Variance Contracts: Fixed Income Security Design

Abstract

This chapter provides a general framework for designing and pricing variance swaps referencing fixed income securities. In a variance swap contract, the seller pays the amount by which the realized variance of some market variable of interest over a fixed time horizon exceeds a threshold—the variance swap rate—agreed upon at the trade’s inception. As such, the variance swap rate represents the market’s expectation of future variance, and in turn serves as the basis of a volatility gauge. Certain conditions are required of the contract design in order for the variance swap rate to be priced with as few modeling assumptions as possible, and ideally lead to a model-independent volatility gauge. Some theoretical and empirical properties of these volatility gauges are presented.

Appendix A: Appendix on Security Design and Volatility Indexing

2.1.1 A.1 Proof of Proposition 2.2

We begin with the following preliminary result, from which Eqs. (2.6)–(2.7) follow for $Y_{t}=N_{t}$. Note that the arguments in this proof rely on spanning arguments similar to those utilized by Bakshi and Madan (2000) and Carr and Madan (2001) in the equity case, although centered around the notion of a market numéraire. We then provide the proof of the proposition with general stochastic multipliers.

Lemma A.1

We have:

$$ \mathbb{E}_{t}^{Q^{N}} \bigl( V^{\mathrm{bp}} ( t,T ) \bigr) = \frac{2}{N_{t}} \biggl( \int_{0}^{X_{t}} \mathrm{Put}_{t} ( K ) dK+\int_{X_{t}}^{\infty} \mathrm{Call}_{t} ( K ) dK \biggr) , $$

(A.1)

and

$$ \mathbb{E}_{t}^{Q^{N}} \bigl( V ( t,T ) \bigr) = \frac {2}{N_{t}} \biggl( \int_{0}^{X_{t}} \frac{\mathrm{Put}_{t} ( K ) }{K^{2}}dK+\int_{X_{t}}^{\infty} \frac{\mathrm {Call}_{t} ( K ) }{K^{2}}dK \biggr) . $$

(A.2)

Proof

We provide the proof of Eq. (A.1), as that of Eq. (A.2) follows as a special case of the arguments leading to Eq. (A.19) and Eq. (A.20) in Appendix A.5 regarding the jump-diffusion case. By Itô’s lemma,

$$ \mathbb{E}_{t}^{Q^{N}} \bigl( V^{\mathrm{bp}} ( t,T ) \bigr) = \mathbb{E}_{t}^{Q^{N}} \bigl( X_{T}^{2}-X_{t}^{2} \bigr) . $$

(A.3)

Moreover, by a Taylor expansion with remainder,

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

(A.4)

Multiplying both sides of the previous equation by $e^{-\int_{t}^{T}r_{u}du}N_{T}$, and taking expectation under the risk-neutral probability leaves

$$\begin{aligned} \mathbb{E}_{t} \bigl( e^{-\int_{t}^{T}r_{u}du}N_{T} \bigl( X_{T}^{2}-X_{t}^{2} \bigr) \bigr) & =2X_{t}\mathbb{E}_{t} \bigl( e^{-\int _{t}^{T}r_{u}du}N_{T} ( X_{T}-X_{t} ) \bigr) \\ &\quad{} +2 \biggl( \int_{0}^{X_{t}}\mathbb{E}_{t} \bigl( e^{-\int _{t}^{T}r_{u}du}N_{T} ( K-X_{T} ) ^{+} \bigr) dK \\ &\quad{}+\int _{X_{t}}^{\infty}\mathbb{E}_{t} \bigl( e^{-\int _{t}^{T}r_{u}du}N_{T} ( X_{T}-K ) ^{+} \bigr) dK \biggr) \\ & =2 \biggl( \int_{0}^{X_{t}}\mathrm{Put}_{t} ( K ) dK+\int _{X_{t}}^{\infty}\mathrm{Call}_{t} ( K ) dK \biggr) , \end{aligned}$$

(A.5)

where the last line follows by a change of probability, from $Q$ to $Q^{N}$,

$$ \frac{dQ^{N}}{dQ}\bigg\vert _{\mathbb{F}_{T}}=\frac{e^{-\int _{t}^{T}r_{u}du}N_{T}}{N_{t}}, $$

the martingale property of $X_{\tau}$ under $Q^{N}$, and the expressions for $\mathrm{Put}_{t} ( K )$ and $\mathrm{Call}_{t} ( K )$ in Definition 2.2. By the assumption that $N_{\tau}$ is the price of a traded asset, and $N_{\tau}>0$, $dQ^{N}$ integrates to one. Similarly, by a change of probability,

$$ \mathbb{E}_{t} \bigl( e^{-\int_{t}^{T}r_{u}du}N_{T} \bigl( X_{T}^{2}-X_{t}^{2} \bigr) \bigr) =N_{t}\mathbb{E}_{t}^{Q^{N}} \bigl( X_{T}^{2}-X_{t}^{2} \bigr) . $$

(A.6)

