Lattice-based hedging schemes under GARCH models

Abstract

An efficient way to implement quadratic hedging schemes for European options when the asset return process follows an asymmetric non-affine GARCH model driven by Gaussian innovations

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 There have been several alternative approaches proposed in the literature to deal with option pricing in non-affine GARCH models. For example, Duan et al. (1999, 2006) provide numerical approximations to European option prices in several asymmetric GARCH models using the Edgeworth expansion, while Duan and Simonato (2001) propose a Markov chain approximation of a GARCH model for pricing American options. Other contributions include the use of dynamic programming (Ben-Ameur et al. 2009), PDE techniques (Breton and Frutos 2010) and quadrature methods (Simonato 2019).

2 Depending on the choice of the objective function in the optimization problem, other hedging criteria have been proposed in a GARCH context; see e.g. Augustyniak et al. (2017) and Rémillard and Rubenthaler (2013) for global risk-minimization and Huang and Guo (2012) for risk-adjusted cost of hedging optimization.

3 In addition to this advantage of the Q-LRM implementation, Augustyniak and Badescu (2020) showed that the Q-LRM hedging ratio admits a semi-closed form solution for affine Gaussian GARCH models; such an explicit solution for the P-LRM does not exist. Ortega (2012) argued that in general there should be no significant differences between the P-LRM and Q-LRM hedging ratios, while Augustyniak et al. (2017) contrasted the effectiveness of local and global quadratic hedging strategies under GARCH models, and investigated the impact of the choice of measure on hedging effectiveness.

4 The LRM hedging scheme described in this paper can be applied to any European contingent claim, but here we only discuss the lattice approximation for European call options. The hedging of other financial derivatives can be implemented in a similar way.

5 The MMM is typically a signed measure in discrete time settings as the process $N_t$ can take negative values, which may lead to negative prices. However, this is rarely the case when the asset dynamics follow the GARCH model in (1(1) $\begin{array}{rl}{y}_t& =y_{t-1}+r+\lambda \sqrt{h_t}-\frac{1}{2}{h}_t+\sqrt{h_t}{ϵ}_t,\\ h_{t+1}& ={\text{β}}_0+{\text{β}}_1{h}_t+{\text{β}}_2{h}_t({ϵ}_t-c{)}^2,\\ {ϵ}_t\mid {\mathcal{F}}_{t-1}& \sim N(0,1).\end{array}$(1) ). Ortega (2012) showed that allowing for bounded innovations within a GARCH setting represents a sufficient condition for the derivative prices to be well-defined.

6 Although the computation of the optimal hedge ratio in (8(8) ${\xi }_{t+1}=\frac{{\mathrm{C}\mathrm{o}\mathrm{v}}^P\left({\mathrm{e}}^{-rT}H\prod _{k=t+2}^T{N}_k,{\tilde{S}}_{t+1}-{\tilde{S}}_t|{\mathcal{F}}_t\right)}{{\mathrm{V}\mathrm{a}\mathrm{r}}^P\left[{\tilde{S}}_{t+1}-{\tilde{S}}_t|{\mathcal{F}}_t\right]}.$(8) ) using Monte Carlo is feasible, its implementation for hedging longer term options under daily rebalancing may be demanding computationally. Moreover, if hedging is performed at a lower frequency than the underlying is observed (e.g. weekly or monthly), so that the hedge ratio is kept unchanged over the period $[t,t+i)$ with i>1 (note that in this case the P-LRM is obtained by minimizing ${\mathbb{E}}^P\left[\right({\tilde{C}}_{t+i}-{\tilde{C}}_t{)}^2\mid {\mathcal{F}}_t]$), then a closed-form expression for $N_t$ is no longer available; thus, one has to use nested Monte Carlo simulations for its evaluation, making the computation of (8(8) ${\xi }_{t+1}=\frac{{\mathrm{C}\mathrm{o}\mathrm{v}}^P\left({\mathrm{e}}^{-rT}H\prod _{k=t+2}^T{N}_k,{\tilde{S}}_{t+1}-{\tilde{S}}_t|{\mathcal{F}}_t\right)}{{\mathrm{V}\mathrm{a}\mathrm{r}}^P\left[{\tilde{S}}_{t+1}-{\tilde{S}}_t|{\mathcal{F}}_t\right]}.$(8) ) untractable (see e.g. Badescu et al. 2014, for details). This makes our LRM lattice implementation even more appealing since it can be easily constructed for different frequencies.

