4.1.1. Simulated Market

First, we evaluate the hedging strategies in a controlled environment. In this test, we assume *St* and *vt* to follow the Heston model, with the key difference that *v*¯ and *γ* are time-dependent mean reverting processes (as in Equation (A1) in Appendix B). This set-up will be informative, as it is in line with historical observations. The processes of *St* and *vt* are discretized by the Quadratic Exponential (QE) scheme, as proposed in Andersen (2008). Furthermore, the processes of *v*¯ and *γ* are discretized by the Milstein scheme, while also taking the correlations with *vt* into account. The details of this simulation scheme can be found in Appendix B.

The performance of the classical Heston Delta-Vega, the adjusted Heston Delta-Vega and the full hedge is compared under the dynamics of this market. The strategies all aim at hedging a short position in a European call option with maturity *T* = 1.0 years and strike *K* = 50. Moreover, at the start date of the option, *t* = 0, we assume

$$\begin{cases} \text{ } \text{S}\_0 = 49, \quad \upsilon\_0 = 0.05, \quad \upsilon\_0 = 0.1, \quad \gamma\_0 = 0.7, \\\ \text{ } \kappa = 1.0, \quad \rho = -0.75, \quad r = 0.01. \end{cases} \tag{31}$$

The adjusted Delta-Vega and full hedge involve additional options. These options depend on the same stock as Option A, but the contract details are different, i.e.,

$$\begin{cases} \begin{array}{ll} \& \mathbb{R} = 50.0, & \text{ $\mathcal{T}$  = 2.0, $} \\ \mathbb{R} = 50.0, & \text{$ \mathcal{T} $ = 3.0,$ } \\ \hat{\mathbb{K}} = 50.0, & \hat{\mathbb{T}} = 4.0. \end{array} \end{cases} \tag{32}$$

Moreover, we assume the following parameters in the dynamics of the *v*¯ and *γ* processes,

$$\begin{cases} \kappa\_{\mathcal{V}} = 1.4, \quad \vartheta\_{\text{Mean}} = 0.1, \quad a\_{\mathcal{V}} = 0.8, \quad \rho\_{\mathcal{V}} = 0.9, \\\ \kappa\_{\mathcal{V}} = 2.1, \quad \gamma\_{\text{Mean}} = 0.7, \quad a\_{\mathcal{V}} = 1.0, \quad \rho\_{\mathcal{V}} = 0.9. \end{cases} \tag{33}$$

In reality, these parameters cannot be freely chosen, as they are implied by the market. By analysing the historical behaviour of *v*¯*t* and *γt*, one is able to estimate these SDE parameters. In this case, however, we assume to know these parameters and use them when determining the hedge strategy.

Ideally, the portfolio value should be equal to zero for each point in time, because the initial portfolio value is equal to zero. Every deviation from zero is thought of as a hedge error. Over the entire life of the option we desire the mean and standard deviation of this error to be as close as possible to zero. To this end, we introduce the following error measures for simulation *j*,

$$E\_{\text{Mean}}^{(j)} = \frac{1}{N+1} \sum\_{i=0}^{N} \Pi\_i^{(j)} \qquad E\_{\text{Std}}^{(j)} = \sqrt{\frac{1}{N} \sum\_{i=0}^{N} \left( \Pi\_i^{(j)} - E\_{\text{Mean}}^{(j)} \right)^2}. \tag{34}$$

The overall hedging performance of the *M* simulations can be judged by

$$\begin{cases} \begin{array}{l} E\_{\text{Mean}} = \frac{1}{\mathcal{M}} \sum\_{j=1}^{M} E\_{\text{Mean}'}^{(j)} \\\ E\_{\text{Std}} = \frac{1}{\mathcal{M}} \sum\_{j=1}^{M} E\_{\text{Std}}^{(j)} \end{array} \end{cases} \tag{35}$$

In the current set-up, these error measures are random variables, as they are determined by means of Monte Carlo simulations. To this end, we analyse the stability of the error measures across the simulated trajectories by the standard error,

$$\text{SE}(E) = \frac{\sqrt{\frac{1}{M-1} \sum\_{j=1}^{M} \left(E - E^{(j)}\right)^2}}{\sqrt{M}},\tag{36}$$

with error measure *E*, its mean *E* ¯ and the simulated trajectories *E*(*j*).

The performance of the strategies under these assumptions based on *M* = 200 simulations is given in Table 1.

**Table 1.** Hedge errors of the different hedging strategies. The standard errors of the estimates are given in parentheses.


These results show that, in this experiment, it is beneficial to take parameter correlations into account when hedging, in terms of both the mean error and the standard deviation. While still not perfect, the dynamic Heston Delta-Vega hedge is better able to remain risk-neutral on average and it deviates less from this average. The full hedge performs even better with a mean approximately equal to zero and a standard deviation equal to or lower than any of the previous strategies, despite the dynamic behaviour of the market.

The purpose of these hedging strategies is to replicate the value of an option. In the case of the classical Delta-Vega hedge, only *St* and *vt* are allowed to change. By respecting the assumptions of the Heston model, we are not fully able to replicate the future option values. On the other hand, when assuming the adjusted Heston model, *v*¯*t* and *γt* are allowed to change as well. Hedging strategies considering this dynamic behaviour, produce more accurate future option price estimates in the current set-up. This indicates the importance of taking dynamic parameters into account when determining future option prices, even when the assumptions of the underlying model are violated. However, note that the comparison is not completely fair, since we specifically assume a Heston market with time-dependent *v*¯ and *γ*. It can therefore be expected that a strategy taking these assumptions into account outperforms one that does not. Therefore, to better assess the *true* performance of these strategies, we also perform an empirical test.
