## 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:

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

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

*These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.*

This is a preview of subscription content, access via your institution.

## Buying options

## Notes

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

- 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*X*−*a*−*bY*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.
*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.
In Ortega (2012) it is proved that the range of GARCH situations in which the minimal martingale measure exists is rather limited.

- 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.
- 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.
- 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.
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.
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.
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.
In these three cases,

*θ*_{ t }^{q}has an analytic form: the pricing equations (3.47) have an explicit solution. - 18.
Alexander and Lazar (2006) prove that there is no real forecasting improvement introducing more than two components in the mixture.

- 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.
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.
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.
- 23.
Nevertheless, Monfort and Pegoraro (2012) proved that a natural extension is possible in the case of the mixture of Gaussian distributions.

- 24.
- 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.
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.
- 28.
See for instance Hsieh and Ritchken (2005) for comparisons of linear and non-linear modelling in the Gaussian case.

- 29.

## References

Abramowitz M, Stegun I (1964). Handbook of mathematical functions with formulas, graphs and mathematical tables. Dover, New York

Aît-Sahalia Y, Lo A (1998) Nonparametric estimation of state-price densities implicit in financial asset prices. J Financ 53:499–547

Aît-Sahalia Y, Lo A (2000) Nonparametric risk management and implied risk aversion. J Econ 94:9–51

Alexander C, Lazar E (2006) Normal mixture GARCH(1,1): applications to exchange rate modelling. J Appl Econ 21:307–336

Back K, Pliska SR (1991) On the fundamental theorem of asset pricing with an infinite state space. J Math Econ 20(1):1–18

Badescu A, Kulperger R (2008) GARCH option pricing: a semiparametric approach. Insur Math Econ 43(1):69–84

Badescu A, Kulperger R, Lazar E (2008) Option valuation with normal mixture GARCH. Stud Nonlinear Dynam Econ 12(2):1–42

Badescu A, Elliott RJ, Siu TK (2009) Esscher transforms and consumption-based models. Insur Math Econ 45(3):337–347

Badescu A, Elliott RJ, Kulperger R, Miettinen J, Siu TK (2011) A comparison of pricing kernels for GARCH option pricing with generalized hyperbolic distributions. Int J Theor Appl Financ 14(5):669–708

Barndorff-Nielsen OE (1995) Normal inverse Gaussian processes and the modelling of stock returns. Research Report 300, Department Theoretical Statistics, Aarhus University

Barndorff-Nielsen OE, Blaesild P (1981) Hyperbolic distributions and ramifications: contributions to theory and applications. In: Taillie C, Patil GP, Baldessari BA (eds) Statistical distributions in scientific work, vol 4. Reidel, Dordrecht, pp 19–44

Barone-Adesi G, Engle RF, Mancini LA (2008) GARCH option pricing model in incomplete markets. Rev Financ Stud 21(3):1223–1258

Bertholon H, Monfort A, Pegoraro F (2008) Econometric asset pricing modelling. J Financ Econ 6(4):407–458

Black F (1975) Fact and fantasy in the use of options. Financ Anal J 31(4):36–41

Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81:637–659

Bouchaud JP, Potters M (2003) Theory of financial risk and derivative pricing: from statistical physics to risk management. Cambridge University Press, Cambridge

Breeden DT, Gibbons MR, Litzenberger RH (1989) Empirical test of the consumption-oriented CAPM. J Financ 44(2):231–262

Brennan MJ (1979) The pricing of contingent claims in discrete time models. J Financ 24(1):53–68

Bühlmann H, Delbaen F, Embrechts P, Shiryaev AN (1996) No-arbitrage change of measure and conditional Esscher transforms. CWI Q 9:291–317

Carr P, Madan D (1999) Option valuation using the fast fourier transform. J Comput Financ 2(4):61–73

Chorro C, Guégan D, Ielpo F (2010) Martingalized historical approach for option pricing. Financ Res Lett 7(1):24–28