7 Note that based on equation (5(5) ${\tilde{V}}_t={\mathbb{E}}^P\left[{\tilde{V}}_{t+1}{N}_{t+1}|{\mathcal{F}}_t\right],$(5) ), one could also use $V_t={\mathrm{e}}^{-r}{\mathbb{E}}^P[V_{t+1}{N}_{t+1}\mid {\mathcal{F}}_t]$ as an alternative and equivalent relationship to equation (10(10) $\begin{array}{rl}{V}_t& ={\mathrm{e}}^{-r}{\mathbb{E}}^P\left[V_{t+1}\mid {\mathcal{F}}_t\right]-{\xi }_{t+1}{S}_t\left({\mathrm{e}}^{\lambda \sqrt{h_{t+1}}}-1\right).\end{array}$(10) ) for computing the lattice-based option price under the MMM.

8 We note that if one chooses to perform LRM with respect to the MMM $Q^{min}$ instead of Duan's LRNVR, then the optimal option prices given in (7(7) $V_t={\mathrm{e}}^{-r(T-t)}{\mathbb{E}}^P\left[H\prod _{k=t+1}^T{N}_k|{\mathcal{F}}_t\right].$(7) ) and (15(15) $\begin{array}{rl}{V}_t& ={\mathrm{e}}^{-r(T-t)}{\mathbb{E}}^Q\left[H\mid {\mathcal{F}}_t\right].\end{array}$(15) ) are the same, but the optimal hedge ratios computed according to (8(8) ${\xi }_{t+1}=\frac{{\mathrm{C}\mathrm{o}\mathrm{v}}^P\left({\mathrm{e}}^{-rT}H\prod _{k=t+2}^T{N}_k,{\tilde{S}}_{t+1}-{\tilde{S}}_t|{\mathcal{F}}_t\right)}{{\mathrm{V}\mathrm{a}\mathrm{r}}^P\left[{\tilde{S}}_{t+1}-{\tilde{S}}_t|{\mathcal{F}}_t\right]}.$(8) ) and (14(14) $\begin{array}{rl}{\xi }_{t+1}& ={\mathrm{e}}^{-r(T-t)}\frac{{\mathbb{E}}^Q\left[H\left(S_{t+1}{\mathrm{e}}^{-r}-S_t\right)\mid {\mathcal{F}}_t\right]}{{\mathrm{V}\mathrm{a}\mathrm{r}}^Q\left[S_{t+1}{\mathrm{e}}^{-r}-S_t\mid {\mathcal{F}}_t\right]},\end{array}$(14) ) are nonetheless different.

9 For the conditional distribution of $y_{t+1}^a$ to have mean ${\mu }_{t+1}^a$ and variance $h_{t+1}^a$, the expectation and variance of the i.i.d. log-price changes over each subinterval of length 1/n in the day must be equal to ${\mu }_{t+1}^a/n$ and $h_{t+1}^a/n$, respectively. The log-price change from t to $t+1/n$ is either $(a{\gamma }_n/n+\eta {\gamma }_n)$, $\left(a{\gamma }_n/n\right)$ or $(a{\gamma }_n/n-\eta {\gamma }_n)$ with respective probabilities $p_u$, $p_m$ and $p_d$. To match a mean of ${\mu }_{t+1}^a/n$ and a variance of $h_{t+1}^a/n$, these probabilities must satisfy the following two equations: $\begin{array}{rl}& a{\gamma }_n/n+\eta {\gamma }_n{p}_u-\eta {\gamma }_n{p}_d={\mu }_{t+1}^a/n,\mathrm{a}\mathrm{n}\mathrm{d}\\ & {\eta }^2{\gamma }_n^2{p}_u+{\eta }^2{\gamma }_n^2{p}_d-{\left(\eta {\gamma }_n{p}_u-\eta {\gamma }_n{p}_d\right)}^2=h_{t+1}^a/n.\end{array}$ Solving these equations for $p_u$ and $p_d$, combined with the fact that $p_m=1-p_u-p_d$, gives the above expressions.

10 The quantity $h_{t+2}^a(\cdot ,k^{\ast }+1)$ is to be interpreted as ∞ when $k^{\ast }=K$.

Additional information

Funding

Maciej Augustyniak and Alexandru Badescu gratefully acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC). Zhiyu Guo gratefully acknowledges financial support from the National Natural Science Foundation of China (Nos. 71532001 and 11631004).