4.1.2. Empirical Market

When hedging in practice, the underlying assumptions are not always respected and the *true* parameters are, of course, unknown. To quantify the effect of these difficulties, we test the hedging strategies on historical data in this section. All hedging ratios depend on the dynamic risk-neutral parameters, hence *vt*, *v*¯, *γ* and *ρ* vary over time, according to the changes in the historically observed implied volatility surfaces. Moreover, the adjusted Heston Delta-Vega hedge depends on the correlation and volatility of *v*¯ and *γ*, which we define as follows

$$\begin{cases} \rho\_{\mathcal{D}} = \rho\_{\mathcal{V}} = 0.95, \\\ a\_{\mathcal{D}} = \sqrt{\frac{\frac{1}{N} \sum\_{i=-N}^{-1} \log\left(\frac{\mathcal{D}\_{t\_{i+1}}}{\mathcal{D}t\_{i}}\right)^{2}}{\Delta t}}, \\\ a\_{\mathcal{V}} = \sqrt{\frac{\frac{1}{N} \sum\_{i=-N}^{-1} \log\left(\frac{\mathcal{T}t\_{i+1}}{\mathcal{T}t\_{i}}\right)^{2}}{\Delta t}}. \end{cases} \tag{37}$$

It can be quite challenging to determine the correlation between the Brownian motions driving the parameters, hence we assume it to be equal for each time-interval. Moreover, the volatility estimator only depends on past observations; the sum indices vary from −*N* to −1 with *t*−*<sup>N</sup>* = −1 year, where *t*0 = 0 indicates the starting date of the option.

In this test, we hedge an at-the-money European call option with one year maturity on the S&P-500 index. All hedging strategies are subjected to daily rebalances that are based on the parameters as seen on that date. The transaction costs are excluded from this test, as we are interested in the performance of the hedging strategies with respect to changes in *v*¯ and *γ*. Including transaction costs would increase the costs of the full dynamic Heston hedging strategy, as it involves more financial assets. This would bias the results and it is therefore best to exclude the transaction costs from the present test.

The test is repeated on a monthly basis from July 2006 to February 2013 and the performance is assessed by the mean error and mean squared error during the life of the option,

$$E\_{\text{Mean}}^{(j)} = \frac{1}{N+1} \sum\_{i=0}^{N} \Pi\_{\stackrel{j}{\star}T'}^{(j)} \quad E\_{\text{MSE}}^{(j)} = \frac{1}{N+1} \sum\_{i=0}^{N} \left(\Pi\_{\stackrel{j}{\star}T}^{(j)}\right)^2. \tag{38}$$

The time intervals of the hedging portfolios overlap in this set-up, since the test is repeated on a monthly basis and the option maturity is one year. However, all strategies depend on different initial conditions and therefore perform differently, despite the overlapping time-intervals. The results of this test are graphically presented in Figure 2.

The hedging performances are similar to the simulation results:


**Figure 2.** Mean error and mean squared error for different hedging strategies performed on monthly historical data.

We can conclude that respecting the assumptions of the underlying model (in this case, the Heston model) does not necessarily lead to more accurate future option prices. By taking changes of the *v*¯ and *γ* parameters into account, we are able to replicate option values more accurately both in a controlled (simulation) and uncontrolled (empirical) environment.

### **5. VIX Heston Model Results**
