Government Bonds and Time-Deposits

Abstract

This chapter builds on the general framework of Chap. 2, and develops indexes of expected volatility in government bond and time deposit markets. Variance contract design and pricing in these markets are plagued by two major complexities that are absent in the interest rate swap (Chap. 3) and equity cases. First, variance pricing in these markets rely on options referencing bond or time deposit futures where the options and futures often have different maturities (e.g. one-month options on two-month futures). Second, typically available for trading are American-style options, whereas model-free variance pricing requires European-style exercise features. We analyze the impact of the maturity mismatch and exercise style of the options on variance pricing. In addition to the usual percentage and basis point volatility formulations, this chapter derives indexes of expected volatility expressed in terms of basis point yield volatility.

Appendix C: Appendix on Government Bonds and Time Deposit Markets

4.1.1 C.1 The Equity VIX with Stochastic Interest Rates

The VIX for equity market volatility maintained by the Chicago Board Options Exchange relies on the assumption that interest rates are constant. This appendix relaxes this assumption and shows that, under certain conditions, equity volatility can be priced under the forward probability. Let $S_{\tau}$ be the price of a stock index at time $\tau$. We assume that it satisfies

$$ \frac{dS_{\tau}}{S_{\tau}}=r_{\tau}dt+\sigma_{\tau}\cdot dW_{\tau}, $$

where $W_{\tau}$ now denotes a vector Brownian motion under the risk-neutral probability, $Q$, and both the short-term rate $r_{\tau}$ and the instantaneous stock index volatility $\sigma_{\tau}$ are adapted to $W_{\tau}$.

Next, let $F_{\tau}^{\mathrm{e}} ( T ) =\frac{S_{\tau }}{P_{\tau } ( T ) }$ denote the forward stock index, where $P_{\tau } ( T )$ is the price at $\tau$ of a zero coupon bond expiring at some $T$, and satisfies

$$ \frac{dP_{\tau} ( T ) }{P_{\tau} ( T ) }=r_{\tau }dt+\sigma_{\tau} ( T ) \cdot dW_{\tau}, $$

and $\sigma_{\tau} ( T )$ is the process of the return volatility of the zero, a vector-valued process adapted to $W_{\tau}$. Naturally, the forward stock index satisfies

$$ \frac{dF_{\tau}^{\mathrm{e}} ( T ) }{F_{\tau}^{\mathrm {e}} ( T ) }=v_{\tau} ( T ) \cdot dW_{\tau}^{F^{T}}, $$

where $W_{\tau}^{F^{T}}$ is a vector of Brownian motions under the forward probability $Q_{F^{T}}$, and the instantaneous forward stock volatility is $v_{\tau} ( T ) \equiv\sigma_{\tau}-\sigma_{\tau} ( T )$.

By standard arguments, the fair value of the variance strike for forward volatility delivery, $\mathbb{P}^{\mathrm{e}} ( t,T )$ say, is

$$ \mathbb{P}^{\mathrm{e}} ( t,T ) =\frac{1}{P_{t} ( T ) }\mathbb{E}_{t} \biggl( e^{-\int_{t}^{T}r_{\tau }d\tau }\int_{t}^{T} \bigl\Vert v_{\tau} ( T ) \bigr\Vert ^{2}d\tau \biggr) . $$

Similar to the arguments relating to the government bond variance swap contract (see Eq. (4.18) in the main text), we have that

$$ -\mathbb{E}_{t}^{Q_{F^{T}}} \biggl( \ln\frac{F_{T}^{\mathrm{e}} ( T ) }{F_{t}^{\mathrm{e}} ( T ) } \biggr) =\frac {1}{2}\mathbb{E}_{t}^{Q_{F^{T}}} \biggl( \int _{t}^{T} \bigl\Vert v_{\tau} ( T ) \bigr\Vert ^{2}d\tau \biggr) =\frac{1}{2}\mathbb{P}^{\mathrm {e}} ( t,T ) . $$

(C.1)

The term on the L.H.S. can be cast in the usual model-free format by spanning the log-contract on the forward through OTM equity options

$$\begin{aligned} &\mathbb{E}_{t}^{Q_{F^{T}}} \biggl( \ln\frac{F_{T}^{\mathrm{e}} ( T ) }{F_{t}^{\mathrm{e}} ( T ) } \biggr) \\ &\quad=-\frac{1}{P_{t} ( T ) } \biggl( \int_{0}^{F_{t}^{\mathrm{e}} ( T ) } \mathrm {Put}_{t}^{\mathrm{e}} ( K,T ) \frac{1}{K^{2}}dK+\int _{F_{t}^{\mathrm{e}} ( T ) }^{\infty}\mathrm {Call}_{t}^{\mathrm{e}} ( K,T ) \frac{1}{K^{2}}dK \biggr) , \end{aligned}$$

(C.2)

with obvious notation.

Combining Eq. (C.1) and Eq. (C.2) delivers

$$ \mathbb{P}^{\mathrm{e}} ( t,T ) =\frac{2}{P_{t} ( T ) } \biggl( \int_{0}^{F_{t}^{\mathrm{e}} ( T ) }\mathrm{Put}_{t}^{\mathrm{e}} ( K,T ) \frac {1}{K^{2}}dK+\int_{F_{t}^{\mathrm{e}} ( T ) }^{\infty} \mathrm{Call}_{t}^{\mathrm{e}} ( K,T ) \frac{1}{K^{2}}dK \biggr) , $$

leading to the following index:

$$ \fbox{$\displaystyle\mathrm{VIX}_{\mathrm{r}} ( t,T ) \equiv\sqrt{\frac {1}{T-t} \mathbb{P}^{\mathrm{e}} ( t,T ) }$} $$

This expression collapses to that underlying the CBOE calculations when rates are constant. Clearly, VIX indexes calculated at different maturities necessitate different discount rates as the yield curve is not necessarily flat when rates are random, implying that the entire yield curve, $P_{t} ( T )$, needs to be used as an input.

We study the bias in an experiment. We assume the short-term rate is generated by the Vasicek (1977) model, where the short-term rate is a solution to Eq. (4.57) under the risk-neutral probability. Define the yield-to-maturity $y_{t} ( T- t ) \equiv-\frac{1}{T-t}\ln P_{t}^{\mathrm{v}} ( T )$, where $P_{t}^{\mathrm{v}} ( T )$ is the price of the zero generated by the Vasicek model.

We assume the current short-term rate is $r_{t}=0.01$, implying that the yield-to-maturity for 1 month is $y_{t} ( \frac{1}{12} ) =-12\cdot \ln P_{t} ( \frac{1}{12} ) =0.0109$. Assume we use the yield-to-maturity for 1 month to determine the VIX for maturities larger than 1 month. The percentage bias underlying these calculations is

$$ \mathrm{Bias} ( t,T ) =\frac{\mathrm{VIX} ( t,T ) -\mathrm{VIX}_{\mathrm{r}} ( t,T ) }{\mathrm{VIX}_{\mathrm {r}} ( t,T ) }=\sqrt{e^{- ( y_{t} ( T-t ) -y_{t} ( \frac {1}{12} ) ) \cdot ( T-t ) }}-1. $$

Figure 4.8 depicts this bias calculated with the same parameter values used in the experiments of Appendix B of Chap. 3: $\kappa=0.3807$, $\bar {r}=0.072$, and $\sigma=3.3107\times 10^{-2}$. In this experiment, we observe mild biases arising while assuming interest rates are constant when in fact they are not.

Fig. 4.8

Percentage bias arising while pricing equity volatility under the incorrect assumption that interest rate are constant. Biases are obtained assuming interest rates are generated by the Vasicek (1977) model

4.1.2 C.2 Naïve Model-Free Methodology and Bias in Vasicek’s Market

The bias. We calculate the bias in Eq. (4.8) assuming that the short-term rate is generated by the Vasicek (1977) model where the short-term rate is the solution to Eq. (4.57). Below, we show that

$$ \mathbb{P} ( t,T,\mathbb{T} ) =\frac{\sigma^{2}}{\kappa ^{2}} \biggl( ( T-t ) + \frac{e^{-2\kappa ( \mathbb {T}-T ) }-e^{-2\kappa ( \mathbb{T}-t ) }-4e^{-\kappa ( \mathbb {T}-T ) }+4e^{-\kappa ( \mathbb{T}-t ) }}{2\kappa} \biggr) , $$

(C.3)

and

$$ \mathrm{Bias} ( t,T ) =-\frac{\sigma^{2}}{\kappa^{2}} \biggl[ T-t-\frac{1}{2\kappa} \bigl( \bigl( 2-e^{-\kappa ( T-t ) } \bigr) ^{2}-1 \bigr) \biggr] . $$

(C.4)

Figure 4.9 depicts $\sqrt{\frac{\mathbb{P} ( t,T,\mathbb{T} ) }{T-t}}$ and $\sqrt{\frac{\mathbb{P}_{\mathrm{VIX}} ( t,T,\mathbb{T} ) }{T-t}}$ in Eq. (4.8), where $\mathbb {P} ( t,T,\mathbb{T} )$ is obtained through Eq. (C.3) and $\mathbb{P}_{{\mathrm{VIX}}} ( t,T,\mathbb {T} )$ through Eq. (4.8) and Eq. (C.4). We consider a 2-year zero-coupon bond, $\mathbb {T}=2$, a 5-year zero-coupon bond, $\mathbb{T}=5$, maturities $T$ ranging from one to 12-months. For the 2-year zero, we use the same parameter values reported in Appendix B of Chap. 3, i.e. $\sigma=3.3107\times10^{-2}$ and $\kappa =0.3807$. For the 5-year zero, we use the same values but halved persistence, $\kappa=2\times0.3807$.

