---
title: "Modelling Overnight-Rate Products"
book: "Rates, Credit, XVA and Risk"
subject: quant
language: en
chapter: 10
exercises: 8
source: https://one-course.com/books/quant/6/en/chapter/10-modelling-overnight-rate-products
---

# Chapter 10 — Modelling Overnight-Rate Products

A company’s floating-rate loan pays, each quarter, SOFR compounded daily over the quarter, known only when the quarter ends. Its treasurer buys a cap, and the desk that sells it notices something the old interbank caps never did: the first caplet, whose period has already started, still has optionality, and every caplet keeps accruing risk after its period begins, until its last overnight fixing. An interbank rate fixed at the start of its period; a compounded overnight rate fixes at the end. The models of chapters 7 and 8 were built for rates known in advance. This chapter extends them to rates known in arrears: in the [Hull–White model](https://one-course.com/books/quant/6/en/chapter/7-short-rate-models#def-rc-short-rate-models-hw) the [backward-looking caplet](#def-rc-modelling-overnight-rate-products-caplet) has a closed form, in the market model the extension is the [generalised forward market model](#def-rc-modelling-overnight-rate-products-gfmm), and with the calendar of central-bank meetings the rate becomes a step function whose risk arrives on meeting days.

## 10.1 Backward-looking rates and their options

**Definition 10.1 (Backward-looking rate, forward-looking term rate).**

A *backward-looking rate* for a period $[S,E]$ is the overnight rate compounded in arrears over the period (One Quant Book 2, chapter 1): $1+\delta R(S,E) = \prod_i(1+r_id_i/360)$, known only at $E$. A *forward-looking term rate* for the same period is published at $S$, as an expectation of that compounded rate implied by futures and swaps (a term SOFR rate) or as an interbank offered rate (Euribor); it is known at the start.

**Definition 10.2 (Backward-looking caplet).**

A *backward-looking caplet* pays $\delta\max(R(S,E)-K,0)$ at $E$ (or a few days later) on the compounded rate of its own period.

![A forward-looking rate carries risk until the start of its period and is known from then on; a compounded overnight rate keeps moving until the last fixing at the end, so an option on it carries risk through its whole accrual period.](https://one-course.com/images/onecourse/chapters/quant-6/rc-modelling-overnight-rate-products/fig-bdee5193fa63.svg)

***Figure 10.1.** A forward-looking rate carries risk until the start of its period and is known from then on; a compounded overnight rate keeps moving until the last fixing at the end, so an option on it carries risk through its whole accrual period.*

**Proposition 10.3 (Backward-looking caplets in Hull–White).**

Approximate daily compounding by $1+\delta R = \exp\int_S^Er_u\,du$. In Hull–White with constant $\sigma$ and $c = 1+\delta K$,

$$
\mathrm{Caplet}_0 = P(0,S)\bigl[\Phi(-d_2) - c\,m\,\Phi(-d_1)\bigr],\quad
m = \frac{P(0,E)}{P(0,S)},\ d_1 = \frac{\ln(mc)+V/2}{\sqrt V},\ d_2 = d_1-\sqrt V,
$$

with $V = V_{\text{before}}+V_{\text{during}}$, $V_{\text{before}} = \sigma^2B(\delta)^2\frac{1-e^{-2\kappa S}}{2\kappa}$ and $V_{\text{during}} = \sigma^2\int_0^\delta B(\tau)^2\,d\tau\approx\sigma^2\delta^3/3$. The forward-looking caplet on a term rate for the same period has the same formula with $V = V_{\text{before}}$ only.

