Credit

Abstract

This chapter builds on the general framework of Chap. 2, and develops indexes of expected volatility in the credit market. An inherent risk in this market that is absent from all others studied in this book is default risk, and this chapter describes how to account for it when pricing credit variance while also handling the exotic nature of options on credit default swap indexes. For example, options to enter into an on-the-run credit default swap index are typically struck at spreads differing from the initial contractual coupons, which calls for strike adjustments that lead to non-standard payoffs. This chapter proposes modifications to the volatility index calculations based on vanilla payoffs that adjust for the non-standard nature of options traded in practice.

Appendix D: Appendix on Credit Markets

5.1.1 D.1 Preliminary Facts Concerning CDS Indexes

Accruals of premiums. We consider an extension of Eq. (5.1), which takes into account accruals of premiums occurring over the reset times of the index. To develop intuition, assume, initially, that the short-term rate $r=0$, so that the premium made available to the protection seller over a generic time interval $[ T_{i-1},T_{i} ]$ is

$$ \frac{1}{b}C_{T}\cdot Z_{T_{i}}, $$

where

$$\begin{aligned} Z_{T_{i}}& \equiv\frac{1}{n}\mathcal{S}_{T_{i-1}} \mathbb{I}_{\mathrm{no}\text{-}\mathrm{defaults}\ \mathrm{in}\ [ T_{i-1},T_{i} ] } \\ &\quad{} +\sum _{j=1}^{n} \biggl( \frac{\tau _{j}-T_{i-1}}{T_{i}-T_{i-1}} \frac{1}{n}\mathcal{S}_{T_{i-1}}+\frac {T_{i}-\tau_{j}}{T_{i}-T_{i-1}}\frac{1}{n} ( \mathcal {S}_{T_{i-1}}-\mathbb{I}_{\tau_{j}\in [ T_{i-1},T_{i} ] } ) \biggr) \mathbb{I}_{\tau_{j}\in [ T_{i-1},T_{i} ] }, \end{aligned}$$

(D.1)

and $\mathbb{I}_{\mathrm{no}\text{-}\mathrm{defaults}\ \mathrm{in}\ [ T_{i-1},T_{i} ] }$ is the indicator of the event that no defaults have occurred over the time interval $[ T_{i-1},T_{i} ]$. Over this interval, either no defaults occur at all or at least one name defaults. If there are no defaults, the protection seller receives $\frac{1}{b}C_{T}$ times the first term on the R.H.S. of Eq. (D.1). If there is at least one default, the protection seller receives $\frac{1}{b}C_{T}$ times the second term on the R.H.S. of Eq. (D.1). The latter is the sum of two components: (i) the premium accrued from $T_{i-1}$ to the time of default $\tau_{j}$ of the $j$-th name, calculated upon the notional value at time $T_{i-1}$, $\frac{1}{n}\mathcal{S}_{T_{i-1}}$, and (ii) the premium calculated upon the outstanding nominal at time $T_{i}$, for the time remaining from $\tau_{j}$ to $T_{i}$.

In the case of stochastic interest rates, define the random present value of $Z_{T_{i}}$

$$\begin{aligned} \hat{Z}_{t} ( T_{i} ) &\equiv e^{-\int _{t}^{T_{i}}r_{\tau}d\tau} \frac{1}{n}\mathcal {S}_{T_{i}}\mathbb{I}_{\mathrm{no}\text{-}\mathrm{defaults}\ \mathrm{in}\ [ T_{i-1},T_{i} ] } \\ &\quad{} +\sum _{j=1}^{n} \biggl( e^{-\int_{t}^{\tau _{j}}r_{\tau }d\tau} \frac{\tau_{j}-T_{i-1}}{T_{i}-T_{i-1}}\frac{1}{n}\mathcal {S}_{T_{i-1}}\\ &\quad{}+e^{-\int_{t}^{T_{i}}r_{\tau}d\tau} \frac {T_{i}-\tau _{j}}{T_{i}-T_{i-1}}\frac{1}{n} ( \mathcal{S}_{T_{i-1}}-\mathbb {I}_{\tau _{j}\in [ T_{i-1},T_{i} ] } ) \biggr) \mathbb{I}_{\tau _{j}\in [ T_{i-1},T_{i} ] }, \end{aligned}$$

so that the value of the premium leg is

$$ \frac{1}{b}C_{t}\cdot v_{1t},\quad v_{1t}=\sum _{i=1}^{bM} \mathbb{E}_{t} \bigl( \hat{Z}_{t} ( T_{i} ) \bigr) . $$