Fig. 4.9

Biases arising while estimating expected volatility with standard model-free methodology. This figure depicts the true expected volatility on a 2-year (left panel) and 5-year (right panel) zero-coupon bond log-return arising in the Vasicek (1977) market (solid line, in percent), along with that estimated by applying a naïve model-free methodology (dashed line, in percent), over a number of months. The right panel depicts the expected volatility arising while keeping the instantaneous basis point volatility of the short-term rate equal to that in the left panel, but with a halved persistence of interest rate shocks

Proof of Eqs. (C.3) and (C.4). Some of the derivations below are taken as given, and will be made available in Appendix C.9 below. In particular, it is easy to show (see Eq. (C.32) below) that the expression for the instantaneous bond return volatility predicted by Vasicek’s model is

$$ \sigma_{\mathrm{v},\tau} ( T ) \equiv-\frac{\sigma ( 1-e^{-\kappa(T-\tau)} ) }{\kappa}. $$

Equation (C.3) follows by evaluating the integral

$$ \mathbb{P} ( t,T,\mathbb{T} ) =\int_{t}^{T} \sigma_{\mathrm{v},\tau}^{2} ( T ) d\tau. $$

Moreover, it can be shown (see Eq. (C.33) below) that

$$\begin{aligned} \mathbb{E}_{t}^{Q_{F^{T}}} ( r_{\tau} ) &= \bar{r}+e^{-\kappa ( \tau-t ) } ( r_{t}-\bar{r} ) \\ &\quad{}-\frac{\sigma^{2}}{2\kappa ^{2}} \bigl[ 2 \bigl( 1-e^{-\kappa ( \tau-t ) } \bigr) +e^{-\kappa ( T-t ) } \bigl( e^{-\kappa ( \tau-t ) }-e^{\kappa ( \tau-t ) } \bigr) \bigr] . \end{aligned}$$

For example, in the special case where $\tau=T\equiv\theta$ (say), we get

$$ \mathbb{E}_{t}^{Q_{F^{\theta}}} ( r_{\theta} ) =\bar {r}+e^{-\kappa ( \theta-t ) } ( r_{t}-\bar{r} ) -\frac{\sigma^{2}}{2\kappa^{2}} \bigl( 1-e^{-\kappa ( \theta -t ) } \bigr) ^{2}, $$

so that

$$ \mathbb{E}_{t}^{Q_{F^{T}}} ( r_{\tau} ) -\mathbb {E}_{t}^{Q_{F^{\tau}}} ( r_{\tau} ) =-\frac{\sigma ^{2}}{2\kappa^{2}} \bigl[ \bigl( 1-e^{-2\kappa ( \tau-t ) } \bigr) \bigl( 1-e^{-\kappa ( T-t ) }e^{\kappa ( \tau-t ) } \bigr) \bigr] . $$

(C.5)

By substituting Eq. (C.5) into Eq. (4.9), and integrating, we are left with the expression in Eq. (C.4). □

4.1.3 C.3 Marking to Market

The value as of time $\tau$ of the percentage variance swap payoff in Eq. (4.15) is

$$\begin{aligned} & \mathbb{E}_{\tau} \bigl( e^{-\int_{\tau}^{T}r_{u}du} \bigl( V ( t,T,\mathbb{T} ) -\mathbb{P} ( t,T,\mathbb{T} ) \bigr) \bigr) \\ &\quad =\mathbb{E}_{\tau} \bigl( e^{-\int_{\tau}^{T}r_{u}du} \bigl( V ( t,\tau, \mathbb{T} ) +V ( \tau,T,\mathbb{T} ) -\mathbb{P} ( t,T,\mathbb{T} ) \bigr) \bigr) \\ &\quad =V ( t,\tau,\mathbb{T} ) P_{\tau} ( T ) +\mathbb {E}_{\tau} \bigl( e^{-\int_{\tau}^{T}r_{u}du}V ( \tau ,T,\mathbb{T} ) \bigr) -\mathbb{P} ( t,T, \mathbb{T} ) P_{\tau} ( T ) . \end{aligned}$$

By Eq. (4.17), we have that

$$ \mathbb{E}_{\tau} \bigl( e^{-\int_{\tau}^{T}r_{u}du}V ( \tau ,T,\mathbb{T} ) \bigr) =\mathbb{P} ( \tau,T,\mathbb{T} ) P_{\tau} ( T ) , $$

so that, by rearranging terms, we obtain the expression for $\mathrm{M}\text{-}\mathrm{Var}_{\tau} ( T,\mathbb{T} )$ in Eq. (4.29). The marking to market updates regarding the basis point variance swap follow by nearly identical arguments applied to the payoff in Eq. (4.22).

4.1.4 C.4 Replication of Variance Swaps

Percentage. We have:

$$\begin{aligned} &\ln\frac{F_{T} ( T,\mathbb{T} ) }{F_{t} ( T,\mathbb {T} ) } \\ &\quad =\frac{F_{T} ( T,\mathbb{T} ) -F_{t} ( T,\mathbb{T} ) }{F_{t} ( T,\mathbb{T} ) } \\ &\qquad{} - \biggl( \int_{0}^{F_{t} ( T,\mathbb{T} ) } \bigl( K-F_{T} ( T,\mathbb{T} ) \bigr) ^{+}\frac{1}{K^{2}}dK+ \int_{F_{t} ( T,\mathbb{T} ) }^{\infty} \bigl( F_{T} ( T,\mathbb{T} ) -K \bigr) ^{+}\frac{1}{K^{2}}dK \biggr) . \end{aligned}$$

(C.6)

By Itô’s lemma,

$$ V ( t,T,\mathbb{T} ) =2\int_{t}^{T} \frac{dF_{s} ( T,\mathbb{T} ) }{F_{s} ( T,\mathbb{T} ) }-2\ln\frac {F_{T} ( T,\mathbb{T} ) }{F_{t} ( T,\mathbb{T} ) }, $$

(C.7)

where, by Eq. (C.6), the second term on the R.H.S. of Eq. (C.7) is the time $T$ payoff delivered by a portfolio set up at $t$, which is two times: (a) short $1/F_{t} ( T,\mathbb{T} )$ units a forward struck at $F_{t} ( T,\mathbb{T} )$, and (b) long a continuum of out-of-the-money options with weights $K^{-2}dK$. By Eq. (4.20), the cost of this position at time $t$ is $P ( t,T,\mathbb{T} ) P_{t} ( T )$, as indicated by row (ii) of Table 4.1 in the main text. We borrow $\mathbb{P} ( t,T,\mathbb{T} ) P_{t} ( T )$ at time $t$, which we repay back at time $T$, as in row (iii) of Table 4.1.

Finally, we derive the self-financing portfolio in row (i) of Table 4.1. The value of this portfolio must be zero at time $t$, and replicate the first term on the R.S.H. of Eq. (C.7). Consider a self-financed strategy investing in (a) the forward, and (b) a money market account that has value $M_{\tau}$ that grows over time according to: $dM_{\tau }=r_{\tau }M_{\tau}d\tau$. The total value of the strategy is

$$ \upsilon_{\tau}=\theta_{\tau}F_{\tau} ( T,\mathbb{T} ) + \psi _{\tau}M_{\tau}, $$

where $\theta_{\tau}$ are the units in the forward, and $\psi_{\tau}$ are the units in the money market account. Let

$$ \hat{\theta}_{\tau}F_{\tau} ( T,\mathbb{T} ) =1,\qquad \hat { \psi}_{\tau}M_{\tau}=\int_{t}^{\tau} \frac{dF_{s} ( T,\mathbb{T} ) }{F_{s} ( T,\mathbb{T} ) }-1, $$

(C.8)

so that $\hat{\upsilon}_{\tau}=\hat{\theta}_{\tau}F_{\tau}+\hat{\psi }_{\tau}M_{\tau}$ satisfies

$$ \hat{\upsilon}_{\tau}=\int_{t}^{\tau} \frac{dF_{s} ( T,\mathbb{T} ) }{F_{s} ( T,\mathbb{T} ) }, $$

(C.9)

with

$$ \hat{\upsilon}_{t}=0,\quad \text{and}\quad \hat{\upsilon}_{T}= \int_{t}^{T}\frac{dF_{s} ( T,\mathbb{T} ) }{F_{s} ( T,\mathbb{T} ) }. $$

Therefore, by going long two portfolios $(\hat{\theta}_{\tau},\hat{\psi }_{\tau})$, the first term on the R.H.S. and by the previous results, the whole of the R.H.S. of Eq. (C.7) can be replicated, provided $(\hat{\theta}_{\tau},\hat{\psi}_{\tau})$ is self-financed. To show that $(\hat{\theta}_{\tau},\hat{\psi}_{\tau})$ is self-financed, we have

$$\begin{aligned} d\hat{\upsilon}_{\tau}& =\hat{\theta}_{\tau}F_{\tau} ( T,\mathbb {T} ) \frac{dF_{\tau} ( T,\mathbb{T} ) }{F_{\tau} ( T,\mathbb{T} ) }+\hat{\psi}_{\tau}M_{\tau} \frac{dM_{\tau }}{M_{\tau}} \\ & = \biggl( \frac{dF_{\tau} ( T,\mathbb{T} ) }{F_{\tau} ( T,\mathbb{T} ) }-r_{\tau}d\tau \biggr) + \biggl( \int _{t}^{\tau}\frac{dF_{\tau} ( T,\mathbb{T} ) }{F_{\tau} ( T,\mathbb{T} ) } \biggr) r_{\tau}d\tau \\ & = \biggl( \frac{dF_{\tau} ( T,\mathbb{T} ) }{F_{\tau} ( T,\mathbb{T} ) }-r_{\tau}d\tau \biggr) +r_{\tau} \hat{\upsilon }_{\tau }d\tau \\ & =\hat{\theta}_{\tau}F_{\tau} ( T,\mathbb{T} ) \biggl( \frac {dF_{\tau} ( T,\mathbb{T} ) }{F_{\tau} ( T,\mathbb {T} ) }-r_{\tau}d\tau \biggr) +r_{\tau}\hat{ \upsilon}_{\tau }d\tau, \end{aligned}$$

(C.10)

where the second line follows by Eq. (C.8), the third by Eq. (C.9), and the fourth, again, by Eq. (C.8). The dynamics of $\hat{\upsilon}_{\tau}$ in Eq. (C.10) are those of a self-financed strategy.

