Assessing the effectiveness of local and global quadratic hedging under GARCH models

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Notes

No potential conflict of interest was reported by the authors.

1 Global risk-minimization under a quadratic penalty function is also known as variance-optimal hedging and mean–variance hedging.

2 Note that our definition of the value of the hedging portfolio at time t (see definition 2) corresponds to the one adopted by Lamberton et al. (1998), and thus differs slightly from the one given in Schweizer (1988). However, both definitions lead to equivalent solutions as shown by Lamberton et al. (1998, section 4.1).

3 In the context of quadratic hedging, it is generally assumed that is square-integrable (e.g. Schweizer 1995). Accordingly, the assumptions on the existence of exponential moments and on second-order stationarity are included because they ensure that moments of are well-defined. However, we note that Gugushvili (2003) and Melnikov and Nechaev (1999) were able to weaken the square-integrability requirements commonly encountered in the quadratic hedging literature, and remove the nondegeneracy condition of Schweizer (1995).

4 The stationary (or unconditional) variance corresponds to

provided that the denominator is positive.

5 Delta hedging under a GARCH model is somewhat of an ambiguous concept because this model is incomplete and defined in discrete time. Different approaches to specify a continuous time limit could, in principle, lead to different definitions of delta hedging. Accordingly, Duan (1995)’s definition of the delta was subject to criticism, notably by Garcia and Renault (1998). Nevertheless, Badescu et al. (2014) show that this delta appears as a component of an approximate locally risk-minimizing strategy by letting the time between two consecutive observations approach zero.

6 This equation is obtained as follows:

7 For a comparison of option prices under different pricing measures, including measures related to the local and global quadratic hedging problems in continuous time, see Henderson and Hobson (2003), Henderson (2005) and Henderson et al. (2005).

8 Ortega (2012) showed that when the drift term in the GARCH model is ‘sufficiently small’ the linear Taylor expansions of the locally risk-minimizing value process in the drift term under measures and coincide. Based on this result, he asserted that ‘carrying out the risk-minimizing program with respect to the physical measure or the equivalent martingale measure ...yields virtually the same results’.

9 Note that in this experiment we assume that the stock price process is generated from a physical probability measure that is distinct from (and ). The hedging strategy is therefore constructed based on a mis-specified Gaussian GARCH model. For an alternative way to assess the robustness of quadratic hedging to model risk, see Di Nunno et al. (2015).

10 The maximum likelihood parameter estimates of the GARCH model based on the S&P 500 price index for the period 31 December 1986–1 April 2008 are: , , , and . These estimates are very close to the ones provided in table 1.

11 This delta hedge is computed under the minimal equivalent martingale measure in Heston’s model, which is known explicitly, and thus corresponds to the locally risk-minimizing delta hedge under the physical probability measure of this model. Although this delta hedge has the advantage of being available in semi-closed-form, it requires us to infer the unobserved instantaneous variance in Heston’s model. To infer this variance, we use a particle filtering algorithm constructed similarly to the one proposed by Christoffersen et al. (2010b, section 3). The particle filtering algorithm is applied on a discretized form of Heston’s model. It aims to approximate the conditional distribution of the instantaneous variance given the observed data (known as the filtering distribution) sequentially in time with a discrete distribution whose support points are called particles.

12 Unlike risk-minimizing delta hedges, the GARCH delta hedge ignores that the stock price and its volatility could be related through a leverage effect. This well-documented stylized fact of stock market returns entails that negative returns are generally followed by an increase in volatility. Therefore, when returns are negative, the call option directly loses value through the stock price decrease and indirectly gains value through an expected higher future volatility, resulting in a natural partial hedge. This feature is not taken into account by the GARCH delta hedge, which explains why it overhedges the call option with a larger stock position than is needed. See also Badescu et al. (2014, p. 21) for a mathematical justification of this observation.

Additional information

Funding

Maciej Augustyniak gratefully acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada [RGPIN-2015-05066]. Frédéric Godin gratefully acknowledges financial support from the Fonds de recherche du Québec – Nature et technologies [2017-NC-197517].