Forward position in CDX indexes. We prove Eq. (5.3). Conditional upon the information set at time $\tau\leq T$, we have that for $i=1,\ldots,bM$,

$$\begin{aligned} & \mathrm{LGD}\frac{1}{n}\mathbb{E}_{\tau} \Biggl( \sum _{j=1}^{n}e^{-\int_{\tau}^{\tau_{j}}r_{s}ds}\mathbb {I}_{ \{ \mathrm{Surv}_{j}\ \mathrm{at}\ \tau \} } \mathbb{I}_{ \{ T\leq\tau_{j}\leq T_{bM} \} } \Biggr) \\ &\quad =\mathrm{LGD}\frac{1}{n}\sum _{j=1}^{n} \mathbb{I}_{ \{ \mathrm{Surv}_{j}\ \mathrm{at}\ \tau \} }\mathbb {E}_{\tau} \bigl( e^{-\int_{\tau}^{\tau_{j}}r_{s}ds} \mathbb{I}_{ \{ T\leq \tau_{j}\leq T_{bM} \} } \bigr) \\ &\quad=\mathrm{LGD}\cdot\mathcal {N}_{\tau} \mathbb{E}_{\tau} \bigl( e^{-\int_{\tau}^{\tau _{\ast}}r_{s}ds}\mathbb{I}_{ \{ T\leq\tau_{\ast}\leq T_{bM} \} } \bigr) , \end{aligned}$$

where the last equality follows by (i) the definition of the outstanding notional value in Eq. (5.4), and (ii) the assumption that all the names in the index have the same credit quality. The previous expression is the first term in Eq. (5.3).

The second term in Eq. (5.3) follows by the following equalities:

$$\begin{aligned} & \mathbb{E}_{\tau} \bigl( e^{-\int_{\tau }^{T_{i}}r_{s}ds}\cdot \mathbb{I}_{ \{ \mathrm{Surv}_{j}\ \mathrm{at}\ T_{i} \} } \bigr) \\ &\quad =\mathbb{E}_{\tau} \bigl( e^{-\int_{\tau }^{T_{i}}r_{s}ds}\cdot \mathbb{I}_{ \{ \mathrm{Surv}_{j}\ \mathrm{at}\ \tau \} } \mathbb{I}_{ \{ \mathrm{Surv}_{j}\ \mathrm {at}\ T_{i}\vert \mathrm{Surv}_{j}\ \mathrm{at}\ \tau \} } \bigr) \\ &\quad=\mathbb{I}_{ \{ \mathrm{Surv}_{j}\ \mathrm{at}\ \tau \} }\mathbb{E}_{\tau} \bigl( e^{-\int_{\tau }^{T_{i}}r_{s}ds}\mathbb{I}_{ \{ \mathrm{Surv}_{j}\ \mathrm{at}\ T_{i}\vert \mathrm{Surv}_{j}\ \mathrm {at}\ \tau \} } \bigr) , \end{aligned}$$

and summing over the reset dates and all names $j$, using the definition of the outstanding notional value in Eq. (5.4), and noting, again, that credit risk is the same for all names.

5.1.2 D.2 Spanning Credit Variance Contracts

Pricing. We derive pricing results by relying on the general framework of Chap. 2. The derivations in this appendix are made to help the reader become acquainted with the specific details arising while pricing credit variance and make the chapter self-contained. To alleviate the notation, we set $\mathrm{CDX}_{\tau}\equiv\mathrm{CDX}_{\tau} ( M )$, although we shall keep on emphasizing the dependence on $M$ of other objects of interest. First, we deal with percentage contracts, and then with basis point contracts.

As for the percentage contracts of Sect. 5.3.1, we have, by the usual Taylor expansion with remainder,

$$\begin{aligned} \ln\frac{\mathrm{CDX}_{T}}{\mathrm{CDX}_{t}}&=\frac{1}{\mathrm{CDX}_{t}} ( \mathrm{CDX}_{T}- \mathrm{CDX}_{t} ) -\int _{0}^{\mathrm{CDX}_{t}} ( K- \mathrm{CDX}_{T} ) ^{+}\frac{1}{K^{2}}dK \\ &\quad{}-\int _{\mathrm{CDX}_{t}}^{\infty} ( \mathrm{CDX}_{T}-K ) ^{+}\frac{1}{K^{2}}dK. \end{aligned}$$

(D.2)

Multiplying both sides of the previous equation by $\frac{1}{b}v_{1t}$, taking expectations under $Q_{\mathrm{sc}}$, and using Eq. (5.9),

$$ v_{1t}\mathbb{E}_{t}^{\mathrm{sc}} \biggl( \ln \frac{\mathrm {CDX}_{T}}{\mathrm{CDX}_{t}} \biggr) =-\int_{0}^{\mathrm {CDX}_{t}} \frac{\mathrm{SW}_{t}^{r} ( K,T;M ) }{K^{2}}dK-\int _{\mathrm{CDX}_{t}}^{\infty} \frac{\mathrm{SW}_{t}^{p} ( K,T;M ) }{K^{2}}dK. $$