**Proof.** The payoff is $(e^{-\int_0^Sr}-c\,e^{-\int_0^Er})^+$. Under the measure with density $e^{-\int_0^Sr}/P(0,S)$ the price is $P(0,S)\E[(1-cL)^+]$ with $L=e^{-\int_S^Er}$, lognormal with mean $P(0,E)/P(0,S)$; its log-variance is the variance of $\int_S^Ex_u\,du$, which splits into the shocks before $S$ (through the bond price at $S$) and those during the period. The expectation is a Black put. ∎

```python
def hw_period_variances(kappa: float, sigma: float, S: float, E: float) -> tuple[float, float]:
    """(variance of int_S^E r accumulated before S, variance accumulated during [S, E]) in Hull-White."""
    before = sigma**2 * _B(kappa, E - S) ** 2 * (1 - math.exp(-2 * kappa * S)) / (2 * kappa)
    during = sigma**2 * _int_B2(kappa, E - S)
    return before, during


def _lognormal_caplet(curve, S: float, E: float, K: float, var: float) -> float:
    """P(0,S) E[(1 - c L)^+] with L lognormal, mean P(0,E)/P(0,S), log-variance var, c = 1 + delta K."""
    delta = E - S
    c = 1 + delta * K
    ps, pe = curve.df_t(S), curve.df_t(E)
    m = pe / ps
    v = math.sqrt(var)
    d1 = (math.log(m * c) + 0.5 * var) / v
    return ps * (_cdf(-(d1 - v)) - c * m * _cdf(-d1))


def hw_backward_caplet(curve, kappa: float, sigma: float, S: float, E: float, K: float) -> float:
    """Caplet paying delta (R - K)^+ at E, R the overnight rate compounded over [S, E]."""
    b, d = hw_period_variances(kappa, sigma, S, E)
    return _lognormal_caplet(curve, S, E, K, b + d)
```

***Listing 10.1.** The two parts of the variance, and the backward-looking caplet in closed form. code/firm/rfrcaplet/firm_rfrcaplet.py*

**Example 10.4 (A two-year SOFR cap).**

On chapter 1’s SOFR curve with $\kappa=3\%$ and $\sigma=90$ basis points, a two-year cap on three-month compounded SOFR at 3.75% on USD 100 million costs USD 288 963: eight caplets from USD 13 623 (the current quarter) to 53 846 ([Figure 10.2](#fig-rc-modelling-overnight-rate-products-caplets)). The same cap on a three-month term rate, whose first caplet has already fixed, costs USD 250 741: the backward-looking cap is USD 38 222 more expensive, of which 13 623 is the first caplet and 24 599 the in-period variance of the other seven. A Monte Carlo of the short rate with 260 steps a year reprices the caplet on the fifth quarter (USD 40 944) to within a fifth of a standard error.

![Caplets of a two-year cap at 3.75% on USD 100 million, on compounded SOFR and on a term rate for the same quarters. The current quarter has a caplet only in the backward-looking cap; each later backward-looking caplet is worth more by its period’s in-arrears variance. Data: chapter 1’s curve, Hull–White =3\%, =90 basis points; the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-6/rc-modelling-overnight-rate-products/fig-19e6ed5d9b5a.svg)

***Figure 10.2.** Caplets of a two-year cap at 3.75% on USD 100 million, on compounded SOFR and on a term rate for the same quarters. The current quarter has a caplet only in the backward-looking cap; each later [backward-looking caplet](#def-rc-modelling-overnight-rate-products-caplet) is worth more by its period’s in-arrears variance. Data: chapter 1’s curve, Hull–White $\kappa=3\%$, $\sigma=90$ basis points; the chapter’s tutorial.*

The in-period share of the variance is 100% for the current quarter, 25% for the next, 7.9% for the fifth and 4.8% for the eighth: close to $(\delta/3)/(S+\delta/3)$, the ratio a linear decay of volatility through the period gives ([Figure 10.3](#fig-rc-modelling-overnight-rate-products-share)).

