---
title: "Convexity Adjustments and Constant-Maturity Products"
book: "Rates, Credit, XVA and Risk"
subject: quant
language: en
chapter: 6
exercises: 8
source: https://one-course.com/books/quant/6/en/chapter/6-convexity-adjustments-and-constant-maturity-products
---

# Chapter 6 — Convexity Adjustments and Constant-Maturity Products

A structured note pays, once a year, the ten-year euro swap rate observed that day. For the coupon ten years from now the forward ten-year swap rate on chapter 2’s curves is 3.05%, and a trader who quoted the coupon at the forward would lose money: its fair value is 3.37%. Nothing is wrong with the forward. It is the fair rate of a swap starting in ten years, an expectation under the measure whose numeraire is that swap’s annuity; the note pays the rate once, a year after it is observed, and under the measure of that payment date the rate’s expectation is higher, by an amount that depends on the volatility of the rate and on how the annuity moves with it. The difference is a convexity adjustment. Book 2 met one on futures (One Quant Book 2, chapter 8); this chapter computes them for rates paid at the wrong time, in the wrong currency or with the wrong maturity, prices them by replication with the swaptions of chapter 5’s cube, and values a note on the difference of two swap rates.

## 6.1 Paying a rate at the wrong time: timing adjustments

A forward rate $F_k$ for $[T_{k-1},T_k]$ is a martingale under the $T_k$-forward measure ([Proposition 4.1](https://one-course.com/books/quant/6/en/chapter/4-vanilla-rates-options#prop-rc-vanilla-rates-options-measures)), whose numeraire matches its natural payment date. Pay it at another date and its expectation changes.

**Definition 6.1 (Timing adjustment).**

The *timing adjustment* of a rate paid at a date $T_p$ other than its natural payment date is $\E^{T_p}[F_k(T_{k-1})]-F_k(0)$: the difference between its expectation under the $T_p$-forward measure and its forward.

**Proposition 6.2 (A rate paid when it fixes).**

If $F_k$ is paid at its fixing date $T_{k-1}$ instead of $T_k$,

$$
\E^{T_{k-1}}\bigl[F_k(T_{k-1})\bigr] = F_k(0) +
\frac{\delta_k\,\Var^{T_k}\bigl(F_k(T_{k-1})\bigr)}{1+\delta_kF_k(0)} ,
$$

which is $F_k(0)+\delta_k\sigma_N^2T_{k-1}/(1+\delta_kF_k(0))$ in the normal model, and $F_k(0)+\delta_kF_k(0)^2\sigma^2T_{k-1}/(1+\delta_kF_k(0))$ to first order in the lognormal model.

**Proof.** The density of $\mathbb Q^{T_{k-1}}$ against $\mathbb Q^{T_k}$ on $\mathcal
F_{T_{k-1}}$ is the ratio of the numeraires normalised at zero, $\frac{P(T_{k-1},T_{k-1})/P(T_{k-1},T_k)}{P(0,T_{k-1})/P(0,T_k)} =
\frac{1+\delta_kF_k(T_{k-1})}{1+\delta_kF_k(0)}$ (One Quant Book 4, chapter 5). So $\E^{T_{k-1}}[F_k] = \E^{T_k}[F_k(1+\delta_kF_k)]/(1+\delta_kF_k(0))$, and $\E^{T_k}[F_k^2] = F_k(0)^2+\Var^{T_k}(F_k)$. ∎

![Where a rate is paid decides the measure of its expectation. Paid at the end of its accrual period, a forward rate needs no adjustment; paid at its fixing, it needs a timing adjustment; a swap rate paid once, on a single date, needs a constant-maturity adjustment.](https://one-course.com/images/onecourse/chapters/quant-6/rc-convexity-adjustments-and-constant-maturity-products/fig-f00f040f6cf3.svg)