(D.3)

Moreover, applying Itô’s lemma to Eq. (5.10) leaves

$$\begin{aligned} d\ln\mathrm{CDX}_{\tau}& =- \bigl( \mathbb{E}_{\tau}^{\mathrm {sc}} \bigl( e^{j_{\tau} ( M ) }-1 \bigr) \eta_{\tau} \bigr) d\tau- \frac {1}{2}\bigl\Vert \sigma_{\tau} ( M ) \bigr\Vert ^{2}d\tau\\ &\quad{} +\sigma _{\tau} ( M ) \cdot dW_{\tau}^{\mathrm{sc}}+j_{\tau} ( M ) dJ_{\tau}^{\mathrm{sc}} \\ & =-\frac{1}{2} \bigl( \bigl\Vert \sigma_{\tau} ( M ) \bigr\Vert ^{2}d\tau+j_{\tau}^{2} ( M ) dJ_{\tau}^{\mathrm {sc}} \bigr) +\sigma_{\tau} ( M ) \cdot dW_{\tau}^{\mathrm {sc}} \\ &\quad{} - \bigl( \mathbb{E}_{\tau}^{\mathrm{sc}} \bigl( e^{j_{\tau } ( M ) }-1 \bigr) \eta_{\tau} \bigr) d\tau+j_{\tau} ( M ) dJ_{\tau}^{\mathrm{sc}}+\frac{1}{2}j_{\tau}^{2} ( M ) dJ_{\tau}^{\mathrm{sc}}. \end{aligned}$$

We have

$$ \mathbb{E}_{\tau}^{\mathrm{sc}} \bigl( e^{j_{\tau} ( M ) }-1 \bigr) dJ_{\tau}^{\mathrm{sc}}= \bigl( \mathbb{E}_{\tau}^{\mathrm {sc}} \bigl( e^{j_{\tau} ( M ) }-1 \bigr) \eta_{\tau} \bigr) d\tau. $$

(D.4)

Therefore, using the definition of $V_{M} ( t,T )$ in Eq. (5.11), and Eq. (D.4), leaves

$$\begin{aligned} & {-}2\mathbb{E}_{t}^{\mathrm{sc}} \biggl( \ln\frac{\mathrm {CDX}_{T}}{\mathrm{CDX}_{t}} \biggr) -2\mathbb{E}_{t}^{\mathrm{sc}} \biggl[ \int _{t}^{T} \biggl( e^{j_{\tau} ( M ) }-1-j_{\tau} ( M ) -\frac{1}{2}j_{\tau}^{2} ( M ) \biggr) dJ_{\tau }^{\mathrm{sc}} \biggr] \\ &\quad =\mathbb{E}_{t}^{\mathrm{sc}} \bigl[ V_{M} ( t,T ) \bigr] \\ &\quad =\frac{1}{v_{1t}}\mathbb{E}_{t} \bigl( e^{-\int _{t}^{T}r_{s}ds} \mathcal{N}_{T}v_{1T}V_{M} ( t,T ) \bigr) \\ &\quad =\frac{\mathbb{F}_{\mathrm{var},M} ( t,T ) }{v_{1t}}, \end{aligned}$$

(D.5)

with $\mathbb{E}_{t}$ denoting the risk-neutral expectation conditional on information available at $t$, and where the second equality follows by a change in probability, and the third follows by the definition of the credit variance forward agreement (Definition 5.1). The approximation of $\mathbb{F}_{\mathrm{var},M} ( t,T )$ in Eq. (5.15) follows by comparing Eq. (D.3) and Eq. (D.5), and disregarding the second term on the L.H.S. of Eq. (D.5), which is of order $O ( ( \frac{d\mathrm{CDX}_{t}}{\mathrm {CDX}_{t}} ) ^{3} )$, as first pointed out by Carr and Wu (2008) in the context of the derivation of an equity VIX. Note that Jiang and Tian (2005) also derive the equity VIX in a jump-diffusion model, proposing approximations similar to ours.

Note that the exact formula prevailing in the presence of jumps is

$$\begin{aligned} \mathbb{F}_{\mathrm{var},M} ( t,T ) & =2 \biggl( \int _{0}^{\mathrm{CDX}_{t}}\frac{\mathrm{SW}_{t}^{r} ( K,T;M ) }{K^{2}}dK+\int_{\mathrm{CDX}_{t}}^{\infty} \frac{\mathrm {SW}_{t}^{p} ( K,T;M ) }{K^{2}}dK \biggr) \\ &\quad{} -2v_{1t}\mathbb{E}_{t}^{\mathrm{sc}} \biggl[ \int _{t}^{T} \biggl( e^{j_{\tau} ( M ) }-1-j_{\tau} ( M ) -\frac {1}{2}j_{\tau }^{2} ( M ) \biggr) dJ_{\tau}^{\mathrm{sc}} \biggr] . \end{aligned}$$