![Share of a backward-looking caplet’s variance that accrues during its own quarter, by the quarter’s start: all of it for the current quarter, a quarter for the next, a few per cent two years out. Data: the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-6/rc-modelling-overnight-rate-products/fig-4bf8ccc697fc.svg)

***Figure 10.3.** Share of a [backward-looking caplet](#def-rc-modelling-overnight-rate-products-caplet)’s variance that accrues during its own quarter, by the quarter’s start: all of it for the current quarter, a quarter for the next, a few per cent two years out. Data: the chapter’s tutorial.*

## 10.2 The generalised forward market model

**Definition 10.5 (Generalised forward market model).**

The *generalised forward market model* (Lyashenko and Mercurio, 2019) models, for each accrual period $[T_{k-1},T_k]$, the forward $R_k(t) = \E^{T_k}_t[R(T_{k-1},T_k)]$ of the [backward-looking rate](#def-rc-modelling-overnight-rate-products-rates), for all $t\le T_k$: before $T_{k-1}$ it coincides with the forward-looking forward of chapter 8, and during the period it keeps moving with the overnight fixings. Its volatility is $\sigma_k(t)g_k(t)$ with $g_k=1$ before $T_{k-1}$ and $g_k$ decaying to zero at $T_k$, for instance linearly.

**Proposition 10.6 (Effective variance with linear decay).**

With constant $\sigma_k$ and $g_k(t) = (T_k-t)/(T_k-T_{k-1})$ on the period, the [backward-looking caplet](#def-rc-modelling-overnight-rate-products-caplet) is priced by Black’s formula with total variance $\sigma_k^2\bigl(T_{k-1}+\tfrac13(T_k-T_{k-1})\bigr)$, against $\sigma_k^2T_{k-1}$ for the forward-looking caplet.

**Proof.** $\int_0^{T_{k-1}}\sigma_k^2\,dt+\int_{T_{k-1}}^{T_k}\sigma_k^2\bigl(\frac{T_k-t}{\delta}\bigr)^2dt =
\sigma_k^2(T_{k-1}+\delta/3)$, and $R_k$ is lognormal under $\mathbb Q^{T_k}$. ∎

The generalised model contains the old one: one process gives both the forward-looking and the [backward-looking rates](#def-rc-modelling-overnight-rate-products-rates), so a book of legacy interbank products and new overnight ones is risk-managed in one model, with the market model’s simulation and calibration machinery.

## 10.3 Low-dimensional Markov models

Market models are high-dimensional; production systems for exotics and exposures prefer models with a handful of state variables whose forward curve has an exact formula.

**Definition 10.7 (Cheyette model).**

A *Cheyette model* (1992) is an HJM model whose forward volatility factorises as $\sigma_f(t,T) = \sigma_r(t,x_t,y_t)\,e^{-\kappa(T-t)}$. Then $f(t,T) = f(0,T)+e^{-\kappa(T-t)}\bigl(x_t+B(t,T)y_t\bigr)$ with two state variables, $dx = (y-\kappa x)\,dt+\sigma_r\,dW$ and $dy = (\sigma_r^2-2\kappa y)\,dt$. With deterministic $\sigma_r$ it is Hull–White; letting $\sigma_r$ depend on $x$ gives a local-volatility (skewed) [short-rate model](https://one-course.com/books/quant/6/en/chapter/7-short-rate-models#def-rc-short-rate-models-short) that is still Markov in $(x,y)$.

The integral of the short rate over an accrual period is one more state variable in such a model, so backward-looking products stay low-dimensional: this is the Markov route to the same caplets, and the one that PDE and tree pricers follow.

## 10.4 Meeting dates and jumps in the short rate

The overnight rate does not diffuse: it sits at the policy rate and moves on the days a central bank decides (One Quant Book 2, chapter 8). Its uncertainty arrives in lumps, on a published calendar.

**As of September 2026 — Meeting calendar and term rates.**

The Federal Reserve’s policy committee meets on 27–28 October and 8–9 December 2026, and on 26–27 January, 16–17 March, 27–28 April, 8–9 June, 27–28 July, 14–15 September, 26–27 October and 7–8 December 2027 (tentative until confirmed at the preceding meeting). The ARRC formally recommended CME Group’s forward-looking term SOFR rates on 29 July 2021, and its best practices do not support their use in derivatives except by end users hedging cash products that reference them.

```python
def meeting_variance(meetings: Sequence[float], jump_sd: float, S: float, E: float, t: float = 0.0) -> float:
    """Variance, seen at t, of the average overnight rate over [S, E] when the rate moves only at meetings,
    by independent normal jumps of standard deviation jump_sd: a meeting at m moves the average by
    J * (E - max(m, S)) / (E - S) if m < E."""
    d = E - S
    return sum((jump_sd * (E - max(m, S)) / d) ** 2 for m in meetings if t < m < E)
```