***Figure 6.1.** Where a rate is paid decides the measure of its expectation. Paid at the end of its accrual period, a forward rate needs no adjustment; paid at its fixing, it needs a [timing adjustment](#def-rc-convexity-adjustments-and-constant-maturity-products-timing); a swap rate paid once, on a single date, needs a constant-maturity adjustment.*

**Example 6.3 (A six-month rate paid in advance).**

A six-month rate with a forward of 3% fixing in five years, with a normal volatility of 75 basis points (a [lognormal volatility](https://one-course.com/books/quant/6/en/chapter/4-vanilla-rates-options#def-rc-vanilla-rates-options-lognormal) of 25%), paid at its fixing: $0.5\times0.0075^2\times5/(1+0.5\times0.03) = 1.39$ basis points. Small for one period, it is paid on every period of an in-arrears swap and grows with the fixing date.

## 6.2 Constant-maturity swaps and their replication by swaptions

**Definition 6.4 (Constant-maturity swap).**

A *constant-maturity swap* (CMS) exchanges a floating leg whose rate at each reset is the par swap rate of a fixed tenor (the “CMS rate”, for instance the ten-year swap rate observed at the reset) against a fixed or another floating leg. A CMS caplet pays $\max(S(T)-K,0)$ on the CMS rate observed at $T$, once, at a date $T_p$.

**Definition 6.5 (CMS convexity adjustment).**

The *CMS convexity adjustment* is $\E^{T_p}[S_{a,b}(T)]-S_{a,b}(0)$: the expectation of the swap rate under the forward measure of its payment date minus its forward, the expectation under the annuity measure.

Change measure from the annuity to the payment date:

$$
\E^{T_p}[S(T)] = \frac{A(0)}{P(0,T_p)}\,\E^{a,b}\Bigl[S(T)\,\frac{P(T,T_p)}{A(T)}\Bigr].
$$

The ratio $P(T,T_p)/A(T)$ is a function of the whole curve at $T$; to price, it is mapped to a function of the swap rate alone.

**Definition 6.6 (Linear terminal swap-rate model).**

The *linear terminal swap-rate model* sets $P(T,T_p)/A(T) = a_0 + a_1S(T)$, with $a_0 = 1/\sum_i\delta_i$ (the value when all rates are zero) and $a_1$ chosen so that the relation holds today: $a_0+a_1S(0) = P(0,T_p)/A(0)$.

**Proposition 6.7 (The CMS rate by replication).**

Under the [linear terminal swap-rate model](#def-rc-convexity-adjustments-and-constant-maturity-products-tsr),

$$
\E^{T_p}[S(T)] = S(0) + \frac{a_1\,\Var^{a,b}(S(T))}{a_0+a_1S(0)},\qquad
\Var^{a,b}(S(T)) = 2\int_{-\infty}^{\infty} O(K)\,dK,
$$

where $O(K)$ is the out-of-the-money swaption price per unit annuity at strike $K$ (a receiver below the forward, a payer above). With a flat normal smile, $\Var^{a,b}(S(T)) = \sigma_N^2T$ (Hagan, 2003).

