Local volatility of volatility for the VIX market

Abstract

Following a trend of sustained and accelerated growth, the VIX futures and options market has become a closely followed, active and liquid market. The standard stochastic volatility models—which focus on the modeling of instantaneous variance—are unable to fit the entire term structure of VIX futures as well as the entire VIX options surface. In contrast, we propose to model directly the VIX index, in a mean-reverting local volatility-of-volatility model, which will provide a global fit to the VIX market. We then show how to construct the local volatility-of-volatility surface by adapting the ideas in Carr (Local variance gamma. Bloomberg Quant Research, New York, 2008) and Andreasen and Huge (Risk Mag 76–79, 2011) to a mean-reverting process.

Appendices

Appendix I

Proof of Proposition 2.1

Denote $\rho (V, T)$ the density of $V_T$. We have

$$\begin{aligned} C(K, T) = e^{-r T} \int \limits _K^{\infty } (V - K) \cdot \rho (V, T) dV \end{aligned}$$

(29)

from where, differentiating with respect to strike $K$ twice

$$\begin{aligned} \frac{\partial C}{\partial K} (K, T)&= - e^{-rT} \int \limits _K^{\infty } \rho (V,T) dV \\ \frac{\partial ^2 C}{\partial K^2} (K, T)&= e^{-rT} \rho (K,T). \end{aligned}$$

Recall Kolmogorov’s forward equation for $\rho (V,T)$, see e.g. Jeanblanc et al. (2009):

$$\begin{aligned} \frac{\partial \rho }{\partial T} (V,T) = - \frac{\partial }{\partial V} \Big [ k \left( \theta (T) - V \right) \cdot \rho (V,T) \Big ] + \frac{1}{2} \frac{\partial ^2}{\partial V^2} \Big [ V^2 \cdot \lambda (V,T)^2 \cdot \rho (V,T) \Big ]. \nonumber \\ \end{aligned}$$

(30)

Differentiating both sides of (29) with respect to maturity $T$ and using (30)

$$\begin{aligned} \frac{\partial C}{\partial T}&= -rC + e^{-rT} \Bigg [ -\int \limits _K^{\infty } (V-K) \cdot \frac{\partial }{\partial V} \Big [ k \left( \theta (T) - V \right) \cdot \rho (V,T) \Big ] dV \nonumber \\&\quad + \frac{1}{2} \int \limits _K^{\infty } (V-K) \cdot \frac{\partial ^2}{\partial V^2} \Big [ V^2 \cdot \lambda (V,T)^2 \cdot \rho (V,T) \Big ] dV \Bigg ] \end{aligned}$$

(31)

Denoting the two terms in the parenthesis by $A$, respectively $B$, we proceed to compute each in turn, using integration by parts. The first term becomes

$$\begin{aligned} A&= - (V-K) \cdot \Big [ k \left( \theta (T) - V \right) \cdot \rho (V,T) \Big ] \Bigg |_K^{\infty } + \int \limits _K^{\infty } k \left( \theta (T) - V \right) \cdot \rho (V,T) dV \\&= \int \limits _K^{\infty } k \left( \theta (T) - K - (V-K) \right) \cdot \rho (V,T) dV \\&= k \left( \theta (T) - K \right) \cdot \int \limits _K^{\infty } \rho (V,T) dV - k \int \limits _K^{\infty } (V-K) \cdot \rho (V,T) dV \\&= e^{rT} \Big [ - k \left( \theta (T) - K \right) \cdot \frac{\partial C}{\partial K} (K,T) - k \cdot C(K,T) \Big ] \end{aligned}$$

where we have assumed that the boundary term, on the first line, vanishes at $K=+\infty$. The second term becomes

$$\begin{aligned} B\;=\;\frac{1}{2} (V\!-\!K) \cdot \frac{\partial }{\partial V} \Big [ V^2 \cdot \lambda (V,T)^2 \!\cdot \! \rho (V,T) \Big ] \Bigg |_K^{\infty } \\&- \frac{1}{2} \int \limits _K^{\infty } \frac{\partial }{\partial V} \Big [ V^2 \cdot \lambda (V,T)^2 \cdot \rho (V,T) \Big ] dV = - \frac{1}{2} V^2 \cdot \lambda (V,T)^2 \cdot \rho (V,T) \Bigg |_K^{\infty }\\&= \frac{1}{2} K^2 \cdot \lambda (K,T)^2 \cdot \frac{\partial ^2 C}{\partial K^2} (K, T). \end{aligned}$$

where we have assumed that the boundary terms vanish at $K=+\infty$. Using the expressions for $A$ and $B$ in (31), we obtain

$$\begin{aligned} \frac{\partial C}{\partial T}&= -rC - k \left( \theta (T) - K \right) \cdot \frac{\partial C}{\partial K} - k \cdot C + \frac{1}{2} K^2 \cdot \lambda (K,T)^2 \cdot \frac{\partial ^2 C}{\partial K^2} \end{aligned}$$

which, upon rearrangement, leads to the statement in Proposition 2.1.