***Listing 10.2.** The variance of a period’s average rate when the rate moves only at meetings. code/firm/rfrcaplet/firm_rfrcaplet.py*

**Example 10.8 (One quarter, two meetings).**

The quarter from 29 September to 29 December 2027 (years 1.0 to 1.25 from spot) contains the meetings of 27 October and 8 December. If the rate moves only at meetings, a decision on 27 October moves the quarter’s average rate by 69% of the jump (the share of the quarter left), one on 8 December by 23%; after 8 December the quarter’s rate is known, three weeks before its end. The generalised model’s linear decay spreads the same risk smoothly ([Figure 10.4](#fig-rc-modelling-overnight-rate-products-profile)): scaled to the same total, the two agree before the quarter and differ within it.

![Remaining standard deviation, seen at t, of the compounded rate of the quarter from year 1.0 to 1.25, in two models scaled to the same total: a smooth decay through the period, or steps at the Federal Reserve’s meeting days (ten meetings before the quarter ends; the last inside it on 8 December 2027). Data: Federal Reserve meeting calendar; the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-6/rc-modelling-overnight-rate-products/fig-1693e00b8ec8.svg)

***Figure 10.4.** Remaining standard deviation, seen at $t$, of the compounded rate of the quarter from year 1.0 to 1.25, in two models scaled to the same total: a smooth decay through the period, or steps at the Federal Reserve’s meeting days (ten meetings before the quarter ends; the last inside it on 8 December 2027). Data: Federal Reserve meeting calendar; the chapter’s tutorial.*

**Remark 10.9 (What the meeting view changes).**

Prices change little, since the total variance is calibrated either way; risk changes a lot. A caplet’s theta and vega are concentrated on meeting days, and a book of short-dated overnight options should be hedged meeting by meeting, with the futures of One Quant Book 2, chapter 8, and options on them.

## 10.5 Forward-looking term rates

Term rates survive for loans: a borrower wants to know the next quarter’s interest at the start. Derived from futures and overnight swaps, a term SOFR rate is a published expectation; options and swaps on it are forward-looking instruments of chapter 4’s kind. The market keeps derivatives on the overnight rate itself, where the liquidity is, and restricts term-rate derivatives to the hedges of the loans that use them, so that the term rate does not come to rest on trading in itself. The desk that sells a term-rate cap to a borrower therefore hedges it with overnight-rate options and carries the basis between the two: the in-period variance of [Example 10.4](#ex-rc-modelling-overnight-rate-products-cap).

## 10.6 Tutorial: backward against forward

**Goal.** Price a two-year cap on compounded SOFR and on a term rate, check the closed form by simulation, and compare smooth and meeting-date risk profiles. **End state:** Figures [10.2](#fig-rc-modelling-overnight-rate-products-caplets) and [10.4](#fig-rc-modelling-overnight-rate-products-profile) and the numbers of [Example 10.4](#ex-rc-modelling-overnight-rate-products-cap).

1. **Variances** : `hw_period_variances(kappa, sigma, S, E)` for each quarter.
2. **Caps** : `cap_table()` and `cap_totals()` .
3. **Simulate** : `mc_check()` , 260 steps a year so that every period date is on the grid.
4. **Meetings** : `remaining_sd_profile()` ; `fig_rc_rfr.py` writes the charts.

**What to change next.** Run the simulation with 250 steps a year and an unaligned grid (remove the check) and see the bias; price a caplet on a period that contains no meeting.

## 10.7 Build: backward-looking caplets

**Purpose.** Caps and floors on compounded overnight rates, the options of the book’s floating-rate loans since the end of the interbank rates.