**Proof.** $\E^{a,b}[S(a_0+a_1S)] = a_0S(0) + a_1\bigl(S(0)^2+\Var^{a,b}(S)\bigr)$ since $S$ is an $\mathbb Q^{a,b}$-martingale; divide by $a_0+a_1S(0) = P(0,T_p)/A(0)$. For the variance, $S^2 = S(0)^2 + 2S(0)(S-S(0)) + 2\int_{-\infty}^{S(0)}(K-S)^+dK +
2\int_{S(0)}^{\infty}(S-K)^+dK$ (the static replication of One Quant Book 5, chapter 14), and take expectations. ∎

```python
def otm_integral(fwd: float, t: float, smile: Callable[[float], float], lo: float | None = None) -> float:
    """Integral over strikes of out-of-the-money options per unit annuity (receivers below the
    forward, payers above), from `lo` (default: far below) to far above: Var^A(S) / 2 when lo is None."""
    ks = _grid(fwd, t, smile)
    if lo is not None:
        ks = np.concatenate([[lo], ks[ks > lo]])
    vals = [bachelier_put(fwd, k, t, smile(k)) if k < fwd else bachelier_call(fwd, k, t, smile(k)) for k in ks]
    return float(np.trapezoid(vals, ks))


def cms_rate(fwd: float, t: float, annuity0: float, df_pay: float, accruals: list[float],
             smile: Callable[[float], float], lo: float | None = None) -> float:
    """E^{Tp}[S(T)] by replication: [F(a0 + a1 F) + 2 a1 * integral of OTM options] / (a0 + a1 F).
    `lo`: lowest strike of the smile's model (a shifted model has no mass below minus its shift)."""
    a0, a1 = linear_tsr(annuity0, df_pay, fwd, accruals)
    num = fwd * (a0 + a1 * fwd) + 2 * a1 * otm_integral(fwd, t, smile, lo)
    return num / (a0 + a1 * fwd)
```

***Listing 6.1.** The CMS rate by replication: out-of-the-money swaptions integrated over the smile. code/firm/cms/firm_cms.py*

The CMS adjustment is therefore a price: a strip of out-of-the-money swaptions, weighted by $2a_1$. It is paid for by the smile. With the cube’s smile the wings are richer than a [flat volatility](https://one-course.com/books/quant/6/en/chapter/4-vanilla-rates-options#def-rc-vanilla-rates-options-flat), the variance is higher, and so is the adjustment ([Figure 6.2](#fig-rc-convexity-adjustments-and-constant-maturity-products-adj)).

**Example 6.8 (Ten-year CMS on the euro cube).**

For the ten-year swap rate observed in one, two, three, five, seven and ten years and paid one year later, the adjustments on chapter 2’s curves are 1.39, 3.49, 5.87, 11.73, 17.40 and 26.58 basis points with each section’s at-the-money volatility held flat, and 1.62, 4.46, 7.92, 15.51, 22.91 and 32.14 with chapter 5’s smile. The ten-year CMS rate observed in ten years has a forward of 3.051% and an expectation of 3.372%: the hook’s coupon.

![Convexity adjustment of the ten-year euro CMS rate paid one year after observation, by replication with a flat volatility and with the smile of chapter 5’s cube. The smile adds 16 to 35 per cent to the adjustment. Data: chapter 2’s curves and chapter 5’s synthetic cube; the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-6/rc-convexity-adjustments-and-constant-maturity-products/fig-6649a0702804.svg)

***Figure 6.2.** Convexity adjustment of the ten-year euro CMS rate paid one year after observation, by replication with a [flat volatility](https://one-course.com/books/quant/6/en/chapter/4-vanilla-rates-options#def-rc-vanilla-rates-options-flat) and with the smile of chapter 5’s cube. The smile adds 16 to 35 per cent to the adjustment. Data: chapter 2’s curves and chapter 5’s synthetic cube; the chapter’s tutorial.*

![Where the ten-year CMS adjustment of the ten-year rate comes from: the contribution of swaptions of each strike, receivers below the forward and payers above, to the 32.14 basis points. Swaptions a percentage point or more away from the money carry a large share of it, which is why the wings of the smile matter. Data: the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-6/rc-convexity-adjustments-and-constant-maturity-products/fig-107760ba9fe0.svg)

***Figure 6.3.** Where the ten-year CMS adjustment of the ten-year rate comes from: the contribution of swaptions of each strike, receivers below the forward and payers above, to the 32.14 basis points. Swaptions a percentage point or more away from the money carry a large share of it, which is why the wings of the smile matter. Data: the chapter’s tutorial.*