(D.6)

Next, we determine the credit variance swap rate of Definition 5.2. We have

$$ 0=\mathbb{E}_{t} \bigl[ e^{-\int_{t}^{T}r_{s}ds} \bigl( V_{M} ( t,T ) \times ( \mathcal{N}_{T}v_{1T} ) -\mathbb {P}_{\mathrm{var},M} ( t,T ) \bigr) \bigr] , $$

or

$$ P_{t} ( T ) \mathbb{P}_{\mathrm{var},M} ( t,T ) = \mathbb{E}_{t} \bigl( e^{-\int_{t}^{T}r_{s}ds}V_{M} ( t,T ) \times \mathcal{N}_{T}v_{1T} \bigr) =\mathbb{F}_{\mathrm{var},M} ( t,T ) , $$

(D.7)

where the last equality holds by the definition of the credit variance forward agreement (Definition 5.1). Equation (5.16) immediately follows. Finally, we derive the standardized credit variance swap rate of Definition 5.3. We have that

$$ 0=\mathbb{E}_{t} \bigl[ e^{-\int_{t}^{T}r_{s}ds} \bigl( V_{M} ( t,T ) -\mathbb{P}_{\mathrm{var},M}^{\ast} ( t,T ) \bigr) \times \mathcal{N}_{T}v_{1T} \bigr] , $$

or, using the definition of the Radon–Nikodym derivative in Eq. (5.8),

$$ \mathbb{P}_{\mathrm{var},M}^{\ast} ( t,T ) v_{1t}=\mathbb {E}_{t} \bigl( e^{-\int_{t}^{T}r_{s}ds}V_{M} ( t,T ) \times \mathcal{N} ( T ) v_{1T} \bigr) =\mathbb{F}_{\mathrm {var},M} ( t,T ) , $$

(D.8)

where the second equality follows, again, by Definition 5.1. The expression of $\mathbb{P}_{\mathrm{var},M}^{\ast} ( t,T )$ in Eq. (5.17) follows immediately.

Next, we proceed with the pricing of basis point contracts introduced in Sect. 5.3.2. By a Taylor’s expansion with remainder, we have

$$\begin{aligned} \mathrm{CDX}_{T}^{2}& =\mathrm{CDX}_{t}^{2}+2 \mathrm{CDX}_{t} ( \mathrm{CDX}_{T}-\mathrm{CDX}_{t} ) \\ &\quad{} +2 \biggl( \int_{0}^{\mathrm{CDX}_{t}} ( K-\mathrm {CDX}_{T} ) ^{+}dK+\int_{\mathrm{CDX}_{t}}^{\infty} ( \mathrm{CDX}_{T}-K ) ^{+}dK \biggr) . \end{aligned}$$

(D.9)

Multiplying both sides of the previous equation by $v_{1t}$, and taking expectations under $Q_{\mathrm{sc}}$, leaves

$$\begin{aligned} &v_{1t}\mathbb{E}_{t}^{\mathrm{sc}} \bigl( \mathrm{CDX}_{T}^{2}-\mathrm {CDX}_{t}^{2} \bigr) \\ &\quad =2 \biggl( \int_{0}^{\mathrm {CDX}_{t}} \mathrm{SW}_{t}^{r} ( K,T;M ) dK+\int _{\mathrm{CDX}_{t}}^{\infty}\mathrm{SW}_{t}^{p} ( K,T;M ) dK \biggr) . \end{aligned}$$

(D.10)

Next, note that by Itô’s lemma for jump-diffusion processes,

$$\begin{aligned} \frac{d\mathrm{CDX}_{\tau}^{2}}{\mathrm{CDX}_{\tau}^{2}}& =-2 \bigl( \mathbb{E}_{\tau}^{\mathrm{sc}} \bigl( e^{j_{\tau} ( M ) }-1 \bigr) \eta_{\tau} \bigr) d\tau+2\sigma_{\tau} ( M ) \cdot dW_{\tau}^{\mathrm{sc}} \\ &\quad{}+\bigl\Vert \sigma_{\tau} ( M ) \bigr\Vert ^{2}d\tau+ \bigl( e^{2j_{\tau} ( M ) }-1 \bigr) dJ_{\tau}^{\mathrm{sc}} \\ & =-2 \bigl( \mathbb{E}_{\tau}^{\mathrm{sc}} \bigl( e^{j_{\tau} ( M ) }-1 \bigr) \eta_{\tau} \bigr) d\tau+2 \bigl( e^{j_{\tau } ( M ) }-1 \bigr) dJ_{\tau}^{\mathrm{sc}}+2\sigma_{\tau} ( M ) \cdot dW_{\tau}^{\mathrm{sc}} \\ &\quad{} +\bigl\Vert \sigma_{\tau} ( M ) \bigr\Vert ^{2}d \tau + \bigl( e^{j_{\tau} ( M ) }-1 \bigr) ^{2}dJ_{\tau}^{\mathrm{sc}}, \end{aligned}$$

