**4. Hedge Test**

Before implementing the dynamic risk-neutral measure (the VIX Heston model) in a risk-management application, we first test its applicability from a theoretical point of view. For example, in theory, one should be able to hedge against future positions using today's implied volatility surface. This no longer applies when one assumes a risk-neutral measure that changes over time. To this end, we perform an experimental hedge test that determines which approach is more accurate in terms of future option prices, a dynamic or constant risk-neutral measure.

The plain Heston model assumes *v*¯ and *γ* to be constant, hence, from a theoretical point of view, it would be redundant to hedge against changes of these parameters. However, due to the dynamic behaviour of the implied volatility surface, *v*¯ and *γ* will change over time (see Figure 1). Thus, from an empirical point of view, the option price dynamics are subject to changes of these parameters. To support this claim, we compare three different hedging strategies. The first strategy is the *classical Delta-Vega hedge*, which does not take any changes of *v*¯ and *γ* into account. The replicating portfolio aims at hedging an option "A", with value *Ct*, by holding a certain amount of stocks and a different option with value *C* ˜ *t* (which is called option "B" from this point onwards), i.e.,

$$\begin{cases} \begin{aligned} \Pi\_t &= -\mathbb{C}\_t + \Delta^{(1)}(t)\mathbb{S}\_t + \Delta^{(2)}(t)\mathbb{C}\_t + B\_{t\prime} \\ \Pi\_0 &= 0, \end{aligned} \end{cases} \tag{19}$$

where *Bt* denotes the risk-free asset (for example, a bank account or a governmen<sup>t</sup> bond), which grows with constant risk-free rate *r*. Note that Option B depends on the same underlying market factors as Option A. Following Bakshi et al. (1997), we impose the so-called minimized variance constraints,

$$\begin{cases} \left< \mathbf{d} \Pi\_{t\prime} \mathbf{d} \mathbf{S}\_{t} \right> = 0, \\ \left< \mathbf{d} \Pi\_{t\prime} \mathbf{d} \mathcal{W}\_{t}^{v} \right> = 0, \end{cases} \tag{20}$$

where ·, · refers to the covariation between the two processes and *Wvt* is the Brownian motion which is independent from *St*, driving random changes in the volatility. By imposing these constraints, one obtains a portfolio that has no covariation with the underlying asset and its volatility. In other words, changes in the asset's value and changes in the asset's volatility will have neither direct nor indirect (through correlations) effect on the portfolio. These constraints give us the following hedge ratios,

$$\begin{cases} \Delta^{(1)}(t) = \frac{\partial \mathbb{C}}{\partial \mathbb{S}\_t} - \Delta^{(2)}(t) \frac{\partial \mathbb{C}}{\partial \mathbb{S}\_t}, \\\ \Delta^{(2)}(t) = \frac{\partial \mathbb{C}/\partial \mathbb{v}\_t}{\partial \mathbb{C}/\partial \mathbb{v}\_t}. \end{cases} \tag{21}$$

Under the assumptions of the Heston model, the portfolio dynamics are given by,

$$\begin{split} \mathrm{d}\Pi\_{l} &= \mathrm{d}t \left( -\frac{\partial \mathcal{C}}{\partial t} - \frac{1}{2} v\_{l} S\_{t}^{2} \frac{\partial^{2} \mathcal{C}}{\partial S\_{t}^{2}} - \frac{1}{2} \gamma^{2} v\_{l} \frac{\partial^{2} \mathcal{C}}{\partial v\_{l}^{2}} - \rho \gamma v\_{l} S\_{l} \frac{\partial^{2} \mathcal{C}}{\partial S\_{l} \partial v\_{l}} + r B\_{l} \\ &+ \Delta^{(2)}(t) \left( \frac{\partial \mathcal{C}}{\partial t} + \frac{1}{2} v\_{l} S\_{t}^{2} \frac{\partial^{2} \mathcal{C}}{\partial S\_{t}^{2}} + \frac{1}{2} \gamma^{2} v\_{l} \frac{\partial^{2} \mathcal{C}}{\partial v\_{l}^{2}} + \rho \gamma v\_{l} S\_{t} \frac{\partial^{2} \mathcal{C}}{\partial S\_{l} \partial v\_{l}} \right) \right), \end{split} \tag{22}$$

Note that the random components have disappeared from the portfolio. Thus, the portfolio should be insensitive to changes in the market, if it respects the assumptions of the Heston model.

