Interest Rate Swaps

Abstract

This chapter builds on the general framework of Chap. 2 and develops indexes of expected volatility for interest rate swaps in a model-free fashion. It illustrates the main empirical and theoretical challenges described in the previous chapters in the context of the interest rate swap market while attempting to be as self-contained as possible. Such challenges include the arduous task of expressing directional volatility views through options-based strategies and the difficulty of insulating the pure volatility component in a variance contract design. The chapter explains how to overcome these issues and contains many empirical illustrations of the resulting indexes of swap rate volatility, as well as extensions regarding their relationship with constant maturity swaps and applications to trading strategies.

Appendix B: Appendix on Interest Rate Swap Markets

3.1.1 B.1 P&L of Option-Based Volatility Trading

We provide details regarding P&L calculations associated with swaption-based volatility trading. First, we provide basic definitions related to the Black (1976) formula. Second, we derive the approximations in Eqs. (3.8) and (3.9). Third, we provide details regarding the experiments in Sect. 3.2.2 on directional volatility trading.

Definitions. The price of a payer swaption in Eq. (3.14) is known in closed-form when the volatility $\sigma_{t}$ in Eq. (3.12), and hence the integrated variance $V_{n} ( t,T )$ in Eq. (3.13), are deterministic. It is given by the Black (1976) formula

$$ \mathrm{Swpn}\_\mathrm{Bl}_{t}^{\mathrm{P}} \bigl( R_{t}^{n},\mathrm {PVBP}_{t},K,T;\bar{V} \bigr) = \mathrm{PVBP}_{t}\cdot Z_{t} \bigl( R_{t}^{n},T,K; \bar{V} \bigr) , $$

(B.1)

where

$$ Z_{t}(R_{t};T,K;\bar{V})=R_{t}\varPhi ( d_{t} ) -K\varPhi (d_{t}-\sqrt{\bar{V}}),\quad d_{t}=\frac{\ln\frac{R_{t}}{K}+\frac {1}{2}\bar{V}}{\sqrt{\bar{V}}}, $$

$\varPhi$ denotes the cumulative standard normal distribution, and to alleviate notation, $R_{t}^{n}\equiv R_{t} ( T_{1},\ldots ,T_{n} )$, $\mathrm{PVBP}_{t}\equiv\mathrm{PVBP}_{t} ( T_{1},\ldots ,T_{n} )$, and $\bar{V}$ is the constant value of $V_{n} ( t,T )$ in Eq. (3.13). By the definition of the forward swap rate in Eq. (3.5), we have

$$\begin{aligned} &\mathrm{Swpn}\_\mathrm{Bl}_{t}^{\mathrm{P}} \bigl( R_{t}^{n},\mathrm {PVBP}_{t},K,T;\bar{V} \bigr) \\ &\quad\equiv \bigl( P_{t} ( T ) -P_{t} ( T_{n} ) \bigr) \varPhi ( d_{t} ) -\mathrm{PVBP}_{t}K\varPhi(d_{t}- \sqrt {\bar{V}}). \end{aligned}$$

(B.2)

Equation (B.2) suggests that the swaption can be hedged through portfolios of zero-coupon bonds: (i) long $\varPhi ( d_{t} )$ units of a portfolio which is long one zero expiring at $T$ and short one zero expiring at $T_{n}$, and (ii) short $K\varPhi(d_{t}-\sqrt{\bar{V}})$ units of the $\mathrm{PVBP}_{t}$ basket.

Next, and against the assumption underlying Eq. (B.2), we assume that the forward swap rate has stochastic volatility, $\sigma_{t}$, as in Eq. (3.12) of the main text. Suppose, further, that we have a view that volatility will rise compared to the current implied volatility, defined as the value

$$ \mathrm{IV}_{t}=\sqrt{\frac{\bar{V}}{T-t}}, $$

(B.3)

so that, once re-normalized again by $T-t$ and inserted into Eq. (B.2), it delivers the swaption market price.

Volatility trading based on delta-hedged swaptions. Consider the strategy of buying the swaption and hedging it with the portfolio underlying Eq. (B.2). We derive the P&L of this strategy. Let $\upsilon$ denote the value of a self-financed hedging portfolio that includes (i) long a zero expiring at $T$ and short a zero expiring at $T_{n}$, (ii) a basket of zeros with value equal to the PVBP. The value of this portfolio satisfies:

$$ \upsilon_{\tau}=a_{\tau} \bigl( P_{\tau} ( T ) -P_{\tau } ( T_{n} ) \bigr) +b_{\tau} \mathrm{PVBP}_{\tau},\quad \tau\in [ t,T ] , $$

where, denoting for simplicity $R_{t}\equiv R_{t}^{n}$,

$$ \begin{aligned} &\upsilon_{t}=\mathrm{Swpn}\_\mathrm{Bl}_{t}^{\mathrm{P}} \bigl( R_{t},\mathrm {PVBP}_{t},K,T, ( T-t ) \mathrm{IV}_{t}^{2} \bigr) , \\ &a_{\tau}=\varPhi \biggl( \frac{\ln\frac{R_{\tau}}{K}+\frac{1}{2} ( T-t ) \mathrm{IV}_{t}^{2}}{\sqrt{T-t}\mathrm{IV}_{t}} \biggr) \quad\text{and}\quad b_{\tau}=-K\varPhi \biggl( \frac{\ln\frac{R_{\tau}}{K}-\frac {1}{2} ( T-t ) \mathrm{IV}_{t}^{2}}{\sqrt{T-t}\mathrm{IV}_{t}} \biggr) . \end{aligned} $$

(B.4)

Because this portfolio is self-financed, $d\upsilon_{\tau} = a_{\tau }d [ P_{\tau} ( T ) -P_{\tau} ( T_{n} ) ] +b_{\tau }d\mathrm{PVBP}_{\tau}$, and, hence

$$\begin{aligned} d\upsilon_{\tau}&= \bigl[ \mu_{\tau}^{b} \upsilon_{\tau}+ \bigl( \mu _{\tau}^{\Delta P}- \mu_{\tau}^{b} \bigr) a_{\tau} \bigl( P_{\tau } ( T ) -P_{\tau} ( T_{n} ) \bigr) \bigr] d\tau \\ &\quad{}+ \bigl[ \sigma _{\tau}^{b}\upsilon_{\tau}+ \bigl( \sigma_{\tau}^{\Delta P}-\sigma _{\tau}^{b} \bigr) a_{\tau} \bigl( P_{\tau} ( T ) -P_{\tau } ( T_{n} ) \bigr) \bigr] dW_{\tau}, \end{aligned}$$

(B.5)

where $W_{\tau}$ is a Wiener process under the physical probability and, accordingly, $\mu_{\tau}^{b}$ and $\sigma_{\tau}^{b}$ denote the drift and instantaneous volatility of $\frac{d\mathrm{PVBP}_{\tau}}{\mathrm {PVBP}_{\tau}}$, and $\mu_{\tau}^{\Delta P}$ and $\sigma_{\tau }^{\Delta P}$ denote the drift and instantaneous volatility of $\frac{d ( P_{\tau } ( T ) -P_{\tau} ( T_{n} ) ) }{P_{\tau} ( T ) -P_{\tau} ( T_{n} ) }$.

Next, consider the swaption price in Eq. (B.1) with $\bar{V}$ replaced by $( T-t ) \mathrm{IV}_{t}^{2}$, as in Eq. (B.3), i.e. $\mathrm{Swpn}\_\mathrm{Bl}_{\tau}^{\mathrm{P}}\equiv \mathrm{Swpn}\_\mathrm{Bl}_{\tau}^{\mathrm{P}} ( R_{\tau},\mathrm {PVBP}_{\tau},K,T, ( T-t ) \mathrm{IV}_{t}^{2} )$. By Itô’s lemma, the definition of the forward swap rate in Eq. (3.5), and the partial differential equation satisfied by the pricing function $Z_{t} ( R,T,K;\bar{V} )$, the swaption price satisfies

$$\begin{aligned} & d\mathrm{Swpn}\_\mathrm{Bl}_{\tau}^{\mathrm{P}} \\ &\quad =\mathrm{PVBP}_{\tau}dZ_{\tau}+Z_{\tau}d \mathrm{PVBP}_{\tau }+dZ_{\tau }d\mathrm{PVBP}_{\tau} \\ &\quad =\mathrm{PVBP}_{\tau}\underset{=0}{ \biggl( \underbrace{ \frac{\partial Z_{\tau}}{\partial\tau}+\frac{1}{2}\frac{\partial^{2}Z_{\tau }}{\partial R^{2}}R_{\tau}^{2} \mathrm{IV}_{t}^{2}} \biggr) }d\tau \\ &\qquad{}+\mathrm {PVBP}_{\tau } \biggl( \frac{1}{2}\frac{\partial^{2}Z_{\tau}}{\partial R^{2}}R_{\tau }^{2} \bigl( \sigma_{\tau}^{2}-\mathrm{IV}_{t}^{2} \bigr) +\frac {\partial Z_{\tau}}{\partial R}\mu_{\tau}^{R}R_{\tau} \biggr) d\tau \\ &\qquad{} + \biggl( \mu_{\tau}^{b}\mathrm{Swpn}\_\mathrm{Bl}_{\tau}^{\mathrm {P}}+\frac{\partial Z_{\tau}}{\partial R} \sigma_{\tau}^{b}\sigma _{\tau } \bigl( P_{\tau} ( T ) -P_{\tau} ( T_{n} ) \bigr) \biggr) d\tau \\ &\qquad{} + \biggl( \frac{\partial Z_{\tau}}{\partial R}\sigma_{\tau } \bigl( P_{\tau} ( T ) -P_{\tau} ( T_{n} ) \bigr) +\sigma _{\tau}^{b}\mathrm{Swpn}\_\mathrm{Bl}_{\tau}^{\mathrm{P}} \biggr) dW_{\tau}, \end{aligned}$$