(D.11)

where the second equality follows by rearranging terms. Note that the first three terms of the second equality in Eq. (D.11) form a martingale under $Q_{\mathrm{sc}}$, due to Eq. (D.4) and the fact that $W_{\tau}^{\mathrm{sc}}$ is obviously a martingale under $Q_{\mathrm{sc}}$. Then, by integrating, taking expectations under $Q_{\mathrm{sc}}$, and using the definition of basis point variance, $V_{M} ( t,T )$ in Eq. (5.12), leaves

$$ \mathbb{E}_{t}^{\mathrm{sc}} \bigl( \mathrm{CDX}_{T}^{2}- \mathrm {CDX}_{t}^{2} \bigr) =\mathbb{E}_{t}^{\mathrm{sc}} \bigl[V_{M}^{\mathrm {bp}} ( t,T ) \bigr]. $$

(D.12)

Substituting Eq. (D.12) into Eq. (D.10) yields

$$\begin{aligned} & 2 \biggl( \int_{0}^{\mathrm{CDX}_{t}}\mathrm{SW}_{t}^{r} ( K,T;M ) dK+\int_{\mathrm{CDX}_{t}}^{\infty}\mathrm {SW}_{t}^{p} ( K,T;M ) dK \biggr) \\ &\quad =v_{1t}\mathbb{E}_{t}^{\mathrm{sc}} \bigl( \mathrm{CDX}_{T}^{2}-\mathrm {CDX}_{t}^{2} \bigr) \\ &\quad =v_{1t}\mathbb{E}_{t}^{\mathrm{sc}} \bigl[V_{M}^{\mathrm{bp}} ( t,T ) \bigr] \\ &\quad =\mathbb{E}_{t} \bigl[ e^{-\int_{t}^{T}r_{s}ds} ( \mathcal{N}_{T}V_{1T} ) \times V_{M}^{\mathrm{bp}} ( t,T ) \bigr] \\ &\quad =\mathbb{F}_{\mathrm{var},M}^{\mathrm{bp}} ( t,T ) , \end{aligned}$$

where the third equality follows by a change of probability and the last by the definition of the credit variance forward agreement (Definition 5.1).

Finally, the expression for the basis point variance swap rates in Eqs. (5.19) follows by arguments nearly identical to those leading to Eq. (D.7) and Eq. (D.8).

Marking to Market. To derive the update in Eq. (5.20), we evaluate the risk-neutral expectation of the discounted payoff $\mathrm{Var}\text{-}\mathrm{Swap}_{M} ( t,T )$ in Eq. (5.13), for any given $\tau\in [ t,T ]$,

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

(D.13)

where $\mathbb{E}_{\tau}$ is the expectation under $Q$, conditional on the information up to time $\tau$. The second equality follows by the definition of the Radon–Nikodym derivative in Eq. (5.8) and the definition of the credit variance forward agreement. Substituting the expression for $\mathbb{P}_{\mathrm{var},n} ( t,T )$ in Eq. (5.16) into Eq. (D.13) gives Eq. (5.20).

Next, we derive Eq. (5.21) by taking the risk-neutral expectation of the discounted payoff, $\mathrm{Var}\text{-}\mathrm{Swap}_{M}^{\ast} ( t,T )$ in Eq. (5.14),

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

where the second equality follows by Eq. (D.8) and the definition of the Radon–Nikodym derivative in Eq. (5.8).

5.1.3 D.3 Hedging

We first derive the portfolio positions in Table 5.1 regarding the replication of the contracts in Definitions 5.1 and 5.2. The additional arguments leading to Table 5.2 are in the main text. If $\mathrm {CDX}_{\tau } ( M )$ in Eq. (5.6) were actually traded and not subject to jumps, we could perfectly hedge these contracts by the following arguments. Itô’s lemma gives us

$$ \mathcal{N}_{T}v_{1T}V_{M} ( t,T ) =2 \mathcal{N}_{T}v_{1T}\int _{t}^{T} \frac{d\mathrm{CDX}_{s}}{\mathrm{CDX}_{s}}-2\mathcal {N}_{T}v_{1T} \biggl( \ln \frac{\mathrm{CDX}_{T}}{\mathrm{CDX}_{t}} \biggr) . $$

(D.14)

By Eq. (D.2), the second term of Eq. (D.14) is the time $T$ payoff of a static portfolio created at $t$, and corresponding to row (ii) of Table 5.1, which is two times: (a) short $1/\mathrm {CDX}_{t}$ units a forward of a FS-CDX, struck at $\mathrm{CDX}_{t}$, and (b) long a continuum of out-of-the-money CDS index options with weights $K^{-2}dK$. The value of this portfolio is $\mathbb{F}_{\mathrm {var},M} ( t,T )$, due to Eqs. (D.2), (D.3), and (5.15), which we finance by borrowing $\mathbb{F}_{\mathrm{var},M} ( t,T )$ at time $t$, in turn repaid at time $T$, as in row (iii) of Table 5.1.