A CMS caplet is replicated the same way. Writing $(S-K)^+(a_0+a_1S) =
(a_0+a_1K)(S-K)^+ + a_1((S-K)^+)^2$ and $((S-K)^+)^2 = 2\int_K^\infty(S-L)^+dL$, its value is $A(0)\bigl[(a_0+a_1K)\,\mathrm{Pay}(K) + 2a_1\int_K^\infty
\mathrm{Pay}(L)\,dL\bigr]$, with $\mathrm{Pay}$ the payer price per unit annuity.

```python
def cms_caplet(strike: float, fwd: float, t: float, annuity0: float, df_pay: float, accruals: list[float],
               smile: Callable[[float], float]) -> float:
    """Value (per unit notional and accrual) of max(S(T) - K, 0) paid at Tp:
    A(0) [ (a0 + a1 K) Pay(K) + 2 a1 * integral_K^inf Pay(L) dL ]."""
    a0, a1 = linear_tsr(annuity0, df_pay, fwd, accruals)
    ks = _grid(fwd, t, smile)
    ks = np.concatenate([[strike], ks[ks > strike]])
    tail = float(np.trapezoid([bachelier_call(fwd, k, t, smile(k)) for k in ks], ks))
    return annuity0 * ((a0 + a1 * strike) * bachelier_call(fwd, strike, t, smile(strike)) + 2 * a1 * tail)
```

***Listing 6.2.** A CMS caplet as payer swaptions at its strike and above. code/firm/cms/firm_cms.py*

**Example 6.9 (A CMS caplet).**

A caplet at 3.50% on the ten-year rate observed in five years (forward 3.098%, smile volatility 75.9 basis points at the strike), paid a year later, is worth 48.5 basis points of notional by replication; discounting a Bachelier call on the forward would give 43.0, 13% too little.

## 6.3 The linear terminal swap-rate model in practice

The model needs only today’s curve: $a_0$ from the accruals, $a_1$ from $P(0,T_p)/A(0)$. It is exact when the curve at $T$ is flat and moves in parallel and is a good approximation otherwise; its alternatives map $P(T,T_p)/A(T)$ through a flat-curve annuity formula (Hagan’s standard model) or through a short-rate model’s bonds (chapter 7). What it misses is the dependence of the mapping on the curve’s shape, which matters for long payment delays and steep curves; what matters more in practice is the smile far from the money, where replication reaches. A desk that marks CMS products therefore extrapolates its cube’s wings with care (chapter 5’s densities) and often calibrates the wing parameters to traded CMS spreads rather than to swaptions.

## 6.4 Quanto adjustments in rates

A euro swap rate paid in dollars is a rate observed in one currency and paid in another. Under the dollar payment measure its drift changes by the covariance with the exchange rate, the quanto adjustment of One Quant Book 5, chapter 17. For a normal rate with volatility $\sigma_N$, an EURUSD volatility $\sigma_X$ (dollars per euro) and correlation $\rho$ between the rate and EURUSD, the expectation moves by $-\rho\,\sigma_N\sigma_X\,T$ to first order.

**Example 6.10 (A euro rate paid in dollars).**

With $\sigma_N = 75$ basis points, $\sigma_X = 8\%$, $\rho = 0.3$ and five years, the adjustment is $-0.3\times0.0075\times0.08\times5 = -9$ basis points: when euro rates rise the euro tends to rise too, so each basis point is paid in more valuable euros than the dollar payment delivers.

## 6.5 CMS spread options

**Definition 6.11 (CMS spread option).**

A *CMS spread option* pays $\max(S_1(T)-S_2(T)-K,0)$ at $T_p$ on the difference of two CMS rates of different tenors observed at the same date, typically the ten-year and the two-year; a strip of them with $K=0$ pays a steepener coupon.

Priced with each rate normal around its own CMS expectation, the spread is normal with volatility $\sigma_s = \sqrt{\sigma_1^2+\sigma_2^2-2\rho\sigma_1\sigma_2}$: the price depends on the correlation between the two rates, which no swaption quotes. The desk that sells a steepener note is short spread volatility and so long correlation; its hedge is correlation-sensitive products, and its mark depends on an estimate.

