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Review of Quantitative Finance and Accounting

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06.05.2016 | Original Research

Retrieving risk neutral moments and expected quadratic variation from option prices

verfasst von: Leonidas S. Rompolis, Elias Tzavalis

Erschienen in: Review of Quantitative Finance and Accounting | Ausgabe 4/2017

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Abstract

This paper derives exact formulas for retrieving risk neutral moments of future payoffs of any order from generic European-style option prices. It also provides an exact formula for retrieving the expected quadratic variation of the stock market implied by European option prices, which nowadays is used as an estimate of the implied volatility, and a formula approximating the jump component of this measure of variation. To implement the above formulas to discrete sets of option prices, the paper suggests a numerical procedure and provides upper bounds of its approximation errors. The performance of this procedure is evaluated through a simulation and an empirical exercise. Both of these exercises clearly indicate that the suggested numerical procedure can provide accurate estimates of the risk neutral moments, over different horizons ahead. These can be in turn employed to obtain accurate estimates of risk neutral densities and calculate option prices, efficiently, in a model-free manner. The paper also shows that, in contrast to the prevailing view, ignoring the jump component of the underlying asset can lead to seriously biased estimates of the new volatility index suggested by the Chicago Board Options Exchange.

Vorheriger Artikel The impact of nominal stock price on ex-dividend price responses

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To improve over their performance, they have been suggested semi-parametric methods, but these depend on the correct specification of the underlying asset price process (Corrado and Su 1996; Melick and Thomas 1997; Steeley 2004).

Note that the original VIX now is known as the VXO.

This fundamental option pricing theorem also implies the opposite. That is, the knowledge of the RNCF of function F(A) enables us to price European-style options written on $A_{S}(T_{0},T)$. See Chourdakis (2004) for a numerical implementation.

Note that several recent papers have relaxed the semimartingale assumption. There are asset pricing models in which absence of arbitrage opportunities exist, even if the asset price process is not a semimartingale (Coviello and Russo 2006, and references therein). In brief, these models include, among others, microstructure noise, discrete trading, and insider trading exploiting private information. The examination of these models is beyond the scope of this paper.

Note that if $r=\delta =0$, then $E_{t}^{Q}\left[ \int _{t+}^{T}\frac{dS_{u}}{ S_{u-}}-\frac{S_{T}-S_{t}}{S_{t}}\right] =1$. In this case, formula (12) reduces to that provided by Britten-Jones and Neuberger (2000) under the assumption that the underlying asset price follows a diffusion process, without discontinuous component.

Note that there is a small difference in the formula employed by the CBOE and Eq. (13). Instead of $S_{t}$, the CBOE formula uses the first strike price below the futures price $F_{t}$, denoted as $K^{*}$, as integration bound. This means that the second term of $IV^{*}$ can be written as

$$\frac{2}{\tau }\left[ \ln \left( \frac{F_{t}}{K^{*}}\right) -\left( \frac{F_{t}}{K^{*}}-1\right) \right] \simeq -\frac{1}{\tau }\left( \frac{ F_{t}}{K^{*}}-1\right) ^{2},$$

so as for the two formulas to be consistent with each other (Jiang and Tian 2007).

Note that interpolation and extrapolation of implied volatilities, instead of option prices themselves, is done in order to avoid numerical difficulties in fitting smooth functions into option prices. To convey the observed option prices into implied volatilities and vise versa, we use the Black–Scholes (BS) formula. This methodology does not require the BS model to be the true option pricing model.

The function L(m) takes the following values for $m=2, \ldots ,6:L(2)=1.47$, $L(3)=3.24$, $L(4)=10.75$, $L(5)=46.88$ and $L(6)=252.67$.

Note that, in our simulation experiments presented in the next section, almost all the log-moneyness intervals are found to be between the values of the endpoints $y_{0}=-1$ and $y_{\infty }=0.57$. In 1000 iterations, it was found only one case where $y_{0}=-1.3,$ which lies outside the above interval $[y_{0}=-1,y_{\infty }=0.57]$. This happened when a linear function was used in the extrapolation scheme, instead of a constant.

This source of error is known as discretization error. By choosing a large number of knot points in the numerical integration technique, this error can be proved negligible. In our simulation study, to calculate the integrals we employ the Gaussian quadrature numerical procedure.