Basis point. We have

$$\begin{aligned} &F_{T}^{2} ( T,\mathbb{T} ) -F_{t}^{2} ( T,\mathbb{T} ) \\ &\quad =2F_{t} ( T,\mathbb{T} ) \bigl( F_{T} ( T,\mathbb{T} ) -F_{t} ( T,\mathbb{T} ) \bigr) \\ &\qquad{} +2 \biggl( \int_{0}^{F_{t} ( T,\mathbb{T} ) } \bigl( K-F_{T} ( T,\mathbb{T} ) \bigr) ^{+}dK+\int _{F_{t} ( T,\mathbb{T} ) }^{\infty} \bigl( F_{T} ( T,\mathbb{T} ) -K \bigr) ^{+}dK \biggr) . \end{aligned}$$

(C.11)

By Itô’s lemma,

$$ V^{\mathrm{bp}} ( t,T,\mathbb{T} ) =-2\int_{t}^{T}F_{s} ( T,\mathbb{T} ) dF_{s} ( T,\mathbb {T} ) + \bigl[ F_{T}^{2} ( T,\mathbb{T} ) -F_{t}^{2} ( T,\mathbb{T} ) \bigr] . $$

(C.12)

By Eq. (C.11), the second term on the R.H.S. of Eq. (C.12) is the payoff at $T$ of a portfolio set up at $t$, which is: (a) long $2F_{t} ( T,\mathbb{T} )$ units a government bond forward struck at $F_{t} ( T,\mathbb{T} )$, and (b) long a continuum of out-of-the-money options with weights $2dK$. It is the static position in row (ii) of Table 4.2 in the main text. By Eq. (4.27), its cost is $\mathbb{P}^{\mathrm{bp}} ( t,T,\mathbb{T} ) P_{t} ( T )$, which we borrow at $t$, to repay it back at $T$, as in row (iii) of Table 4.2. To obtain the self-financed portfolio to be shorted in row (i) of Table 4.2, we proceed as in row (i) of Table 4.1, but with the portfolio,

$$ \hat{\psi}_{\tau}M_{\tau}=\int_{t}^{\tau}2F_{s} ( T,\mathbb{T} ) dF_{s} ( T,\mathbb{T} ) -1, $$

replacing that in Eq. (C.8).

4.1.5 C.5 Estimates Based on Forward Price Approximations

Let $K_{0}$ be the first strike below $F_{t} ( T,\mathbb {T} )$, as defined in the main text. We expand $\ln F_{T} ( T,\mathbb{T} )$ around $K_{0}$, as follows:

$$\begin{aligned} &\ln\frac{F_{T} ( T,\mathbb{T} ) }{F_{t} ( T,\mathbb {T} ) } \\ &\quad=\ln\frac{K_{0}}{F_{t} ( T,\mathbb{T} ) }+\frac {F_{T} ( T,\mathbb{T} ) -K_{0}}{K_{0}} \\ &\qquad{} - \biggl( \int_{0}^{K_{0}} \bigl( K-F_{T} ( T,\mathbb {T} ) \bigr) ^{+}\frac{1}{K^{2}}dK+ \int_{K_{0}}^{\infty} \bigl( F_{T} ( T,\mathbb{T} ) -K \bigr) ^{+}\frac{1}{K^{2}}dK \biggr) , \end{aligned}$$

(C.13)

so that the fair value of the government bond variance swap is

$$\begin{aligned} \mathbb{P} ( t,T,\mathbb{T} ) & =-2\mathbb {E}_{t}^{Q_{F^{T}}} \biggl( \ln\frac{F_{T} ( T,\mathbb{T} ) }{F_{t} ( T,\mathbb {T} ) } \biggr) \\ & =-2 \biggl( \ln\frac{K_{0}}{F_{t} ( T,\mathbb{T} ) }+\frac {F_{t} ( T,\mathbb{T} ) -K_{0}}{K_{0}} \biggr) \\ &\quad{} +\frac{2}{P_{t} ( T ) } \biggl( \int_{0}^{K_{0}} \mathrm{Put}_{t} ( K,T,\mathbb{T} ) \frac {1}{K^{2}}dK+\int _{K_{0}}^{\infty}\mathrm{Call}_{t} ( K,T, \mathbb{T} ) \frac{1}{K^{2}}dK \biggr) . \end{aligned}$$

(C.14)

Consider a second order expansion of the function $-\ln\frac{F_{t}}{K_{0}}$ about $K_{0}$,

$$ -\ln\frac{F_{t}}{K_{0}}\approx-\frac{1}{K_{0}} ( F_{t}-K_{0} ) +\frac{1}{2}\frac{1}{K_{0}^{2}} ( F_{t}-K_{0} ) ^{2}. $$

Substituting this approximation into the first term on the R.H.S. of Eq. (C.14) leaves the expression of $\mathbb{P}_{o} ( t,T,\mathbb{T} )$ in Eq. (4.30) of the main text.

Next, consider the correction applying to the basis point index. Similarly as for Eq. (C.13), expand $F_{T}^{2} ( T,\mathbb {T} )$ around $K_{0}$,

$$\begin{aligned} &F_{T}^{2} ( T,\mathbb{T} ) -F_{t}^{2} ( T,\mathbb{T} ) \\ &\quad=K_{0}^{2}-F_{t}^{2} ( T,\mathbb{T} ) +2K_{0} \bigl( F_{T} ( T,\mathbb{T} ) -K_{0} \bigr) \\ &\qquad{} +2 \biggl[ \int_{0}^{K_{0}} \bigl( K-F_{T} ( T,\mathbb {T} ) \bigr) ^{+}dK+\int _{K_{0}}^{\infty} \bigl( F_{T} ( T,\mathbb{T} ) -K \bigr) ^{+}dK \biggr] , \end{aligned}$$

so that the fair value of the basis point government bond variance swap is

$$\begin{aligned} \mathbb{P}^{\mathrm{bp}} ( t,T,\mathbb{T} ) & =K_{0}^{2}-F_{t}^{2} ( T,\mathbb{T} ) +2K_{0} \bigl( F_{t} ( T,\mathbb{T} ) -K_{0} \bigr) \\ &\quad{} +\frac{2}{P_{t} ( T ) } \biggl( \int_{0}^{K_{0}} \mathrm{Put}_{t} ( K,T,\mathbb{T} ) dK+\int_{K_{0}}^{\infty} \mathrm{Call}_{t} ( K,T,\mathbb{T} ) dK \biggr) . \end{aligned}$$

(C.15)

A second order expansion of the function $F_{t}^{2}$ about $K_{0}$ yields

$$ F_{t}^{2}-K_{0}^{2} \approx2K_{0} ( F_{t}-K_{0} ) + ( F_{t}-K_{0} ) ^{2}. $$

Substituting this approximation into Eq. (C.15) yields the approximation in Eq. (4.31).

4.1.6 C.6 Certainty Equivalence, and Existence of Basis Point Yield Volatility

We show that our model-free measures of yields exist both in the case of the regular indexes of Sect. 4.2, and the sandwich combinations of Sect. 4.5. Finally, we provide details regarding the certainty equivalent bond prices introduced in Sect. 4.2.8.

Regular indexes. We show that $y_{\mathcal{B}} ( t,T,\mathbb{T} )$ in Eq. (4.35) exists and is positive. We have

$$\begin{aligned} &\mathrm{GB}\text{-}\mathrm{VI}^{\mathrm{bp}} ( t,T,\mathbb {T} ) \\ &\quad \equiv\sqrt{\frac{1}{T-t}\frac{1}{P_{t} ( T ) }\mathbb{E}_{t} \biggl( e^{-\int_{t}^{T}r_{\tau }d\tau}\int_{t}^{T}F_{\tau}^{2} ( T,\mathbb{T} ) \bigl\Vert v_{\tau } ( T,\mathbb{T} ) \bigr\Vert ^{2}d\tau \biggr) } \\ &\quad \leq\sqrt{\frac{1}{T-t}\frac{1}{P_{t} ( T ) }\mathbb {E}_{t} \biggl( e^{-\int_{t}^{T}r_{\tau}d\tau}\int_{t}^{T} \Bigl( \sup_{\tau\in ( t,T ) }F_{\tau }^{2} ( T,\mathbb{T} ) \Bigr) \bigl\Vert v_{\tau} ( T,\mathbb{T} ) \bigr\Vert ^{2}d\tau \biggr) } \\ &\quad \leq\sqrt{\frac{1}{T-t}\frac{1}{P_{t} ( T ) }\mathbb {E}_{t} \biggl( e^{-\int_{t}^{T}r_{\tau}d\tau}\int_{t}^{T} \bigl\Vert v_{\tau} ( T,\mathbb{T} ) \bigr\Vert ^{2}\cdot \hat{P}^{2} ( 0 ) \biggr) d\tau} \\ &\quad =\hat{P} ( 0 ) \cdot\mathrm{GB}\text{-}\mathrm {VI} ( t,T,\mathbb{T} ) , \end{aligned}$$

where $\hat{P}$ is the function defined in Eq. (4.36), and the third line follows as $F_{\tau} ( T,\mathbb{T} ) =\mathbb {E}_{\tau }^{Q_{F^{T}}} ( B_{T} ( \mathbb{T} ) ) \leq\hat {P} ( 0 )$, for all $\tau\in [ t,T ]$, with the last inequality holding as the price of a coupon-bearing bond is bounded by the (undiscounted) sum of the coupons plus principal, i.e., by $\hat {P} ( 0 )$. That is, using Eq. (4.34),

$$ \frac{\mathrm{G\mathrm{B}}\text{-}\mathrm{VI}^{\mathrm {bp}} ( t,T,\mathbb{T} ) }{\mathrm{GB}\text{-}\mathrm {VI} ( t,T,\mathbb{T} ) }=\mathcal{B} ( t,T,\mathbb {T} ) \leq \hat{P} ( 0 ) . $$

Because $\hat{P} ( y )$ is monotonically decreasing in $y$ with $\lim_{y\rightarrow\infty}\hat{P} ( y ) =0$, the previous equality shows that there exists a value $y_{\mathcal{B}} ( t,T,\mathbb {T} ) \geq0$ such that Eq. (4.35) holds true.