As for the self-financing portfolio of row (i) in Table 5.1, we want it to be worthless at time $t$, and to replicate the first term of Eq. (D.14). Because we are assuming that $\mathrm{CDX}_{t}$ is traded, we can consider a self-financed strategy that is long (a) the FS-CDX, and (b) a money market account, with value $M_{\tau}$ satisfying $dM_{\tau }=r_{\tau }M_{\tau}d\tau$. The strategy at time $\tau$ is worth: $\upsilon _{\tau }=\theta_{\tau}\mathrm{CDX}_{\tau}+\psi_{\tau}M_{\tau}$, where $\theta _{\tau}$ are the units invested in the FS-CDX, and $\psi_{\tau}$ are the units invested in the money market account. Let

$$ \hat{\theta}_{\tau}\mathrm{CDX}_{\tau}=\mathcal{N}_{\tau}v_{1\tau },\qquad \hat{\psi}_{\tau}M_{\tau}=\mathcal{N}_{\tau}v_{1\tau} \biggl( \int _{t}^{\tau}\frac{d\mathrm{CDX}_{s}}{\mathrm{CDX}_{s}}-1 \biggr) , $$

(D.15)

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

$$ \hat{\upsilon}_{\tau}=\mathcal{N}_{\tau}v_{1\tau}\int _{t}^{\tau }\frac{d\mathrm{CDX}_{s}}{\mathrm{CDX}_{s}}, $$

(D.16)

and

$$ \hat{\upsilon}_{t}=0,\quad\text{and}\quad\hat{\upsilon}_{T}= \mathcal {N}_{T}v_{1T}\int_{t}^{T} \frac{d\mathrm{CDX}_{s}}{\mathrm{CDX}_{s}}. $$

Therefore, we can replicate the first term on the R.H.S. of Eq. (D.14) (and, then, by the previous results, the R.H.S. of Eq. (D.14)) through a long position in the two portfolios $(\hat {\theta}_{\tau},\hat{\psi}_{\tau})$, provided $(\hat{\theta}_{\tau },\hat{\psi}_{\tau})$ is self-financed. To check that $(\hat{\theta }_{\tau},\hat{\psi}_{\tau})$ is self-financed, note that

$$\begin{aligned} d\hat{\upsilon}_{\tau}& =\hat{\theta}_{\tau} \mathrm{CDX}_{\tau}\frac {d\mathrm{CDX}_{\tau}}{\mathrm{CDX}_{\tau}}+\hat{\psi}_{\tau}M_{\tau } \frac{dM_{\tau}}{M_{\tau}} \\ & =\mathcal{N}_{\tau}v_{1\tau} \biggl( \frac{d\mathrm{CDX}_{\tau }}{\mathrm{CDX}_{\tau}}-r_{\tau}d \tau \biggr) + \biggl( \mathcal {N}_{\tau}v_{1\tau }\int _{t}^{\tau}\frac{d\mathrm{CDX}_{s}}{\mathrm {CDX}_{s}} \biggr) r_{\tau}d\tau \\ & =\mathcal{N}_{\tau}v_{1\tau} \biggl( \frac{d\mathrm{CDX}_{\tau }}{\mathrm{CDX}_{\tau}}-r_{\tau}d \tau \biggr) +r_{\tau}\hat{\upsilon }_{\tau}d\tau \\ & =\hat{\theta}_{\tau}\mathrm{CDX}_{\tau} \biggl( \frac{d\mathrm {CDX}_{\tau}}{\mathrm{CDX}_{\tau}}-r_{\tau}d\tau \biggr) +r_{\tau }\hat{ \upsilon}_{\tau}d\tau, \end{aligned}$$

(D.17)

where the second line follows by Eqs. (D.15), the third by Eq. (D.16), and the fourth, again, by Eq. (D.15). The dynamics of $\hat{\upsilon}_{\tau}$ in Eq. (D.17) are indeed those of a self-financed strategy.

Next, we derive the portfolio positions in Table 5.3, which leads to the replication of the contracts in Definitions 5.4-(a) and 5.4-(b), with the additional arguments concerning Table 5.4 found in the main text.