**Example 6.12 (The spread’s volatility).**

Over 2016–2026 daily changes of the two- and ten-year Treasury par yields had a correlation of 0.769 (chapter 3’s data, a proxy for swap rates). With the cube’s at-the-money volatilities for the one-year expiry, the spread of the ten- and two-year euro rates has a normal volatility of 45.1 basis points, well below either rate’s.

![Fair participation of a ten-year steepener note (the weekend problem) as a function of the correlation between the ten- and two-year swap rates: the higher the correlation, the less volatile the spread and the cheaper each unit of its floor, so the more of it the note can pay. Data: the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-6/rc-convexity-adjustments-and-constant-maturity-products/fig-28e667308390.svg)

***Figure 6.4.** Fair participation of a ten-year steepener note (the weekend problem) as a function of the correlation between the ten- and two-year swap rates: the higher the correlation, the less volatile the spread and the cheaper each unit of its floor, so the more of it the note can pay. Data: the chapter’s tutorial.*

## 6.6 Tutorial: replicating a CMS rate

**Goal.** Compute CMS adjustments by replication on the cube, price a CMS caplet and a steepener note. **End state:** Figures [6.2](#fig-rc-convexity-adjustments-and-constant-maturity-products-adj), [6.3](#fig-rc-convexity-adjustments-and-constant-maturity-products-contrib) and [6.4](#fig-rc-convexity-adjustments-and-constant-maturity-products-steep).

1. **Set-up** : `setup(e, tenor)` returns the forward swap rate, its annuity, the payment discount factor and the accruals from chapter 2’s curves.
2. **Adjustments** : `cms_table()` with a flat smile ( `cms_rate_hagan` ) and with the cube ( `cms_rate` ); the cube’s shifted model has no mass below $-3\%$ , so the integral starts there.
3. **Where it comes from** : `replication_contributions()` .
4. **The note** : `steepener()` ; `fig_rc_cms.py` writes the charts.

**What to change next.** Replace the cube’s smile by its flat at-the-money volatility in the steepener and compare; move the payment date to the end of the underlying swap and watch the adjustment grow.

## 6.7 Build: CMS pricing

**Purpose.** Pricing of every product paying a rate at the wrong time, in the wrong currency or with a constant maturity; the CMS legs of the book’s structured notes.

**Interface.** `timing_adjustment_black`, `timing_adjustment_normal`; `linear_tsr(annuity0, df_pay, fwd, accruals)`; `otm_integral`, `cms_rate`, `cms_caplet`, `cms_rate_hagan`; `quanto_adjustment_normal`; `spread_vol`, `cms_spread_option`.

**Rules.** The smile is any function from strike to normal volatility (the cube of chapter 5 in the book); the integral is bounded below by the smile model’s floor; numpy only.

**Acceptance tests.** `code/firm/cms/tests/`: with a flat smile the replicated adjustment equals the closed form; with no slope in the annuity mapping there is no adjustment; a deep in-the-money CMS caplet is the discounted CMS forward; [timing adjustments](#def-rc-convexity-adjustments-and-constant-maturity-products-timing) agree between the Black and normal forms at matched volatilities; perfect correlation removes the spread option’s time value.

**Stretch.** Hagan’s standard (flat-curve annuity) mapping; [CMS spread options](#def-rc-convexity-adjustments-and-constant-maturity-products-spread) with a copula of the two smiles; a CMS swap’s full leg with payment delays.

Sources and further reading

- P. S. Hagan, “Convexity conundrums: pricing CMS swaps, caps, and floors”, *Wilmott Magazine* , March 2003.
- L. Andersen and V. Piterbarg, *Interest Rate Modeling* , volume 3, Atlantic Financial Press, 2010, chapters 16–17.
- US Department of the Treasury, *Daily Treasury Par Yield Curve Rates* (the correlation estimate).