Sandwich combinations. Next, we show that the index in Eq. (4.139) is well-defined. We have

$$\begin{aligned} V_{t}^{\mathrm{perc}} ( T_{i} ) =&\frac{1}{T_{i}-t} \frac {1}{P_{t} ( T_{i} ) }\mathbb{E}_{t} \biggl( e^{-\int _{t}^{T_{i}}r_{\tau}d\tau}\int _{t}^{T_{i}}F_{\tau }^{2} ( T_{i},\mathbb{T} ) d\tau \biggr) , \\ V_{t}^{\mathrm{bp}} ( T_{i} ) =&\frac{1}{T_{i}-t} \frac {1}{P_{t} ( T_{i} ) }\mathbb{E}_{t} \biggl( e^{-\int _{t}^{T_{i}}r_{\tau}d\tau}\int _{t}^{T_{i}}F_{\tau }^{2} ( T_{i},\mathbb{T} ) \bigl\Vert v_{\tau} ( T_{i}, \mathbb{T} ) \bigr\Vert ^{2}d\tau \biggr) , \end{aligned}$$

so that

$$ x_{t}V_{t}^{\mathrm{bp}} ( T_{i} ) + ( 1-x_{t} ) V_{t}^{\mathrm{bp}} ( T_{i+1} ) \leq \bigl( x_{t}V_{t}^{\mathrm {perc}} ( T_{i} ) + ( 1-x_{t} ) V_{t}^{\mathrm {perc}} ( T_{i+1} ) \bigr) \hat{P}^{2} ( 0 ) . $$

Therefore, $\frac{\mathcal{I}_{t}^{\mathrm{bp}}}{\mathcal {I}_{t}^{\mathrm{perc}}}\leq\hat{P} ( 0 )$, and it follows that the index in Eq. (4.139) exists and is positive by the same arguments used to demonstrate the existence of a regular index.

Yield volatility in the post-issuance case. We provide details regarding the case in which the maturity of the forward exceeds that of the first date the bond pays off a coupon after it is issued. Define

$$ x_{\mathcal{B}} ( t,T,\mathbb{T} ) :\mathcal{B} ( t,T,\mathbb{T} ) = \frac{\mathrm{GB}\text{-}\mathrm {VI}^{\mathrm{bp}} ( t,T,\mathbb{T} ) }{\mathrm{GB}\text{-}\mathrm{VI} ( t,T,\mathbb{T} ) }=\hat{P}_{T} \bigl( x_{\mathcal{B}} ( t,T,\mathbb{T} ) \bigr) , $$

(C.16)

where $\mathrm{GB}\text{-}\mathrm{VI} ( t,T,\mathbb{T} )$ and $\mathrm{GB}\text{-}\mathrm{VI}^{\mathrm{bp}} ( t,T,\mathbb {T} )$ are as in Eq. (4.21) and Eq. (4.28), respectively, and, denoting by $N_{T}$ the number of payments that still have to take place,

$$ \hat{P}_{T} ( x ) =\sum_{i=1}^{N_{T}} \frac {C_{t_{i}}}{n} ( 1+x ) ^{-\frac{t_{i}-T}{365}}+100 ( 1+x ) ^{-\frac{t_{N_{T}}-T}{365}}, $$

(C.17)

and the sequence of $t_{i}$ tracks dates at which the coupon payments are still to take place. The modified duration of the guaranteed price is

$$\begin{aligned} &\hat{D}_{\mathcal{B}} ( t,T,\mathbb{T} ) \equiv \frac{1}{1+x_{\mathcal{B}} ( t,T,\mathbb{T} ) } \Biggl( \sum _{i=1}^{N_{T}} \omega_{t_{i}} \frac{t_{i}-T}{365}+\hat{\omega }_{t_{N_{T}}} \frac{t_{N_{T}}-T}{365} \Biggr) \\ &\quad\omega_{t_{i}}\equiv\frac{ \frac{C_{t_{i}}}{n} / ( 1+x_{\mathcal{B}} ( t,T,\mathbb{T} ) ) ^{\frac {t_{i}-T}{365}}}{\mathcal{B} ( t,T,\mathbb{T} ) },\qquad \hat { \omega}_{t_{N_{T}}} \equiv\frac{ 100 / ( 1+x_{\mathcal {B}} ( t,T,\mathbb{T} ) ) ^{\frac{t_{N_{T}}-T}{365}}}{\mathcal {B} ( t,T,\mathbb{T} ) } \end{aligned}$$

(C.18)

where $x_{\mathcal{B}} ( t,T,\mathbb{T} )$ is defined as in Eq. (C.16). Accordingly, our model-free duration-based yield volatility is still as in Eq. (4.38), with $\hat{D}_{\mathcal{B}} ( t,T,\mathbb{T} )$ replacing $D_{\mathcal{B}} ( t,T,\mathbb {T} )$, so that, using the definitions of $\hat{P}_{T}$ in Eq. (C.17) and that of $\hat{D}_{\mathcal{B}}$ in Eqs. (C.18):

$$\begin{aligned} &\mathrm{GB}\text{-}\mathrm{VI}_{\mathrm{Yd}}^{\mathrm{bp}} ( t,T,\mathbb{T} ) \\ &\quad=\frac{100\times ( 1+\hat{P}_{T}^{-1} [ \frac{\mathrm {GB}\text{-}\mathrm{VI}^{\mathrm{bp}} ( t,T,\mathbb {T} ) }{\mathrm{GB}\text{-}\mathrm{VI} ( t,T,\mathbb {T} ) } ] ) \times\mathrm{GB}\text{-}\mathrm{VI}^{\mathrm{bp}} ( t,T,\mathbb{T} ) }{\sum_{i=1}^{N_{T}}\frac {C_{t_{i}}}{n} ( 1+\hat{P}_{T}^{-1} [ \frac{\mathrm{GB}\text{-}\mathrm{VI}^{\mathrm{bp}} ( t,T,\mathbb{T} ) }{\mathrm{GB}\text{-}\mathrm{VI} ( t,T,\mathbb{T} ) } ] ) ^{-\frac{t_{i}-T}{365}}\frac{t_{i}-T}{365}+100 ( 1+\hat{P}_{T}^{-1} [ \frac{\mathrm{GB}\text{-}\mathrm {VI}^{\mathrm{bp}} ( t,T,\mathbb{T} ) }{\mathrm{GB}\text{-}\mathrm{VI} ( t,T,\mathbb{T} ) } ] ) ^{-\frac{t_{N_{T}}-T}{365}}\frac{t_{N_{T}}-T}{365}} \end{aligned}$$

(C.19)

where $\hat{P}_{T}^{-1}$ denotes the inverse function of $\hat{P}_{T}$ in Eq. (C.17).

Similarly, the yield-based yield volatility index in Eq. (4.40) of the main text can be replaced with

$$ \mathrm{GB}\text{-}\mathrm{VI}_{\mathrm{Y}}^{\mathrm{bp}} ( t,T, \mathbb{T} ) =100\times\hat{P}_{T}^{-1} \biggl[ \frac{\mathrm {GB}\text{-}\mathrm{VI}^{\mathrm{bp}} ( t,T,\mathbb {T} ) }{\mathrm{GB}\text{-}\mathrm{VI} ( t,T,\mathbb {T} ) } \biggr] \times \mathrm{GB}\text{-}\mathrm{VI} ( t,T, \mathbb{T} ) , $$

(C.20)

where $\hat{P}_{T}^{-1}$ is the inverse function of $\hat{P}_{T}$ in Eq. (C.17).

Finally, we can generalize Eq. (4.110) and Eq. (4.111) in Sect. 4.4 of the main text, as follows:

$$\begin{aligned} &\mathrm{GB}\text{-}\mathrm{VI}_{\mathrm{Yd}}^{\mathrm{bp}} ( t,T,S,\mathbb{T} ) \\ &\quad=\frac{100\times ( 1+\hat{P}_{T}^{-1} [ \frac{\mathrm {GB}\text{-}\mathrm{VI}^{\mathrm{bp}} ( t,T,S,\mathbb {T} ) }{\mathrm{GB}\text{-}\mathrm{VI} ( t,T,S,\mathbb {T} ) } ] ) \mathrm{GB}\text{-}\mathrm{VI}^{\mathrm{bp}} ( t,T,S,\mathbb{T} ) }{\sum_{i=1}^{N_{T}}\frac {C_{t_{i}}}{n} ( 1+\hat{P}_{T}^{-1} [ \frac{\mathrm{GB}\text{-}\mathrm{VI}^{\mathrm{bp}} ( t,T,S,\mathbb{T} ) }{\mathrm{GB}\text{-}\mathrm{VI} ( t,T,S,\mathbb{T} ) } ] ) ^{-\frac{t_{i}-T}{365}}\frac{t_{i}-T}{365}+100 ( 1+\hat{P}_{T}^{-1} [ \frac{\mathrm{GB}\text{-}\mathrm {VI}^{\mathrm{bp}} ( t,T,S,\mathbb{T} ) }{\mathrm{GB}\text{-}\mathrm{VI} ( t,T,S,\mathbb{T} ) } ] ) ^{-\frac{t_{N_{T}}-T}{365}}\frac{t_{N_{T}}-T}{365}}, \end{aligned}$$

(C.21)

and

$$ \mathrm{GB}\text{-}\mathrm{VI}_{\mathrm{Y}}^{\mathrm{bp}} ( t,T,S, \mathbb{T} ) =100\times\hat{P}_{T}^{-1} \biggl[ \frac{\mathrm {GB}\text{-}\mathrm{VI}^{\mathrm{bp}} ( t,T,S,\mathbb {T} ) }{\mathrm{GB}\text{-}\mathrm{VI} ( t,T,S,\mathbb {T} ) } \biggr] \times\mathrm{GB}\text{-}\mathrm{VI} ( t,T,S, \mathbb{T} ) , $$

(C.22)

where $\mathrm{GB}\text{-}\mathrm{VI} ( t,T,S,\mathbb{T} )$ and $\mathrm{GB}\text{-}\mathrm{VI}^{\mathrm{bp}} ( t,T,S,\mathbb{T} )$ are as in Eq. (4.103) and Eq. (4.108), respectively.

