VIX Computation Based on Affine Stochastic Volatility Models in Discrete Time

Abstract

We propose a class of discrete-time stochastic volatility models that, in a parsimonious way, capture the time-varying higher moments observed in financial series. Three desirable results are obtained. First, we have a recursive procedure for the log-price characteristic function which allows a semi-analytical formula for option prices as in Heston and Nandi (Rev Financ Stud 13(3):585–625, 2000). Second, we reproduce some features of the VIX Index. Finally, we derive a simple formula for the VIX index and use it for option pricing.

A Appendix

1.1 A.1 Conditional Moment Generating Function

Following the approach proposed in Heston and Nandi (2000) we derive a system of recursive equations for the time dependent coefficients of the conditional m.g.f. of the random variable ln(S _T ) given the available information at time t. We want to prove that the conditional m.g.f. is given by the following formula:

$$\displaystyle{ E_{t}\left [\left.\exp \left (c\ln \left (S_{T}\right )\right )\right \vert \mathcal{F}_{t}\right ] = S_{t}^{c}\exp \left [A\left (t;T,c\right ) + B\left (t;T,c\right )h_{ t+1}\right ]. }$$

(A.1)

We use the mathematical induction method.

• 1. We observe that relation (A.1) holds at time T since A(T; T, c) = 0 and B(T; T, c) = 0. 

• 2. We suppose the relation (A.1) holds at time t + 1 and, by the law of iterated expectations, we prove it at time t.

$$\displaystyle{ \begin{array}{l} E\left [\left.E\left [\left.S_{T}^{c}\right \vert \mathcal{F}_{t+1}\right ]\right \vert \mathcal{F}_{t}\right ] = E\left [\left.\exp \left [A\left (t + 1;T,c\right ) + B\left (t + 1;T,c\right )h_{t+2}\right ]\right \vert \mathcal{F}_{t}\right ] \\ = E\left [\exp \left [c\ln \left (S_{T}\right ) + cr + A(t + 1;T,c)\right.\right. \\ + \left.\left.c\lambda _{0}h_{t+1} + c\lambda _{1}V _{t+1} + c\sigma \sqrt{V _{t+1}}Z_{t+1}+\right.\right. \\ +\alpha _{0}B\left (t + 1;T,c\right ) +\alpha _{1}B\left (t + 1;T,c\right )V _{t+1} +\beta B\left (t + 1;T,c\right )h_{t+1}\left.\left.\right ]\left.\right \vert \mathcal{F}_{t}\right ] \\ = S_{t}^{c}\exp \left [cr + A\left (t + 1;T,c\right ) +\alpha _{0}B\left (t + 1;T,c\right ) + \left (c\lambda _{0} +\beta B\left (t + 1;T,c\right )\right )h_{t+1}\right ]{\ast} \\ {\ast} E\left [\left.\exp \left [\left (c\lambda _{1} +\alpha _{1}B\left (t + 1;T,c\right ) + \frac{c^{2}\sigma ^{2}} {2} \right )V _{t+1}\right ]\right \vert \mathcal{F}_{t}\right ].\end{array} }$$

(A.2)

Using the conditional m.g.f. of the r.v. V _t+1, Eq. (A.2) becomes:

$$\displaystyle{ \begin{array}{lll} E\left [\left.E\left [\left.S_{T}^{c}\right \vert \mathcal{F}_{t+1}\right ]\right \vert \mathcal{F}_{t}\right ] & = & S_{t}^{c}\exp \left [cr + A\left (t + 1; T,c\right ) +\alpha _{0}B\left (t + 1; T,c\right )+\right. \\ & & + \left.\left (c\lambda _{0} +\beta B\left (t + 1; T,c\right ) + f\left (c\lambda _{1} +\alpha _{1}B\left (t + 1; T,c\right ) + \frac{c^{2}\sigma ^{2}} {2},\theta \right )\right )h_{t+1}\right ] \end{array} }$$

(A.3)

By comparing the expression obtained in Eq. (A.3) with (A.1) we obtain the following recursive system:

$$\displaystyle{ \left \{\begin{array}{ll} A(t; T,c) & = cr + A(t + 1; T,c) +\alpha _{0}B(t + 1; T,c) \\ B(t; T,c) & = c\lambda _{0} +\beta B(t + 1; T,c)+ \\ & f(c\lambda _{1} +\alpha _{1}B(t + 1; T,c) + \frac{c^{2}\sigma ^{2}} {2},\theta ) \end{array} \right. }$$

(A.4)

with A(T; T, c) = 0 and B(T; T, c) = 0.