## 6.8 Exercises

**Exercise 6.1 ★.**

A three-month rate of 2.5% fixing in four years, with 80 basis points of normal volatility, is paid at its fixing. Compute its [timing adjustment](#def-rc-convexity-adjustments-and-constant-maturity-products-timing).

**Solution of Exercise 6.1.**

$\delta\sigma_N^2T/(1+\delta F) = 0.25\times0.008^2\times4/(1+0.25\times0.025) =
0.64$ basis points.

**Exercise 6.2 ★.**

Why is the CMS adjustment positive for a rate paid shortly after it is observed? What sign does $a_1$ have when rates are positive?

**Solution of Exercise 6.2.**

Paid soon after observation, the payment measure’s numeraire $P(T,T_p)$ is nearly one while the annuity falls when rates rise, so $P/A$ rises with $S$: $a_1>0$, and the payment measure weights high-rate states more than the annuity measure does. With positive rates $a_0 + a_1S(0) = P(0,T_p)/A(0) >
1/\sum\delta_i = a_0$, hence $a_1 > 0$.

**Exercise 6.3 ★.**

For a ten-year annual swap with accruals of 1.0139 each, compute $a_0$.

**Solution of Exercise 6.3.**

$a_0 = 1/(10\times1.0139) = 0.09863$.

**Exercise 6.4 ★★.**

With a flat smile, the five-year adjustment of [Example 6.8](#ex-rc-convexity-adjustments-and-constant-maturity-products-cmsadj) is 11.73 basis points. How large would it be at a normal volatility 20% higher?

**Solution of Exercise 6.4.**

With a flat smile the adjustment is proportional to $\sigma_N^2$: $11.73\times1.2^2 = 16.89$ basis points.

**Exercise 6.5 ★★.**

Read [Figure 6.3](#fig-rc-convexity-adjustments-and-constant-maturity-products-contrib): roughly what share of the adjustment comes from swaptions more than two percentage points from the forward? What does that say about how the cube’s wings are marked?

**Solution of Exercise 6.5.**

About 29%: swaptions more than two percentage points out of the money carry nearly a third of the ten-year adjustment. The cube’s wings, which the swaption market quotes thinly, must be marked with care, bounded by densities (chapter 5), and are often calibrated to traded CMS spreads.

**Exercise 6.6 ★★.**

In [Example 6.10](#ex-rc-convexity-adjustments-and-constant-maturity-products-quanto), what happens to the adjustment if the correlation is $-0.3$, and why?

**Solution of Exercise 6.6.**

$+9$ basis points: when euro rates rise the euro tends to fall, so each euro basis point is worth fewer dollars in the states where rates are high; the dollar payment measure shifts the rate’s expectation up.

**Exercise 6.7 ★★★.**

*Coding.* Compute the ten-year CMS adjustment of the rate observed in five years when it is paid at the end of the underlying swap (fifteen years) instead of one year after observation. Explain the sign.

**Solution of Exercise 6.7.**

$-19.24$ basis points. Paid at the swap’s end, $P(T,T_p)/A(T)$ falls when rates rise (the long discount factor falls faster than the annuity), so $a_1<0$ and the payment measure weights low-rate states more: the adjustment is negative.

**Exercise 6.8 ★★★.**

*Find the flaw.* “A CMS swap is linear in rates, so we value its coupons at the forward swap rates and hedge them with forward-starting swaps.” Correct it.

**Solution of Exercise 6.8.**

Each CMS coupon is a swap rate paid once, on a date that does not match the annuity numeraire under which the forward swap rate is a martingale; its value is the forward plus a convexity adjustment worth a strip of swaptions (32 basis points for the ten-year rate in ten years here). Valued at forwards the coupons are underpriced, and a hedge with forward swaps leaves the book short volatility; hedge the adjustment with swaptions across strikes.