Additional details regarding certainty equivalent prices. We prove that Eq. (4.41) holds true. By Eq. (4.34), we have that $\mathcal{B} ( t,T,\mathbb{T} )$ satisfies

$$\begin{aligned} \mathcal{B}^{2} ( t,T,\mathbb{T} ) &=\frac{\int_{t}^{T}\mathbb{E}_{t}^{Q_{F^{T}}} ( F_{\tau}^{2} ( T,\mathbb {T} ) \Vert v_{\tau} ( T,\mathbb{T} ) \Vert ^{2} ) d\tau}{\int_{t}^{T}\mathbb{E}_{t}^{Q_{F^{T}}} ( \Vert v_{\tau} ( T,\mathbb{T} ) \Vert ^{2} ) d\tau }\\ &= \frac{\int_{t}^{T}\mathbb{E}_{t}^{Q_{F^{T}}} ( \Vert v_{\tau } ( T,\mathbb{T} ) \Vert ^{2} ) \mathbb {E}_{t}^{Q_{v^{\tau}}} ( F_{\tau}^{2} ( T,\mathbb{T} ) ) d\tau}{\int_{t}^{T}\mathbb{E}_{t}^{Q_{F^{T}}} ( \Vert v_{\tau} ( T,\mathbb{T} ) \Vert ^{2} ) d\tau}, \end{aligned}$$

where $\mathbb{E}_{t}^{Q_{v^{\tau}}}$ is the conditional expectation taken under the realized variance probability defined through the Radon–Nikodym derivative in Eq. (4.42). This is Eq. (4.41) with weighting function

$$ \omega_{\tau}\equiv\frac{\mathbb{E}_{t}^{Q_{F^{T}}} ( \Vert v_{\tau} ( T,\mathbb{T} ) \Vert ^{2} ) }{\int_{t}^{T}\mathbb{E}_{t}^{Q_{F^{T}}} ( \Vert v_{\tau } ( T,\mathbb{T} ) \Vert ^{2} ) d\tau}. $$

(C.23)

4.1.7 C.7 Illustrations with a Stochastic Volatility Model

Solution of the stochastic volatility model, Eq. (4.45). By standard arguments (see, e.g., Mele 2014, Chap. 12), the pricing function in Eq. (4.45) holds with

$$ P_{\tau} \bigl( r_{\tau},v_{\tau}^{2},T \bigr) =e^{A_{T} ( \tau ) - ( T-\tau ) r_{\tau}+C_{T} ( \tau ) v_{\tau }^{2}}, $$

(C.24)

where

$$ A_{T} ( \tau ) =\int_{\tau}^{T} ( s-T ) \theta _{s}ds, $$

(C.25)

and $C_{T} ( \cdot )$ is the solution to Eq. (4.46). Matching the instantaneous forward rate predicted by the model at $t$, say $f_{t} ( r_{t},v_{t}^{2},T )$, to the hypothetically observed instantaneous forward rate at the same time $t$, $f_{\$} ( t,T )$, leaves

$$ f_{\$} ( t,T ) =f_{t} \bigl( r_{t},v_{t}^{2},T \bigr) \equiv -\frac{\partial\ln P_{t} ( r_{t},v_{t}^{2},T ) }{\partial T}=\int_{t}^{T} \theta_{\tau}d\tau+r_{t}-\frac{\partial C_{T} ( t ) }{\partial T}v_{t}^{2}, $$

(C.26)

where the last equality follows by differentiating the price function in Eq. (C.24), and the expression for $A_{T} ( \tau )$ in Eq. (C.25). Differentiating Eq. (C.26) with respect to $T$ leaves $\frac{\partial f_{\$} ( t,T ) }{\partial T}=\theta _{T}-\frac{\partial^{2}C_{T} ( t ) }{\partial T^{2}}v_{t}^{2}$ and Eq. (4.44) of the main text. Finally, substituting Eq. (4.44) into Eq. (C.25) leaves the expression for the integral in the exponent of the price function in Eq. (4.45) of the main text.

It is easy to verify that $\theta_{\tau}$ in Eq. (4.44) is indeed the infinite-dimensional parameter we are searching for. Substitute Eq. (4.44) into Eq. (C.26), and readily check that the result holds as an identity. Moreover, let us evaluate Eq. (4.45) when $\tau=t$,

$$ P_{t} \bigl( r_{t},v_{t}^{2},T \bigr) =e^{\ell_{T} ( r_{t},v_{t}^{2}, ( \theta_{\tau} ) _{\tau\in [ t,T ] },T ) }, $$

where,

$$\begin{aligned} \ell_{T} \bigl( r_{t},v_{t}^{2}, ( \theta_{\tau} ) _{\tau\in [ t,T ] },T \bigr) &\equiv\int_{t}^{T} ( \tau -T ) \theta_{\tau}d\tau- ( T-t ) r_{t}+C_{T} ( t ) v_{t}^{2}\\ &=-\int_{t}^{T}f_{\$} ( t,\tau ) d\tau, \end{aligned}$$

and the last equality follows after integrating Eq. (C.26), establishing that Eq. (4.47) holds true.

Dynamics of the forward in the stochastic volatility model, Eq. (4.48). We prove that Eq. (4.48) holds true. By Itô’s lemma, the price of a zero coupon bond expiring at $S$ predicted by the model satisfies

$$ \left\{ \textstyle\begin{array}{l} \displaystyle\frac{dP_{\tau} ( S ) }{P_{\tau} ( S ) } = r_{\tau }d\tau+v_{\tau} \bigl( - ( S-\tau ) dW_{1,\tau}+\xi C_{S} ( \tau ) dW_{2,\tau} \bigr) \\ dv_{\tau}^{2} = \xi v_{\tau}dW_{2,\tau} \end{array}\displaystyle \right. $$

By standard arguments, we can define two Brownian motions under the forward probability $Q_{F^{T}}$, $dW_{1,\tau}^{F^{T}}=dW_{1,\tau}+v_{\tau } ( T-\tau ) d\tau$ and $dW_{2,\tau}^{F^{T}}=dW_{2,\tau}-\xi C_{T} ( \tau ) v_{\tau}d\tau$, and use Itô’s lemma to conclude that the price of the forward on a zero coupon bond expiring at time $\mathbb{T}$ is the solution to Eq. (4.48) in the main text.

Extension to mean-reverting stochastic volatility. Consider the following extension of the model in Eqs. (4.43), one for which the instantaneous basis point variance of the short-term rate is mean-reverting under the risk-neutral probability, similar to the Heston (1993) model for equity returns:

$$ \left\{ \textstyle\begin{array}{l} dr_{\tau} = \theta_{\tau}d\tau+v_{\tau}dW_{1,\tau} \\ dv_{\tau}^{2} = k ( m-v_{\tau}^{2} ) d\tau+\xi v_{\tau } \bigl( \rho dW_{1,\tau}+\sqrt{1-\rho^{2}}dW_{2,\tau} \bigr) \end{array}\displaystyle \right. $$

where $k$ is the speed of adjustment of $v_{\tau}^{2}$ towards its long term mean, $m$, and $\rho$ is a constant correlation between the conditional changes, $dr_{\tau}$ and $dv_{\tau}^{2}$.

It is straightforward to show that the price of a zero-coupon bond is given by the following expressions generalizing Eq. (4.45) in the main text:

$$ P_{\tau} \bigl( r_{\tau},v_{\tau}^{2},T \bigr) \equiv e^{ \int _{\tau}^{T} ( s-T ) \theta_{s}ds+km\int _{\tau }^{T}C_{T} ( s ) ds- ( T-\tau ) r_{\tau}+C_{T} ( \tau ) v_{\tau}^{2}}, $$

where $C_{T} ( \cdot )$ is the solution to the Riccati equation

$$ \dot{C}_{T} ( \tau ) = \bigl( k+\rho\xi ( T-\tau ) \bigr) C_{T} ( \tau ) - \biggl( \frac{1}{2} ( T-\tau ) ^{2}+ \frac{1}{2}\xi^{2}C_{T}^{2} ( \tau ) \biggr) ,\quad C_{T} ( T ) =0, $$

and the infinite-dimensional parameter satisfies

$$ \theta_{\tau}=\frac{\partial f_{\$} ( t,\tau ) }{\partial \tau}+km\int_{\tau}^{T} \frac{\partial^{2}C_{\tau} ( u ) }{\partial\tau^{2}}du+\frac{\partial^{2}C_{\tau} ( t ) }{\partial \tau^{2}}v_{t}^{2}. $$

Details regarding realized variance probability. We show that Eq. (4.50) holds true. We need to show that Eq. (4.41) collapses to Eq. (4.50) when the forward price solves Eqs. (4.48). By Eq. (4.49), and the second of Eqs. (4.48), we have that, for all $\tau$,

$$ \mathbb{E}_{t}^{Q_{F^{T}}} \bigl( \bigl\Vert v_{\tau} ( T,\mathbb {T} ) \bigr\Vert ^{2} \bigr) =\bar{\phi}_{\tau} ( T,\mathbb {T} ) \cdot v_{t}^{2}, $$

(C.27)

where $\bar{\phi}_{\tau} ( T,\mathbb{T} )$ is as in the main text, so that Eq. (C.23) collapses to Eq. (4.50), as claimed in the main text.

We are left to prove that under $Q_{v^{\tau}}$, the forward price variance satisfies Eq. (4.51). Define the following density process, relying on the Radon–Nikodym derivative in Eq. (4.42):

$$\begin{aligned} \rho ( s;T ) & \equiv\frac{\mathbb{E}_{s}^{Q_{F^{T}}} ( \Vert v_{\tau} ( T,\mathbb{T} ) \Vert ^{2} ) }{\mathbb{E}_{t}^{Q_{F^{T}}} ( \Vert v_{\tau} ( T,\mathbb {T} ) \Vert ^{2} ) } \\ & =\frac{\bar{\phi}_{\tau} ( s,T,\mathbb{T} ) }{\bar{\phi }_{\tau } ( t,T,\mathbb{T} ) }\cdot\frac{v_{s}^{2}}{v_{t}^{2}} \\ & =e^{-\xi^{2}\int_{t}^{s}C_{T} ( u ) du}\frac {v_{s}^{2}}{v_{t}^{2}} \\ & =e^{-\frac{1}{2}\int_{t}^{s} ( \frac{\xi}{v_{u}} ) ^{2}du-\int_{t}^{s} ( -\frac{\xi}{v_{u}} ) dW_{2,u}^{F^{T}}},\quad s\in [ t,\tau ] , \end{aligned}$$

so that

$$ \frac{d\rho ( s;T ) }{\rho ( s;T ) }=- \biggl( -\frac {\xi }{v_{s}} \biggr) dW_{2,s}^{F^{T}}, \quad s\in [ t,\tau ] . $$

By Girsanov’s theorem, we have that $dW_{2,s}^{v^{\tau}}=dW_{2,s}^{F^{T}}-\frac{\xi}{v_{s}}ds$ is a Brownian motion under $Q_{v^{\tau}}$, and Eq. (4.51) follows.