Secondly, we assume a model which is similar to the classical Delta-Vega hedge, but with adjusted hedge ratios. We call this strategy the *adjusted Delta-Vega hedge*. Assuming a dynamic model (see Appendix B for more details), we can apply Ito's lemma to obtain

$$\mathbf{d}\,\mathbf{C}\_{t}^{\text{Dynamic}} = \frac{\partial \mathbf{C}}{\partial t}\mathbf{d}t + \sum\_{p\_t} \frac{\partial \mathbf{C}}{\partial p\_t} \mathbf{d}p\_t + \frac{1}{2} \sum\_{p\_t} \sum\_{q\_t} \frac{\partial^2 \mathbf{C}}{\partial p\_t \partial q\_t} \left< \mathbf{d}p\_t, \mathbf{d}q\_t \right>,\tag{23}$$

with *pt*, *qt* ∈ {*St*, *vt*, *<sup>v</sup>*¯*t*, *<sup>γ</sup>t*}. For notational purposes, we rewrite this expression as

$$\mathbf{d}\mathbf{C}\_{t}^{\text{Dynamic}} = c\_{1}\mathbf{d}t + c\_{2}\mathbf{d}\mathcal{W}\_{t}^{\text{S}} + c\_{3}\mathbf{d}\mathcal{W}\_{t}^{\text{v}} + c\_{4}\mathbf{d}\mathcal{W}\_{t}^{\text{v}} + c\_{5}\mathbf{d}\mathcal{W}\_{t}^{\text{v}}.\tag{24}$$

where the coefficients are defined as 

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *c*1 = *∂C ∂t* + *rSt ∂C ∂St* + *<sup>κ</sup>*(*v*¯*t* − *vt*) *∂C ∂vt* + *<sup>κ</sup>v*¯(*v*¯Mean − *<sup>v</sup>*¯*t*) *∂C ∂v*¯*t* <sup>+</sup>*κγ*(*<sup>γ</sup>*Mean − *<sup>γ</sup>t*) *∂C∂γt* + 1 2 ∑*pt* ∑*qt ∂*2*C ∂pt∂qt* d*pt*, d*qt*, *c*2 = √*vtSt ∂C ∂St* + *ργt* √*vt ∂C ∂vt* + *ρρv*¯*av*¯*v*¯*<sup>t</sup> ∂C ∂v*¯*t* + *ρργaγγt ∂C∂γt* , *c*3 = *γt vt*(<sup>1</sup> − *ρ*<sup>2</sup>) *∂C ∂vt* + *ρv*¯*av*¯*v*¯*<sup>t</sup>* 1 − *ρ*2 *∂C ∂v*¯*t* + *ργaγγt* 1 − *ρ*2 *∂C∂γt* , *c*4 = *av*¯*v*¯*<sup>t</sup>* 1 − *ρ*2 *v*¯ *∂C ∂v*¯*t* , *c*5 = *<sup>a</sup>γγt* 1 − *ρ*2 *γ ∂C∂γt* . (25)

The hedge ratios now take the correlated components of *v*¯ and *γ* into account, due to the minimized variance constraints of Equation (20). Using Equation (24), the hedge ratios can be derived, giving 

$$\begin{cases} \Delta\_{\text{Adjusted}}^{(1)}(t) = \frac{c\_2}{\sqrt{v\_l S\_l}} - \frac{\mathcal{E}\_2 c\_3}{\sqrt{v\_l S\_l} c\_3}, \\ \Delta\_{\text{Adjusted}}^{(2)}(t) = \frac{c\_3}{\mathcal{E}\_3}, \end{cases} \tag{26}$$

where *c*˜2 and *c*˜3 are defined as in Equation (25) for Option B. The additional stochastic variables *v*¯*t* and *γt* follow mean reverting processes (see Equation (A1) in Appendix B), such that the portfolio dynamics are found to be,

