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From Time Series of Returns to Option Prices: The Stochastic Discount Factor Approach

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Abstract

In the perfect and unrealistic Black and Scholes (J Polit Econ 81:637–659, 1973) world, the dynamics \((S_{t})_{t\in [0,T]}\) of the risky asset, under the historical probability \(\mathbb{P}\), is given by the following stochastic differential equation:

$$\displaystyle{ dS_{t} =\mu S_{t}dt +\sigma S_{t}dW_{t} }$$

where \((W_{t})_{t\in [0,T]}\) is a standard Brownian motion under \(\mathbb{P}\). In this case, there is no ambiguity in the definition the arbitrage-free price of any European contingent claim with maturity T. In fact, in this complete market which is set in continuous time, this value is none other than the value of any replicating portfolio. Moreover, prices may be expressed in terms of conditional expectations under a unique equivalent martingale measure Q whose density with respect to the historical probability is given by the Girsanov theorem

$$\displaystyle{ \frac{dQ} {d\mathbb{P}} = e^{-\frac{\mu -r} {\sigma } W_{T}-\left (\frac{\mu -r} {\sigma } \right )^{2} \frac{T} {2} } }$$

where r is the constant and continuously compound risk-free rate. Unfortunately, as we have seen in Sect. 2.1, the restrictive underlying hypotheses (constant volatility, independent increments, Gaussian log-returns, etc…) are questioned by many empirical studies and GARCH models appear as excellent alternative solutions to potentially overcome some well-documented systematic biases associated with the Black and Scholes model.

Keywords

  • Option Price
  • Risky Asset
  • GARCH Model
  • Option Price Model
  • Scholes Model

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Notes

  1. 1.

    In their paper (see also Sect. 2.7), Heston and Nandi also provided a particular GARCH structure that may be seen as a discrete time counterpart of the so-called Heston (1993) stochastic volatility model replicating one of key features of this continuous time model: the fact that the characteristic function of the log-returns under the risk-neutral distribution had a closed-form expression (see Sect. 3.7). This result is particularly interesting because it underlines that GARCH option pricing models may also be seen as competitive discrete time approximations of the classical models used in continuous time finance with the great advantage that they are easy to estimate because the resulting filtering problem is simple. In particular, they may represent interesting and efficient alternative options to Euler approximation schemes (see Duan et al. 2006 and Lindner 2009).

  2. 2.

    For example, it is now well-documented that the Black and Scholes model underprices out of the money and short maturity options (see Black 1975) and is not able to reproduce the so-called U-shaped relationship between implied volatility and strike price as remarked in Rubinstein (1985).

  3. 3.

    In this case, the information filtration \((\mathcal{F}_{t})_{t\in \{1,\ldots,T\}}\) is also generated by the log-returns \((Y _{t})_{t\in \{1,\ldots,T\}}\) and the weak and strong market efficiency hypotheses coincide (see Elliott and Madan 1998).

  4. 4.

    In the case of the discrete time economy introduced in Sect. 3.1, we can take c t  = Y t and h(c, s) = se c.

  5. 5.

    It means that, in this case, the relative risk aversion coefficient \(c_{t+1}\rho _{t}(c_{t+1})\) is a constant.

  6. 6.

    This may be seen as a conditional version of the following elementary result concerning two-dimensional Gaussian vectors: When (X, Y ) is a Gaussian vector, \(\exists (a,b) \in \mathbb{R}^{2}\) such that XabY is a centered Gaussian random variable independent of Y. The proof derives easily from the fact that the orthogonal projection \(\Pi (X)\) of X on the vector space generated by the random variables 1 and Y is an affine function of Y and that \(X - \Pi (X)\) is, by construction, a Gaussian random variable independent of Y.

  7. 7.

    As introduced in Remark A.1 of the Appendix, σ(e Z ) is the smallest σ-algebra on \(\Omega \) that makes the random variable e Z measurable. In particular, a real random variable X defined on \(\Omega \) is σ(e Z ) measurable if and only if it is of the form X = h(e Z ) where \(h: \mathbb{R} \rightarrow \mathbb{R}\) is Borelian.

  8. 8.

    In Ortega (2012) it is proved that the range of GARCH situations in which the minimal martingale measure exists is rather limited.

  9. 9.

    As remarked in Heston and Nandi (2000) for Gaussian innovations, this exponential affine parameterization is equivalent to obtaining of Black and Scholes prices for call options with 1 day to expiration.

  10. 10.

    We will elaborate on this point later on for particular distributions using the intermediate value theorem (see Propositions 3.4.5 and 3.4.8).

  11. 11.

    When X is a centered random variable such that the mapping \(\Psi (u) = \mathit{log}(E[e^{\mathit{uX}}])\) is regular we have \(\Psi ^{{\prime}}(0) = E[X] = 0\), \(\Psi ^{{\prime\prime}}(0) = \mathit{Var}[X]\), \(\frac{\Psi ^{{\prime\prime\prime}}(0)} {(\Psi ^{{\prime\prime}}(0))^{\frac{3} {2} }} = \mathit{sk}[X]\) and \(\frac{\Psi ^{{\prime\prime\prime}{\prime}}(0)} {(\Psi ^{{\prime\prime}}(0))^{2}} = k[X] - 3\).

  12. 12.

    See Example 3.4.1 or Remark 3.4.1 for an intuitive interpretation of this modified Sharpe ratio in terms of risk aversion.

  13. 13.

    Sometimes it is interesting to work with an unconstrained GH distribution and it is sufficient to consider \(\tilde{z}_{t} = \frac{z_{t}-E_{\mathbb{P}}[z_{t}]} {\sqrt{\mathit{Var } _{\mathbb{P} } [z_{t } ]}}\) instead of z t in (3.4). This is the point of view of Badescu et al. (2011). The only difference between the two approaches is a modification in the parameterization of the model.