## 6.9 Problem: The Steepener Note

**Problem 6.1.**

Weekend problem — pricing a ten-year steepener

A bank issues a ten-year euro note that pays each year $k=1,\dots,10$ a coupon of $p\times\max(S_{10}(k)-S_2(k),0)$, the ten-year minus the two-year swap rate observed at year $k$, paid at $k+1$. The bank’s funding target is to pay coupons worth as much as a fixed 2% coupon on the same dates. Rates are those of chapter 2, smiles those of chapter 5, and the correlation between the two rates is estimated at 0.769.

**Part I — The pieces.**

1. What is each coupon, as an option?
2. Why does each rate need a CMS adjustment, and which one is larger?
3. Give the value of the fixed 2% leg per unit notional.
4. Give the value of the coupon leg with $p=1$ .
5. Give the fair participation $p$ .

**Part II — Correlation.**

6. Give the fair participation at correlations of 0.6, 0.8 and 0.95.
7. Why does it rise with correlation?
8. Which way is the issuing bank exposed to correlation after selling the note?
9. Give the normal volatility of the first year’s spread.
10. Why is Treasury data only a proxy here?

**Part III — Risk.**

11. Which swaptions hedge the CMS adjustments?
12. What happens to the note’s value if the smile’s wings rise?
13. Is the note’s value sensitive to the level of rates, the slope, or both?
14. What happens to the investor if the curve inverts?
15. Why do dealers’ books of such notes become large and one-sided?

**Part IV — Judgement.**

16. What participation would you quote, and why not 2.96?
17. How would you mark correlation in the absence of quotes?
18. What would you tell an investor who says the curve “always” steepens back?
19. State the *named result* : the fair participation at the estimated correlation, and its range between correlations of 0.6 and 0.95.
20. In one sentence: what does a steepener investor buy?

**Solution of Problem 6.1.**

**1.** A [CMS spread option](#def-rc-convexity-adjustments-and-constant-maturity-products-spread) struck at zero on the ten-year minus two-year rate, paid a year after observation, times $p$. **2.** Each is a swap rate paid once, a year after observation: each needs a CMS adjustment; the ten-year’s is larger, its annuity being more sensitive to rates and its volatility higher per unit of $a_1$. **3.** 0.17118 per unit notional. **4.** 0.05785. **5.** $0.17118/0.05785 = 2.96$. **6.** 2.44, 3.10 and 4.47. **7.** Higher correlation, lower spread volatility, cheaper spread floors struck at zero (the spread’s forward being positive): each unit of coupon costs less, so more units fit in the same budget. **8.** It is short the spread options it pays, so short spread volatility and long correlation: it loses if correlation falls. **9.** 45.1 basis points. **10.** Swap rates and Treasury yields differ by swap spreads, which move with their own dynamics; the correlation that matters is under the pricing measure and over the note’s horizon, not daily historical changes. **11.** Out-of-the-money payer and receiver swaptions on each rate’s underlying swap, across strikes, in the replication weights. **12.** The CMS adjustments rise, raising the CMS forwards of both rates; the effect on the spread depends on which rises more, and the spread options gain from the higher volatility of each leg. **13.** Both: the level through the CMS adjustments and the annuity mapping, the slope directly through the spread’s forward. **14.** Coupons fall to zero: the floor at zero protects the investor from negative coupons, not from receiving nothing for years. **15.** Many investors buy the same trade from few dealers, who end up short the same correlation and spread volatility and can hedge it only with each other; the book grows with issuance and is hard to reduce. **16.** Lower than 2.96: the bank charges for its hedging costs, correlation reserve and funding; a margin of a few tenths of participation is usual for illiquid correlation. **17.** From historical estimates of several windows and regimes, from the correlations implied by any traded spread options, with a reserve for the spread between them; stress it in the risk report. **18.** That the note pays only if the curve is steeper than zero at each fixing; a flat or inverted curve for years pays nothing, and the forwards already price the expected steepening. **19.** Named result: *the steepener note* can pay 2.96 times the ten-year minus two-year spread at a correlation of 0.769, between 2.44 and 4.47 as the correlation goes from 0.6 to 0.95. **20.** A strip of options on the curve’s slope, paid for by giving up a fixed coupon.