Combining Eqs. (A.5) and (A.6) with Eq. (A.3) yields Eq. (A.1). □

Next, we prove the claims of Proposition 2.2 regarding basis point variance, $V^{\mathrm{bp}} ( t,T )$; those for percentage variance $V ( t,T )$ follow through a mere change in notation. We only prove the “only if” part, as the “if” part is trivial from the derivation of the proof to follow. Consider the Radon–Nikodym derivative of $Q^{Y}$ against $Q^{N}$,

$$ \zeta_{T}\equiv\frac{dQ^{Y}}{dQ^{N}}\bigg\vert _{\mathbb{F}_{T}}, $$

and suppose on the contrary that there exists a stochastic multiplier $Y_{T}$ such that

$$ \mathrm{cov}^{Q^{N}} \bigl( V^{\mathrm{bp}} ( t,T ) ,\zeta _{T} \bigr) \neq0, $$

and that at the same time,

$$ \mathbb{E}_{t}^{Q^{Y}} \bigl( V^{\mathrm{bp}} ( t,T ) \bigr) = \mathbb{E}_{t}^{Q^{N}} \bigl( V^{\mathrm{bp}} ( t,T ) \bigr) . $$

In this case, Eqs. (A.3) and (A.5) would imply that the conclusions of Lemma A.1 hold. However, we also have:

$$ \mathbb{E}_{t}^{Q^{Y}} \bigl( V^{\mathrm{bp}} ( t,T ) \bigr) = \mathbb{E}_{t}^{Q^{N}} \bigl( V^{\mathrm{bp}} ( t,T ) \bigr) + \mathrm{cov}^{Q^{N}} \bigl( V^{\mathrm{bp}} ( t,T ) ,\zeta _{T} \bigr) . $$

Then, $\mathrm{cov}^{Q^{N}} ( V^{\mathrm{bp}} ( t,T ) ,\zeta _{T} ) =0$, a contradiction. Proposition 2.2 follows by Proposition 2.1 in the main text and Lemma A.1. □

2.1.2 A.2 A Stochastic Multiplier Beyond the Market NumÉraire

Consider the following stochastic multiplier, $Y_{T}=N_{T}\epsilon_{T}$, where $\epsilon_{T}$ is $\mathbb{F}_{T}$-measurable, and such that $\mathrm{cov}^{Q^{N}} ( V^{\mathrm{bp}} ( t,T ) ,\epsilon _{T} ) =0$. We have,

$$ \zeta_{T}\equiv\frac{dQ^{Y}}{dQ^{N}}\bigg\vert _{\mathbb {F}_{T}}=c_{t} \epsilon_{T},\quad\text{with }c_{t}\equiv\frac {\mathbb{E}_{t} ( e^{-\int_{t}^{T}r_{u}du}N_{T} ) }{\mathbb{E}_{t} ( e^{-\int_{t}^{T}r_{u}du}N_{T}\epsilon _{T} ) }. $$

Heuristically,

$$ \zeta_{T}=\biggl( \frac{dQ^{Y}}{dQ}:\frac{dQ^{N}}{dQ} \biggr) \bigg\vert _{\mathbb{F}_{T}}=\frac{e^{-\int _{t}^{T}r_{u}du}Y_{T}}{\mathbb{E}_{t} ( e^{-\int _{t}^{T}r_{u}du}Y_{T} ) }:\frac{e^{-\int _{t}^{T}r_{u}du}N_{T}}{\mathbb{E}_{t} ( e^{-\int _{t}^{T}r_{u}du}N_{T} ) }= \epsilon_{T}c_{t}. $$

Next, we claim that

$$ \mathrm{cov}^{Q^{N}} \bigl( V^{\mathrm{bp}} ( t,T ) ,\zeta _{T} \bigr) =c_{t}\cdot\mathrm{cov}^{Q^{N}} \bigl( V^{\mathrm{bp}} ( t,T ) ,\epsilon_{T} \bigr) =0, $$

as in the class of multipliers identified by Proposition 2.2. Indeed, we have:

$$ \mathbb{E}_{t}^{Q^{Y}} \bigl( V^{\mathrm{bp}} ( t,T ) \bigr) =c_{t}\mathbb{E}_{t}^{Q^{N}} ( \epsilon_{T} ) \mathbb {E}_{t}^{Q^{N}} \bigl( V^{\mathrm{bp}} ( t,T ) \bigr) =\mathbb {E}_{t}^{Q^{N}} \bigl( V^{\mathrm{bp}} ( t,T ) \bigr) , $$

where the first equality follows by the fact that $\epsilon_{T}$ is uncorrelated with $V^{\mathrm{bp}} ( t,T )$, and the second follows by the definition of $c_{t}$,

$$ \mathbb{E}_{t}^{Q^{N}} ( \epsilon_{T} ) = \frac {1}{N_{t}}\mathbb{E}_{t} \bigl( e^{-\int _{t}^{T}r_{u}du}N_{T} \epsilon_{T} \bigr) =\frac{1}{c_{t}}. $$