**Interface.** `hw_period_variances`; `hw_backward_caplet`, `hw_forward_caplet`, `hw_backward_caplet_mc`; `gfmm_effective_variance`, `black_caplet`; `meeting_variance`, `bachelier_caplet`.

**Rules.** Continuous compounding of the short rate as the model of daily compounding; simulation grids must contain every period date (enforced); numpy only.

**Acceptance tests.** `code/firm/rfrcaplet/tests/`: a [backward-looking caplet](#def-rc-modelling-overnight-rate-products-caplet) exceeds its forward-looking twin by its in-period variance, which is $\sigma^2\delta^3/3$ for a period starting now; the closed form agrees with Monte Carlo on an aligned grid and an unaligned grid is refused; the linear-decay variance and the meeting variance have their closed forms.

**Stretch.** Daily compounding with business-day calendars; lookback and payment delay; a meeting-date [short-rate model](https://one-course.com/books/quant/6/en/chapter/7-short-rate-models#def-rc-short-rate-models-short) calibrated to futures and their options.

Sources and further reading

- A. Lyashenko and F. Mercurio, “Looking forward to backward-looking rates: a modeling framework for term rates replacing LIBOR”, 2019.
- O. Cheyette, “Term structure dynamics and mortgage valuation”, *Journal of Fixed Income* 1(4), 1992.
- ARRC, “ARRC formally recommends term SOFR”, 29 July 2021.
- Federal Reserve, *Meeting calendars and information* .

## 10.8 Exercises

**Exercise 10.1 ★.**

Why does a cap on compounded SOFR include a caplet on the current quarter, when an interbank cap never did?

**Solution of Exercise 10.1.**

An interbank rate for the current period was fixed at its start, so its caplet had a known payoff. A compounded overnight rate for the current quarter is known only at the quarter’s end: the fixings still to come make the caplet a genuine option.

**Exercise 10.2 ★.**

With linear decay, by how much is the total variance of a [backward-looking caplet](#def-rc-modelling-overnight-rate-products-caplet) on $[2,2.25]$ larger than that of its forward-looking twin?

**Solution of Exercise 10.2.**

By $\sigma^2\delta/3 = \sigma^2\times0.0833$: total variance $\sigma^2\times2.0833$ against $\sigma^2\times2$, 4.2% more.

**Exercise 10.3 ★.**

In [Example 10.8](#ex-rc-modelling-overnight-rate-products-meetings), compute the weights 69% and 23% from the meeting dates.

**Solution of Exercise 10.3.**

27 October 2027 is 1.077 years from spot and 8 December 1.192; the quarter ends at 1.25: $(1.25-1.077)/0.25 = 69\%$ and $(1.25-1.192)/0.25 = 23\%$.

**Exercise 10.4 ★★.**

Using $V_{\text{during}}\approx\sigma^2\delta^3/3$ and $V_{\text{before}}\approx\sigma^2\delta^2S$, estimate the in-period share of variance of the second quarter and compare with the 25% of the text.

**Solution of Exercise 10.4.**

$\frac{\delta^3/3}{\delta^2S+\delta^3/3} = \frac{\delta/3}{S+\delta/3} = \frac{0.0833}{0.3333} = 25\%$ for $S=0.25$, as in the text (25.2% with mean reversion).

**Exercise 10.5 ★★.**

Read [Figure 10.4](#fig-rc-modelling-overnight-rate-products-profile). On which days does a short caplet on the quarter lose most of its time value under the meeting view, and what does that mean for its theta?

**Solution of Exercise 10.5.**

On the meeting days: its remaining uncertainty falls in steps on 27 October and 8 December, and after 8 December it has none. Its theta is concentrated on those days (small in between), unlike the smooth decay of the generalised model; a trader must hedge and mark it meeting by meeting.

**Exercise 10.6 ★★.**

Why do the best practices restrict derivatives on term SOFR, and what basis does a desk that sells term-rate caps carry?