Itô’s lemma gives us

$$ \mathcal{N}_{T}v_{1T}V_{M}^{\mathrm{bp}} ( t,T ) =-2\mathcal {N}_{T}v_{1T}\int_{t}^{T} \mathrm{CDX}_{s}d\mathrm {CDX}_{s}+\mathcal{N}_{T}v_{1T} \bigl( \mathrm{CDX}_{T}^{2}-\mathrm {CDX}_{t}^{2} \bigr) . $$

(D.18)

By Eq. (D.9), the second term on the R.H.S. of Eq. (D.18) is the payoff at $T$ of a portfolio set up at $t$, which is: (a) long $2\mathrm{CDX}_{t}$ units a FS-CDX struck at $\mathrm{CDX}_{t}$, and (b) long a continuum of out-of-the-money CDS index options with weights $2dK$. It is the static position (ii) in Table 5.3 of the main text. By Eq. (5.18), its cost is $\mathbb{F}_{\mathrm{var},M}^{\mathrm {bp}} ( t,T )$, which we borrow at $t$, to repay it back at $T$, as in row (iii) of Table 5.3. The self-financed portfolio to be shorted, as indicated by row (i) of Table 5.3, is obtained similarly as the portfolio in row (i) of Table 5.1, but with the portfolio

$$ \hat{\psi}_{\tau}M_{\tau}=\mathcal{N}_{\tau}v_{1\tau} \biggl( \int _{t}^{\tau}2\mathrm{CDX}_{s}d \mathrm{CDX}_{s}-1 \biggr) , $$

replacing that in Eq. (D.15).

When $\mathrm{CDX}_{\tau} ( M )$ is not traded, we may replicate its approximation based on $\widehat{\mathrm{CDX}}_{\tau} ( M )$ in Eq. (5.22) of the main text, based on the assumption in the main text, and at least up to the jump components possibly affecting $v_{1\tau}$, as follows. First, relying on the third equality in (5.23), consider a self-financed strategy into (i) a continuum of defaultable zero coupon bonds, say $\theta_{\tau} ( x )$ for a continuum of maturities $x\in [ T,T_{bM} ]$, (ii) in the defaultable annuity $v_{1\tau}$, say $\theta_{1\tau}$, and (iii) in a money market account, say $\theta_{2\tau}$. The value of this strategy satisfies

$$ dV_{\tau}=\int_{T}^{T_{bM}} \theta_{\tau} ( x ) d_{\tau }P_{\mathrm{def},\tau} ( x ) dx+ \theta_{1\tau}dv_{1\tau }+\theta _{2\tau}dM_{\tau}. $$

On the other hand, $\mathcal{C}_{\tau}\equiv\widehat{\mathrm {CDX}}_{\tau } ( M )$ satisfies

$$ d\mathcal{C}_{\tau}=\mathrm{LGD}\cdot\lambda\frac{\int _{T}^{T_{bM}}d_{\tau}P_{\mathrm{def},\tau} ( x ) dx}{v_{1\tau}}- \frac{\mathcal{C}_{\tau}}{v_{1\tau}}dv_{1\tau }+\mathrm{VC}_{\tau}^{cr}d \tau, $$

for some $\tau$-measurable process $\mathrm{VC}_{\tau}^{cr}$. Therefore, by neglecting jumps that $v_{1\tau}$ possibly experiences, we have that with

$$ \theta_{\tau} ( x ) =\frac{\mathrm{LGD}\cdot\lambda }{v_{1\tau}},\quad \text{for each }x\in [ T,T_{bM} ] ,\qquad \theta_{1\tau}=-\frac{\widehat{\mathrm{CDX}}_{\tau} ( M ) }{v_{1\tau}}, $$

one has that

$$ dV_{\tau}-d\mathcal{C}_{\tau}= \bigl( r_{\tau} \theta_{2\tau}M_{\tau }-\mathrm{VC}_{\tau}^{cr} \bigr) d\tau, $$

where $\theta_{2\tau}$ is chosen to replicate $\widehat{\mathrm {CDX}}_{\tau} ( M )$; that is,

$$ \theta_{2\tau}M_{\tau}=\widehat{\mathrm{CDX}}_{\tau} ( M ) -\int_{T}^{T_{bM}}\theta_{\tau} ( x ) d_{\tau }P_{\mathrm{def},\tau} ( x ) dx-\theta_{\tau1}v_{1\tau }= \widehat{\mathrm{CDX}}_{\tau} ( M ) . $$

While this portfolio replicates $\widehat{\mathrm{CDX}}_{\tau} ( M )$, it leads to a hedging cost equal to $\varepsilon_{\tau }^{cr}\equiv\widehat{\mathrm{CDX}}_{\tau} ( M ) -V_{\tau}$, which satisfies $d\varepsilon_{\tau}^{cr}=(\mathrm{VC}_{\tau }^{cr}-r_{\tau}\widehat{\mathrm{CDX}}_{\tau} ( M ) )d\tau$.