Chorro C, Guégan D, Ielpo F (2012) Option pricing for GARCH type models with generalized hyperbolic innovations. Quant Financ 12(7):1079–1094

Christoffersen P, Heston SL, Jacobs K (2006) Option valuation with conditional skewness. J Econ 131:253–284

Christoffersen P, Jacobs K, Ornthanalai C, Wang Y (2008) Option valuation with long-run and short-run volatility components. J Financ Econ 90:272–297

Christoffersen P, Elkamhi R, Feunou B, Jacobs K (2010) Option valuation with conditional heteroskedasticity and non-normality. Rev Financ Stud 23(5):2139–2183

Christoffersen P, Jacobs K, Ornthanalai C (2012) Dynamic jump intensities and risk premiums: evidence from S&P500 returns and options. J Financ Econ 106(3):447–472

Christoffersen P, Heston SL, Jacobs K (2013) Capturing option anomalies with a variance-dependent pricing kernel. Rev Financ Stud 26(8):1963–2006

Cochrane J (2001) Asset pricing. Princeton University Press, Princeton

Dalang RC, Morton A, Willinger W (1990) Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stochast Stochast Rep 29(2):185–201

Delbaen F, Schachermayer W (2006) The mathematics of arbitrage. Springer finance series. Springer, Berlin

Duan JC (1995) The GARCH option pricing model. Math Financ 5:13–32

Duan JC, Simonato JG (1998) Empirical martingale simulation for asset prices. Manag Sci 44:1218–1233

Duan JC, Gauthier G, Simonato JG (2001) Asymptotic distribution of the EMS option price estimator. Manag Sci 47:1122–1132

Duan JC, Ritchken P, Sun Z (2005) Jump starting GARCH: pricing and hedging options with jumps in returns and volatilities. Working Paper, University of Toronto

Duan JC, Ritchken P, Sun Z (2006) Approximating GARCH-jump models, jump diffusion processes, and option pricing. Math Financ 16:21–52

Eberlein E, Hammerstein EAV (2003) Generalized hyperbolic and inverse Gaussian distributions: limiting cases and approximation of processes. In: Proceedings of the 4th Ascona conference. Birkhauser, Boston

Eberlein E, Prause K (2002) The generalized hyperbolic model: financial derivatives and risk measures. In: Geman H, Madan D, Pliska S, Vorst T (eds) Mathematical finance-bachelier congress 2000. Springer, Heidelberg, pp 245–267

Elliott R, Madan D (1998) A discrete time equivalent martingale measure. Math Financ 2(8):127–152

Elliott R, Siu TK, Chan L (2006) Option pricing for GARCH models with Markov switching. Int J Theor Appl Financ 9(6):825–841

Embretchs P, Frey R, McNeil AJ (2005) Quantitative risk management. Princeton University Press, Princeton

Esscher F (1932) On the probability function in the collective theory of risk. Skandinavisk Aktuarietidskrift 15:175–195

Föllmer H, Schweiser M (1991) Hedging of contingent claims under incomplete information. In: Davis MHA, Elliott RJ (eds) Applied stochastic analysis. Gordon and Breach, London, pp 389–414

Gerber HU, Shiu SW (1994a) Option pricing by Esscher transforms. Trans Soc Actuar 46:99–191

Gerber HU, Shiu SW (1994b) Martingale approach to pricing perpetual American options. In: Proceedings of the 4th AFIR international colloqium, transaction of Society of Actuaries, Orlando, pp 659–689

Gouriéroux C, Monfort A (2007) Econometric specification of stochastic discount factor models. J Econ 136(2):509–530

Guégan D, Ielpo F, Lalaharison H (2013) Option pricing with discrete time jump processes. J Econ Dyn Control 37(12):2417–2445

Hansen LP, Richard SF (1987) The role of conditioning information in deducing testable restrictions implied by dynamic asset pricing models. Econometrica 55(3):587–613

Harrison JM, Kreps DM (1979) Martingales and arbitrage in multiperiod securities markets. J Econ Theory 20(3):381–408