(B.6)

where $\mu_{\tau}^{R}$ is the drift of $\frac{dR_{\tau}}{R_{\tau}}$ under the physical probability, and equals

$$ \mu_{\tau}^{R}=\mu_{\tau}^{\Delta p}- \mu_{\tau}^{b}-\sigma_{\tau }\sigma_{\tau}^{b}, \quad \sigma_{\tau}=\sigma_{\tau}^{\Delta p}-\sigma _{\tau}^{b}, $$

(B.7)

and the second equality in Eqs. (B.7) follows by Itô’s lemma. By the definition of the forward swap rate in Eq. (3.5), Eqs. (B.7), and the relation $a_{\tau}=\frac{\partial Z_{\tau }}{\partial R}$, we can integrate the difference between $d\mathrm{Swpn}\_\mathrm{Bl}_{\tau }^{\mathrm{P}}$ in Eq. (B.6) and $d\upsilon_{\tau}$ in Eq. (B.5) to obtain the P&L of the hedged swaption at the swaption maturity:

$$\begin{aligned} \mathrm{Swpn}\_\mathrm{Bl}_{T}^{\mathrm{P}}- \upsilon_{T}& =\mathrm {PVBP}_{T}\cdot [ R_{T}-K ] ^{+}-\upsilon_{T} \\ & =\frac{1}{2}\int_{t}^{T} \frac{\partial^{2}Z_{\tau }}{\partial R^{2}}R_{\tau}^{2} \bigl( \sigma_{\tau}^{2}- \mathrm{IV}_{t}^{2} \bigr) ( \varLambda_{\tau,T} \mathrm{PVBP}_{\tau} ) d\tau \\ &\quad{}+\int_{t}^{T} \varLambda_{\tau,T}\sigma_{\tau}^{b} \bigl( \mathrm{Swpn}\_\mathrm{Bl}_{\tau}^{\mathrm{P}}-\upsilon_{\tau} \bigr) dW_{\tau}, \end{aligned}$$

(B.8)

where we have used the first relation in Eqs. (B.4), $\mathrm{Swpn}\_\mathrm{Bl}_{\tau}^{\mathrm{P}}=\upsilon_{\tau}$ for $\tau=t$, and defined $\varLambda_{\tau,T}=e^{\int_{\tau}^{T}\mu_{s}^{b}ds}$. The approximation in Eq. (3.8), which merely aims to simplify the presentation, relies on: (i) $\mathrm{PVBP}_{T}\approx\varLambda_{\tau ,T}\mathrm{PVBP}_{\tau}$; (ii) $\widetilde{\frac{\Delta R_{t}}{R_{t}}}\approx dW_{t}$; and (iii) disregarding the term $\varLambda_{\tau,T}$ inside the stochastic integral in Eq. (B.8).

Volatility trading based on straddles. Here we derive the P&L regarding the straddle. It is useful in the sequel to note that the parity for payer and receiver swaptions is

$$ \mathrm{Swpn}_{t}^{\mathrm{P}} ( K,T;T_{1}, \ldots,T_{n} ) =\mathrm{IRS}_{t} ( K;T_{1}, \ldots,T_{n} ) +\mathrm{Swpn}_{t}^{\mathrm{R}} ( K,T;T_{1},\ldots,T_{n} ) . $$

(B.9)

Consider the value of a straddle, $\mathrm{Straddle}_{\tau}=\mathrm {Swpn}_{\tau}^{\mathrm{P}}+\mathrm{Swpn}_{\tau}^{\mathrm{R}}$. By the Black (1976) formula, the price of the receiver swaption is

$$ \begin{aligned} &\mathrm{Swpn}\_\mathrm{Bl}_{t}^{\mathrm{R}} \bigl( R_{t}^{n},\mathrm {PVBP}_{t},K,T;\bar{V} \bigr) =\mathrm{PVBP}_{t}\cdot\hat{Z}_{t} \bigl( R_{t}^{n},T,K;\bar{V} \bigr) , \\ &\hat{Z}_{t} ( R_{t},T,K;\bar{V} ) =K \bigl( 1- \varPhi(d_{t}-\sqrt {\bar{V}}) \bigr) -R_{t} \bigl( 1- \varPhi(d_{t}) \bigr) ,\quad d_{t}=\frac {\ln\frac{R_{t}}{K}+\frac{1}{2}\bar{V}}{\sqrt{\bar{V}}}. \end{aligned} $$

(B.10)

Therefore, the dynamics of $\mathrm{Swpn}\_\mathrm{Bl}_{\tau}^{\mathrm{R}}$ are the same as $\mathrm{Swpn}\_\mathrm{Bl}_{\tau}^{\mathrm{P}}$ in Eq. (B.6), but with $\hat{Z}_{\tau}$ replacing $Z_{t}$. Moreover, we have, by Eq. (B.9) and Eq. (3.6) of the main text,

$$ \frac{\partial\mathrm{Straddle}_{\tau}}{\partial R}=\mathrm {PVBP}_{\tau } \biggl( 1+2\frac{\partial\hat{Z}_{\tau}}{\partial R} \biggr) ,\quad\frac{\partial\hat{Z}_{\tau}}{\partial R}=\frac{\partial Z_{\tau }}{\partial R}-1, $$

where $Z_{\tau}$ is as in Eq. (B.1). Assuming the straddle delta is sufficiently small, which it is by the previous equation when $2\frac{\partial Z_{\tau}}{\partial R}\approx1$, the value of the straddle is, by Eq. (B.6) and the previous arguments concerning the dynamics of $\mathrm{Swpn}\_\mathrm{Bl}_{\tau}^{\mathrm{R}}$,

$$\begin{aligned} & \mathrm{Straddle}_{T}-\mathrm{Straddle}_{t} \\ &\quad =\int_{t}^{T}\frac{\partial^{2}Z_{\tau}}{\partial R^{2}}R_{\tau }^{2} \bigl( \sigma_{\tau}^{2}-\mathrm{IV}_{t}^{2} \bigr) ( \varLambda _{\tau,T}\mathrm{PVBP}_{\tau} ) d\tau+\int _{t}^{T}\varLambda _{\tau,T} \sigma_{\tau}^{b}\mathrm{Straddle}_{\tau}dW_{\tau}. \end{aligned}$$

(B.11)

The approximation in Eq. (3.9) relies on the same arguments leading to Eq. (3.8). Note that the approximation $2\frac{\partial Z_{\tau}}{\partial R}\approx1$ is quite inadequate as the forward swap rate drifts away from ATM—a very well-known feature discussed, for example, by Mele (2014, Chap. 10) in the case of equity straddles.

Directional volatility trades. The P&L displayed in Fig. 3.3 are calculated using daily data from January 1998 to December 2009, and comprise yield curve data as well as implied volatilities for swaptions. Yield curve data are needed to calculate forward swap rates for a fixed 5-year tenor and their realized volatilities, and implied volatilities are needed to compute P&Ls.

As for the straddle, the strategy is to go long an ATM swaption straddle. Let $t$ denote the beginning of the holding period (1 month or 3 months). The terminal P&L of the straddle, say as of time $T$, is

$$\begin{aligned} &\mathrm{PVBP}_{T} ( T_{1},\ldots,T_{n} ) \bigl( \bigl[ R_{T} ( T_{1},\ldots,T_{n} ) -K \bigr] ^{+}+ \bigl[ K-R_{T} ( T_{1},\ldots ,T_{n} ) \bigr] ^{+} \bigr) \\ &\quad{}-\mathrm{Straddle}_{t} ( T_{1},\ldots ,T_{n} ) , \end{aligned}$$

where $\mathrm{Straddle}_{t} ( T_{1},\ldots,T_{n} )$ is the cost of the straddle at $t$, and $T-t$ is either 1 month (as in the top panel of Fig. 3.3) or 3 months (as in the bottom panel).

Instead, the terminal P&L of the variance swap contract is calculated consistently with Definition 3.2 and Eq. (3.22) as

$$ \frac{1}{T-t} \biggl( \mathrm{PVBP}_{T} ( T_{1}, \ldots,T_{n} ) V_{n} ( t,T ) -\frac{\mathbb{F}_{\mathrm{var},n}^{\ast} ( t,T ) }{P_{t} ( T ) } \biggr) , $$

where $\mathbb{F}_{\mathrm{var},n}^{\ast} ( t,T )$ approximates the unnormalized strike of the contract to be entered at $t$. Its exact value, $\mathbb{F}_{\mathrm{var},n} ( t,T )$, is that in Eq. (3.21). By Eq. (3.38),

$$ \frac{1}{T-t}\mathbb{F}_{\mathrm{var},n} ( t,T ) =\mathrm {PVBP}_{t} ( T_{1},\ldots,T_{n} ) \cdot \mathrm{IRS}\text{-}\mathrm{VI}_{n}^{2} ( t,T ) , $$

which we approximate with

$$ \frac{1}{T-t}\mathbb{F}_{\mathrm{var},n}^{\ast} ( t,T ) \equiv \mathrm{PVBP}_{t} ( T_{1},\ldots,T_{n} ) \cdot \mathrm {ATM}_{n}^{2} ( t,T ) , $$

where $\mathrm{ATM}_{n} ( t,T )$ is the ATM implied volatility.

