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Review of Derivatives Research

Erschienen in:

02.08.2017

Risk-adjusted option-implied moments

verfasst von: Felix Brinkmann, Olaf Korn

Erschienen in: Review of Derivatives Research | Ausgabe 2/2018

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Abstract

This paper provides a new way of converting risk-neutral moments into the corresponding physical moments, which are required for many applications. The main theoretical result is a new analytical representation of the expected payoffs of put and call options under the physical measure in terms of current option prices and a representative investor’s preferences. This representation is then used to derive analytical expressions for a variety of ex-ante physical return moments, showing explicitly how moment premiums depend on current option prices and preferences. As an empirical application of our theoretical results, we provide option-implied estimates of the representative stock market investor’s disappointment aversion using S&P 500 index option prices. We find that disappointment aversion has a procyclical pattern. It is high in times of high index levels and declines when the index falls. We confirm the view that investors with high risk aversion and disappointment aversion leave the stock market during times of turbulence and reenter it after a period of high returns.

Nächster Artikel Optimal discrete hedging of American options using an integrated approach to options with complex embedded decisions

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For example, the volatility index VIX uses a model-free approach similar to the one proposed in Britten-Jones and Neuberger (2000) and serves itself as the underlying of a fast-growing derivatives market (Drimus and Farkas 2013).

These papers use a representative investor with a specific utility function. Ross (2015) develops an alternative method by imposing restrictions on the dynamics of the stochastic discount factor. However, Borovicka et al. (2015) point out that the latter approach suffers from identification problems.

See Theorem 2 of Bakshi et al. (2003).

See Theorem 1 of Bakshi and Madan (2006).

As Bliss and Panigirtzoglou (2004), p. 409, state, “the origins of the development of this result have proven difficult to trace.” A derivation of the result can be found, for example, in Aït-Sahalia and Lo (2000), p. 13ff.

The utility function has to satisfy only mild conditions. It must be twice continuously differentiable with \(U'>0\) and \(U''<0\), a condition fulfilled by many common utility functions, such as the class discussed by Brockett and Golden (1987) and the hyperbolic absolute risk aversion (HARA) class.

The corresponding risk-neutral model-free implied moments are given in Kozhan et al. (2013).

See Christoffersen et al. (2012) for a presentation of the corresponding risk-neutral model-free implied moments.

See Jiang and Tian (2005) for the corresponding result under the risk-neutral measure.

See the survey articles by Poon and Granger (2003) and Christoffersen et al. (2012).

For the standard definition of skewness, it is unclear what a reasonable realized moment would be.

Digital options on the S&P 500 index trade on the CBOE since July 2008. These digital options, however, are very illiquid.

It is important to note that, at this point, we still do not have to assume any particular option pricing model, such as the Black–Scholes model. Given that our volatility curve is continuously differentiable, the prices obtained from a duplication strategy consisting of k long call options with strike price K and k short call options with strike price \(K+(1/k)\) will converge to the prices we use for \(k\rightarrow \infty \).

See Kozhan et al. (2013) for details.

The utility function with disappointment aversion is not differentiable at \(W=F_{t,t+\tau }\), which violates the requirements of Proposition 1. It is no problem, however, to approximate the utility function in a small interval around \(F_{t,t+\tau }\) with a twice continuously differentiable function. That is how we proceed.

In general, the reference point is the implicitly defined certainty-equivalent wealth at time \(t+\tau \) that depends on the endogenously determined portfolio of the investor. Since the representative investor holds the market, the certainty equivalent equals the forward price.

The main results of increasing call payoffs with larger risk aversion and disappointment aversion as well as decreasing put payoffs with larger risk aversion are also stable over time and do not only reflect the particular situation on January 20, 2004. Moreover, the effects are generally stronger for at-the-money options than for out-of-the-money options.

Mehra and Prescott (1985), p. 154, cite a number of papers arguing that relative risk aversion falls in the range between 1 and 2. However, they allow for values of up to 10 in their own study and a more recent paper by Azar (2006) estimates a value of 4.5. Ang et al. (2005) vary the coefficient of relative risk averison between 2 and 10 and consider disappointment aversion coefficients between 1.0 and 0.6. We allow for an even broader range of parameters in our optimization to be on the safe side.

See Proposition 19.13 of Alsmeyer (2003). Special cases of this result in which \(\mu \) is a probability measure and f is a random variable can be found in many textbooks, such as Lemma 6.1. of Feller (1971).

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Titel: Risk-adjusted option-implied moments
verfasst von: Felix Brinkmann
Olaf Korn
Publikationsdatum: 02.08.2017
Verlag: Springer US
Erschienen in: Review of Derivatives Research / Ausgabe 2/2018
Print ISSN: 1380-6645
Elektronische ISSN: 1573-7144
DOI: https://doi.org/10.1007/s11147-017-9136-4