**Solution of Exercise 10.6.**

A term rate is computed from derivatives on the overnight rate; if derivatives on the term rate became large, it would rest partly on trading in itself and could be moved by the flows it is meant to measure. Restricting its derivatives to end users’ hedges keeps the liquidity in overnight products. A desk that sells term-rate caps hedges with overnight-rate options and is left with the in-period variance and the difference between the published term rate and the realised compounded rate.

**Exercise 10.7 ★★★.**

*Coding.* With `cap_table()`, split the USD 38 222 difference between the two caps into the first caplet and the in-period variance of the others.

**Solution of Exercise 10.7.**

The first caplet, USD 13 623, exists only in the compounded cap; the other seven caplets are worth USD 24 599 more in total than their forward-looking twins, from their in-period variance: 13 623 plus 24 599 is the 38 222.

**Exercise 10.8 ★★★.**

*Find the flaw.* “We moved our cap pricer from LIBOR to SOFR by changing the index name: it still skips the first caplet and uses the variance to each period’s start.”

**Solution of Exercise 10.8.**

Both assumptions are wrong for compounded rates: the current period’s caplet is still an option and must be priced, and each caplet’s variance runs to the end of its period, not its start. The pricer undervalues every cap, here by USD 38 222 on USD 100 million over two years, and its vega is on the wrong dates.

## 10.9 Problem: The SOFR Cap on a Leveraged Loan

**Problem 10.1.**

Weekend problem — a borrower chooses its cap

A private-equity-owned company has a USD 100 million two-year loan paying three-month compounded SOFR plus a margin; its lenders require a cap at 3.75%. Its bank offers a cap on compounded SOFR and a cap on three-month term SOFR. Use chapter 1’s curve and Hull–White with $\kappa=3\%$ and $\sigma=90$ basis points.

**Part I — The two caps.**

1. Give the price of each cap.
2. Why is the compounded-SOFR cap more expensive?
3. Give the value of its current-quarter caplet.
4. Which cap hedges the loan exactly, and what does the other leave?
5. Give the in-period share of variance for the second and the eighth quarters.

**Part II — Checks.**

6. Give the fifth quarter’s caplet by the closed form and by simulation.
7. What went wrong when the simulation used 250 steps a year?
8. What does the [generalised forward market model](#def-rc-modelling-overnight-rate-products-gfmm) ’s linear decay predict for the fifth quarter’s extra variance, and how does it compare with Hull–White’s?
9. Why do both models give nearly the same answer?
10. Which inputs matter most for the price?

**Part III — Meetings.**

11. Which meetings fall in the fifth quarter?
12. When is its rate known under the meeting view?
13. What is the caplet’s vega on the day after the December meeting?
14. How would the bank hedge the caplet meeting by meeting?
15. Does the meeting view change the price or the risk?

**Part IV — Judgement.**

16. Which cap would you advise the company to buy?
17. What basis risk does the bank take if it sells the term-rate cap?
18. Why do lenders accept either?
19. State the *named result* : the two cap prices and the gap, with its split.
20. In one sentence: what does “in arrears” add to an option?

**Solution of Problem 10.1.**

**1.** USD 288 963 on compounded SOFR, USD 250 741 on the term rate. **2.** It includes the current quarter’s caplet and each caplet carries risk until the end of its period. **3.** USD 13 623. **4.** The compounded-SOFR cap, since the loan pays compounded SOFR; the term-rate cap leaves the difference between the loan’s rate and the term rate. **5.** 25.2% and 4.8%. **6.** USD 40 944 by the closed form; the simulation agrees to within a fifth of a standard error. **7.** The time grid missed the period’s dates (312.5 steps truncated to 312), so the simulated period was shorter than the quarter and the rate too low: a 6% bias with a tight standard error. **8.** A variance larger by $\delta/(3S) = 8.3\%$; Hull–White gives an in-period share of 7.9%, i.e. 8.6% more, almost the same. **9.** Hull–White’s in-period variance is close to $\sigma^2\delta^3/3$, exactly what a linear decay of the rate’s volatility produces. **10.** The volatility level and, for this out-of-the-money strike, the forward rates. **11.** 27 October and 8 December 2027. **12.** After the 8 December decision, three weeks before the quarter ends. **13.** Zero: nothing uncertain remains. **14.** With futures on the meeting months and options on them, sized to each meeting’s weight in the quarter. **15.** The risk: when theta is earned and where the vega sits. **16.** The compounded-SOFR cap, which matches the loan; the extra USD 38 222 buys protection on the current quarter and removes basis risk. **17.** The difference between term SOFR and compounded SOFR over each quarter, including the in-period variance it cannot hedge in term-rate options. **18.** Both limit the loan’s interest; the term-rate cap leaves a small basis they tolerate. **19.** Named result: *the SOFR cap*: USD 288 963 on compounded SOFR against USD 250 741 on the term rate; the USD 38 222 gap is the current quarter’s caplet (13 623) and in-period variance (24 599). **20.** Risk that keeps accruing until the end of the period.

