Monte Carlo Pricing of Range Accrual Note Options

Summary

The document frames a Monte Carlo pricing problem for an option on a range accrual note under a Heath-Jarrow-Morton interest-rate model. It specifies a note with quarterly coupons and a three-year maturity. Coupon amounts are formed from an average of daily Libor rates over each coupon period, with the rate either set as a fixed level or as Libor plus a spread. The option payoff is the positive part of the note value, constructed from coupon amounts and zero-coupon bond prices, less a strike.

The author asks whether QuantLib contains an implementation and proposes adapting a Monte Carlo treatment of Libor-rate options described in a reference on simulation methods. The document provides the payoff and coupon setup but no simulation algorithm, calibration procedure, code location, or pricing results. Its equations also contain apparent indexing or notation inconsistencies, so the accrual dates, rate definitions, discounting, and precise note structure would need to be checked before implementing or relying on a model.

Key ideas

• A range accrual note option can be represented by a payoff based on the present value of its coupon cash flows relative to a strike.
• The example forms quarterly coupons from averages of daily Libor rates.
• Monte Carlo simulation under an HJM interest-rate model is proposed as a pricing approach.
• The setup does not provide implementation details or numerical evidence, and its notation should be validated before use.

Tags

# Monte Carlo approach to RAN bonds in Quantlib or suggestions


# Monte Carlo approach to RAN bonds in Quantlib or suggestions












This is a problem from Schlogl's book in the chapter on the HJM model:

Price option of the RAN instrument with 3 month coupons and maturity 3 years using Monte Carlo(Exercise 4 Range Accrual Note).

Is the code in the Quantlib library? If so can you tell me its location, thank you. If not can you give me some suggestions on how to approach it.

The background

A bond's value and payoff are of the form:

$$V=\sum_{i}^{N} c_{i} B(T_{i-1},T_{i}) \text{ and } [V-K]^{+},$$ where K is the strike, $c_{i}$ are the coupons and $B(T_{i-1},T_{i})$ is the price of a zero bond at time $T_{i-1}$ with maturity $T_{i+1}$.

In the FRN the coupons $c_{i}$ are determined by some asset or floating rate. Specifically in R AN with 3-month coupons (90 days) and interest compounded daily we have:

$$c_{i}=\bar{r}\frac{1}{90}\sum_{k=90\cdot i}^{90\cdot (i+1)}L(t_{k},t_{k+1}),$$

where the Libor rate is $L(t_{k},t_{k+1})=\frac{1}{t_{k+1}-t_{k+1}}(\frac{1}{B(t_{k},t_{k+1})}-1)$ and $\bar{r}$ is a fixed number or the Libor rate plus a spread s.

So we are searching to price the option with payoff $(\sum_{i}^{N} c_{i} B(T_{i-1},T_{i})-K)^{+}$ with the above $c_{i}$.

Glasserman in his Monte Carlo book section 3.7, describes how to go about the Libor rate option. I will try that as well.

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.