Finally, the realized variance rescaled by $T-t$, $\frac {1}{T-t}V_{n} ( t,T )$, is calculated as

$$ \overline{\mathrm{RV}}_{t}^{m}= \biggl( \frac{21\cdot m}{252} \biggr) ^{-1}\mathrm{RV}_{t}^{m},\quad\text{where } \mathrm{RV}_{t}^{m}=\sum_{i=t-21\cdot m}^{t} \ln^{2}\frac{R_{i} ( T_{1},\ldots ,T_{n} ) }{R_{i-1} ( T_{1},\ldots,T_{n} ) }, $$

with $m\in \{ 1,3 \}$ and $T_{n}-t=5$. The variance risk premium is defined as $\overline{\mathrm{RV}}_{t}^{m}-\mathrm {ATM}_{n}^{2} ( t-21\cdot m,T )$, where $T-t$ is either $\frac{1}{12}$ (for 1 month) or $\frac{3}{12}$ (for 3 months). Finally, to make the P&Ls of the volatility swap contract and the straddle line up to the same order of magnitude, the values in Fig. 3.3 are obtained by re-multiplying the P&Ls of the variance contracts by $\frac{1}{12}$ (top panel) and $\frac {3}{12}$ (bottom panel).

3.1.2 B.2 Spanning IRS Variance Contracts

The pricing results of this chapter are a special case of those in Chap. 2. However, we derive them in the context of the interest rate swap market of this chapter for three reasons. First, these are the results that we first derived in Mele and Obayashi (2012); second, the derivations in this appendix make this chapter self-contained; third, some of the derivations and notation in the following proof reveal useful for hedging variance swaps.

Because the reader may be more familiar with the pricing of percentage variance swaps, we first provide details for these contracts, and then details regarding basis point variance swaps.

Pricing I: Percentage. To alleviate notation, we set $R_{t}\equiv R_{t} ( T_{1},\ldots,T_{n} )$. By the usual Taylor expansion with remainder,

$$ \ln\frac{R_{T}}{R_{t}}=\frac{1}{R_{t}} ( R_{T}-R_{t} ) - \int_{0}^{R_{t}} ( K-R_{T} ) ^{+} \frac{1}{K^{2}}dK-\int_{R_{t}}^{\infty} ( R_{T}-K ) ^{+}\frac{1}{K^{2}}dK. $$

(B.12)

Multiplying both sides of the previous equation by $\mathrm {PVBP}_{t} ( T_{1},\ldots,T_{n} )$, and taking expectations under the annuity probability $Q_{\mathrm{sw}}$ (see Eq. (3.11) in the main text):

$$\begin{aligned} &\mathrm{PVBP}_{t} ( T_{1},\ldots,T_{n} ) \mathbb {E}_{t}^{Q_{\mathrm{sw}}} \biggl( \ln\frac{R_{T}}{R_{t}} \biggr) \\ &\quad=-\int _{0}^{R_{t}}\frac{\mathrm{Swpn}_{t}^{\mathrm{R}} ( K,T;T_{n} ) }{K^{2}}dK-\int _{R_{t}}^{\infty}\frac {\mathrm{Swpn}_{t}^{\mathrm{P}} ( K,T;T_{n} ) }{K^{2}}dK, \end{aligned}$$

(B.13)

where we have used (i) the fact that by Eq. (3.12) of the main text, the forward swap rate is a martingale under $Q_{\mathrm{sw}}$, and (ii) the pricing Equations (3.14) and (3.15) of the main text.

By Eq. (3.12) and a change of measure obtained through Eq. (3.11),

$$\begin{aligned} &{-}2\mathbb{E}_{t}^{Q_{\mathrm{sw}}} \biggl( \ln\frac{R_{T}}{R_{t}} \biggr) \\ &\quad =\mathbb{E}_{t}^{Q_{\mathrm{sw}}} \biggl( \int_{t}^{T} \sigma _{s}^{2} ( T_{1},\ldots,T_{n} ) ds \biggr) \\ &\quad =\frac{1}{\mathrm{PVBP}_{t} ( T_{1},\ldots,T_{n} ) } \\ &\qquad{}\times\mathbb {E}_{t} \biggl[ e^{-\int_{t}^{T}r_{s}ds} \biggl( \mathrm {PVBP}_{T} ( T_{1},\ldots,T_{n} ) \int _{t}^{T}\sigma_{s}^{2} ( T_{1},\ldots,T_{n} ) ds \biggr) \biggr] , \end{aligned}$$

(B.14)

where $\mathbb{E}_{t}$ denotes the risk-neutral expectation conditional on information available at time-$t$. Combining Eq. (B.13) and Eq. (B.14) gives:

$$\begin{aligned} & \mathbb{E}_{t} \biggl[ e^{-\int_{t}^{T}r_{s}ds} \biggl( \mathrm {PVBP}_{T} ( T_{1},\ldots,T_{n} ) \int _{t}^{T}\sigma _{s}^{2} ( T_{1},\ldots,T_{n} ) ds \biggr) \biggr] \\ &\quad =2 \biggl( \int_{0}^{R_{t}}\frac{\mathrm{Swpn}_{t}^{\mathrm {R}} ( K,T;T_{1},\ldots,T_{n} ) }{K^{2}}dK+ \int_{R_{t}}^{\infty}\frac{\mathrm{Swpn}_{t}^{\mathrm{P}} ( K,T;T_{1},\ldots ,T_{n} ) }{K^{2}}dK \biggr) . \end{aligned}$$

(B.15)

The L.H.S. of the previous equation is the value of the IRV forward agreement in Definition 3.1 (in percentage), and Eq. (3.21) is its approximation.

To derive the IRV swap rate $\mathbb{P}_{\mathrm{var},n} ( t,T )$ in the first of Eqs. (3.22), note that by Eq. (3.16), it solves

$$ 0=\mathbb{E}_{t} \bigl( e^{-\int_{t}^{T}r_{s}ds} \bigl( V_{n} ( t,T ) \times\mathrm{PVBP}_{T} ( T_{1},\ldots,T_{n} ) - \mathbb{P}_{\mathrm{var},n} ( t,T ) \bigr) \bigr) , $$

which yields the expression of $\mathbb{P}_{\mathrm{var},n} ( t,T )$ in the first of Eqs. (3.22) after rearranging terms. As for the standardized IRV swap rate $\mathbb{P}_{\mathrm{var},n}^{\ast} ( t,T )$ in the second of Eqs. (3.22), note that by Eq. (3.18), it satisfies:

$$ 0=\mathbb{E}_{t} \bigl( e^{-\int_{t}^{T}r_{s}ds} \bigl( V_{n} ( t,T ) -\mathbb{P}_{\mathrm{var},n}^{\ast} ( t,T ) \bigr) \times \mathrm{PVBP}_{T} ( T_{1},\ldots,T_{n} ) \bigr) , $$

which yields the expression of $\mathbb{P}_{\mathrm{var},n}^{\ast} ( t,T )$ in the second of Eqs. (3.22) by the definition of the Radon–Nikodym derivative in Eq. (3.11) of the main text after rearranging terms.

Pricing II: Basis point. We price contracts for which the variance payoff is given by

$$ V_{n}^{\mathrm{bp}} ( t,T ) \times\mathrm{PVBP}_{T} ( T_{1},\ldots,T_{n} ) , $$

where $V_{n}^{\mathrm{bp}} ( t,T )$ is the realized variance of the forward swap rate changes during the interval $[ t,T ]$ (see the first of Eqs. (3.13)).