$$\begin{split} \mathbf{d}\Pi\_{t} &= \left(-\mathbf{c}\_{1} + \frac{\mathbf{c}\_{3}}{\tilde{\mathbf{c}}\_{3}}\tilde{\mathbf{c}}\_{1} + \frac{\tilde{\mathbf{c}}\_{2}\mathbf{r}}{\sqrt{\mathbf{v}\_{t}}} - \frac{\tilde{\mathbf{c}}\_{2}\mathbf{c}\_{3}\mathbf{r}}{\sqrt{\mathbf{v}\_{t}}\tilde{\mathbf{c}}\_{3}} + r\mathcal{B}\_{t}\right) \mathbf{d}t \\ &+ \left(\frac{\mathbf{c}\_{3}}{\tilde{\mathbf{c}}\_{3}}\tilde{\mathbf{c}}\_{4} - \mathbf{c}\_{4}\right) \mathbf{d}\mathcal{W}\_{t}^{\mathcal{D}} + \left(\frac{\mathbf{c}\_{3}}{\tilde{\mathbf{c}}\_{3}}\tilde{\mathbf{c}}\_{5} - \mathbf{c}\_{5}\right) \mathbf{d}\mathcal{W}\_{t}^{\mathcal{I}}.\end{split} \tag{27}$$

The portfolio still depends on the randomness associated with *v*¯ and *γ*. However, the randomness associated with *St* and *vt* have disappeared, including the random components of *v*¯ and *γ* that are correlated to *St* and *vt*.

We also consider a strategy that aims at completely hedging against any changes of *v*¯ and *γ*, by introducing two additional options,

$$\begin{cases} \begin{aligned} \Pi\_{t} &= -\mathbb{C}\_{t} + \Delta\_{\text{Full}}^{(1)}(t)\mathbb{S}\_{t} + \Delta\_{\text{Full}}^{(2)}(t)\bar{\mathbb{C}}\_{t} + \Delta\_{\text{Full}}^{(3)}(t)\bar{\mathbb{C}}\_{t} + \Delta\_{\text{Full}}^{(4)}(t)\hat{\mathbb{C}}\_{t}, \\\ \Pi\_{0} &= 0. \end{aligned} \end{cases} \tag{28}$$

Again, all options depend on the same underlying market factors, but they have different contract details. In this case, we require to be protected against any changes of *St*, *vt*, *v*¯*t* and *γt*, hence we impose

$$\begin{cases} \langle \mathbf{d} \Pi\_{t\prime} \mathbf{d} \mathbf{S}\_{t} \rangle = 0, \\ \langle \mathbf{d} \Pi\_{t\prime} \mathbf{d} \mathbf{W}\_{t}^{\mathrm{p}} \rangle = 0, \\ \langle \mathbf{d} \Pi\_{t\prime} \mathbf{d} \mathbf{W}\_{t}^{\mathrm{p}} \rangle = 0, \\ \langle \mathbf{d} \Pi\_{t\prime} \mathbf{d} \mathbf{W}\_{t}^{\mathrm{p}} \rangle = 0. \end{cases} \tag{29}$$

Substituting these constraints leads to a system of equations, which is solved by

$$
\begin{bmatrix}
\Delta\_{\text{Full}}^{(1)}(t) \\
\Delta\_{\text{Full}}^{(2)}(t) \\
\Delta\_{\text{Full}}^{(3)}(t) \\
\Delta\_{\text{Full}}^{(4)}(t)
\end{bmatrix} = \begin{bmatrix}
\sqrt{\upsilon\_t}S\_t & \varepsilon\_2 & \varepsilon\_2 & \varepsilon\_2 \\
0 & \varepsilon\_3 & \varepsilon\_3 & \varepsilon\_3 \\
0 & \varepsilon\_4 & \varepsilon\_4 & \varepsilon\_4 \\
0 & \varepsilon\_5 & \varepsilon\_5 & \varepsilon\_5
\end{bmatrix}^{-1} \begin{bmatrix}
c\_2 \\ c\_3 \\ c\_4 \\ c\_5
\end{bmatrix} \tag{30}
$$

By imposing Equation (29), one removes all randomness associated with *St*, *vt*, *v*¯*t* and *γt*. Hence, the portfolio dynamics only depend on deterministic changes under the assumed market dynamics. From this point onwards, we refer to this strategy as the *full hedge* strategy.