1.2 A.2 Martingale Condition

We want to prove that $\forall s \leq t$:

$$\displaystyle{ \lambda _{0} = -f(\lambda _{1} + \frac{\sigma ^{2}} {2};\theta )\ \stackrel{(1)}{\Longrightarrow}\ E\left [\left.\frac{S_{t}} {e^{r}} \right \vert \mathcal{F}_{t-1}\right ] = S_{t-1}\stackrel{(2)}{\Longrightarrow}E\left [\left. \frac{S_{t}} {e^{r(t-s)}}\right \vert \mathcal{F}_{s}\right ] = S_{s}.\ }$$

(A.5)

($\stackrel{(1)}{\Longrightarrow}$)

We assume r to be constant but the proof holds even assuming r to be a predictable process. By simple calculus, we obtain:

$$\displaystyle{ E\left [\left.\frac{S_{t}} {e^{r}} \right \vert \mathcal{F}_{t-1}\right ] = S_{t-1}\exp \left [\left (\lambda _{0} + f\left (\lambda _{1} + \frac{\sigma ^{2}} {2};\theta \right )\right )h_{t-1}\right ] }$$

(A.6)

substituting $\lambda _{0} = -f(\lambda _{1} + \frac{\sigma ^{2}} {2};\theta )$ in (A.6) we obtain the result.

($\stackrel{(2)}{\Longrightarrow}$)

By the iterated law of conditional expectation we have:

$$\displaystyle\begin{array}{rcl} E\left [\left. \frac{S_{t}} {e^{r(t-s)}}\right \vert \mathcal{F}_{s}\right ]& =& E\left [\left.E\left [\left. \frac{S_{t}} {e^{r(t-s)}}\right \vert \mathcal{F}_{t-1}\right ]\right \vert \mathcal{F}_{s}\right ] {}\\ & =& E\left [\left. \frac{1} {e^{r(t-s-1)}}\mathop{\underbrace{ E\left [\left.\frac{S_{t}} {e^{r}} \right \vert \mathcal{F}_{t-1}\right ]}}\limits _{S_{t-1}}\right \vert \mathcal{F}_{s}\right ] {}\\ & =& \ldots \ = E\left [\left.\frac{S_{s+1}} {e^{r}} \right \vert \mathcal{F}_{s}\right ]\ = S_{s}. {}\\ \end{array}$$

1.3 A.3 VIX Index: Derivation Formula

We derive an analytical formula for the VIX index when the dynamics of S&P 500 belongs to our class. Defined S ^∗ as the forward price of S _t with maturity T − t, we start from the VIX definition:

$$\displaystyle{ \left (\frac{V IX_{t}} {100} \right )^{2} = \frac{2e^{r(T-t)}} {T - t} \left [\mathop{\underbrace{E^{Q}\left [\left.\frac{S_{T}-S^{{\ast}}} {S^{{\ast}}} \right \vert \mathcal{F}_{t}\right ]}}\limits _{({\ast})} -\mathop{\underbrace{ E^{Q}\left [\left.\ln \left (\frac{S_{T}} {S^{{\ast}}}\right )\right \vert \mathcal{F}_{t}\right ]}}\limits _{({\ast}{\ast})}\right ]. }$$

The quantity in (∗) is 0 since:

$$\displaystyle{ E^{Q}\left [\left.\frac{S_{T} - S^{{\ast}}} {S^{{\ast}}} \right \vert \mathcal{F}_{t}\right ] = \frac{1} {S_{t}e^{r(T-t)}}E^{Q}\left [\left.S_{ T}\right \vert \mathcal{F}_{t}\right ] - 1 = 0. }$$