By a Taylor expansion with remainder, we have

$$ R_{T}^{2}=R_{t}^{2}+2R_{t} ( R_{T}-R_{t} ) +2 \biggl( \int_{0}^{R_{t}} ( K-R_{T} ) ^{+}dK+\int_{R_{t}}^{\infty} ( R_{T}-K ) ^{+}dK \biggr) . $$

(B.16)

Multiplying both sides of this equation by $\mathrm{PVBP}_{t} ( T_{1},\ldots,T_{n} )$ and taking expectations under $Q_{{\mathrm {sw}}}$ leads to

$$\begin{aligned} & \mathrm{PVBP}_{t} ( T_{1},\ldots,T_{n} ) \mathbb {E}_{t}^{Q_{\mathrm{sw}}} \bigl( R_{T}^{2}-R_{t}^{2} \bigr) \\ &\quad =2 \biggl( \int_{0}^{R_{t}}\mathrm{Swpn}_{t}^{\mathrm{R}} ( K,T;T_{1},\ldots,T_{n} ) dK+\int_{R_{t}}^{\infty } \mathrm{Swpn}_{t}^{\mathrm{P}} ( K,T;T_{1}, \ldots,T_{n} ) dK \biggr) . \end{aligned}$$

(B.17)

Moreover, by Eq. (3.12) and Itô’s lemma,

$$ \mathbb{E}_{t}^{Q_{\mathrm{sw}}} \bigl( R_{T}^{2}-R_{t}^{2} \bigr) =\mathbb{E}_{t}^{Q_{\mathrm{sw}}} \bigl( V_{n}^{\mathrm{bp}} ( t,T ) \bigr) . $$

Substituting this expression into Eq. (B.17) yields

$$\begin{aligned} & 2 \biggl( \int_{0}^{R_{t}}\mathrm{Swpn}_{t}^{\mathrm{R}} ( K,T;T_{1},\ldots,T_{n} ) dK+\int_{R_{t}}^{\infty } \mathrm{Swpn}_{t}^{\mathrm{P}} ( K,T;T_{1}, \ldots,T_{n} ) dK \biggr) \\ &\quad =\mathrm{PVBP}_{t} ( T_{1},\ldots,T_{n} ) \mathbb {E}_{t}^{Q_{\mathrm{sw}}} \bigl( R_{T}^{2}-R_{t}^{2} \bigr) \\ &\quad =\mathrm{PVBP}_{t} ( T_{1},\ldots,T_{n} ) \mathbb {E}_{t}^{Q_{\mathrm{sw}}} \bigl( V_{n}^{\mathrm{bp}} ( t,T ) \bigr) \\ &\quad =\mathbb{E}_{t} \bigl[ e^{-\int_{t}^{T}r_{s}ds} \bigl( \mathrm{PVBP}_{T} ( T_{1},\ldots,T_{n} ) V_{n}^{\mathrm {bp}} ( t,T ) \bigr) \bigr] , \end{aligned}$$

(B.18)

where the last line follows by the Radon–Nikodym derivative defined in Eq. (3.11) of the main text. The R.H.S. of this equation is the value of the IRV forward agreement in Definition 3.1 in basis points.

Volatility indexes and Black’s Implied volatilities. The percentage standardized IRV swap rate in Definition 3.3 leads to an index of interest rate swap volatility, Eq. (3.38) (Sect. 3.5). This expression can be simplified by the Black (1976) formula, as claimed in the main text. Indeed, substitute Eq. (3.21) into Eq. (3.38), and use the Black formulae in Eqs. (B.1) and (B.10). The result is

$$\begin{aligned} & \mathrm{IRS}\text{-}\mathrm{VI}_{n} ( t,T ) \\ &\quad =\sqrt{\frac{2}{T-t} \biggl( \sum_{i:K_{i}< R_{t}} \frac{\hat {Z}_{t} ( R_{t}^{n},T,K_{i}; ( T-t ) \cdot\mathrm {IV}_{i,t}^{2} ) }{K_{i}^{2}}\Delta K_{i} +\sum _{i:K_{i}\geq R_{t}} \frac{Z_{t} ( R_{t}^{n},T,K_{i}; ( T-t ) \cdot\mathrm{IV}_{i,t}^{2} ) }{K_{i}^{2}}\Delta K_{i} \biggr) }, \end{aligned}$$

(B.19)

where the expressions for $Z_{t}$ and $\hat{Z}_{t}$ are given in Eqs. (B.1) and (B.10), $R_{t}^{n}$ is the current forward swap rate for maturity $T$ and tenor length $T_{n}-T$, $R_{t}^{n}\equiv R_{t} ( T_{1},\ldots,T_{n} )$, and, finally, $\mathrm{IV}_{i,t}$ denotes the time $t$ implied percentage volatility for swaptions with strike equal to $K_{i}$.

Similarly, the BP volatility index in Eq. (3.37) can be simplified while calculating the L.H.S. of Eq. (B.18) as follows:

$$\begin{aligned} &\mathrm{IRS}\text{-}\mathrm{VI}_{n}^{\mathrm{bp}} ( t,T ) \\ &\quad =\sqrt{\frac{2}{T-t} \biggl( \sum_{i:K_{i}< R_{t}}\hat {Z}_{t} \bigl( R_{t}^{n},T,K_{i}; ( T-t ) \cdot\mathrm{IV}_{i,t}^{2} \bigr) \Delta K_{i}+\sum_{i:K_{i}\geq R_{t}}Z_{t} \bigl( R_{t}^{n},T,K_{i}; ( T-t ) \cdot \mathrm{IV}_{i,t}^{2} \bigr) \Delta K_{i} \biggr).} \end{aligned}$$

(B.20)

Connections with local volatility surfaces. We develop arguments that hinge upon those Dumas (1995) and Britten-Jones and Neuberger (2000) put forth for the equity case. Consider the price of the payer swaption in Eq. (3.14) of the main text, $\mathrm{Swpn}_{t}^{\mathrm{P}} ( K,T;T_{n} )$. For simplicity, let $R_{t}\equiv R_{t} ( T_{1},\ldots,T_{n} )$. We have

$$ \frac{\partial\mathrm{Swpn}_{t}^{\mathrm{P}}}{\partial T}=\frac {\partial \ln\mathrm{PVBP}_{t}}{\partial T}\cdot\mathrm{Swpn}_{t}^{\mathrm {P}}+ \mathrm{PVBP}_{t}\cdot\frac{d\mathbb{E}_{t}^{Q_{\mathrm{sw}}} ( R_{T}-K ) ^{+}}{dT}. $$

(B.21)

Since the forward swap rate is a martingale under $Q_{\mathrm{sw}}$, with instantaneous volatility $\sigma_{\tau} ( \cdot )$ as in Eq. (3.12) of the main text, we have

$$ \frac{d\mathbb{E}_{t}^{Q_{\mathrm{sw}}} ( R_{T}-K ) ^{+}}{dT}=\frac{1}{2}\mathbb{E}_{t}^{Q_{\mathrm{sw}}} \bigl[ \delta ( R_{T}-K ) R_{T}^{2} \sigma_{T}^{2} \bigr] , $$

(B.22)

where $\delta ( \cdot )$ is the Dirac delta. We can elaborate on the R.H.S. of Eq. (B.22), obtaining

$$\begin{aligned} \mathbb{E}_{t}^{Q_{\mathrm{sw}}} \bigl[ \delta ( R_{T}-K ) R_{T}^{2}\sigma_{T}^{2} \bigr] & =\iint \delta ( R_{T}-K ) R_{T}^{2}\sigma_{T}^{2} \underset{\equiv\, \mathrm{joint}\ \mathrm{density}\ \mathrm{of}\ ( \sigma_{T},R_{T} ) }{\underbrace{\phi_{T}^{\mathrm{c}} ( \sigma_{T} \vert R_{T} ) \phi_{T}^{\mathrm{m}} ( R_{T} ) }}dR_{T}d\sigma_{T} \\ & =K^{2}\phi_{T}^{\mathrm{m}} ( K ) \mathbb{E}_{t}^{Q_{\mathrm {sw}}} \bigl( \sigma_{T}^{2}\big\vert R_{T}=K \bigr) \\ & =K^{2}\frac{\frac{\partial^{2}\mathrm{Swpn}_{t}^{\mathrm {P}}}{\partial K^{2}}}{\mathrm{PVBP}_{t}}\mathbb{E}_{t}^{Q_{\mathrm{sw}}} \bigl( \sigma_{T}^{2}\big\vert R_{T}=K \bigr) , \end{aligned}$$

where $\phi_{T}^{\mathrm{c}} ( \sigma_{T}\vert R_{T} )$ denotes the conditional density of $\sigma_{T}$ given $R_{T}$ under $Q_{\mathrm{sw}}$, $\phi_{T}^{\mathrm{m}} ( R_{T} )$ denotes the marginal density of $R_{T}$ under $Q_{\mathrm{sw}}$, and the third line follows by a well-known property of option-like prices. By substituting this result into Eq. (B.22) and then into Eq. (B.21), we obtain

$$ \frac{\partial\mathrm{Swpn}_{t}^{\mathrm{P}}}{\partial T}=\frac {\partial \ln\mathrm{PVBP}_{t}}{\partial T}\cdot\mathrm{Swpn}_{t}^{\mathrm {P}}+ \frac{1}{2}K^{2}\frac{\partial^{2}\mathrm{Swpn}_{t}^{\mathrm {P}}}{\partial K^{2}}\mathbb{E}_{t}^{Q_{\mathrm{sw}}} \bigl( \sigma_{T}^{2}\big\vert R_{T}=K \bigr) . $$

Rearranging this equation, we obtain

$$ \sigma_{\mathrm{loc}}^{2} ( K,T ) \equiv\mathbb {E}_{t}^{Q_{\mathrm{sw}}} \bigl( \sigma_{T}^{2}\big\vert R_{T}=K \bigr) =2\frac{\frac{\partial\mathrm{Swpn}_{t}^{\mathrm {P}}}{\partial T}-\frac{\partial\ln\mathrm{PVBP}_{t}}{\partial T}\mathrm{Swpn}_{t}^{\mathrm{P}}}{K^{2}\frac{\partial^{2}\mathrm {Swpn}_{t}^{\mathrm{P}}}{\partial K^{2}}}. $$

(B.23)

Next, assume that the volatility of the forward swap rate in Eq. (3.12) of the main text is only a function of the forward swap rate and calendar time, $\sigma_{s}=\sigma ( R_{s},s )$. Define

$$ \frac{dR_{\tau}}{R_{\tau}}=\sigma_{\mathrm{loc}} ( R_{\tau},\tau ) dW_{\tau}^{\mathrm{sw}},\quad \tau\in [ t,T ] , $$

where $\sigma_{\mathrm{loc}} ( R_{\tau},\tau )$ is as in Eq. (B.23). This model can then match the cross-section of swaptions prices (without errors) and then be used to price all of the non-traded swaptions in Eq. (3.21) through Monte Carlo integration.