4.1.8 C.8 The Future Price in Vasicek’s Model

We derive the second equality of Eq. (4.58). By the first equality of Eq. (4.58), and the expression of the coupon-bearing bond price predicted by Vasicek’s model, Eq. (4.63), we have:

$$ \tilde{F}_{t} ( r_{t};T,\mathbb{T} ) =\sum _{i=i_{T}}^{N}\bar{C}_{i}\cdot \mathbb{E}_{t} \bigl( P_{T} ( r_{T},T_{i} ) \bigr) \equiv\sum_{i=i_{T}}^{N} \bar{C}_{i}\cdot \tilde{f}_{t} ( r_{t};T_{i} ) , $$

where the index $i_{T}$ and the coupon series $\bar{C}_{i}$ are defined as in the main text and, by Eq. (4.64):

$$ \tilde{f}_{t} ( r_{t};T_{i} ) \equiv \mathbb{E}_{t} \bigl( e^{a_{T} ( T_{i} ) -b_{T} ( T_{i} ) \cdot r_{T}} \bigr) . $$

Note that conditionally upon the information set as of time $t$, the short-term rate is normally distributed, with expectation $\mathbb{E}_{t} ( r_{T} )$ and variance $\mathbb{V}_{t} ( r_{T} )$ given by:

$$ \mathbb{E}_{t} ( r_{T} ) =\bar{r}+e^{-\kappa(T-t)} ( r_{t}-\bar{r} ), \qquad\mathbb{V}_{t} ( r_{T} ) =\frac{\sigma^{2}}{2\kappa} \bigl( 1-e^{-2\kappa(T-t)} \bigr) , $$

(C.28)

so that

$$ \tilde{f}_{t} ( r_{t};T_{i} ) =e^{a_{T} ( T_{i} ) -b_{T} ( T_{i} ) \cdot\mathbb{E}_{t} ( r_{T} ) +\frac {1}{2}b_{T}^{2} ( T_{i} ) \cdot\mathbb{V}_{t} ( r_{T} ) }=e^{a_{t}^{F} ( T,T_{i} ) -b_{t}^{F} ( T,T_{i} ) r_{t}}, $$

where $a_{t}^{F} ( T,T_{i} )$ and $b_{t}^{F} ( T,T_{i} )$ are defined in Eq. (4.59), and the second equality follows by utilizing the expressions for $\mathbb{E}_{t} ( r_{T} )$ and $\mathbb{V}_{t} ( r_{T} )$ in Eqs. (C.28).

4.1.9 C.9 Future and Forward LIBOR Options in Vasicek’s Model

Future Libor. We derive the second equality of Eq. (4.81). By the first equality of Eq. (4.81), and the expression of the zero-coupon bond price predicted by Vasicek’s model, Eq. (4.64), we have

$$\begin{aligned} \tilde{z}_{t} ( r_{t};T,\Delta ) &=\mathbb{E}_{t} \bigl( l_{T} ( r_{T},\Delta ) \bigr) \equiv \frac{1}{\Delta}\mathbb{E}_{t} \biggl( \frac{1}{P_{T} ( r_{T};T+\Delta ) }-1 \biggr) \\ &=\frac{1}{\Delta } \bigl( \mathbb{E}_{t} \bigl( e^{-a_{T} ( T+\Delta ) +b_{T} ( T+\Delta ) \cdot r_{T}} \bigr) -1 \bigr) . \end{aligned}$$

We know that conditionally upon the information set as of time $t$, the short-term rate is normally distributed, with expectation $\mathbb {E}_{t} ( r_{T} )$ and variance $\mathbb{V}_{t} ( r_{T} )$ given by Eqs. (C.28), so that

$$\begin{aligned} \tilde{z}_{t} ( r_{t};T,\Delta ) &=\frac{1}{\Delta} \bigl( e^{-a_{T} ( T+\Delta ) +b_{T} ( T+\Delta ) \cdot \mathbb{E}_{t} ( r_{T} ) +\frac{1}{2}b_{T}^{2} ( T+\Delta ) \cdot \mathbb{V}_{t} ( r_{T} ) }-1 \bigr) \\ &=\frac{1}{\Delta} \bigl( e^{a_{t}^{Z} ( T,\Delta ) +b_{t}^{Z} ( T,\Delta ) r_{t}}-1 \bigr) , \end{aligned}$$

where $a_{t}^{Z} ( T,\Delta )$ and $b_{t}^{Z} ( T,\Delta )$ are defined in Eq. (4.82), and the second equality follows by utilizing the expressions for $\mathbb{E}_{t} ( r_{T} )$ and $\mathbb{V}_{t} ( r_{T} )$ in Eqs. (C.28).

Forward Libor options. We prove Eq. (4.85). At time $T$, the forward price predicted by Vasicek’s model collapses to:

$$\begin{aligned} Z_{T} ( r_{T};T,T+\Delta ) & =100\times \bigl( 1-l_{T} ( r_{T};\Delta ) \bigr) \\ & =100\times \biggl( 1-\frac{1}{\Delta} \bigl( e^{a_{T}^{Z} ( T,\Delta ) +b_{T}^{Z} ( T,\Delta ) r_{T}}-1 \bigr) \biggr) \\ & =100\times\frac{1}{\Delta} \bigl( 1+\Delta-\tilde{B} ( r_{T} ) \bigr) , \end{aligned}$$

where

$$ \tilde{B} ( r_{T} ) \equiv e^{-a_{\Delta}+b_{\Delta}r_{T}}, $$

(C.29)

and $a_{\Delta}\equiv a_{T} ( T+\Delta )$ and $b_{\Delta }\equiv b_{T} ( T+\Delta )$, due to the expressions of $a_{t}^{Z} ( T,\Delta )$ and $b_{t}^{Z} ( T,\Delta )$ in Eqs. (4.82), and where $a_{T} ( T+\Delta )$ and $b_{T} ( T+\Delta )$ are as in Eqs. (4.64). Therefore, the price of the call on the forward can be expressed as

$$\begin{aligned} \mathrm{Call}_{t}^{z} ( K,T,\Delta ) & = \mathbb{E}_{t} \bigl( e^{-\int_{t}^{T}r_{\tau}d\tau} \bigl( Z_{T} ( r_{T};T,T+\Delta ) -K \bigr) ^{+} \bigr) \\ & =100\times\frac{1}{\Delta}\mathbb{E}_{t} \bigl( e^{-\int _{t}^{T}r_{\tau}d\tau} \bigl( K^{o} ( K ) -\tilde {B} ( r_{T} ) \bigr) ^{+} \bigr) \\ & =100\times P_{t} ( T ) \frac{1}{\Delta}\mathbb {E}_{t}^{Q_{F^{T}}} \bigl( K^{o} ( K ) -\tilde{B} ( r_{T} ) \bigr) ^{+}, \end{aligned}$$

(C.30)

where $K^{o} ( K ) \equiv1+ ( 1-\frac{K}{100} ) \Delta$, the third equality follows by a change of probability and, finally, $Q_{F^{T}}$ denotes the forward probability as in the main text.

The Brownian motion under the forward probability is

$$ W_{\tau}^{F^{T}}=W_{\tau}-\int_{t}^{\tau} \sigma_{u} ( T ) du, $$

where $\sigma_{\tau} ( T )$ is the vector of instantaneous volatilities for a zero in Eq. (4.1), and $W_{\tau}$ a Brownian motion under the risk-neutral probability.

Therefore, for the Vasicek model, the short-term rate is a solution to

$$ dr_{\tau}= \bigl[ \kappa ( \bar{r}-r_{\tau} ) +\sigma\cdot \sigma_{\mathrm{v},\tau} ( T ) \bigr] d\tau+\sigma dW_{\tau }^{F^{T}}, \quad\bar{r}\equiv\mu-\frac{\lambda\sigma}{\kappa}, $$

(C.31)

where notation is as in Eq. (4.57) of the main text, and $\sigma _{\mathrm{v},\tau} ( T )$ is the bond return volatility predicted by Vasicek’s model; by Itô’s lemma,

$$ \sigma_{\mathrm{v},\tau} ( T ) \equiv-\sigma b_{\tau} ( T ) =- \frac{\sigma ( 1-e^{-\kappa(T-\tau)} ) }{\kappa}, $$

(C.32)

where we have used the expression for $b_{t} ( T )$ in Eq. (4.64). By tedious but straightforward calculations, we have that

$$\begin{aligned} \mathbb{E}_{t}^{Q_{F^{T}}} ( r_{\tau} ) &= \bar{r}+e^{-\kappa ( \tau-t ) } ( r_{t}-\bar{r} ) \\ &\quad{}-\frac{\sigma^{2}}{2\kappa ^{2}} \bigl[ 2 \bigl( 1-e^{-\kappa ( \tau-t ) } \bigr) +e^{-\kappa ( T-t ) } \bigl( e^{-\kappa ( \tau-t ) }-e^{\kappa ( \tau-t ) } \bigr) \bigr] . \end{aligned}$$

(C.33)

In particular, under the forward probability, and conditionally upon the information set at time $t$, the short-term rate at time $T$ is normally distributed with expectation equal to

$$ \mathbb{E}_{t}^{Q_{F^{T}}} ( r_{T} ) = \bar{r}+e^{-\kappa ( T-t ) } ( r_{t}-\bar{r} ) -\frac{\sigma^{2}}{2\kappa ^{2}} \bigl( 1-e^{-\kappa ( T-t ) } \bigr) ^{2}, $$

and variance $\mathbb{V}_{t} ( r_{T} )$ given by the expression in Eqs. (C.28).