## 10.10 Interview questions

**Interview question 10.1 ★ trader, bank.**

What is the difference between a caplet on Euribor and a caplet on compounded €STR?

**Solution of Interview question 10.1.**

Euribor fixes at the start of the period, so its caplet’s risk ends then; compounded €STR fixes at the end, so its caplet carries risk through the accrual period, is worth more, and the current period’s caplet is an option too.

*What the interviewer is looking for: the timing of the fixing and its consequence for variance.*

**Interview question 10.2 ★★ researcher.**

How does the market model need to change for [backward-looking rates](#def-rc-modelling-overnight-rate-products-rates)?

**Solution of Interview question 10.2.**

Model the forward of the [backward-looking rate](#def-rc-modelling-overnight-rate-products-rates) for each period up to the period’s end, with a volatility that decays to zero during the period (the [generalised forward market model](#def-rc-modelling-overnight-rate-products-gfmm)); before the period it equals the forward-looking forward, so one process gives both kinds of rate.

*What the interviewer is looking for: forwards alive to the end of the period, and decaying volatility.*

**Interview question 10.3 ★★ researcher, developer.**

Price a [backward-looking caplet](#def-rc-modelling-overnight-rate-products-caplet) in Hull–White. Where does the extra variance come from?

**Solution of Interview question 10.3.**

Write the payoff as $(e^{-\int_0^Sr}-c\,e^{-\int_0^Er})^+$, change to the measure with density $e^{-\int_0^Sr}/P(0,S)$, and price a Black put on $e^{-\int_S^Er}$, lognormal with mean $P(0,E)/P(0,S)$. Its variance is the bond-price variance at $S$ plus the variance of the rate’s integral over the period, $\approx\sigma^2\delta^3/3$.

*What the interviewer is looking for: the measure change and the two variance terms.*

**Interview question 10.4 ★★ trader.**

How do central-bank meetings change the risk of short-dated overnight-rate options?

**Solution of Interview question 10.4.**

The overnight rate moves only on meeting days, so an option’s uncertainty resolves in steps: vega and theta concentrate on meetings, and after the last meeting in a period the period’s rate is known. Hedging is by meeting, with futures and their options.

*What the interviewer is looking for: step risk and meeting-by-meeting hedging.*

**Interview question 10.5 ★★★ developer, risk.**

Your Monte Carlo price of a compounded-rate caplet is 6% below the closed form with a tight standard error. What do you check first?

**Solution of Interview question 10.5.**

Discretisation before statistics: whether the simulation grid contains the period’s start and end (a truncated step count shortens the period), the compounding convention, then convergence in the step size, the curve’s forwards used for the drift, and the discounting; compare with the forward-looking case, which has a closed form too.

*What the interviewer is looking for: bias versus noise, and the grid.*

**Interview question 10.6 ★★★ researcher, bank.**

Why is term SOFR a problem as a derivatives underlying, and what do banks do instead?

**Solution of Interview question 10.6.**

It is computed from overnight-rate derivatives; large derivative activity on it would be circular and thin the underlying market. Official guidance limits its derivative use to end users hedging term-rate loans; banks hedge those with overnight-rate instruments and carry the basis.

*What the interviewer is looking for: circularity and where the liquidity is.*