Marking to market. First, we derive the updates in Eq. (3.23). For a given $\tau\in ( t,T )$, we need to derive the following conditional expectation of the payoff $\mathrm {Var}\text{-}\mathrm{Swap}_{n} ( t,T )$ in Eq. (3.16):

$$\begin{aligned} & \mathbb{E}_{\tau}^{Q} \bigl( e^{-\int_{\tau }^{T}r_{u}du}\mathrm{Var} \text{-}\mathrm{Swap}_{n} ( t,T ) \bigr) \\ &\quad =\mathbb{E}_{\tau}^{Q} \bigl[ e^{-\int_{\tau }^{T}r_{u}du} \bigl( V_{n} ( t,\tau ) +V_{n} ( \tau ,T ) \bigr) \times \mathrm{PVBP}_{T} ( T_{1},\ldots,T_{n} ) \bigr] \\ &\qquad{} -P_{\tau} ( T ) \mathbb{P}_{\mathrm{var},n} ( t,T ) \\ &\quad =V_{n} ( t,\tau ) \mathrm{PVBP}_{\tau} ( T_{1}, \ldots ,T_{n} ) +\mathbb{F}_{\mathrm{var},n} ( \tau,T ) -P_{\tau } ( T ) \mathbb{P}_{\mathrm{var},n} ( t,T ) , \end{aligned}$$

(B.24)

where we have used the definition of $\mathbb{F}_{\mathrm{var},n} ( \cdot,T )$ in Definition 3.1. Equation (3.23) follows after plugging the expression for $\mathbb{P}_{\mathrm{var},n} ( t,T )$ in the first of Eqs. (3.22) into Eq. (B.24).

Next, we derive Eq. (3.24). Utilizing the expression of $\mathrm{Var}\text{-}\mathrm{Swap}_{n}^{\ast} ( t,T )$ in Eq. (3.18), we obtain

$$\begin{aligned} & \mathbb{E}_{\tau}^{Q} \bigl( e^{-\int_{\tau }^{T}r_{u}du}\mathrm{Var} \text{-}\mathrm{Swap}_{n}^{\ast} ( t,T ) \bigr) \\ &\quad =\mathbb{E}_{\tau}^{Q} \bigl[ e^{-\int_{\tau }^{T}r_{u}du} \bigl( V_{n} ( t,\tau ) +V_{n} ( \tau ,T ) -\mathbb{P}_{\mathrm{var},n}^{\ast} ( t,T ) \bigr) \times\mathrm{PVBP}_{T} ( T_{1}, \ldots,T_{n} ) \bigr] \\ &\quad =\mathrm{PVBP}_{\tau} ( T_{1},\ldots,T_{n} ) \bigl( V_{n} ( t,\tau ) +\mathbb{P}_{\mathrm{var},n}^{\ast} ( \tau,T ) -\mathbb{P}_{\mathrm{var},n}^{\ast} ( t,T ) \bigr) , \end{aligned}$$

where the second equality follows by the definition of $\mathbb{F}_{\mathrm{var},n} ( \cdot,T )$ in Definition 3.1 and the expression of $\mathbb{P}_{\mathrm{var},n}^{\ast} ( t,T )$ in the second of Eqs. (3.22).

3.1.3 B.3 Hedging

We provide details relating to hedging IRV contracts. One element in these proofs involves the construction of a portfolio that replicates the forward swap rate. Accordingly, we first clarify how this portfolio is constructed, and proceed with the proofs in the second step.

Replication of the forward swap rate. We provide details regarding how to replicate the forward swap rate. Consider the forward swap rate at $t$, as defined in Eqs. (3.5) and (3.3), which we repeat for the reader’s convenience:

$$ R_{t}=\frac{P_{t} ( T_{0} ) -P_{t} ( T_{n} ) }{\mathrm {PVBP}_{t}},\quad\mathrm{PVBP}_{t}=\sum _{i=1}^{n}\delta _{i-1}P_{t} ( T_{i} ) , $$

where $R_{t}\equiv R_{t} ( T_{1},\ldots,T_{n} )$ and $\mathrm {PVBP}_{t}\equiv\mathrm{PVBP}_{t} ( T_{1},\ldots,T_{n} )$ to simplify notation.

We aim to set up a portfolio that replicates the forward swap rate, $R_{t}$. Note that $R_{t}$ is a function of three freely tradable assets:

$$ R_{t}=\varphi \bigl( P_{t} ( T_{0} ) ,P_{t} ( T_{n} ) ,\mathrm{PVBP}_{t} \bigr) ,\qquad\varphi ( x_{1},x_{2},x_{3} ) \equiv \frac{x_{1}-x_{2}}{x_{3}}, $$

(B.25)

suggesting a portfolio that comprises these three assets. Let $V_{t}$ be the value of this portfolio, which satisfies

$$ dV_{t}=\theta_{1t}dP_{t} ( T_{0} ) + \theta_{2t}dP_{t} ( T_{n} ) +\theta_{3t}d \mathrm{PVBP}_{t}+\theta_{4t}dM_{t}, $$

(B.26)

where $M_{t}$ denotes the value of a money market account at time $t$, and $\theta_{\cdot t}$ are the units of assets in the portfolio at time $t$. Next, note that by Itô’s lemma, and Eq. (B.25), the forward swap rate is the solution to

$$\begin{aligned} dR_{t}& =\varphi_{1} \bigl( P_{t} ( T_{0} ) ,P_{t} ( T_{n} ) ,\mathrm{PVBP}_{t} \bigr) dP_{t} ( T_{0} ) \\ &\quad{} +\varphi_{2} \bigl( P_{t} ( T_{0} ) ,P_{t} ( T_{n} ) ,\mathrm{PVBP}_{t} \bigr) dP_{t} ( T_{n} ) \\ &\quad{} +\varphi_{3} \bigl( P_{t} ( T_{0} ) ,P_{t} ( T_{n} ) ,\mathrm{PVBP}_{t} \bigr) d \mathrm{PVBP}_{t} \\ &\quad{} +\mathrm{VC}_{t}dt, \end{aligned}$$

(B.27)

where subscripts denote partial derivatives,

$$\begin{aligned} \mathrm{VC}_{t}& \equiv\frac{1}{2}\varphi_{11} \bigl( P_{t} ( T_{0} ) ,P_{t} ( T_{n} ) ,\mathrm{PVBP}_{t} \bigr) \bigl\Vert \sigma_{P_{t}} ( T_{0} ) \bigr\Vert ^{2} \\ &\quad{} +\frac{1}{2}\varphi_{22} \bigl( P_{t} ( T_{0} ) ,P_{t} ( T_{n} ) ,\mathrm{PVBP}_{t} \bigr) \bigl\Vert \sigma_{P_{t}} ( T_{n} ) \bigr\Vert ^{2} \\ &\quad{} +\frac{1}{2}\varphi_{33} \bigl( P_{t} ( T_{0} ) ,P_{t} ( T_{n} ) ,\mathrm{PVBP}_{t} \bigr) \bigl\Vert \sigma_{\mathrm {PVBP}_{t}} ( T_{0} ) \bigr\Vert ^{2} \\ &\quad{} +\varphi_{12} \bigl( P_{t} ( T_{0} ) ,P_{t} ( T_{n} ) ,\mathrm{PVBP}_{t} \bigr) \sigma_{P_{t}} ( T_{0} ) \cdot \sigma _{P_{t}} ( T_{n} ) \\ &\quad{} +\varphi_{13} \bigl( P_{t} ( T_{0} ) ,P_{t} ( T_{n} ) ,\mathrm{PVBP}_{t} \bigr) \sigma_{P_{t}} ( T_{0} ) \cdot \sigma_{\mathrm{PVBP}_{t}} \\ &\quad{} +\varphi_{23} \bigl( P_{t} ( T_{0} ) ,P_{t} ( T_{n} ) ,\mathrm{PVBP}_{t} \bigr) \sigma_{P_{t}} ( T_{n} ) \cdot \sigma_{\mathrm{PVBP}_{t}}, \end{aligned}$$

with $\sigma_{P_{t}} ( T_{j} )$ and $\sigma_{\mathrm {PVBP}_{t}}\$ denoting the instantaneous volatility vector of $P_{t} ( T_{j} )$ and $\mathrm{PVBP}_{t}$.