Next, we elaborate on the expectation of Eq. (C.30). Let $\mathbb{I}_{ \{ \mathcal{E} \} }$ denote the indicator function of the event ℰ, i.e. the function taking a value equal to one if the event ℰ is true and zero, otherwise. We have

$$\begin{aligned} &\mathbb{E}_{t}^{Q_{F^{T}}} \bigl( K^{o} ( K ) - \tilde{B} ( r_{T} ) \bigr) ^{+} \\ &\quad =\mathbb{E}_{t}^{Q_{F^{T}}} \bigl( \bigl( K^{o} ( K ) -\tilde{B} ( r_{T} ) \bigr) \mathbb {I}_{ \{ \tilde{B} ( r_{T} ) \leq K^{o} ( K ) \} } \bigr) \\ &\quad =K^{o} ( K ) \mathrm{Pr}_{t}^{Q_{F^{T}}} \bigl( \tilde{B} ( r_{T} ) -K^{o} ( K ) \leq0 \bigr) -\mathbb {E}_{t}^{Q_{F^{T}}} \bigl( \tilde{B} ( r_{T} ) \mathbb {I}_{ \{ \tilde{B} ( r_{T} ) -K^{o} ( K ) \leq0 \} } \bigr) , \end{aligned}$$

(C.34)

where $\mathrm{Pr}_{t}^{Q_{F^{T}}}$ denotes the probability under the forward measure given the information set at time $t$. Note that

$$ \tilde{B} ( r_{T} ) \equiv e^{-a_{\Delta}+b_{\Delta }r_{T}}=e^{-a_{\Delta}+b_{\Delta}\mathbb{E}_{t}^{Q_{F^{T}}} ( r_{T} ) +\frac{1}{2}y_{t}^{2}}\cdot e^{y_{t}\tilde{N}_{T}-\frac {1}{2}y_{t}^{2}}, $$

(C.35)

where

$$ \tilde{N}_{T}\equiv\frac{r_{T}-\mathbb{E}_{t}^{Q_{F^{T}}} ( r_{T} ) }{\sqrt{\mathbb{V}_{t} ( r_{T} ) }},\qquad y_{t}\equiv b_{\Delta}\sqrt{\mathbb{V}_{t} ( r_{T} ) }, $$

(C.36)

so that

$$ \tilde{B} ( r_{T} ) -K^{o} ( K ) =e^{-a_{\Delta }+b_{\Delta}\mathbb{E}_{t}^{Q_{F^{T}}} ( r_{T} ) +\frac {1}{2}y_{t}^{2}}\cdot \bigl( e^{y_{t}\tilde{N}_{T}-\frac {1}{2}y_{t}^{2}}-\hat{K}_{t}^{o} ( K ) \bigr) , $$

(C.37)

where

$$ \hat{K}_{t}^{o} ( K ) \equiv e^{a_{\Delta}-b_{\Delta}\mathbb {E}_{t}^{Q_{F^{T}}} ( r_{T} ) -\frac{1}{2}y_{t}^{2}}\cdot K^{o} ( K ) . $$

(C.38)

Conditionally upon the information at $t$, $\tilde{N}_{T}$ is standard normally distributed under the forward probability. Therefore, by the expression of $\tilde{B} ( r_{T} )$ in Eq. (C.35), and by Eq. (C.37),

$$ \mathrm{Pr}_{t}^{Q_{F^{T}}} \bigl( \tilde{B} ( r_{T} ) -K^{o} ( K ) \leq0 \bigr) =\varPhi ( \delta_{t} ) ,\quad \delta _{t}\equiv\frac{\ln\hat{K}_{t}^{o} ( K ) +\frac {1}{2}y_{t}^{2}}{y_{t}}. $$

(C.39)

Moreover, again by Eq. (C.35) and Eq. (C.37), and setting $\tilde{n}_{T}\equiv-\tilde{N}_{T}$,

$$\begin{aligned} &\mathbb{E}_{t}^{Q_{F^{T}}} \bigl( \tilde{B} ( r_{T} ) \mathbb {I}_{ \{ \tilde{B} ( r_{T} ) -K^{o} ( K ) \leq 0 \} } \bigr) \\ &\quad =e^{-a_{\Delta}+b_{\Delta}\mathbb {E}_{t}^{Q_{F^{T}}} ( r_{T} ) +\frac{1}{2}y_{t}^{2}}\cdot\mathbb{E}_{t}^{Q_{F^{T}}} \bigl( e^{y_{t}\tilde{N}_{T}-\frac{1}{2}y_{t}^{2}}\mathbb{I}_{ \{ \tilde {N}_{T}\leq\delta_{t} \} } \bigr) \\ &\quad \equiv e^{-a_{\Delta}+b_{\Delta}\mathbb{E}_{t}^{Q_{F^{T}}} ( r_{T} ) +\frac{1}{2}y_{t}^{2}}\cdot\mathbb{E}_{t}^{Q_{F^{T}}} \bigl( e^{-y_{t}\tilde{n}_{T}-\frac{1}{2}y_{t}^{2}}\mathbb{I}_{ \{ \tilde {n}_{T}\geq-\delta_{t} \} } \bigr) \\ &\quad =e^{-a_{\Delta}+b_{\Delta}\mathbb{E}_{t}^{Q_{F^{T}}} ( r_{T} ) +\frac{1}{2}y_{t}^{2}}\cdot\varPhi ( \delta _{t}-y_{t} ) . \end{aligned}$$

(C.40)

Substituting Eqs. (C.39) and (C.40) into Eq. (C.34) leaves

$$ \mathbb{E}_{t}^{Q_{F^{T}}} \bigl( K^{o} ( K ) - \tilde{B} ( r_{T} ) \bigr) ^{+}=K^{o} ( K ) \varPhi ( \delta _{t} ) -e^{-a_{\Delta}+b_{\Delta}\mathbb{E}_{t}^{Q_{F^{T}}} ( r_{T} ) +\frac{1}{2}y_{t}^{2}}\cdot\varPhi ( \delta _{t}-y_{t} ) . $$

Substituting this expression into Eq. (C.30) leaves

$$ \mathrm{Call}_{t}^{z} ( K,T,\Delta ) =100\times P_{t} ( T ) e^{-a_{\Delta}+b_{\Delta}\mathbb{E}_{t}^{Q_{F^{T}}} ( r_{T} ) +\frac{1}{2}y_{t}^{2}}\frac{1}{\Delta} \bigl( \hat {K}^{o} ( K ) \varPhi ( \delta_{t} ) -\varPhi ( \delta _{t}-y_{t} ) \bigr) . $$

Equation (4.85) in the main text follows by the definitions of (i) $y_{t}$ in the second of Eqs. (C.36), (ii) $\hat{K}_{t}^{o} ( K )$ in Eq. (C.38) and $K^{o} ( K )$ and, finally, by the property of the Forward LIBOR that

$$ K^{o} ( Z_{t} ) =1+ \biggl( 1-\frac{Z_{t}}{100} \biggr) \Delta =e^{-a_{\Delta}+b_{\Delta}\mathbb{E}_{t}^{Q_{F^{T}}} ( r_{T} ) +\frac{1}{2}y_{t}^{2}}, $$

a property that we show next.

Forward Libor. We determine the current forward price predicted by Vasicek’s model, $Z_{t} ( r_{t};T,T+\Delta )$ in Eq. (4.88), by using Eq. (4.69):

$$ Z_{t} ( r_{t};T,T+\Delta ) =100\times \bigl( 1-f_{t} ( r_{t};T,T+\Delta ) \bigr) , $$

where

$$\begin{aligned} f_{t} ( r_{t};T,T+\Delta ) & =\mathbb{E}_{t}^{Q_{F^{T}}} \bigl( l_{T} ( r_{T};\Delta ) \bigr) \\ & =\frac{1}{\Delta}\mathbb{E}_{t}^{Q_{F^{T}}} \biggl( \frac{1}{P_{T} ( r_{T};T+\Delta ) }-1 \biggr) \\ & =\frac{1}{\Delta} \bigl( \mathbb{E}_{t}^{Q_{F^{T}}} \bigl( e^{-a_{\Delta }+b_{\Delta}\cdot r_{T}} \bigr) -1 \bigr) \\ & =\frac{1}{\Delta} \bigl( e^{-a_{\Delta}+b_{\Delta}\cdot\mathbb {E}_{t}^{Q_{F^{T}}} ( r_{T} ) +\frac{1}{2}b_{\Delta}^{2}\cdot \mathbb{V}_{t} ( r_{T} ) }-1 \bigr) , \end{aligned}$$

and $a_{\Delta}$ and $b_{\Delta}$ are as in Eq. (C.29).

4.1.10 C.10 The Impact of Early Exercise Premiums and Maturity Mismatch

We implement numerical experiments to gauge the quality of the approximation of the interest rate volatility indexes in this chapter based on simplifying assumptions. We rely on the government bond case and illustrate the order of magnitude of these approximations. First, we analyze the impact of early exercise premiums and future corrections while calculating an index with American rather than European options (see Sect. 4.2.10). Second, we assess the impact of maturity mismatch arising when using European options and forwards that have different maturities (see Sect. 4.4).

Early exercise premiums. We convert values of American options on futures into values of European options on forwards based on the algorithm of Sect. 4.2.10 Step 2.a of the main text. We consider a coupon-bearing bond that pays off an annual coupon of $4, just as in Sect. 4.2.11, but take a maturity equal to 7 years. We assume that $\kappa=0.3807$, $\mu =0.01$, $\sigma=0.033107$ and $\lambda=-0.7$, and calculate hypothetical American option prices predicted by the Vasicek model, the benchmark in this exercise, while assuming that the initial value of the short-term rate is $r=0.01$. All options are taken to have a maturity equal to 1 month, the same as the future and the forward, and strikes that are equidistant at 50 points (10 strikes below and 10 strikes above the ATM), as in the hypothetical example of Table 4.3 in the main text, with the ATM strike set equal to the current value of the future.

As regards the weighting function $\omega$ in Eq. (4.61), we experimented with various functional forms in a Monte Carlo study described below, and found that the optimization problem in Eq. (4.62) performs accurately once we fix

$$ \omega ( K ) =\frac{1}{O^{\$} ( K ) }. $$

In the Monte Carlo experiment, we fix $( \kappa,\mu,\sigma )$ to equal the values indicated above, and calculate the price of American options and future as explained in the main text: through a Longstaff and Schwartz (2001) approximation of $O ( K;\lambda ) \equiv \mathcal{C}_{\tau} ( r_{\tau};K ) \vert _{\tau=t}$ (price of American options), where $\mathcal{C}_{\tau} ( r_{\tau };K )$ satisfies Eq. (4.60); and $\tilde{F}_{t} ( r_{t};T,\mathbb{T} )$ in Eq. (4.58) (price of future). For reference, the price of the forward is 95.557306, obtained through Eq. (4.68). This price is much lower than that in Table 4.3 in the main text (based on hypothetical market conditions on April 27, 2012). The reason is that we are using a risk-premium parameter, $\lambda$, which is as high as 0.7 in absolute value. This value implies that the medium-long end of the yield curve is much higher than that prevailing in late April 2012, when market conditions were such that the yield curve was substantially flat at almost zero.

