resolve this problem for some time but I cannot get the correct answer. The problem is the following one.

> Compute the initial value of a forward-starting swap that begins at $t=1$, with maturity $T=10$ and a fixed rate of 4.5%. (The first payment then takes place at $t=2$ and the final payment takes place at $t=11$ as we are assuming, as usual, that payments take place in arrears.) You should assume a swap notional of 1 million and assume that you receive floating and pay fixed.)

We also know that

- $r_{0,0}=5\%$

- $u=1.1$

- $d=0.9$

- $q=1−q=1/2$

Using forward equations from $t=1$ to $t=9$, I cannot resolve the problem:

Here is what I have done in Excel with a final result of -31076 but it is not the correct answer:

## Answer by Artem Oboturov (score 2)

https://quant.stackexchange.com/a/28243

You have to use T=1...10 because last payment is discounted to year 10. So your short rate lattice is incomplete.

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