2.1.3 A.3 Vega and Gamma in Gaussian Markets

We first show Eq. (2.9) (“Constant volatility”), then prove that the statement in (2.21) is true (“Constant vega”) and, finally, validate our claims regarding the implications on gamma of option portfolios (“Constant gamma exposure”). In what follows, we assume that the portfolio weightings and their first order derivative are integrable with respect to the Gaussian distribution, (i) $E ( \vert \omega ( \tilde {y} ) \vert ) <\infty$ and $E(\int_{-\infty }^{0}\vert \omega ( u ) u\vert \varPhi ( u ) du)<\infty$, and (ii) $E ( \vert \omega^{\prime} ( \tilde{y} ) \vert ) <\infty$, where $\tilde{y}$ is Gaussian. The first two conditions are necessary for the existence of an abstract volatility index based on at-the-money and out-of-the-money options in a Gaussian market (see Eqs. (A.7) and (A.8) below). The second is a regularity condition needed while dealing with boundedness of the sensitivity of vega with respect to the forward risk, $X_{t}$.

We shall rely on the following result. Consider the market in Sects. 2.2 and 2.3. If the forward risk $X_{t}$ is a solution to Eq. (2.8), the prices of put and call options are given by the “Bachelier formulae” (see Eq. (2.24) in the main text):

$$ \begin{aligned} \mathcal{O}_{t}^{\mathrm{P}} ( X_{t},K,T,\hat{ \sigma}_{\mathrm {n}} ) &\equiv N_{t}\cdot\mathcal{Z}_{t}^{\mathrm{P}} ( X_{t},K,T,\hat{\sigma}_{\mathrm{n}} ) ,\\ \mathcal {O}_{t}^{\mathrm{C}} ( X_{t},K,T,\hat{ \sigma}_{\mathrm{n}} ) &\equiv N_{t}\cdot\mathcal {Z}_{t}^{\mathrm{C}} ( X_{t},K,T,\hat{\sigma}_{\mathrm{n}} ) , \end{aligned} $$

(A.7)

where $\hat{\sigma}_{\mathrm{n}}\equiv\sqrt{\Vert \sigma _{\mathrm{n}}\Vert ^{2}}$,

$$\begin{aligned} \mathcal{Z}_{t}^{\mathrm{P}} ( X,K,T,\sigma ) =& ( K-X ) \varPhi \biggl( \frac{K-X}{\sigma\sqrt{T-t}} \biggr) +\sigma\sqrt{T-t}\phi \biggl( \frac{X-K}{\sigma\sqrt{T-t}} \biggr) , \\ \mathcal{Z}_{t}^{\mathrm{C}} ( X,K,T,\sigma ) =& ( X-K ) \varPhi \biggl( \frac{X-K}{\sigma\sqrt{T-t}} \biggr) +\sigma\sqrt{T-t}\phi \biggl( \frac{X-K}{\sigma\sqrt{T-t}} \biggr) , \end{aligned}$$

and $\phi$ denotes the standard normal density.

Constant volatility. Plugging Eq. (A.7) into Eq. (2.6) leaves, after substituting the expressions $\mathrm{Put}_{t} ( K ) \equiv\mathcal{O}_{t}^{\mathrm{P}} ( X,K,T,\hat{\sigma}_{\mathrm {n}} )$ and $\mathrm{Call}_{t} ( K ) \equiv\mathcal {O}_{t}^{\mathrm{C}} ( X,K,T,\hat{\sigma}_{\mathrm{n}} )$, and extending the left limit of the first integral in Eq. (2.6) to $-\infty$,

$$\begin{aligned} \frac{1}{2}K_{Y}& =\sigma^{2} ( T-t ) +\int _{-\infty }^{X} ( K-X ) \varPhi \biggl( \frac{K-X}{\sigma\sqrt{T-t}} \biggr) dK \\ &\quad{}+\int_{X}^{\infty} ( X-K ) \varPhi \biggl( \frac {X-K}{\sigma\sqrt{T-t}} \biggr) dK \\ & =\sigma^{2} ( T-t ) \biggl[ 1+2\int_{-\infty }^{0}u \varPhi ( u ) du \biggr] \\ & =\sigma^{2} ( T-t ) \biggl[ 1+2 \biggl( -\frac{1}{2}\int _{-\infty}^{0}u^{2}\phi ( u ) du \biggr) \biggr] \\ & =\frac{1}{2}\sigma^{2} ( T-t ) , \end{aligned}$$

where the first equality follows by the property of the normal distribution that, $\int_{-\infty}^{\infty}\phi ( \frac{X-K}{\nu } ) dK=\nu$ for $\nu>0$, and the second holds by a change in variables, the third by an integration by parts, and the fourth by a basic property of the standard normal distribution.