Harrison JM, Pliska SR (1981) Martingales and stochastic integrals in the theory of continuous trading. Stoch Process Appl 11:215–260

Heston SL (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6(2):327–343

Heston SL, Nandi S (2000) A closed-form GARCH option valuation. Rev Financ Stud 13:585–625

Hsieh K, Ritchken P (2005) An empirical comparison of GARCH option pricing models. Rev Deriv Res 8:129–150

Huang SF (2014) A modified empirical martingale simulation for financial derivative pricing. Commun Stat Theory Methods 43(2):328–334

Huang SF, Tu YT (2014) Asymptotic distribution of the EPMS estimator for financial derivatives pricing. Comput Stat Data Anal 73:129–145

Jackwerth J (2000) Recovering risk aversion from option prices and realized returns. Rev Financ Stud 13:433–451

Jacod J, Shiryaev A (1998) Local martingales and the fundamental asset pricing theorems in the discrete-time case. Financ Stochast 2:259–273

Kreps DM (1981) Arbitrage and equilibrium in economics with infinitely many commodities. J Math Econ 8:15–35

Lindner AM (2009) Continuous time approximations to GARCH and stochastic volatility models. In: Andersen TG, Davis RA, Kreiss JP, Mikosch T (eds) Handbook of financial time series. Springer, Heidelberg, pp 481–496

Lucas RE (1978) Asset prices in an exchange economy. Econometrica 46:1429–1445

Mainardi F, Rogosin S (2006) The origin of infinitely divisible distributions: from de Finetti’s problem to Lévy-Khintchine formula. Math Methods Econ Financ 1:37–55

Menn C, Rachev ST (2005) A GARCH option pricing model with

*α*-stable innovations. Eur J Oper Res 163:201–209Mercuri L (2008) Option pricing in a garch model with tempered stable innovations. Financ Res Lett 5(3):172–182

Monfort A, Pegoraro F (2012) Asset pricing with second-order Esscher transforms. J Bank Financ 36(6):1678–1687

Ortega JP (2012) GARCH options via local risk minimization. Quant Financ 12(7):1095–1110

Pratt JW (1964) Risk aversion in the small and in the large. Econometrica 32:122–136

Rosenberg JV, Engle RF (2002) Empirical pricing kernels. J Financ Econ 64:341–372

Rubinstein M (1976) The valuation of uncertain income streams and the pricing of options. Bell J Econ 7:407–425

Rubinstein M (1985) Nonparametric tests of alternative option pricing models using all reported trades and quotes on the 30 most active CBOE option classes from August 23, 1976 through August 31, 1978. J Financ 40:455–480

Schachermayer W (1992) A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time. Insur Math Econ 11(4):249–257

Schoutens W (2003) Lévy processes in finance: pricing financial derivatives. Wiley, New York

Schroder M (2004) Risk neutral parameter shifts and derivatives pricing in discrete time. J Financ 59:2375–2402

Siu TK, Tong H, Yang H (2004) On pricing derivatives under GARCH models: a dynamic Gerber-Shiu approach. N Am Actuar J 8:17–31

Walker JS (1996) Fast fourier transforms, 2nd edn. CRC Press, Boca Raton

Yuan ZS, Chen GM (2009a) Strong consistency of the empirical martingale simulation option price estimator. Acta Math Appl Sin (English Series) 25(3):355–368

Yuan ZS, Chen GM (2009b) Asymptotic normality for EMS option price estimator with continuous or discontinuous payoff functions. Manag Sci 55:1438–1450

## Author information

### Authors and Affiliations

## Rights and permissions

## Copyright information

© 2015 Springer-Verlag Berlin Heidelberg

## About this chapter

### Cite this chapter

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

### Download citation

DOI: https://doi.org/10.1007/978-3-662-45037-6_3

Published:

Publisher Name: Springer, Berlin, Heidelberg

Print ISBN: 978-3-662-45036-9

Online ISBN: 978-3-662-45037-6

eBook Packages: Business and EconomicsEconomics and Finance (R0)