Estimates based on forward premium approximations. Let $K_{0}$ be the first strike below $\mathrm{CDX}_{t}$, as defined in the main text. We expand $\ln\mathrm{CDX}_{T}$ around $K_{0}$, as follows:

$$\begin{aligned} \ln\frac{\mathrm{CDX}_{T}}{\mathrm{CDX}_{t}}& =\ln\frac{K_{0}}{\mathrm {CDX}_{t}}+\frac{\mathrm{CDX}_{T}-K_{0}}{K_{0}} \\ &\quad{} - \biggl( \int_{0}^{K_{0}} ( K- \mathrm{CDX}_{T} ) ^{+}\frac{1}{K^{2}}dK+\int _{K_{0}}^{\infty} ( \mathrm {CDX}_{T}-K ) ^{+}\frac{1}{K^{2}}dK \biggr) , \end{aligned}$$

(D.19)

so that the fair value of the standardized credit variance swap rate of Definition 5.3 satisfies:

$$\begin{aligned} & \frac{\mathbb{F}_{\mathrm{var},M} ( t,T ) }{v_{1t}}+2\mathbb {E}_{t}^{\mathrm{sc}} \biggl[ \int _{t}^{T} \biggl( e^{j_{\tau } ( M ) }-1-j_{\tau} ( M ) -\frac{1}{2}j_{\tau}^{2} ( M ) \biggr) dJ_{\tau}^{\mathrm{sc}} \biggr] \\ &\quad =-2\mathbb{E}_{t}^{\mathrm{sc}} \biggl( \ln\frac{\mathrm {CDX}_{T}}{\mathrm{CDX}_{t}} \biggr) \\ &\quad =-2 \biggl( \ln\frac{K_{0}}{\mathrm{CDX}_{t}}+\frac{\mathrm {CDX}_{t}-K_{0}}{K_{0}} \biggr) \\ &\qquad{} +\frac{2}{v_{1t}} \biggl( \int_{0}^{K_{0}} \mathrm {SW}_{t}^{r} ( K,T,M ) \frac{1}{K^{2}}dK+\int _{K_{0}}^{\infty }\mathrm{SW}_{t}^{p} ( K,T,M ) \frac{1}{K^{2}}dK \biggr) . \end{aligned}$$

(D.20)

Next, consider a second order expansion of the function $-\ln\frac {\mathrm{CDX}_{t}}{K_{0}}$ about $K_{0}$:

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

By substituting this approximation into the first term in the R.H.S. of Eq. (D.20), disregarding the jump component terms, and proceeding as usual with a discretization, leaves the expression of $\mathbb {P}_{o,\mathrm{var},M}^{\ast} ( t,T )$ in Eq. (5.26).

Next, consider the correction applying to the basis point index. Similarly as for Eq. (D.19), expand $\mathrm{CDX}_{T}^{2}$ around $K_{0}$,

$$\begin{aligned} \mathrm{CDX}_{T}^{2}-\mathrm{CDX}_{t}^{2}& =K_{0}^{2}-\mathrm {CDX}_{t}^{2}+2K_{0} ( \mathrm{CDX}_{T}-K_{0} ) \\ &\quad{} +2 \biggl( \int_{0}^{K_{0}} ( K- \mathrm{CDX}_{T} ) ^{+}dK+\int_{K_{0}}^{\infty} ( \mathrm{CDX}_{T}-K ) ^{+}dK \biggr) , \end{aligned}$$

so that the standardized BP-variance swap rate of Definition 5.4-(c) is

$$\begin{aligned} \frac{\mathbb{F}_{\mathrm{var},M}^{\mathrm{bp}} ( t,T ) }{v_{1t}}& =\mathbb{E}_{t}^{\mathrm{sc}} \bigl( \mathrm {CDX}_{T}^{2}-\mathrm{CDX}_{t}^{2} \bigr) \\ & =K_{0}^{2}-\mathrm{CDX}_{t}^{2}+2K_{0} ( \mathrm {CDX}_{t}-K_{0} ) \\ &\quad{} +\frac{2}{v_{1t}} \biggl( \int_{0}^{K_{0}} \mathrm {SW}_{t}^{r} ( K,T,M ) dK+\int_{K_{0}}^{\infty} \mathrm{SW}_{t}^{p} ( K,T,M ) dK \biggr) . \end{aligned}$$

(D.21)

A second order expansion of the function $\mathrm{CDX}_{t}^{2}$ about $K_{0}$ yields

$$ \mathrm{CDX}_{t}^{2}-K_{0}^{2} \approx2K_{0} ( \mathrm {CDX}_{t}-K_{0} ) + ( \mathrm{CDX}_{t}-K_{0} ) ^{2}. $$

Substituting this approximation into Eq. (D.21) and discretizing the integral as usual yields the approximation in Eq. (5.27).