Proof of Proposition 2.2

We will find it convenient to work with the undiscounted Black function. Also, noting that, for $T \in [T_{i-1}, T_i]$, the futures price $F_0^T$ can be written as

$$\begin{aligned} F_0^T = \theta _i + e^{-k(T-T_{i-1})} \left( F_0^{T_{i-1}} - \theta _i \right) \end{aligned}$$

the definition of the Black implied volatility surface becomes

$$\begin{aligned} e^{rT} \widetilde{C} (x, T) \stackrel{\triangle }{=} C^B \left( \theta _i + e^{-k(T-T_{i-1})} \left( F_0^{T_{i-1}} - \theta _i \right), V_0 e^x, T, \sigma (x, T) \right). \end{aligned}$$

(32)

For easier reference, we recall below the Black greeks which will be needed in our subsequent calculations:

$$\begin{aligned} \begin{array}{lll} \frac{\partial C^B}{\partial F} = N(d_1)&\frac{\partial C^B}{\partial K} = - N(d_2)&\frac{\partial C^B}{\partial T} = \frac{F \sigma n(d_1)}{2 \sqrt{T}} \\ \frac{\partial C^B}{\partial \sigma } = F \sqrt{T} n(d_1)&\frac{\partial ^2 C^B}{\partial K^2} = \frac{F n(d_1)}{\sigma \sqrt{T} K^2}&\frac{\partial ^2 C^B}{\partial \sigma \partial K} = \frac{F d_1 n(d_1)}{\sigma K} \\ \frac{\partial ^2 C^B}{\partial \sigma ^2} = F \sqrt{T} n(d_1) \frac{d_1 d_2}{\sigma }&\,&\end{array} \end{aligned}$$

From (32)

$$\begin{aligned} e^{rT} \frac{\partial \widetilde{C}}{\partial T}&= \!-\!re^{rT} \widetilde{C} \!+\! \frac{\partial C^B}{\partial F} \cdot \frac{\partial F_0^T}{\partial T} \!+\! \frac{\partial C^B}{\partial T} \!+\! \frac{\partial C^B}{\partial \sigma } \cdot \frac{\partial }{\partial T} \sigma (x, T) \nonumber \\&= \!-\!re^{rT} \widetilde{C} \!-\! k e^{\!-\!k(T\!-\!T_{i\!-\!1})} \left( F_0^{T_{i\!-\!1}} \!-\! \theta _i \right) \cdot N(d_1) \!+\! \frac{F_0^T \sigma (x,T) n(d_1)}{2 \sqrt{T}} \nonumber \\&\!+\! F_0^T \sqrt{T} n(d_1) \frac{\partial }{\partial T} \sigma (x, T). \end{aligned}$$

(33)

$$\begin{aligned} e^{rT} \frac{\partial \widetilde{C}}{\partial x}&= \frac{\partial C^B}{\partial K} \cdot V_0 e^{x} \!+\! \frac{\partial C^B}{\partial \sigma } \cdot \frac{\partial }{\partial x} \sigma (x, T) \nonumber \\&= \!-\! N(d_2) \cdot V_0 e^{x} \!+\! F_0^T \sqrt{T} n(d_1) \frac{\partial }{\partial x} \sigma (x, T). \end{aligned}$$

(34)

$$\begin{aligned} e^{rT} \frac{\partial ^2 \widetilde{C}}{\partial x^2}&= \left( \frac{\partial ^2 C^B}{\partial K^2} \cdot V_0 e^{x} \!+\! \frac{\partial ^2 C^B}{\partial \sigma \partial K} \cdot \frac{\partial }{\partial x} \sigma (x, T) \right) V_0 e^{x} \!+\! \frac{\partial C^B}{\partial K} \cdot V_0 e^{x} \nonumber \\&\!+\! \left( \frac{\partial ^2 C^B}{\partial \sigma \partial K} \cdot V_0 e^{x} \!+\! \frac{\partial ^2 C^B}{\partial \sigma ^2} \cdot \frac{\partial }{\partial x} \sigma (x, T) \right) \frac{\partial }{\partial x} \sigma (x, T) \!+\! \frac{\partial C^B}{\partial \sigma } \frac{\partial ^2 \sigma }{\partial x^2}(x,T) \nonumber \\&= \frac{F_0^T n(d_1)}{\sigma (x, T) \sqrt{T}} \!+\! 2 \frac{F_0^T d_1 n(d_1)}{\sigma (x,T)} \frac{\partial }{\partial x} \sigma (x, T) \!+\! F_0^T \sqrt{T} n(d_1) \frac{d_1 d_2}{\sigma (x, T)} \left( \frac{\partial }{\partial x} \sigma (x, T) \right)^2 \nonumber \\&\!-\! N(d_2) V_0 e^{x} \!+\! F_0^T \sqrt{T} n(d_1) \frac{\partial ^2}{\partial x^2} \sigma (x, T). \end{aligned}$$