Note that the optimization procedure requires the value of the short-term rate to be backed up from the price of the future through Eq. (4.58), which therefore changes over different trial values of $\lambda$ over the optimization of the criterion function in Eq. (4.62). (The seeds utilized to implement the Longstaff and Schwartz algorithm are the same over different trial values of $\lambda$.) Table 4.7 below reports key figures regarding the empirical distribution of the estimated risk premiums over 100 runs for an experiment in which $\lambda$ is estimated searching over a fine grid on $[ -1,0 ]$.

Table 4.7 Minimum, first quartile, median, mean, third quartile and maximum values of the estimated values of $\lambda$ over Monte Carlo experiments with 100 runs. Truth is $\lambda=-0.7$

Figure 4.10 depicts the government bond volatility index as a function of the risk-premium parameter $\lambda$, calculated using both (i) European options on forwards, which represents the (square of the) true, fair value of a government bond variance swap, and (ii) American options on futures. The American-option based government bond volatility index overstates the truth, due to the presence of an early exercise premium. The American-based approximation is about one relative percentage point higher than the truth when $\lambda=-0.7$, which is quantitatively at least as important as the forward price adjustments in the equity space as documented by the VIX white paper (see Chicago Board Options Exchange 2009).

Fig. 4.10

Government bond volatility index calculated through European options on forwards (solid line) and American options on futures (dashed line), as a function of the risk-premium parameter $\lambda$ in the Vasicek (1977) model. Forward looking horizon is one month

Maturity mismatch. We implement an exercise with the aim of assessing the severity of the maturity mismatch issue described in Sect. 4.4. Recall that under the $T$-forward probability, the forward price on a coupon-bearing bond, $F_{\tau} ( S,\mathbb{T} )$, satisfies Eq. (4.130), and that by Eq. (4.134), the fair value of a government bond variance swap is

$$\begin{aligned} &\mathbb{P} ( t,T,S,\mathbb{T} ) \\ &\quad =2 \bigl( 1-\mathbb {E}_{t}^{Q_{F^{T}}} \bigl( e^{\tilde{\ell} ( t,T,S,\mathbb{T} ) }-\tilde{\ell} ( t,T,S,\mathbb{T} ) \bigr) \bigr) \\ &\qquad{} +2 \biggl( \int_{0}^{F_{t} ( S,\mathbb{T} ) }\frac {\mathrm{Put}_{t} ( K,T,S,\mathbb{T} ) }{P_{t} ( T ) } \frac{1}{K^{2}}dK+\int_{F_{t} ( S,\mathbb{T} ) }^{\infty} \frac{\mathrm{Call}_{t} ( K,T,S,\mathbb{T} ) }{P_{t} ( T ) }\frac{1}{K^{2}}dK \biggr) , \end{aligned}$$

(C.41)

where $\tilde{\ell} ( t,T,S,\mathbb{T} )$ is as in Eq. (4.132), and the remaining notation is obvious.

We assume that $T-t=\frac{1}{12}$ and $S-t=\frac{3}{12}$, and take the Vasicek (1977) model as the benchmark. By tedious but straightforward calculations, we find that the value of $\tilde{\ell} ( t,T,S,\mathbb{T} )$ predicted by this model is

$$ \tilde{\ell} ( t,T,S,\mathbb{T} ) =\sigma^{2}\int_{t}^{T} \sum_{i=1}^{N} \omega_{\tau}^{i} ( S,\mathbb{T} ) \varPsi_{\tau} ( T,S,T_{i} ) d\tau, $$

where

$$\begin{aligned} &\omega_{\tau}^{i} ( S,\mathbb{T} ) \equiv\bar{C}_{i} \frac {F_{\tau}^{z} ( S,T_{i} ) }{F_{\tau} ( S,\mathbb{T} ) } \\ &\quad F_{\tau}^{z} ( S,T_{i} ) \equiv \frac{P_{\tau} ( r_{\tau },T_{i} ) }{P_{\tau} ( r_{\tau},S ) } \qquad F_{\tau} ( S,\mathbb{T} ) \equiv\sum _{i=1}^{N}\frac{C_{i}}{n}F_{\tau }^{z} ( S,T_{i} ) +F_{\tau}^{z} ( S,\mathbb{T} ) ,\quad \mathbb{T}\equiv T_{N} \\ &\varPsi_{\tau} ( T,S,T_{i} ) \equiv \bigl( \varphi_{1\tau } ( T_{i} ) -\varphi_{1\tau} ( S ) \bigr) \bigl( \varphi _{1\tau} ( T ) -\varphi_{1\tau} ( S ) \bigr) ,\\ & \varphi_{1\tau} ( \mathcal{Y} ) \equiv-\frac{1-e^{-\kappa (\mathcal{Y}-\tau ) }}{\kappa},\quad\text{for a generic }\mathcal{Y} \end{aligned}$$

and $P_{\tau} ( r_{\tau},T_{i} )$ is the price of a zero-coupon bond predicted by Vasicek’s formula (see Eq. (4.64) in the main text).

To calculate the first term of the R.H.S. of Eq. (C.41), we need to take the expectation of $e^{\tilde{\ell} ( t,T,S,\mathbb{T} ) }-\tilde{\ell} ( t,T,S,\mathbb{T} )$ under the forward probability. This is accomplished through Monte Carlo simulations, in which the short-term rate is as in Eq. (C.31), viz

$$ dr_{\tau}=\kappa \biggl( \bar{r}-r_{\tau}-\sigma^{2} \frac{1-e^{-\kappa ( T-\tau ) }}{\kappa} \biggr) d\tau+\sigma dW_{\tau }^{F^{T}},\quad \tau\in [ t,T ] , $$

(C.42)

where $W_{\tau}^{F^{T}}$ is a Brownian motion under the $T$-forward probability, and the time-varying term inside the drift captures the tilt caused by the change from the risk-neutral to the forward probability.

To calculate the second term of the R.H.S. of Eq. (C.41), we need to calculate the price of 1-month European options written on 3-month forwards. The latter could be determined by generalizing the Jamshidian (1989) formula, Eqs. (4.66) and (4.67) in the main text. However, these prices can be calculated quite rapidly as a by-product of the previous Monte Carlo simulations by approximating the following expectations:

$$ \begin{aligned} &\frac{\mathrm{Put}_{t} ( K,T,S,\mathbb{T} ) }{P_{t} ( T ) }=\mathbb{E}_{t}^{Q_{F^{T}}} \bigl( \bigl( K-F_{T} ( S,\mathbb{T} ) \bigr) ^{+} \bigr) ,\\ &\frac{\mathrm{Call}_{t} ( K,T,S,\mathbb{T} ) }{P_{t} ( T ) }=\mathbb {E}_{t}^{Q_{F^{T}}} \bigl( \bigl( F_{T} ( S,\mathbb{T} ) -K \bigr) ^{+} \bigr) . \end{aligned} $$

(C.43)

We proceed as follows:

• Simulate values of $r_{\tau}$ from Eq. (C.42), which are used to calculate:

  • $F_{\tau}^{z} ( S,T_{i} )$ and $F_{\tau} ( S,\mathbb{T} )$ for $\tau\in [ t,T )$, needed to calculate the first term of the R.H.S. of Eq. (C.41);

  • $F_{T} ( S,\mathbb{T} )$, needed to calculate Eqs. (C.43) and, hence, the second term of the R.H.S. of Eq. (C.41).

• Calculate the two volatility indexes:

  $$ \begin{aligned} &\mathrm{GB}\text{-}\mathrm{VI}_{\mathrm{Mc}} ( t,T,S,\mathbb{T} ) \equiv\sqrt{\frac{1}{T-t}\mathbb{P}_{\mathrm {Mc}} ( t,T,S,\mathbb{T} ) } \quad\text{and}\\ &\mathrm {GB}\text{-}\mathrm{VI}_{\mathrm{Mc}}^{\ast} ( t,T,S,\mathbb{T} ) \equiv\sqrt{\frac{1}{T-t}\mathbb{P}_{\mathrm {Mc}}^{\ast} ( t,T,S,\mathbb{T} ) }, \end{aligned} $$

  (C.44)

  where

  $$\begin{aligned} \mathbb{P}_{\mathrm{Mc}}^{\ast} ( t,T,S,\mathbb{T} ) &\equiv 2 \biggl( \int_{0}^{F_{t} ( S,\mathbb{T} ) }\frac {\mathrm{Put}_{\mathrm{Mc,}t} ( K,T,S,\mathbb{T} ) }{P_{t} ( T ) } \frac{1}{K^{2}}dK\\ &\quad{}+\int_{F_{t} ( S,\mathbb {T} ) }^{\infty} \frac{\mathrm{Call}_{\mathrm{Mc,}t} ( K,T,S,\mathbb{T} ) }{P_{t} ( T ) }\frac{1}{K^{2}}dK \biggr) , \end{aligned}$$

  and $\mathbb{P}_{\mathrm{Mc}} ( t,T,S,\mathbb{T} )$ is the Monte Carlo estimate of $\mathbb{P} ( t,T,S,\mathbb{T} )$ in Eq. (C.41), and $\mathrm{Put}_{\mathrm{Mc,}t} ( K,T,S,\mathbb{T} )$ and $\mathrm{Call}_{\mathrm{Mc,}t} ( K,T,S,\mathbb{T} )$ are the Monte Carlo estimates of $\mathrm{Put}_{t} ( K,T,S,\mathbb {T} )$ and $\mathrm{Call}_{t} ( K,T,S,\mathbb{T} )$ in Eq. (C.43).

Table 4.8 compares the value of the two indexes in Eqs. (C.44) based on the parameter values underlying the experiments in Figure 4.10, and for various levels of the risk-premium parameter, $\lambda$. The two indexes are substantially the same in this experiment.

Table 4.8 Volatility indexes and risk-premium assumptions