Constant vega. We prove that in Gaussian markets, a portfolio with all out-of-the-money and at-the-money European options has constant vega if and only if these options are equally weighted. The value of the portfolio we consider is a special case of Eq. (2.19),

$$\begin{aligned} &\pi_{t} ( X,T,\sigma ) \\ &\quad=N_{t} \biggl( \int _{-\infty}^{X}\omega ( K ) \mathcal{Z}_{t}^{\mathrm{P}} ( X,K,T,\sigma ) dK+\int_{X}^{\infty}\omega ( K ) \mathcal{Z}_{t}^{\mathrm {C}} ( X,K,T,\sigma ) dK \biggr) , \end{aligned}$$

(A.8)

where $\mathcal{Z}_{t}^{\mathrm{P}}$ and $\mathcal{Z}_{t}^{\mathrm{C}}$ are as in Eqs. (A.7).

Note that the vega of a put is the same as the vega of a call, and equals $N_{t}\nu_{t}^{\mathcal{O}} ( X,K,T,\sigma )$, where:

$$\begin{aligned} \nu_{t}^{\mathcal{O}} ( X,K,T,\sigma ) &\equiv\frac{\partial \mathcal{Z}_{t}^{\mathrm{P}} ( X,K,T,\sigma ) }{\partial \sigma}= \frac{\partial\mathcal{Z}_{t}^{\mathrm{C}} ( X,K,T,\sigma ) }{\partial\sigma}\\ &=\sqrt{T-t}\phi \biggl( \frac{X-K}{\sigma \sqrt{T-t}} \biggr) , \end{aligned}$$

so that the vega of the portfolio is:

$$ \nu_{t} ( X,T,\sigma ) \equiv\frac{\partial\pi_{t} ( X,T,\sigma ) }{\partial\sigma}=N_{t} \sqrt{T-t}\int\omega ( K ) \phi \biggl( \frac{X-K}{\sigma\sqrt{T-t}} \biggr) dK. $$

(A.9)

As for the “if” part in (2.21), let $\omega ( K ) =\mathrm{const.}$, so that by Eq. (A.9), and the fact that the Gaussian density $\phi$ integrates to one, we have that the vega is independent of $X$:

$$ \nu_{t} ( X,T,\sigma ) =N_{t}\sqrt{T-t}\cdot \mathrm{const.} $$

As for the “only if” part, let us differentiate $\nu_{t} ( X,T,\sigma )$ in Eq. (A.9) with respect to $X$:

$$ \frac{\partial\nu_{t} ( X,T,\sigma ) }{\partial X}=-\frac {N_{t}}{\sigma^{2}\sqrt{T-t}}\int\omega ( K ) \phi \biggl( \frac{X-K}{\sigma\sqrt{T-t}} \biggr) ( X-K ) dK. $$

We claim that the constant weighting is the only function $\omega$ independent of $X$, and such that $\frac{\partial\nu_{t} ( X,T,\sigma ) }{\partial X}=0$. Suppose not, and note that $\frac{\partial\nu _{t} ( X,T,\sigma ) }{\partial X}$ is zero if and only if,

$$ X\int\omega ( K ) \phi \biggl( \frac{X-K}{\sigma\sqrt {T-t}} \biggr) dK=\int K\omega ( K ) \phi \biggl( \frac {X-K}{\sigma\sqrt{T-t}} \biggr) dK. $$

Let us moreover define a random variable $\tilde{y}\sim N ( \mu ,\sigma ^{2} ( T-t ) )$. In terms of $\tilde{y}$, the previous equality is, by Stein’s Lemma,

$$\begin{aligned} \mu E \bigl[ \omega ( \tilde{y} ) \bigr] &=E \bigl[ \tilde {y}\omega ( \tilde{y} ) \bigr] =\mu E \bigl[ \omega ( \tilde {y} ) \bigr] +\mathrm{cov} \bigl[ \tilde{y},\omega ( \tilde{y} ) \bigr]\\ & =\mu E \bigl[ \omega ( \tilde{y} ) \bigr] +E \bigl[ \omega^{\prime } ( \tilde{y} ) \bigr] \sigma^{2} ( T-t ) , \end{aligned}$$

which is a contradiction unless $\omega ( \cdot )$ is constant.

Remark A.1

By Proposition 2.2, and previous results in this appendix, the normalized value of the portfolio in Eq. (A.8) is $\sigma^{2}$, in the special case $\omega ( K ) =2$,

$$ \sqrt{\frac{\pi_{t} ( X_{t},T,\sigma ) }{N_{t}}}=\sigma\sqrt{T-t}. $$

We illustrate this fact numerically. We set $X_{t}=5~\%$, $T-t=1$ (one year) and $\sigma=150~\mbox{bps}$, and approximate the integral in Eq. (A.8),

$$\begin{aligned} &100^{2}\times\sqrt{\frac{\hat{\pi}_{t} ( X_{t},T,\sigma ) }{N_{t}}}\\ &\quad\approx100^{2}\times \sqrt{2 \biggl( \sum _{i:K_{i}< X_{t}}\mathcal{Z}_{t}^{\mathrm{P}} ( X_{t},K_{i},T,\sigma ) +\sum _{i:K_{i}\geq X_{t}} \mathcal{Z}_{t}^{\mathrm{C}} ( X_{t},K_{i},T, \sigma ) \biggr) \Delta K}\\ &\quad\approx150.3907, \end{aligned}$$

where $\min_{i} \{ K_{i} \} =0$, $\max_{i} \{ K_{i} \} =10~\%$, $\Delta K=0.0001$.