(35)

Using the expressions in (33), (34) and (35), we obtain for the numerator of Eq. (5)

$$\begin{aligned}&\frac{\partial \widetilde{C}}{\partial T} + k \left( \theta _i - V_0 e^{x} \right) \frac{1}{V_0 e^{x}} \frac{\partial \widetilde{C}}{\partial x} + (r+k) \widetilde{C} = e^{-rT} \Bigg [ k \theta _i \Big ( N(d_1) - N(d_2) \Big ) \nonumber \\&+ F_0^T n(d_1) \Bigg ( \frac{\sigma (x, T) }{2 \sqrt{T}} + \sqrt{T} \cdot \frac{\partial }{\partial T} \sigma (x, T)+\frac{k \left( \theta _i - V_0 e^x \right) \sqrt{T}}{V_0 e^x} \cdot \frac{\partial }{\partial x} \sigma (x, T) \Bigg ) \Bigg ] \nonumber \\&\end{aligned}$$

(36)

and for the denominator of Eq. (5)

$$\begin{aligned}&\frac{1}{2} \cdot \left( \frac{\partial ^2 \widetilde{C}}{\partial x^2} - \frac{\partial \widetilde{C}}{\partial x} \right) \!=\! e^{-rT} \frac{F_0^T}{2} n(d_1) \Bigg [ \frac{1}{\sigma (x, T) \sqrt{T}} + \left( \frac{2d_1}{\sigma (x, T)} - \sqrt{T} \right) \cdot \frac{\partial }{\partial x} \sigma (x, T) \nonumber \\&+ \frac{\sqrt{T} d_1 d_2}{\sigma (x, T)} \cdot \left( \frac{\partial }{\partial x} \sigma (x, T) \right)^2 + \sqrt{T} \cdot \frac{\partial ^2}{\partial x^2} \sigma (x, T) \Bigg ]. \end{aligned}$$

(37)

Finally, combining (36) and (37), we arrive at the statement in Proposition 2.2. $\square$

Proof of Lemma 3.1

The butterfly spread condition (11) is immediate, as the payoff of a butterfly is strictly positive. For the calendar spread condition (10), we consider the VIX futures of maturity $T_2$, given by $F_t^{T_2} = E \left( V_{T_2} | \mathcal F _t \right)$, which is a martingale on $[0, T_2]$. By the conditional form of Jensen’s inequality

$$\begin{aligned} E \left( \left( F_{T_2}^{T_2} - K \right)_+ \Big | \mathcal F _{T_1} \right)&\ge \left( E\left(F_{T_2}^{T_2} \Big | \mathcal F _{T_1} \right) - K \right)_+ \ge \left( F_{T_1}^{T_2} - K \right)_+ \end{aligned}$$

which in turn implies

$$\begin{aligned} E \left( F_{T_2}^{T_2} - K \right)_+&\ge E \left( F_{T_1}^{T_2} - K \right)_+ . \end{aligned}$$

Using that

$$\begin{aligned} F_{T_1}^{T_2}&= e^{-k(T_2-T_1)} \cdot V_{T_1} + e^{-k T_2} \int \limits _{T_1}^{T_2}k e^{kt} \theta (t) dt \\ F_{T_2}^{T_2}&= V_{T_2} \end{aligned}$$

where the first relation follows by applying Itô to the process $e^{kt} V_t$ on $[T_1, T_2]$, we obtain

$$\begin{aligned} E \left( V_{T_2} - K \right)_+&\ge E \left( e^{-k(T_2-T_1)} \cdot V_{T_1} + e^{-k T_2} \int \limits _{T_1}^{T_2}k e^{kt} \theta (t) dt - K \right)_+ \\&= e^{-k(T_2-T_1)} \cdot E \left[ V_{T_1} - \left( K e^{k(T_2-T_1)} - e^{-k T_1} \int \limits _{T_1}^{T_2} k e^{kt} \theta (t) dt \right) \right]_+ . \end{aligned}$$