The full-specification of the diffusion-jump asset price process $(S_{u})_{u\in [t,T]}$ under the risk neutral measure Q for the SVJ model is given as follows:

$$\begin{aligned} dS_{t}& = S_{t}\left[ (r-\delta -\lambda \overline{\mu })dt+\sqrt{V_{t}} dW_{t,1}+J_{t}dN_{t}\right] \\ \text {with }V_{t}& = \kappa (\theta -V_{t})dt+\sigma \sqrt{V_{t}}dW_{t,2} \\ prob(dN_{t}& = 1)=\lambda dt,\ln (1+J_{t})\sim N\left( \mu _{J},\sigma _{J}^{2}\right) , \end{aligned}$$

where $\mu _{J}=\ln (1+\overline{\mu })-\frac{1}{2}\sigma _{J}^{2}$, and $W_{t,1}$ and $W_{t,2}$ are two correlated Brownian motions with correlation coefficient $\rho$.

Note that the perturbated implied volatilities or their associated option prices do not violate the arbitrage conditions, i.e. monotonicity and convexity.

The moment-generating function of the SVJ model is given as:

$$\phi (u)=e^{(r-\delta -\lambda \overline{\mu })u\tau +L(u)+M(u)V_{t}+J(u)},$$

where $\overline{\mu }=e^{\mu _{J}+\sigma _{j}^{2}/2}-1$ and

$$\begin{aligned} L(u)& = \left[ \frac{\kappa \theta }{^{\sigma ^{2}}}(\kappa -\rho \sigma u+d)\tau -2\ln \left( \frac{1-qe^{d\tau }}{1-q}\right) \right] \\ M(u)& = \frac{\kappa -\rho \sigma u+d}{\sigma ^{2}}\left( \frac{1-qe^{d\tau }}{ 1-q}\right) \\ J(u)& = \lambda \left( {\small e}^{u\mu _{J}+\sigma _{j}^{2}u^{2}/2}{-1} \right) \tau , \end{aligned}$$

with

$$q=\frac{\kappa -\rho \sigma u+d}{\kappa -\rho \sigma u-d}\ \ \text {and} \ d= \sqrt{{\small (}\rho \sigma u-\kappa )^{2}-\sigma ^{2}u(u-1)}$$

(Bates 1996). Based on this moment-generating function, we derived the theoretical values of the jth-order RNM by numerically calculating the j th-order derivative of $\phi (u)$ at $u=0$, i.e.

$$\mu _{\rho ,j}=\left[ \frac{d^{j}\phi (u)}{du^{j}}\right] _{u=0}.$$

The annualized expected quadratic variation implied by the SVJ can be calculated analytically based on the following formula:

$$\frac{1}{\tau }E_{t}^{Q}\left[ \left\langle X,X\right\rangle _{t,T}\right] =2(r-\delta )-\frac{2}{\tau }\mu _{\rho ,1}+2\lambda \left( \mu _{J}+\frac{1 }{2}\sigma _{J}^{2}+\frac{1}{2}\mu _{J}^{2}-e^{\mu _{J}+\sigma _{J}^{2}/2}+1\right) .$$

Thus the jump component is given as:

$$JT=2\lambda \left( \mu _{J}+\frac{1}{2}\sigma _{J}^{2}+\frac{1}{2}\mu _{J}^{2}-e^{\mu _{J}+\sigma _{J}^{2}/2}+1\right) .$$

Note here that JT is defined as an expectation under risk-neutral measure Q. A value of JT different from zero indicates that either investors anticipate the occurrence of jumps without requiring a premium for them or jump risk is priced. For both of these cases, however, jumps are expected to occur in the future.

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Titel: Retrieving risk neutral moments and expected quadratic variation from option prices
verfasst von: Leonidas S. Rompolis
Elias Tzavalis
Publikationsdatum: 06.05.2016
Verlag: Springer US
Erschienen in: Review of Quantitative Finance and Accounting / Ausgabe 4/2017
Print ISSN: 0924-865X
Elektronische ISSN: 1573-7179
DOI: https://doi.org/10.1007/s11156-016-0575-z

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