  14. 14.

    In general, we are able to obtain an explicit solution only in particular cases (see Proposition 3.4.7 below). Nevertheless, we can compute it efficiently using refined bracketing methods.

  15. 15.

    For Gaussian innovations (see Example 3.4.1) only the first order conditional moment is impacted by the measure change. Here, θ t q induces not only a shift in the conditional skewness of the GH distribution but also an excess kurtosis (exact values for the associated skewness and excess kurtosis are provided in Barndorff-Nielsen and Blaesild 1981).

  16. 16.

    In this paper, the authors assume that in Eq. (3.4), \(z_{t} = \frac{x_{t}-\frac{a}{b}} {\sqrt{ \frac{a}{b^{2 } }}}\) where x t follows a G a (a, b) distribution having the density \(\frac{b^{a}x^{a-1}e^{-\mathit{bx}}} {\Gamma (a)} 1_{\mathbb{R}_{+}}(x)\).

  17. 17.

    In these three cases, θ t q has an analytic form: the pricing equations (3.47) have an explicit solution.

  18. 18.

    Alexander and Lazar (2006) prove that there is no real forecasting improvement introducing more than two components in the mixture.

  19. 19.

    First, in this model, the Gaussian and the jump components are intrinsically linked: they have up to a constant the same conditional variance because

    $$\displaystyle{\mathit{Var}_{\mathbb{P}}\left [\sqrt{h_{t}}z_{t}^{0}\mid \mathcal{F}_{ t-1}\right ] = h_{t}\ \mbox{ and}\ \mathit{Var}_{\mathbb{P}}\left [\sqrt{h_{t}}\sum \limits _{i=1}^{N_{t}}z_{t}^{i}\mid \mathcal{F}_{ t-1}\right ] = h_{t}\lambda (\mu ^{2} +\sigma ^{2}).}$$

    We refer the reader to Christoffersen et al. (2012) for a multiple-shock version of the model where the Gaussian and the jump parts have distinct volatility dynamics. Finally, in Duan et al. (2005), the excess return m t has a very particular form that is chosen for tractability in order to make the solution of the pricing equations (3.47) explicit. Contrary to this approach, we will take in the empirical part a more realistic expression for m t even if pricing equations have to be solved numerically.

  20. 20.

    Even if it tends to contradict the monotonic pattern assumed in existing theory for the SDF, recent empirical studies show evidences on possible U-shapes speaking for quadratic parameterizations (Christoffersen et al. 2013).

  21. 21.

    Let \(f: \mathbb{R}^{2} \rightarrow \mathbb{R}\) be a continuously differentiable function such that f(x 0, y 0) = 0 and \(\frac{\partial f} {\partial y}(x_{0},y_{0})\not =0\). The implicit function theorem ensures that there exists an open set \(U_{x_{0}}\) containing x 0, an open set \(V _{y_{0}}\) containing y 0 and a continuously differentiable function \(g: U_{x_{0}} \rightarrow V _{y_{0}}\) such that

    $$\displaystyle{\left \{(x,g(x))\mid x \in U_{x_{0}}\right \} = \left \{(x,y) \in U_{x_{0}} \times V _{y_{0}}\mid f(x,y) = 0\right \}.}$$

    In our setting, we just take

    $$\displaystyle{f(x,y) = \mathbb{G}_{(Y _{t},Y _{t}^{2})\mid \mathcal{F}_{t-1}}^{\mathbb{P}}(x + 1,y) - e^{r}\mathbb{G}_{ (Y _{t},Y _{t}^{2})\mid \mathcal{F}_{t-1}}^{\mathbb{P}}(x,y).}$$
  22. 22.

    Even if it is not possible to estimate the free parameter π only using the log-returns (π not being a parameter of the physical dynamics) we will see in Sect. 3.7 that π may be efficiently extracted from option prices or in Chap. 4 that it may be estimated from the so-called VIX.

  23. 23.

    Nevertheless, Monfort and Pegoraro (2012) proved that a natural extension is possible in the case of the mixture of Gaussian distributions.

  24. 24.

    In fact, from (3.78), we deduce the empirical call-put parity

    $$\displaystyle{\frac{e^{-r(T-t)}} {n} \sum \limits _{i=1}^{n}(\tilde{S}_{ T,i} - K)_{+} + S_{t} = \mathit{Ke}^{-r(T-t)} + \frac{e^{-r(T-t)}} {n} \sum \limits _{i=1}^{n}(K -\tilde{ S}_{ T,i})_{+}}$$

    that is equivalent to (3.80).

  25. 25.

    Using the fact that prices may equivalently be expressed using expectations under the historical probability involving the stochastic discount factor (see (3.12)), Huang (2014) and Huang and Tu (2014) proposed and studied an historical version of the EMS when a risk-neutral model is not conveniently obtained. The price to pay is a computational cost that may be heavy if extra techniques are not used.

  26. 26.

    Here, there is no particular assumption on the shape of the SDF but relation (3.78) may be seen as an empirical pricing equation.

  27. 27.

    To be more precise, Heston and Nandi (2000) proved an explicit backward-recursive equation to compute efficiently the moment generating function of the log-returns (see Proposition 3.7.1).

  28. 28.

    See for instance Hsieh and Ritchken (2005) for comparisons of linear and non-linear modelling in the Gaussian case.

  29. 29.

    In Christoffersen et al. (2006) and Mercuri (2008), the authors use the same idea with some non-Gaussian innovations combined with very particular affine GARCH specifications.

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Chorro, C., Guégan, D., Ielpo, F. (2015). From Time Series of Returns to Option Prices: The Stochastic Discount Factor Approach. In: A Time Series Approach to Option Pricing. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45037-6_3

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