Constant gamma exposure. Our claim of a constant gamma exposure in a Gaussian market follows because the option price in Eq. (A.7) satisfies,

$$ \frac{\partial^{2}\mathcal{O}_{t}^{\mathrm{U}} ( X,K,T,\sigma ) }{\partial X^{2}}=N_{t}\frac{1}{\sigma ( T-t ) }\nu _{t}^{\mathcal{O}} ( X,K,T,\sigma ) , $$

where $\mathcal{O}_{t}^{\mathrm{U}} ( X,K,T,\sigma )$, $\mathrm{U}\in \{ \mathrm{P},\mathrm{C} \}$ denotes an out-of-the-money option price (see Eq. (A.7)). Therefore, we have

$$\begin{aligned} \frac{\partial^{2}\pi_{t} ( X,T,\sigma ) }{\partial X^{2}}& =\int\omega ( K ) \frac{\partial^{2}\mathcal{O}_{t}^{\mathrm {U}} ( X,K,T,\sigma ) }{\partial X^{2}}dK \\ & =\frac{1}{\sigma ( T-t ) }\int\omega ( K ) \frac {\partial\mathcal{O}_{t}^{\mathrm{U}} ( X,K,T,\sigma ) }{\partial \sigma}dK \\ & =\frac{1}{\sigma ( T-t ) }\frac{\partial\pi_{t} ( X,T,\sigma ) }{\partial\sigma}. \end{aligned}$$

(A.10)

The L.H.S. of this equation is independent of $X$ if and only if the R.H.S. is. That is, the gamma exposure of the portfolio is constant if and only if the portfolio vega is independent of $X$, as claimed in the main text.

2.1.4 A.4 Proof of Proposition 2.3

For simplicity, we suppress the dependence of all the variables and functions on $t$ and $T$. Assuming the zero homogeneity assumption is satisfied by the implied volatility $\sigma_{K}\equiv\sigma ( X,K )$, we have that the two Black pricers, $\mathcal{P} ( X,K,\sigma_{K} ) \equiv\mathrm{Put} ( K )$ and $\mathcal{C} ( X,K,\sigma_{K} ) \equiv\mathrm{Call} ( K )$, collapse to

$$ \mathcal{P} ( X,K,\sigma_{K} ) =K\varphi_{\mathrm {p}}^{1} ( u ) -X\varphi_{\mathrm{p}}^{2} ( u ) ,\qquad\mathcal {C} ( X,K, \sigma_{K} ) =X\varphi_{\mathrm{c}}^{1} ( u ) -K\varphi _{\mathrm{c}}^{2} ( u ) , $$

(A.11)

for some functions $\varphi_{\mathrm{p}}^{i} ( x )$ and $\varphi _{\mathrm{c}}^{i} ( x )$ of the moneyness $u\equiv\ln ( \frac{K}{X} )$. Substituting the two expressions in Eqs. (A.11) into Eq. (2.6) of Proposition 2.2, and making the change of variable $K\mapsto u$, leaves,

$$\begin{aligned} &\mathrm{{V}}^{\mathrm{bp}}=X^{2}\cdot\xi,\\ &\quad \xi \equiv \frac{2}{N} \biggl( \int_{-\infty}^{0} \bigl( e^{u}\varphi _{\mathrm{p}}^{1} ( u ) - \varphi_{\mathrm{p}}^{2} ( u ) \bigr) e^{u}du+\int _{0}^{\infty} \bigl( \varphi_{\mathrm {c}}^{1} ( u ) -e^{u}\varphi_{\mathrm{c}}^{2} ( u ) \bigr) e^{u}du \biggr) , \end{aligned}$$

where $N$ is the market numéraire at $t$, and the function $\xi$ is independent of $X$, establishing Part (i) of the proposition. Part (ii) is similar. Substituting $\mathcal{P} ( X,K,\sigma_{K} )$ and $\mathcal{C} ( X,K,\sigma_{K} )$ in Eq. (A.11) into Eq. (2.7) of Proposition 2.2, and by changing the variable of integration to $u=\ln \frac{K}{X}$, leaves

$$ \mathrm{{V}}=\frac{2}{N} \biggl( \int_{-\infty}^{0} \bigl( \varphi_{\mathrm{p}}^{1} ( u ) -e^{-u} \varphi_{\mathrm {p}}^{2} ( u ) \bigr) du+\int_{0}^{\infty} \bigl( e^{-u}\varphi_{\mathrm{c}}^{1} ( u ) - \varphi_{\mathrm {c}}^{2} ( u ) \bigr) du \biggr) , $$

which is independent of $X$. □

2.1.5 A.5 Approximating Indexes

We derive Eq. (2.34) and Eq. (2.35). We begin with Eq. (2.35). Substituting Eq. (A.11) into Eq. (2.33) yields, for $\ell\in ( 0,X )$,