Given the spot price S _t , we have $S_{T} = S_{t}\exp \left (\sum _{d=t+1}^{T}X_{d}\right )$ and by substituting in (∗∗) we get the following expression for VIX squared:

$$\displaystyle{ \left (\frac{V IX_{t}} {100} \right )^{2} = -\frac{2e^{r(T-t)}} {T - t} \mathop{\underbrace{E\left [\left.\sum _{d=t+1}^{T}\lambda _{1}V _{d} +\lambda _{0}h_{d}\right \vert \mathcal{F}_{t}\right ]}}\limits _{(\Delta )}. }$$

(A.7)

In order to compute the quantity $(\Delta )$ in (A.7) we use the mathematical induction method. $\forall \:l = t,\ldots,T$ we assume that:

$$\displaystyle{ E\left [\left.\sum _{d=t+1}^{T}\lambda _{ 1}V _{d} +\lambda _{0}h_{d}\right \vert \mathcal{F}_{l}\right ] = C(l; T) + D(l; T)h_{l+1} +\sum _{ d=t+1}^{l}\lambda _{ 1}V _{d} +\lambda _{0}h_{d} }$$

(A.8)

with C(T; T) = 0 and D(T; T) = 0. First, we notice that all the quantities on the right side of (A.8) are known given the information at time l.

• 1. Since V _t and h _t are respectively adapted and predictable process our assumption is true for l = T if C(T; T) = 0 and D(T; T) = 0.

• 2. We suppose the relation holds at time l + 1 and we prove it for time l using the property of iterated expectations.

$$\displaystyle{ E\left [\left.\sum _{d=t+1}^{T}\lambda _{ 1}V _{d} +\lambda _{0}h_{d}\right \vert \mathcal{F}_{l}\right ] = E\left [\left.E\left [\left.\sum _{d=t+1}^{T}\lambda _{ 1}V _{d} +\lambda _{0}h_{d}\right \vert \mathcal{F}_{l+1}\right ]\right \vert \mathcal{F}_{l}\right ]. }$$

(A.9)

The quantity on the right hand side of Eq. (A.9) is equal to:

$$\displaystyle{ E\left [\left.C(l + 1; T) + D(l + 1; T)h_{l+2} +\sum _{ d=t+1}^{l+1}\lambda _{ 1}V _{d} +\lambda _{0}h_{d}\right \vert \mathcal{F}_{l}\right ]. }$$

(A.10)

Substitute in (A.10) the definition of h _l+2 and get:

$$\displaystyle{ \left.\begin{array}{l} C(l + 1; T) +\alpha _{0}D(l + 1; T) + (\beta D(l + 1; T) +\lambda _{0})h_{t+1} +\sum _{ d=t+1}^{l}(\lambda _{1}V _{d} +\lambda _{0}h_{d}) \\ + E\left [\left.\left (\alpha _{1}D(l + 1; T) +\lambda _{1}\right )V _{l+1}\right \vert \mathcal{F}_{l}\right ]. \end{array} \right. }$$

From (7.5) we get:

$$\displaystyle{ \left.\begin{array}{l} C(l + 1; T) +\alpha _{0}D(l + 1; T) + \left [\left (\lambda _{0} +\lambda _{1}g(\theta )\right ) + \left (\beta +\alpha _{1}g(\theta )\right )D(l + 1; T)\right ]h_{t+1} \\ +\sum _{ d=t+1}^{l}\lambda _{1}V _{d} +\lambda _{0}h_{d} \end{array} \right. }$$

and, by comparison with (A.8), we get the following system:

$$\displaystyle{ \left \{\begin{array}{ll} C(l; T) & = C(l + 1; T) + D(l + 1; T)\alpha _{0} \\ D(l; T) & = \left [\lambda _{1}g(\theta ) +\lambda _{0}\right ] + \left (\alpha _{1}g(\theta )+\beta \right )D(l + 1; T)\end{array} \right. }$$

(A.11)

with final conditions C(T; T) = 0 and D(T; T) = 0.