We have by Eqs. (B.26) and (B.27):

$$\begin{aligned} dV_{t}-dR_{t}& = \bigl( \theta_{1t}- \varphi_{1} \bigl( P_{t} ( T_{0} ) ,P_{t} ( T_{n} ) ,\mathrm{PVBP}_{t} \bigr) \bigr) dP_{t} ( T_{0} ) \\ &\quad{} + \bigl( \theta_{2t}-\varphi_{2} \bigl( P_{t} ( T_{0} ) ,P_{t} ( T_{n} ) ,\mathrm{PVBP}_{t} \bigr) \bigr) dP_{t} ( T_{n} ) \\ &\quad{} + \bigl( \theta_{3t}-\varphi_{3} \bigl( P_{t} ( T_{0} ) ,P_{t} ( T_{n} ) , \mathrm{PVBP}_{t} \bigr) \bigr) d\mathrm {PVBP}_{t} \\ &\quad{} + ( r_{t}\theta_{4t}M_{t}- \mathrm{VC}_{t} ) dt \end{aligned}$$

where we have used the dynamics of the money market account, $dM_{t}=r_{t}M_{t}dt$, with $r_{t}$ denoting the instantaneous short-term rate. Therefore, we replicate the forward swap rate through a portfolio comprising the following proportions of assets

$$\begin{aligned} \theta_{1t}& =\varphi_{1} \bigl( P_{t} ( T_{0} ) ,P_{t} ( T_{n} ) ,\mathrm{PVBP}_{t} \bigr) =\frac{1}{\mathrm{PVBP}_{t}}, \\ \theta_{2t}& =\varphi_{2} \bigl( P_{t} ( T_{0} ) ,P_{t} ( T_{n} ) ,\mathrm{PVBP}_{t} \bigr) =-\frac{1}{\mathrm{PVBP}_{t}}, \\ \theta_{3t}& =\varphi_{3} \bigl( P_{t} ( T_{0} ) ,P_{t} ( T_{n} ) ,\mathrm{PVBP}_{t} \bigr) =-\frac{R_{t}}{\mathrm{PVBP}_{t}}. \end{aligned}$$

For the portfolio to replicate the forward swap rate, one chooses

$$ \theta_{4t}M_{t}=R_{t}-\theta_{1t}P_{t} ( T_{0} ) -\theta _{2t}P_{t} ( T_{n} ) -\theta_{3t}\mathrm{PVBP}_{t}=R_{t}. $$

By construction, the portfolio replicates the forward swap rate $R_{t}$, although it gives rise to a hedging cost $\varepsilon_{t}\equiv R_{t}-V_{t}$ at time $t$, which satisfies: $d\varepsilon_{t}= ( \mathrm {VC}_{t}-r_{t}R_{t} ) dt$.

Hedging percentage volatility contracts. We provide the details leading to Table 3.1; those for Table 3.2 are nearly identical. By Itô’s lemma:

$$\begin{aligned} & \mathrm{PVBP}_{T} ( T_{1},\ldots,T_{n} ) \int _{t}^{T}\sigma_{s}^{2} ( T_{1},\ldots,T_{n} ) ds \\ &\quad =2\mathrm{PVBP}_{T} ( T_{1},\ldots,T_{n} ) \int _{t}^{T}\frac{dR_{s} ( T_{1},\ldots,T_{n} ) }{R_{s} ( T_{1},\ldots ,T_{n} ) } \\ &\qquad{}-2\mathrm{PVBP}_{T} ( T_{1},\ldots,T_{n} ) \ln \frac{R_{T} ( T_{1},\ldots,T_{n} ) }{R_{t} ( T_{1},\ldots ,T_{n} ) }. \end{aligned}$$

(B.28)

By Eq. (B.12), the value of the interest rate swap in Eq. (3.7) and the swaption premiums in Eqs. (3.14)–(3.15), the second term on the R.H.S. is the payoff at $T$ of a portfolio set up at $t$, which is two times: (a) short $1/R_{t} ( T_{1},\ldots ,T_{n} )$ units of a fixed interest payer swap struck at $R_{t} ( T_{1},\ldots,T_{n} )$, and (b) long a continuum of OTM swaptions with weights $K^{-2}dK$. This portfolio is the static position (ii) in Table 3.1. By previous results in this appendix (see Eqs. (B.12)–(B.15)), its cost is $\mathbb{F}_{\mathrm {var},n} ( t,T )$. We borrow $\mathbb{F}_{\mathrm{var},n} ( t,T )$ at time $t$, and repay it back at time $T$, as in row (iii) of Table 3.1.

Next, we derive the portfolio of zero-coupon bonds (i) in Table 3.1. This portfolio needs to be worthless at time $t$ and replicate the first term on the R.H.S. of Eq. (B.28). First, we replicate the forward swap through a portfolio of zero-coupon bonds, as in derivations following Eq. (B.25). Note that this replication entails a hedging cost as explained above. For simplicity, we refer to this portfolio as the “forward swap rate.” Then, we consider a self-financed strategy in (a) this portfolio and (b) a money market account (MMA, in the sequel) worth $M_{\tau}\equiv e^{\int_{t}^{\tau }r_{s}ds}$ at time $\tau\geq t$. The value of this portfolio is

$$ \upsilon_{\tau}=\theta_{\tau}R_{\tau} ( T_{1},\ldots ,T_{n} ) +\psi_{\tau}M_{\tau}, $$

where $\theta_{\tau}$ are the units in the forward swap rate and $\psi_{\tau}$ are the units in the MMA. Consider the following portfolio:

$$ \begin{aligned} &\hat{\theta}_{\tau}R_{\tau} ( T_{1}, \ldots,T_{n} ) =\mathrm{PVBP}_{\tau} ( T_{1},\ldots,T_{n} ) ,\\ &\hat {\psi}_{\tau}M_{\tau}= \mathrm{PVBP}_{\tau} ( T_{1},\ldots ,T_{n} ) \biggl( \int_{t}^{\tau}\frac{dR_{s} ( T_{1},\ldots ,T_{n} ) }{R_{s} ( T_{1},\ldots,T_{n} ) }-1 \biggr) . \end{aligned} $$

(B.29)

We have that $\hat{\upsilon}_{\tau}=\hat{\theta}_{\tau}R_{\tau}+\hat {\psi}_{\tau}M_{\tau}$ satisfies

$$ \hat{\upsilon}_{\tau}=\mathrm{PVBP}_{\tau} ( T_{1}, \ldots ,T_{n} ) \int_{t}^{\tau} \frac{dR_{s} ( T_{1},\ldots ,T_{n} ) }{R_{s} ( T_{1},\ldots,T_{n} ) }, $$

(B.30)

so that

$$ \hat{\upsilon}_{t}=0,\quad \text{and}\quad\hat{\upsilon}_{T}= \mathrm {PVBP}_{T} ( T_{1},\ldots,T_{n} ) \int _{t}^{T}\frac {dR_{s} ( T_{1},\ldots,T_{n} ) }{R_{s} ( T_{1},\ldots ,T_{n} ) }. $$

Therefore, by going long two portfolios $(\hat{\theta}_{\tau},\hat{\psi}_{\tau})$, the first term on the R.H.S. of Eq. (B.28)—and by the previous results, the entire R.H.S. of Eq. (B.28)—can be replicated, provided $(\hat{\theta}_{\tau},\hat{\psi}_{\tau})$ is self-financed. A hedging cost is only incurred in the replication of the forward swap rate. We are only left to show that $(\hat{\theta}_{\tau},\hat{\psi}_{\tau })$ is self-financed. We have

$$\begin{aligned} d\hat{\upsilon}_{\tau} & =\hat{\theta}_{\tau}R_{\tau} ( T_{1}, \ldots,T_{n} ) \frac {dR_{\tau} ( T_{1},\ldots,T_{n} ) }{R_{\tau} ( T_{1},\ldots ,T_{n} ) }+\hat{\psi}_{\tau}M_{\tau} \frac{dM_{\tau}}{M_{\tau}} \\ & =\mathrm{PVBP}_{\tau} ( T_{1},\ldots,T_{n} ) \biggl( \frac {dR_{\tau} ( T_{1},\ldots,T_{n} ) }{R_{\tau} ( T_{1},\ldots ,T_{n} ) }-r_{\tau}d\tau \biggr) \\ &\quad{}+ \biggl( \mathrm{PVBP}_{\tau } ( T_{1},\ldots,T_{n} ) \int _{t}^{\tau}\frac{dR_{s} ( T_{1},\ldots,T_{n} ) }{R_{s} ( T_{1},\ldots,T_{n} ) } \biggr) r_{\tau}d\tau \\ & =\mathrm{PVBP}_{\tau} ( T_{1},\ldots,T_{n} ) \biggl( \frac {dR_{\tau} ( T_{1},\ldots,T_{n} ) }{R_{\tau} ( T_{1},\ldots ,T_{n} ) }-r_{\tau}d\tau \biggr) +r_{\tau}\hat{ \upsilon}_{\tau }d\tau \\ & =\hat{\theta}_{\tau}R_{\tau} ( T_{1}, \ldots,T_{n} ) \biggl( \frac{dR_{\tau} ( T_{1},\ldots,T_{n} ) }{R_{\tau} ( T_{1},\ldots,T_{n} ) }-r_{\tau}d\tau \biggr) +r_{\tau}\hat {\upsilon}_{\tau}d\tau, \end{aligned}$$

(B.31)

where the second line follows by the expressions for $(\hat{\theta}_{\tau},\hat{\psi}_{\tau})$ in Eqs. (B.29), the third line holds by Eq. (B.30), and the fourth follows, again, by the expression for $\hat{\theta}_{\tau}R_{\tau}$ in Eq. (B.29). It is easy to check that the dynamics of $\hat{\upsilon}_{\tau}$ in Eq. (B.31) are those of a self-financed strategy.