$$ \mathrm{{V}}_{\ell}=\frac{2}{N} \biggl( \int _{\ln ( \frac{X-\ell}{X} ) }^{0} \bigl( \varphi_{\mathrm{p}}^{1} ( u ) -e^{-u}\varphi_{\mathrm{p}}^{2} ( u ) \bigr) du+\int _{0}^{\ln ( \frac{X+\ell}{X} ) } \bigl( e^{-u} \varphi_{\mathrm{c}}^{1} ( u ) -\varphi_{\mathrm {c}}^{2} ( u ) \bigr) du \biggr) , $$

so that,

$$\begin{aligned} \frac{\partial\mathrm{{V}}_{\ell}}{\partial X}&=\frac{2}{N} \biggl( - \bigl( \varphi_{\mathrm{p}}^{1} ( u ) -e^{-u}\varphi _{\mathrm{p}}^{2} ( u ) \bigr) \big\vert _{u=\ln ( \frac{X-\ell}{X} ) }\cdot \frac{\ell}{X ( X-\ell ) }\\ &\quad{}+\bigl( e^{-u}\varphi_{\mathrm{c}}^{1} ( u ) -\varphi_{\mathrm {c}}^{2} ( u ) \bigr) \big\vert _{u=\ln ( \frac {X+\ell}{X} ) }\cdot\frac{-\ell}{X ( X+\ell ) } \biggr) . \end{aligned}$$

Utilizing the expressions in Eq. (A.11) delivers Eq. (2.35). The proof of Eq. (2.34) proceeds similarly: substitute Eq. (A.11) into Eq. (2.32) to obtain, for $\ell\in ( 0,X )$,

$$ \mathrm{{V}}_{\ell}^{\mathrm{{BP}}}=X^{2}\times \xi_{\ell,X}, $$

(A.12)

where,

$$ \xi_{\ell,X}\equiv\frac{2}{N} \biggl( \int_{\ln ( \frac {X-\ell}{X} ) }^{0} \bigl( e^{u}\varphi_{\mathrm{p}}^{1} ( u ) - \varphi_{\mathrm{p}}^{2} ( u ) \bigr) e^{u}du+\int _{0}^{\ln ( \frac{X+\ell}{X} ) } \bigl( \varphi _{\mathrm{c}}^{1} ( u ) -e^{u}\varphi_{\mathrm{c}}^{2} ( u ) \bigr) e^{u}du \biggr) . $$

We have

$$\begin{aligned} \frac{\partial\xi_{\ell,X}}{\partial X}& =\frac{2}{N} \biggl( - \bigl( e^{u} \varphi_{\mathrm{p}}^{1} ( u ) -\varphi_{\mathrm {p}}^{2} ( u ) \bigr) e^{u}\big\vert _{u=\ln ( \frac {X-\ell}{X} ) }\cdot\frac{\ell}{X ( X-\ell ) }\\ &\quad{}+ \bigl( \varphi_{\mathrm{c}}^{1} ( u ) -e^{u} \varphi_{\mathrm {c}}^{2} ( u ) \bigr) \big\vert _{u=\ln ( \frac {X+\ell}{X} ) }\cdot \frac{-\ell}{X ( X+\ell ) } \biggr) \\ & =-\frac{2\ell}{NX^{3}} \bigl( \mathrm{Put} \bigl( X,K,\sigma ( X,K ) \bigr) \big\vert _{{K=X-\ell}}+\mathrm {Call} \bigl( X,K,\sigma ( X,K ) \bigr) \big\vert _{{K=X+\ell}} \bigr) , \end{aligned}$$

where the second equality follows by the expressions in Eq. (A.11). Equation (2.34) follows by straightforward differentiation of Eq. (A.12).

Next we show that Eqs. (2.34) and (2.36) are mutually consistent. We have

$$\begin{aligned} & \frac{\partial}{\partial X}\int_{X-\ell}^{X}\mathrm {Put} \bigl( X,K,\sigma ( X,K ) \bigr) dK \\ &\quad =\mathrm{Put} \bigl( X,K,\sigma ( X,K ) \bigr) \big\vert _{{K=X}}-\mathrm{Put} \bigl( X,K,\sigma ( X,K ) \bigr) \big\vert _{{K=X-\ell}} \\ &\qquad{}+\int_{X-\ell }^{X} \partial_{X}\mathrm{Put} \bigl( X,K,\sigma ( X,K ) \bigr) dK \\ &\quad = \mathrm{Put} \bigl( X,K,\sigma ( X,K ) \bigr) \big\vert _{{K=X}}-\mathrm{Put} \bigl( X,K,\sigma ( X,K ) \bigr) \big\vert _{{K=X-\ell}} \\ &\qquad{}+\frac{1}{X}\int_{X-\ell }^{X} \mathrm{Put} \bigl( X,K,\sigma ( X,K ) \bigr) dK \\ &\qquad{} -\frac{1}{X}\int_{X-\ell}^{X}K\cdot \partial_{K}\mathrm {Put} \bigl( X,K,\sigma ( X,K ) \bigr) dK, \end{aligned}$$