## 6.10 Interview questions

**Interview question 6.1 ★ researcher, trader.**

What is a convexity adjustment, in one sentence, and give two examples.

**Solution of Interview question 6.1.**

The difference between a rate’s expectation under the measure of its actual payment and its forward, caused by the correlation between the rate and the ratio of numeraires. Examples: futures against forwards (daily margining), a rate paid at its fixing date (timing), a CMS rate, a rate paid in another currency (quanto).

*What the interviewer is looking for: the measure-change explanation and several instances.*

**Interview question 6.2 ★★ researcher.**

Derive the [timing adjustment](#def-rc-convexity-adjustments-and-constant-maturity-products-timing) of a rate paid at its fixing date.

**Solution of Interview question 6.2.**

Change from the $T_k$- to the $T_{k-1}$-forward measure with density $(1+\delta F_k(T_{k-1}))/(1+\delta F_k(0))$; then $\E^{T_{k-1}}[F] =
F_0+\delta\Var^{T_k}(F)/(1+\delta F_0)$, i.e. $\delta\sigma_N^2T/(1+\delta F_0)$ in the normal model.

*What the interviewer is looking for: the Radon–Nikodym density between forward measures.*

**Interview question 6.3 ★★ researcher, bank.**

How do you price a CMS caplet from swaptions?

**Solution of Interview question 6.3.**

Write the payoff under the annuity measure: value $=A(0)\E^A[(S-K)^+P(T,T_p)/A(T)]$; map $P/A$ to $a_0+a_1S$ ([linear terminal swap-rate model](#def-rc-convexity-adjustments-and-constant-maturity-products-tsr)); expand $(S-K)^+(a_0+a_1S) = (a_0+a_1K)(S-K)^+ + 2a_1\int_K^\infty(S-L)^+dL$ and price each payer swaption on the smile.

*What the interviewer is looking for: annuity mapping plus static replication.*

**Interview question 6.4 ★★ trader.**

You are short CMS coupons. Which way do you want implied volatility to move, and which strikes matter most?

**Solution of Interview question 6.4.**

Down: the CMS coupons you owe are worth the forward plus a strip of out-of-the-money swaptions, so you are short volatility, including the wings; strikes around and well away from the forward matter, nearly a third of the adjustment coming from more than two percentage points away.

*What the interviewer is looking for: short vega through the replication, and the wings.*

**Interview question 6.5 ★★★ researcher, risk.**

A [CMS spread option](#def-rc-convexity-adjustments-and-constant-maturity-products-spread) depends on a correlation that is not quoted. How do you mark and hedge it?

**Solution of Interview question 6.5.**

Mark from a range of historical estimates and any implied correlations from traded spread options or dealer polls, with a reserve for the uncertainty; hedge the correlation exposure with offsetting spread products where possible, otherwise run it within limits and stress it; report the correlation sensitivity separately.

*What the interviewer is looking for: marks with reserves, and honesty about unhedgeable risk.*

**Interview question 6.6 ★★★ developer.**

Your CMS replication gives a different number every time the strike grid changes. What do you check?

**Solution of Interview question 6.6.**

Grid width (enough standard deviations, and no strikes below the model’s floor), spacing near the forward where the integrand is steep, whether the strike and the forward are grid nodes, the smile’s behaviour in the wings (extrapolation, NaNs), and convergence as the grid is refined; compare with the closed form under a flat smile.

*What the interviewer is looking for: numerical integration hygiene and a benchmark.*