We show that if the following two conditions are satisfied:

• α ₁ g(θ) + β < 1

• λ ₁ g(θ) + λ ₀ ≤ 0

the right hand of the Eq. (7.12) is positive, coherently with the fact of being equal to the squared VIX value. We notice that D(l; T) is a linear difference equation whose solution at time $l = t,\ \forall t \leq T$ is given by:

$$\displaystyle{ D(t; T) =\mathop{\underbrace{ \left [\lambda _{1}g(\theta ) +\lambda _{0}\right ]}}\limits _{\leq 0}\mathop{\underbrace{ \frac{1-\left [\alpha _{1}g(\theta )+\beta \right ]^{T-t}} {1-\left [\alpha _{1}g(\theta )+\beta \right ]} }}\limits _{>0}. }$$

(A.12)

The solution (A.12) and the positivity of α ₀ imply negative values for C(t; T):

$$\displaystyle\begin{array}{rcl} C(t; T)& =& \mathop{\underbrace{C(T; T)}}\limits _{=0} +\mathop{\underbrace{ D(T; T)}}\limits _{=0} +\alpha _{0}\sum _{l=t+1}^{T-1}\mathop{\underbrace{ D(l; T)}}\limits _{ <0} {}\\ & =& \alpha _{0}\left [\lambda _{1}g(\theta ) +\lambda _{0}\right ]\left \{\frac{T - t - 1 - [\alpha _{1}g(\theta )+\beta ]\frac{1-\left [\alpha _{1}g(\theta )+\beta \right ]^{(T-t)-1}} {1-\left [\alpha _{1}g(\theta )+\beta \right ]} } {1 -\left [\alpha _{1}g(\theta )+\beta \right ]} \right \}. {}\\ \end{array}$$

1.4 A.4 VIX Index: Autoregressive Model

In Eq. (7.6), we substitute the expression for h _t+1 and h _t using the VIX adjusted as in (7.14). We obtain

$$\displaystyle\begin{array}{rcl} & \frac{V IX_{t}^{adj}-C_{30}} {D_{30}} =\alpha _{0} + (\alpha _{1}g(\theta )+\beta )\frac{V IX_{t-1}^{adj}-C_{30}} {D_{30}} +\alpha _{1}(V _{t} - g(\theta )h_{t})\ \Rightarrow & {}\\ & V IX_{t}^{adj} =\alpha _{0}D_{30} + C_{30}\left [1 -\left (\alpha _{1}g(\theta )+\beta \right )\right ] + \left (\alpha _{1}g(\theta )+\beta \right )V IX_{t-1}^{adj} +\alpha _{1}D_{30}(V _{t} - g(\theta )h_{t}).& {}\\ \end{array}$$

We can easily observe that V IX _t ^adj is an AR(1) and it can be written as:

$$\displaystyle{ V IX_{t}^{adj} = int + slopeV IX_{ t-1}^{adj} +\tau _{t}. }$$

Trivially we have:

$$\displaystyle\begin{array}{rcl} int& =& \alpha _{0}D_{30} + C_{30}\left [1 -\left (\alpha _{1}g(\theta )+\beta \right )\right ] {}\\ slope& =& (\alpha _{1}g(\theta )+\beta ) {}\\ \tau _{t}& =& \alpha _{1}D_{30}(V _{t} - g(\theta )h_{t}). {}\\ \end{array}$$

Using the explicit solution (7.13) for C ₃₀ and D ₃₀ and by rearranging, we get a simple expression for int:

$$\displaystyle{ \begin{array}{lll} int& = & \alpha _{0}\left (\lambda _{1}g(\theta ) +\lambda _{0}\right )\frac{1-slope^{30}} {1-slope} +\alpha _{0}\left (\lambda _{1}g(\theta ) +\lambda _{0}\right )\left (29 - slope\frac{1-slope^{29}} {1-slope} \right ) \\ & = & 30\alpha _{0}\left (\lambda _{1}g\left (\theta \right ) +\lambda _{0}\right ).\end{array} }$$