(A.13)

where the second equality follows by the assumption the implied volatilities are homogenous of degree zero in $( X,K )$, so that $\mathrm {Put} ( \cdot ) =X\cdot\partial_{X}\mathrm{Put} ( \cdot) +K\cdot\partial_{K}\mathrm{Put} ( \cdot )$. An integration by parts of the last term in Eq. (A.13) produces

$$\begin{aligned} & \int_{X-\ell}^{X}K\cdot\partial_{K} \mathrm{Put} \bigl( X,K,\sigma ( X,K ) \bigr) dK \\ &\quad =X\cdot\mathrm{Put} \bigl( X,K,\sigma ( X,K ) \bigr) \big\vert _{{K=X}}- ( X-\ell ) \cdot\mathrm {Put} \bigl( X,K,\sigma ( X,K ) \bigr) \big\vert _{{K=X-\ell }} \\ &\qquad{}-\int_{X-\ell}^{X} \mathrm{Put} \bigl( X,K,\sigma ( X,K ) \bigr) dK. \end{aligned}$$

Substituting this term into Eq. (A.13) leaves

$$\begin{aligned} &\frac{\partial}{\partial X}\int_{X-\ell}^{X}\mathrm {Put} \bigl( X,K,\sigma ( X,K ) \bigr) dK \\ &\quad=\frac{2}{X}\int _{X-\ell }^{X}\mathrm{Put} \bigl( X,K,\sigma ( X,K ) \bigr) dK- \frac {\ell}{X}\mathrm{Put} \bigl( X,K,\sigma ( X,K ) \bigr) \big\vert _{{K=X-\ell}}. \end{aligned}$$

(A.14)

Similarly,

$$\begin{aligned} &\frac{\partial}{\partial X}\int_{X}^{X+\ell}\mathrm {Call} \bigl( X,K,\sigma ( X,K ) \bigr) dK \\ &\quad=\frac{2}{X}\int _{X}^{X+\ell }\mathrm{Call} \bigl( X,K,\sigma ( X,K ) \bigr) dK- \frac{\ell }{X}\mathrm{Call} \bigl( X,K,\sigma ( X,K ) \bigr) \big\vert _{{K=X+\ell}}. \end{aligned}$$

(A.15)

Equation (2.34) now follows by taking derivatives with respect to $X$ in Eq. (2.32), using Eqs. (A.14)–(A.15), and rearranging terms.

2.1.6 A.6 Jumps

We derive the expression for the fair value of $K_{Y}$ in Proposition 2.2 under the assumption that the forward risk $X_{t}$ is a solution to the jump-diffusion process in Eq. (2.39).

Basis point. Apply Itô’s lemma for jump-diffusion processes to Eq. (2.39), obtaining,

$$\begin{aligned} \frac{dX_{\tau}^{2}}{X_{\tau}^{2}}& =-2 \bigl( \mathbb{E}_{\tau }^{Q^{N}} \bigl( e^{j_{\tau}}-1 \bigr) \eta_{\tau} \bigr) d\tau +2\sigma _{\tau}\cdot dW_{\tau}+\Vert \sigma_{\tau} \Vert ^{2}d\tau + \bigl( e^{2j_{\tau}}-1 \bigr) dJ_{\tau} \\ & =-2 \bigl( \mathbb{E}_{\tau}^{Q^{N}} \bigl( e^{j_{\tau}}-1 \bigr) \eta _{\tau} \bigr) d\tau+2 \bigl( e^{j_{\tau}}-1 \bigr) dJ_{\tau}+2\sigma _{\tau}\cdot dW_{\tau}+\Vert \sigma_{\tau} \Vert ^{2}d\tau \\ &\quad{}+ \bigl( e^{j_{\tau}}-1 \bigr) ^{2}dJ_{\tau}. \end{aligned}$$

By integrating, taking expectations under $Q^{N}$, and using the definition of basis point variance, $V_{J}^{\mathrm{bp}} ( t,T )$ in Eq. (2.41), leaves:

$$\begin{aligned} & \mathbb{E}_{t}^{Q^{N}} \bigl( X_{T}^{2}-X_{t}^{2} \bigr) \\ &\quad = \underset{=0}{\underbrace{-2\mathbb{E}_{t}^{Q^{N}} \biggl( \int _{t}^{T}X_{\tau}^{2} \bigl( \mathbb{E}_{\tau}^{Q^{N}} \bigl( e^{j_{\tau}}-1 \bigr) \eta_{\tau} \bigr) d\tau \biggr) +2\mathbb {E}_{t}^{Q^{N}} \biggl( \int_{t}^{T}X_{\tau}^{2} \bigl( e^{j_{\tau }}-1 \bigr) dJ_{\tau} \biggr) }} \\ &\qquad{} + \underset{=0}{\underbrace{2\mathbb{E}_{t}^{Q^{N}} \biggl( \int _{t}^{T}X_{\tau}^{2} \sigma_{\tau}^{2}\cdot dW_{\tau} \biggr) }} +\mathbb{E}_{t}^{Q^{N}}\bigl[V_{J}^{\mathrm{bp}} ( t,T ) \bigr], \end{aligned}$$