Hedging basis point volatility contracts. We provide the details leading to Table 3.3 only, as those for Table 3.4 are nearly identical. Itô’s lemma gives us

$$\begin{aligned} & \mathrm{PVBP}_{T} ( T_{1},\ldots,T_{n} ) V_{n}^{\mathrm {bp}} ( t,T ) \\ &\quad =-2\mathrm{PVBP}_{T} ( T_{1},\ldots,T_{n} ) \int_{t}^{T}R_{s} ( T_{1}, \ldots,T_{n} ) dR_{s} ( T_{1}, \ldots,T_{n} ) \\ & \qquad{}+\mathrm{PVBP}_{T} ( T_{1},\ldots,T_{n} ) \bigl( R_{T}^{2} ( T_{1},\ldots,T_{n} ) -R_{t}^{2} ( T_{1},\ldots,T_{n} ) \bigr) . \end{aligned}$$

(B.32)

By Eq. (B.16), the second term on the R.H.S. of Eq. (B.32) is the payoff at $T$ of a portfolio set up at $t$, which is: (a) long $2R_{t} ( T_{1},\ldots,T_{n} )$ units a fixed interest payer swap struck at $R_{t} ( T_{1},\ldots,T_{n} )$, and (b) long a continuum of OTM swaptions with weights $2dK$. It is the static position (ii) in Table 3.3. By previous results in this appendix (see Eqs. (B.16)–(B.18)), its cost is $\mathbb {F}_{\mathrm{var},n}^{\mathrm{bp}} ( t,T )$, which we borrow at $t$ and repay back at $T$, as in row (iii) of Table 3.3. The portfolio to be short in row (i) of Table 3.3 is obtained similarly as the portfolio in row (i) of Table 3.1, but with

$$ \hat{\psi}_{\tau}M_{\tau}=\mathrm{PVBP}_{\tau} ( T_{1},\ldots ,T_{n} ) \biggl( \int_{t}^{\tau}2R_{s} ( T_{1},\ldots ,T_{n} ) dR_{s} ( T_{1},\ldots,T_{n} ) -1 \biggr) , $$

replacing the percentage counterparts in Eq. (B.29).

3.1.4 B.4 Constant Maturity Swaps

Consider the fair value of a single payment of a Constant Maturity Swap (CMS) (see Sect. 3.3.6), say the first one occurring at time $T_{0}+\kappa$. To simplify notation, set $S ( T_{0} ) \equiv R_{T_{0}} ( T_{1},\ldots,T_{n} )$. The current value of $S ( T_{0} )$ to be paid at time $T_{0}+\kappa$ is the same as the current value of $S ( T_{0} ) P_{T_{0}} ( T_{0}+\kappa )$ to be paid at $T_{0}$, so that

$$ \mathrm{cms} ( t,T_{0}+\kappa ) \equiv\mathbb{E}_{t} \bigl( e^{-\int_{t}^{T_{0}}r_{u}du}S ( T_{0} ) P_{T_{0}} ( T_{0}+\kappa ) \bigr) =P_{t} ( T_{0}+\kappa ) \mathbb {E}_{t}^{Q_{\mathrm{sw}}} \biggl( S ( T_{0} ) \frac{\mathcal {G}_{T_{0}}}{\mathcal{G}_{t}} \biggr) , $$

(B.33)

where $\mathcal{G}_{\tau}\equiv\frac{P_{\tau} ( T_{0}+\kappa ) }{\mathrm{PVBP}_{\tau} ( T_{1},\ldots,T_{n} ) }$. We determine $\mathcal{G}_{\tau}$, by discounting through the reset times, $T_{i}-T_{0}$ using the flat rate formula, $P_{T_{0}} ( T_{i} ) = ( 1+\delta S ( T_{0} ) ) ^{-i}$, where $\delta$ is the year fraction between the reset times, so that

$$\begin{aligned} \mathcal{G}_{\tau}&\equiv\frac{P_{\tau} ( T_{0}+\kappa ) }{\mathrm{PVBP}_{\tau} ( T_{1},\ldots,T_{n} ) }\approx\frac { ( 1+\delta S ( \tau ) ) ^{- ( T_{0}+\kappa-\tau ) }}{\delta\sum_{i=1}^{n} ( 1+\delta S ( \tau ) ) ^{- ( T_{i}-\tau ) }}= \frac{S ( \tau ) ( 1+\delta S ( \tau ) ) ^{-\kappa}}{1-\frac{1}{ ( 1+\delta S ( \tau ) ) ^{n}}} \\ &\equiv G \bigl( S ( \tau ) \bigr) . \end{aligned}$$

(B.34)

The previous derivations closely follow those in Hagan (2003), and are provided for completeness. We now make the connection between the value $\mathrm{cms} ( t,T_{0}+\kappa )$ in Eq. (B.33) and the value of the IRV forward contract in BP of Definition 3.1. Replacing the approximation in Eq. (B.34) into Eq. (B.33) leaves

$$\begin{aligned} & \mathrm{cms} ( t,T_{0}+\kappa ) \\ &\quad =P_{t} ( T_{0}+\kappa ) \mathbb{E}_{t}^{Q_{\mathrm {sw}}} \biggl( S ( T_{0} ) \frac{G ( S ( T_{0} ) ) }{G ( R_{t} ) } \biggr) \\ &\quad \approx P_{t} ( T_{0}+\kappa ) \mathbb{E}_{t}^{Q_{\mathrm {sw}}} \biggl[ S ( T_{0} ) \biggl( 1+\frac{G^{\prime} ( R_{t} ) }{G ( R_{t} ) } \bigl( S ( T_{0} ) -R_{t} \bigr) \biggr) \biggr] \\ &\quad =P_{t} ( T_{0}+\kappa ) \mathbb{E}_{t}^{Q_{\mathrm {sw}}} \bigl( S ( T_{0} ) \bigr) \\ &\qquad{} +\mathrm{PVBP}_{t} ( T_{1},\ldots ,T_{n} ) G^{\prime} ( R_{t} ) \mathbb{E}_{t}^{Q_{\mathrm {sw}}} \bigl[ S ( T_{0} ) \bigl( S ( T_{0} ) -R_{t} \bigr) \bigr] \\ &\quad =P_{t} ( T_{0}+\kappa ) R_{t}+G^{\prime} ( R_{t} ) \mathrm{PVBP}_{t} ( T_{1}, \ldots,T_{n} ) \mathbb {E}_{t}^{Q_{\mathrm{sw}}} \bigl( S^{2} ( T_{0} ) -R_{t}^{2} \bigr) \\ &\quad =P_{t} ( T_{0}+\kappa ) R_{t}+G^{\prime} ( R_{t} ) \mathbb{F}_{\mathrm{var},n}^{\mathrm{bp}} ( t,T_{0} ) , \end{aligned}$$

where the second line follows by a first order Taylor approximation of the function $G$ around $R_{t}$, the third from the definition of $G ( R_{t} )$, the fourth from the martingale property of the forward swap rate under $Q_{\mathrm{sw}}$, $\mathbb{E}_{t}^{Q_{\mathrm{sw}}} ( S ( T_{0} ) ) =R_{t}$ and the fifth line from Eqs. (B.17)–(B.18) and the definition of the value of the IRV forward agreement (see Definition 3.1), $\mathbb{F}_{\mathrm{var},n}^{\mathrm {bp}} ( t,T )$. Equation (3.25) of the main text follows by summing every single CMS payment, $\mathrm{cms} ( t,T_{j}+\kappa )$ say, for $j=0,\ldots,N-1$.

As mentioned in the main text, it has been well known since at least Hagan (2003) and Mercurio and Pallavicini (2006) that a CMS is linked to the entire skew. However, our representation of the fair value of a CMS in Eq. (3.25) differs non-trivially from previous ones. For example, Mercurio and Pallavicini (2006) utilize spanning arguments different from ours, which we now explain.

Consider a Taylor expansion with remainder of the function $f ( R_{T} ) \equiv R_{T}^{2}$ around some point $R_{o}$:

$$ R_{T}^{2}=R_{o}^{2}+2R_{o} ( R_{T}-R_{o} ) +2 \biggl( \int_{0}^{R_{o}} ( K-R_{T} ) ^{+}dK+\int_{R_{o}}^{\infty} ( R_{T}-K ) ^{+}dK \biggr) . $$

(B.35)

Mercurio and Pallavicini (2006) consider the point $R_{o}=0$, in which case Eq. (B.35) collapses to

$$ R_{T}^{2}=2\int_{0}^{\infty} ( R_{T}-K ) ^{+}dK, $$

so that

$$ \mathbb{E}_{t}^{Q_{\mathrm{sw}}} \bigl( R_{T}^{2} \bigr) =\frac {2}{\mathrm{PVBP}_{t} ( T_{1},\ldots,T_{n} ) }\int_{0}^{\infty} \mathrm{Swpn}_{t}^{\mathrm{P}} ( K,T;T_{n} ) dK. $$

Our approach differs as we take $R_{o}=R_{t}$ in Eq. (B.35) (see Eq. (B.16)), so that

$$\begin{aligned} \mathbb{E}_{t}^{Q_{\mathrm{sw}}} \bigl( R_{T}^{2}-R_{t}^{2} \bigr) & =2\mathbb{E}_{t}^{Q_{\mathrm{sw}}} \biggl( \int _{0}^{R_{t}} ( K-R_{T} ) ^{+}dK+ \int_{R_{t}}^{\infty} ( R_{T}-K ) ^{+}dK \biggr) \\ & =\frac{2}{\mathrm{PVBP}_{t} ( T_{1},\ldots,T_{n} ) } \\ &\quad{}\times\biggl( \int_{0}^{R_{t}} \mathrm{Swpn}_{t}^{\mathrm{R}} ( K,T;T_{n} ) dK+\int _{R_{t}}^{\infty}\mathrm{Swpn}_{t}^{\mathrm{P}} ( K,T;T_{n} ) dK \biggr) . \end{aligned}$$

3.1.5 B.5 The Contract and Index in the Vasicek Market

We use the Vasicek (1977) model as the basis for the experiments in Sect. 3.5.3 by assuming that the short-term rate follows a Gaussian process:

$$ dr_{t}=\kappa ( \bar{r}-r_{t} ) dt+\sigma_{v}d \hat{W}_{t}, $$

(B.36)

where $\hat{W}_{t}$ is a Wiener process under the risk-neutral probability, and $\kappa$, $\bar{r}$ and $\sigma_{v}$ are three positive constants. The price of a zero-coupon bond predicted by this model is $P_{t} ( r_{t};T ) =A ( T-t ) e^{-B ( T-t ) r_{t}}$ for two functions $A$ and $B$, with obvious notation.