Equivalently, this can be written as

$$\begin{aligned} e^{r T_2} \cdot C (K, T_2) \ge e^{-k(T_2-T_1)} \cdot e^{rT_1} \cdot C \left( K e^{k(T_2-T_1)} - e^{-k T_1} \int \limits _{T_1}^{T_2} k e^{kt} \theta (t) dt , T_1 \right) \end{aligned}$$

which leads to the statement in Lemma 3.1.$\square$

Appendix II: VIX futures and options quotes

We include below CBOE’s VIX futures and options quotes as of the close of trading on Jul-05-2011, when 6 maturities were listed: Jul-20-2011, Aug-17-2011, Sep-21-2011, Oct-19-2011, Nov-16-2011 and Dec-21-2011. The VIX spot closed at 16.06 and the term structure of VIX futures as well as the VIX Put / Call quotes are given in the following tables.

VIX futures term structure
Maturity	07/20/2011	08/17/2011	09/21/2011	10/19/2011	11/16/2011	12/21/2011
VIX futures	16.95	18.10	20.00	21.00	21.60	21.85

Maturity Jul-20-2011
Strike	15	16	17	18	19	20	21	24	25	29
Bid	0.05	0.4	1	0.7	0.5	0.4	0.35	0.15	0.15	0.05
Offer	0.1	0.45	1.05	0.75	0.55	0.5	0.4	0.25	0.2	0.1
Type	P	P	P	C	C	C	C	C	C	C

Maturity Aug-17-2011
Strike	14	15	16	17	18	19	20	21	24	25	30	35	40	50
Bid	0.05	0.25	0.6	1.15	1.8	1.6	1.4	1.2	0.8	0.75	0.45	0.25	0.15	0.05
Offer	0.1	0.3	0.65	1.25	1.9	1.7	1.45	1.3	0.9	0.8	0.5	0.35	0.2	0.1
Type	P	P	P	P	P	C	C	C	C	C	C	C	C	C

Maturity Sep-21-2011
Strike	15	16	17	18	19	20	21	24	25	30	35	40	45	50
Bid	0.15	0.45	0.8	1.35	1.9	2.55	2.2	1.6	1.45	0.85	0.55	0.3	0.2	0.1
Offer	0.25	0.5	0.9	1.4	2	2.65	2.4	1.75	1.55	1	0.65	0.4	0.25	0.2
Type	P	P	P	P	P	P	C	C	C	C	C	C	C	C

Maturity Oct-19-2011
Strike	16	17	18	19	20	21	24	25	30	35	40	50
Bid	0.35	0.7	1.1	1.6	2.15	2.8	2	1.85	1.15	0.75	0.45	0.15
Offer	0.45	0.75	1.2	1.7	2.3	2.9	2.15	1.95	1.25	0.8	0.55	0.25
Type	P	P	P	P	P	C	C	C	C	C	C	C

Maturity Nov-16-2011
Strike	16	17	18	19	20	21	24	25	30	35	40	45	50	70
Bid	0.35	0.6	1	1.45	2	2.65	2.25	2.05	1.25	0.75	0.5	0.3	0.2	0.05
Offer	0.45	0.75	1.1	1.6	2.15	2.75	2.45	2.2	1.4	0.9	0.6	0.4	0.3	0.1
Type	P	P	P	P	P	P	C	C	C	C	C	C	C	C

Maturity Dec-21-2011
Strike	16	17	18	19	20	21	24	25	26	29	30	40	70
Bid	0.35	0.7	1.1	1.5	2.05	2.6	2.5	2.25	2.05	1.5	1.35	0.55	0.05
Offer	0.55	0.9	1.25	1.7	2.25	2.85	2.7	2.45	2.25	1.7	1.55	0.7	0.15
Type	P	P	P	P	P	P	C	C	C	C	C	C	C

We include below CBOE’s VIX futures and options quotes as of the close of trading on Sep-06-2012, when 6 maturities were listed: Sep-19-2012, Oct-17-2012, Nov-21-2012, Dec-19-2012, Jan-16-2013 and Feb-13-2013. The VIX spot closed at 15.60 and the term structure of VIX futures as well as the VIX Put / Call quotes are given in the following tables.