(A.16)

where the first term is zero as,

$$\begin{aligned} \mathbb{E}_{t}^{Q^{N}} \biggl( \int_{t}^{T}X_{\tau}^{2} \bigl( e^{j_{\tau}}-1 \bigr) dJ_{\tau} \biggr) & = \mathbb{E}_{t}^{Q^{N}} \biggl( \int_{t}^{T} \mathbb{E}_{\tau}^{Q^{N}} \bigl( X_{\tau }^{2} \bigl( e^{j_{\tau}}-1 \bigr) dJ_{\tau} \bigr) \biggr) \\ & =\mathbb{E}_{t}^{Q^{N}} \biggl( \int_{t}^{T}X_{\tau}^{2} \bigl( \mathbb{E}_{\tau}^{Q^{N}} \bigl( e^{j_{\tau}}-1 \bigr) \eta_{\tau } \bigr) d\tau \biggr) . \end{aligned}$$

(A.17)

Comparing Eq. (A.16) with Eqs. (A.5) and (A.6) and then (2.6) leads to the conclusions of the main text, and in particular to Eq. (2.43).

Percentage. First, apply Itô’s lemma to Eq. (2.39), obtaining Eq. (2.40), viz

$$\begin{aligned} d\ln X_{\tau}& =- \bigl( \mathbb{E}_{\tau}^{Q^{N}} \bigl( e^{j_{\tau }}-1 \bigr) \eta_{\tau} \bigr) d\tau-\frac{1}{2} \Vert \sigma _{\tau }\Vert ^{2}d\tau+\sigma_{\tau} \cdot dW_{\tau}+j_{\tau }dJ_{\tau} \\ & =-\frac{1}{2} \bigl( \Vert \sigma_{\tau} \Vert ^{2}d\tau +j_{\tau}^{2}dJ_{\tau} \bigr) + \sigma_{\tau}\cdot dW_{\tau}- \bigl( \mathbb{E}_{\tau}^{Q^{N}} \bigl( e^{j_{\tau}}-1 \bigr) \eta_{\tau } \bigr) d \tau \\ &\quad{}+j_{\tau}dJ_{\tau}+\frac{1}{2}j_{\tau}^{2}dJ_{\tau}, \end{aligned}$$

(A.18)

so that, by the definition of $V_{J} ( t,T )$ in Eq. (2.42) and (by arguments similar to those leading to Eq. (A.17)),

$$ \mathbb{E}_{\tau}^{Q^{N}} \bigl( e^{j_{\tau}}-1 \bigr) dJ_{\tau}= \bigl( \mathbb{E}_{\tau}^{Q^{N}} \bigl( e^{j_{\tau}}-1 \bigr) \eta_{\tau } \bigr) d\tau, $$

we obtain,

$$ -2\mathbb{E}_{t}^{Q^{N}} \biggl( \ln\frac{X_{T}}{X_{t}} \biggr) -2\mathbb {E}_{t}^{Q^{N}} \biggl[ \int_{t}^{T} \biggl( e^{j_{\tau }}-1-j_{\tau}-\frac{1}{2}j_{\tau}^{2} \biggr) dJ_{\tau} \biggr] =\mathbb{E}_{t}^{Q^{N}} \bigl( V_{J} ( t,T ) \bigr) . $$

(A.19)

Next, consider the standard Taylor’s expansion with remainder,

$$ \ln\frac{X_{T}}{X_{t}}=\frac{1}{X_{t}} ( X_{T}-X_{t} ) - \biggl( \int_{0}^{X_{t}}\frac{1}{K^{2}} ( K-X_{T} ) ^{+}dK+\int _{X_{t}}^{\infty} \frac{1}{K^{2}} ( X_{T}-K ) ^{+}dK \biggr) . $$

Taking the expectation under $Q^{N}$ yields,

$$ \mathbb{E}_{t}^{Q^{N}} \biggl( \ln\frac{X_{T}}{X_{t}} \biggr) =- \frac {1}{N_{t}} \biggl( \int_{0}^{X_{t}} \frac{1}{K^{2}}\mathrm {Put}_{t} ( K ) dK+\int_{X_{t}}^{\infty} \frac{1}{K^{2}}\mathrm {Call}_{t} ( K ) dK \biggr) , $$

(A.20)

where we have made use of the martingale property of $X_{\tau}$ under $Q^{N}$, and the expressions for $\mathrm{Put}_{t} ( K )$ and $\mathrm{Call}_{t} ( K )$ in Definition 2.2. Combining Eq. (A.19) and Eq. (A.20), and using Proposition 2.1, leaves the expression for $K_{J,Y}\equiv\mathbb{E}_{t}^{Q^{N}} [ V_{J} ( t,T ) ]$ in Eq. (2.44) of the main text.