The model predicts that the forward swap rate is

$$\begin{aligned} &R_{t} ( r_{t};T_{1},\ldots,T_{n} ) = \frac{P_{t} ( r_{t};T_{0} ) -P_{t} ( r_{t};T_{n} ) }{\mathrm {PVBP}_{t} ( r_{t};T_{1},\ldots,T_{n} ) }, \\ &\quad\mathrm{PVBP}_{t} ( r_{t};T_{1}, \ldots,T_{n} ) =\sum_{i=1}^{n} \delta _{i-1}P_{t} ( r_{t};T_{i} ) , \end{aligned}$$

(B.37)

the counterpart to $R_{t} ( T_{1},\ldots,T_{n} )$, Eq. (3.10).

The instantaneous percentage bond return volatility is

$$ \mathrm{Vol}\text{-}\mathrm{P}_{t} ( r_{t};T ) \equiv -B ( T-t ) \sigma_{v}. $$

(B.38)

Finally, the instantaneous percentage forward swap rate volatility is

$$\begin{aligned} \sigma_{t} ( r_{t};T_{1}, \ldots,T_{n} ) & =\frac{\mathrm{Vol}\text{-}\mathrm{P}_{t} ( r_{t};T_{0} ) P_{t} ( r_{t};T_{0} ) -\mathrm{Vol}\text{-}\mathrm{P}_{t} ( r_{t};T_{n} ) P_{t} ( r_{t};T_{n} ) }{P_{t} ( r_{t};T_{0} ) -P_{t} ( r_{t};T_{n} ) } \\ &\quad{}-\frac{\sum _{i=1}^{n}\delta_{i-1}P_{t} ( r_{t};T_{i} ) \mathrm {Vol}\text{-}\mathrm{P}_{t} ( r_{t};T_{i} ) }{\mathrm {PVBP}_{t} ( r_{t};T_{1},\ldots,T_{n} ) }, \end{aligned}$$

(B.39)

the counterpart to $\sigma_{t} ( T_{1},\ldots,T_{n} )$, Eq. (3.12).

We simulate Eq. (B.36) using a Milstein approximation method with initial values of $r\$in the interval $[ 0.01,0.10 ]$ and the following parameter values: $\kappa=0.3807$, $\bar{r}=0.072$, taken from Veronesi (2010, Chap. 15), and $\sigma_{v}=\sqrt{0.02}\times 0.2341$. We generate values of $\sigma_{t} ( r_{t};T_{1},\ldots,T_{n} )$ by plugging in the simulated values of $r_{t}$. The value of $\sigma_{v}$ implies that the instantaneous BP volatility is $100\times\sigma _{v}\approx3~\mbox{bps}$, which is consistent with standard average estimates reported in the literature (see, e.g., Chap. 12 in Mele 2014).

The instantaneous BP forward swap rate volatility is

$$ \sigma_{\mathrm{bp},t} ( r_{t};T_{1},\ldots,T_{n} ) =R_{t} ( r_{t};T_{1},\ldots,T_{n} ) \sigma_{t} ( r_{t};T_{1},\ldots ,T_{n} ) . $$

(B.40)

The two expectations,

$$\begin{aligned} & \mathbb{E}_{t}^{Q_{\mathrm{sw}}}\bigl(V_{n}^{{\mathrm{bp}}} ( r;t,T ) \bigr) \\ & \quad\equiv\frac{1}{\mathrm{PVBP}_{t} ( r_{T};T_{1},\ldots ,T_{n} ) } \\ &\qquad{}\times\mathbb{E}_{t} \biggl[ e^{-\int _{t}^{T}r_{s}ds} \biggl( \mathrm{PVBP}_{T} ( r_{T};T_{1},\ldots ,T_{n} ) \int_{t}^{T} \sigma_{\mathrm{bp},s}^{2} ( r_{s};T_{1}, \ldots,T_{n} ) ds \biggr) \biggr] \end{aligned}$$

(B.41)

and

$$ \mathbb{E}_{t}\bigl(V_{n}^{{\mathrm{bp}}} ( r;t,T ) \bigr) \equiv\mathbb {E}_{t} \biggl( \int_{t}^{T} \sigma_{\mathrm{bp},s}^{2} ( r_{s};T_{1}, \ldots,T_{n} ) ds \biggr) , $$

(B.42)

are estimated through Monte Carlo integration. The expectations regarding percentage volatility are also estimated through Monte Carlo integration. The forward swap rates in Figs. 3.7 and 3.8 (and in Tables 3.7 and 3.8 below) are obtained by plugging the initial values of the short-term rate into Eq. (B.37). We assume reset dates are quarterly and set $\delta_{i}=\frac{1}{4}$.

Table 3.7 The model prediction of the BP volatility index, $\mathrm{IRS}\text{-}\mathrm{VI}_{n}^{\mathrm{bp}} ( t,T )$ in Eq. (3.37) (labeled $\mbox{IRS-VI}^{\mathrm{bp}}$), and future expected volatility in a risk-neutral market, $\sqrt{\frac{1}{T-t}\mathbb{E}_{t}(V_{n}^{{\mathrm {bp}}} ( r_{t};t,T ) )}$, where $V_{n}^{{\mathrm{bp}}} ( r_{t};t,T )$ denotes the model-implied realized BP variance (labeled $\mbox{E-Vol}^{\mathrm{bp}}$), for (i) time-to maturity $T-t$ equal to 1, 6 and 9 months, and 1, 2 and 3 years, and (ii) tenor length $n$ equal to 1, 2, 3 and 5 years, and for time-to maturity $T-t$ equal to 1, 6 and 12 months, and (iii) tenor length $n$ equal to 10, 20 and 30 years. The index and expected volatilities are computed under the assumption the short-term interest rate is as in the Vasicek model of Eq. (B.36), with parameter values $\kappa=0.3807$, $\bar{r}=0.072$ and $\sigma _{v}=0.0331$

Table 3.8 The model prediction of the percentage volatility index, $\mathrm{IRS}\text{-}\mathrm{VI}_{n} ( t,T )$ in Eq. (3.38) (labeled IRS-VI), and future expected volatility in a risk-neutral market, $\sqrt{\frac{1}{T-t}\mathbb{E}_{t}(V_{n} ( r_{t};t,T ) )}$, where $V_{n} ( r_{t};t,T )$ denotes the model-implied realized percentage variance (labeled E-Vol), for (i) time-to maturity $T-t$ equal to 1, 6 and 9 months, and 1, 2 and 3 years, and (ii) tenor length $n$ equal to 1, 2, 3 and 5 years, and for time-to maturity $T-t$ equal to 1, 6 and 12 months, and (iii) tenor length $n$ equal to 10, 20 and 30 years. The index and expected volatilities are computed under the assumption the short-term interest rate is as in the Vasicek model of Eq. (B.36), with parameter values $\kappa=0.3807$, $\bar{r}=0.072$ and $\sigma _{v}=0.0331$

Regarding basis point variance, we calculate the IRS-VI index in Eq. (3.45) as $\sqrt{\frac{1}{T-t}\mathbb{E}_{t}^{Q_{\mathrm {sw}}}(V_{n}^{{\mathrm{bp}}} ( r_{t};t,T ) )}$ and future expected volatility in a risk-neutral market as $\sqrt{\frac{1}{T-t}\mathbb {E}_{t}(V_{n}^{{\mathrm{bp}}} ( r_{t};t,T ) )}$ (see Eqs. (B.41)–(B.42)). Percentage variance is dealt with similarly.

The left panel of Fig. 3.7 and Table 3.7 report results from experiments regarding the calculation of BP volatility corresponding to the index $\mathrm{IRS}\text{-}\mathrm{VI}_{n}^{\mathrm{bp}} ( t,T )$ in Eq. (3.45), as well as future expected volatility in a risk-neutral market. The left panel of Fig. 3.8 and Table 3.8 report results from experiments regarding the percentage case. The right panels of Figs. 3.7 and 3.8 depict the instantaneous forward swap rate volatilities, calculated through Eqs. (B.40) and (B.39), respectively.