VIX futures term structure
Maturity	09/19/2012	10/17/2012	11/21/2012	12/19/2012	01/16/2013	02/13/2012
VIX futures	16.10	18.55	20.60	21.00	24.05	25.35

Maturity Sep-19-2012
Strike	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
Bid	0.05	0.3	0.75	0.6	0.4	0.3	0.25	0.2	0.15	0.15	0.1	0.1	0.1	0.1	0.05	0.05	0.05
Offer	0.1	0.35	0.9	0.7	0.5	0.4	0.35	0.25	0.25	0.2	0.2	0.15	0.15	0.15	0.1	0.1	0.1
Type	P	P	P	C	C	C	C	C	C	C	C	C	C	C	C	C	C

Maturity Oct-17-2012
Strike	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29
Bid	0.05	0.1	0.25	0.6	1.05	1.6	1.75	1.5	1.25	1.1	0.9	0.8	0.65	0.6	0.5	0.45	0.4
Offer	0.1	0.15	0.35	0.7	1.15	1.7	1.9	1.55	1.35	1.2	1	0.9	0.8	0.7	0.6	0.55	0.5
Type	P	P	P	P	P	P	C	C	C	C	C	C	C	C	C	C
Strike	30	32.5	35	37.5	40	42.5	45
Bid	0.35	0.25	0.15	0.1	0.1	0.05	0.05
Offer	0.4	0.35	0.3	0.25	0.2	0.15	0.15
Type	C	C	C	C	C	C	C

Maturity Nov-21-2012
Strike	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29
Bid	0.05	0.1	0.3	0.6	0.95	1.45	1.95	2.5	2.75	2.4	2.15	1.9	1.7	1.5	1.35	1.2	1.05
Offer	0.1	0.2	0.4	0.7	1.1	1.55	2.1	2.7	2.9	2.55	2.3	2.05	1.85	1.65	1.45	1.3	1.15
Type	P	P	P	P	P	P	P	P	C	C	C	C	C	C	C	C	C
Strike	30	32.5	35	37.5	40	42.5	45	47.5	50	55	60
Bid	0.95	0.7	0.55	0.4	0.3	0.2	0.15	0.1	0.05	0.05	0.05
Offer	1.05	0.85	0.65	0.5	0.4	0.35	0.3	0.25	0.2	0.15	0.1
Type	C	C	C	C	C	C	C	C	C	C	C

Maturity Dec-19-2012
Strike	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
Bid	0.1	0.3	0.55	0.9	1.25	1.7	2.25	2.85	3.3	3	2.75	2.5	2.25	2.05	1.85	1.7	1.55
Offer	0.2	0.4	0.65	1	1.4	1.85	2.4	3	3.6	3.2	2.9	2.65	2.4	2.2	2	1.8	1.65
Type	P	P	P	P	P	P	P	P	C	C	C	C	C	C	C	C	C
Strike	32.5	35	37.5	40	42.5	45	47.5	50	55	60
Bid	1.2	0.95	0.75	0.65	0.5	0.4	0.3	0.25	0.15	0.1
Offer	1.35	1.1	0.9	0.75	0.6	0.45	0.4	0.35	0.25	0.2
Type	C	C	C	C	C	C	C	C	C	C

Maturity Jan-16-2013
Strike	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
Bid	0.05	0.2	0.35	0.55	0.9	1.25	1.65	2.15	2.7	3.3	3.9	3.6	3.2	2.95	2.7	2.45	2.25
Offer	0.15	0.25	0.5	0.7	1.05	1.4	1.8	2.3	2.85	3.4	4.1	3.8	3.5	3.2	2.9	2.65	2.45
Type	P	P	P	P	P	P	P	P	P	P	P	C	C	C	C	C	C
Strike	32.5	35	37.5	40	42.5	45	47.5	50	55	60
Bid	1.8	1.45	1.15	0.95	0.75	0.6	0.5	0.4	0.25	0.15
Offer	2	1.6	1.35	1.1	0.85	0.75	0.6	0.5	0.4	0.3
Type	C	C	C	C	C	C	C	C	C	C

Maturity Feb-13-2013
Strike	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
Bid	0.05	0.15	0.3	0.5	0.75	1.05	1.45	1.85	2.35	2.9	3.4	4	4	3.6	3.3	3.1	2.85
Offer	0.2	0.3	0.35	0.6	0.9	1.2	1.6	2.05	2.55	3.1	3.7	4.3	4.2	3.9	3.6	3.4	3.1
Type	P	P	P	P	P	P	P	P	P	P	P	P	C	C	C	C	C
Strike	32.5	35	37.5	40	42.5	45	47.5	50	55	60	65	70
Bid	2.35	1.9	1.55	1.25	1.05	0.85	0.7	0.6	0.4	0.25	0.15	0.15
Offer	2.55	2.1	1.75	1.45	1.2	1	0.85	0.75	0.55	0.4	0.3	0.2
Type	C	C	C	C	C	C	C	C	C	C	C	C