Abstract
This chapter is dedicated to generalising the ACE approach and its results. These extensions are performed in several directions, which offer some practical and/or some mathematical interest. First we describe the generic ACE methodology solving the direct problem at an arbitrary order. We then apply this algorithm to compute meaningful IATM differentials, all located within the second and third layers, which we can then exploit and interpret. Next we discuss alternative baselines to Black’s model, and how to transfer IATM information between them. We then introduce a major structural extension, by considering the multi-dimensional framework. We follow the evolution of the inverse and direct problems, and illustrate these results with the case of generic baskets.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
See [4], for instance.
- 2.
Principal Component Analysis.
- 3.
This would be a priori useful in FX markets, for instance, where the smile level is quoted w.r.t. Straddles, and the curvatures in terms of Strangles.
- 4.
Note that mid-curve options are already treatable within our framework, but nowhere near liquid enough: see p. 276.
- 5.
We note for future reference that there are several other available classes of baselines, in particular in the factor model class. There is, for instance, substantial literature on various quadratic Gaussian parametrisations, but transferring our methodology to this environment requires a brand new derivation.
- 6.
For instance, the boundary conditions associated to the uniqueness of a solution when \(0<\beta \le \frac{1}{2}\), or the existence/integrability constraints when \(\beta >1\).
- 7.
Typically the machine epsilon and more generally underflow issues.
- 8.
For instance, float vs double.
- 9.
Selecting a good cumulative Normal approximation is usually important.
- 10.
Obviously single-dimensional, typically using a Gaussian quadrature.
- 11.
More precisely, that the unit of \(\widetilde{\varSigma }^{(n)}_{y^n}(\star )\) should be \(\Vert \overrightarrow{\sigma }_t\Vert ^{1-n}\). Certainly in the bi-dimensional setting, as was shown in Sect. 3.2, this property is verified for \(\widetilde{\varSigma }{}^{'}_{y}(\star )\), \(\widetilde{\varSigma }{}^{''}_{yy}(\star )\), \(\widetilde{\varSigma }{}^{'''}_{yyy}(\star )\) and \(\widetilde{\varSigma }{}^{(4')}_{y^4}(\star )\).
- 12.
Pun intended.
- 13.
Consumer Price Index.
- 14.
Indeed, as we will recall in Chap. 7, a par swap rate can be written as a basket of Libor rates, but with stochastic (albeit relatively stable) weights.
- 15.
Actually, a series of smiles for benchmark expiries and therefore the associated marginal distributions.
- 16.
All sharing the same initial value as the underlying.
- 17.
Whereas the tensorial inner product, such as \(\mathbf{A}_{2,t} \overrightarrow{c_{2,t}}\), is left implicit.
- 18.
By denoting \(dS^i_t=\overrightarrow{\gamma _{i,t}}^{\bot } d\overrightarrow{W}_t\) and \(dM_t = \overrightarrow{\gamma _t}^{\bot } d\overrightarrow{W}_t\).
- 19.
We specify this property in the weak sense, but dynamically: in other words, all components follow the same SDE but respond to independent drivers, generating the same, but orthogonal, marginal laws.
References
Osajima, Y.: The asymptotic expansion formula of implied volatility for dynamic SABR model and FX hybrid model. University of Tokyo, Graduate School of Mathematical Sciences, Report (2006)
Fournie, E., Lebuchoux, J., Touzi, N.: Small noise expansion and importance sampling. Asymptotic Anal. 14(4), 361–376 (1997)
Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, Cambridge (1990)
Lewis, A.L.: Option Valuation Under Stochastic Volatility. Finance Press, Newport Beach (2000)
Hagan, P.S., Kumar, D., Lesniewski, A.S., Woodward, D.E.: Managing smile risk. Wilmott Mag. 1(September), 84–108 (2002)
Piterbarg, V.V.: Stochastic volatility model with time-dependent skew. Appl. Math. Finance 12, 147–185 (2005)
Durrleman, V.: From implied to spot volatilities. Finance Stochast. 14(2), 157–177 (2010)
Nicolay, D.: Volatility dynamics. Ph.D. thesis, Ecole Polytechnique (2011)
Cont, R., da Fonseca, J.: Dynamics of implied volatility surfaces. Quant. Finance 2, 45–60 (2002)
Fouque, J.-P., Papanicolau, G., Sircar, K.R.: Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press, Cambridge (2000)
Beckers, S.: The constant elasticity of variance model and its implications for option pricing. J. Finance 35(3), 661–673 (1980)
Schroder, M.: Computing the constant elasticity of variance option pricing formula. J. Finance 44(1), 211–219 (1989)
Hull, J.C.: Options, Futures, and Other Derivatives, 5th edn. Finance Series. Prentice Hall, Upper Saddle River (2003)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C, 2nd edn. Cambridge University Press, Cambridge (1992)
Piterbarg, V.V.: Markovian projection for volatility calibration. Risk Mag. 20, 84–89 (2007)
Beresticki, H., Busca, J., Florent, I.: Asymptotics and calibration of local volatility models. Quant. Finance 2, 61–69 (2002)
Hagan, P.S., Woodward, D.E.: Equivalent Black volatilities. Appl. Math. Finance 6, 147–157 (1999)
Durrleman, V., El Karoui, N.: Coupling smiles. Quant. Finance 8, 573–590 (2008)
d’Aspremont, A., El Ghaoui, L.: Static arbitrage bounds on basket option prices. Math. Program. 106, 467–489 (2006)
Laurence, P., Wang, T.H.: What’s a basket worth? Risk Mag. 17, 73–77 (2004)
Brigo, D., Mercurio, F., Rapisarda, F., Scotti, R.: Approximated moment-matching dynamics for basket-options simulation. Working paper, Banca IMI (2002)
Posner, S.E., Milevsky, M.A.: Valuing exotic options by approximating the SPD with higher moments. J. Financial Eng. 7, 109–125 (1998)
Avellaneda, M., Boyer-Olson, D., Busca, J., Fritz, P.: Reconstructing the smile. Risk Mag. 15(10), 84–108 (2002)
d’Aspremont, A.: Interest Rate model calibration and risk-management using semidefinite programming. Ph.D. thesis, Ecole Polytechnique (2003)
Piterbarg, V.V.: Mixture of models: a simple recipe for a ... hangover? Wilmott Mag. 13, 72–77 (2005)
Brigo, D., Mercurio, F., Rapisarda, F.: Lognormal-mixture dynamics and calibration to market volatility smiles. Int. J. Theor. Appl. Finance 5, 427–446 (2002)
Gatarek, D.: LIBOR market model with stochastic volatility. Technical Report, Deloitte and Touche (2003)
Hull, J., White, A.: The pricing of options on assets with stochastic volatilities. J. Finance 42, 281–300 (1987)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2014 Springer-Verlag London
About this chapter
Cite this chapter
Nicolay, D. (2014). Volatility Dynamics for a Single Underlying: Advanced Methods. In: Asymptotic Chaos Expansions in Finance. Springer Finance(). Springer, London. https://doi.org/10.1007/978-1-4471-6506-4_3
Download citation
DOI: https://doi.org/10.1007/978-1-4471-6506-4_3
Published:
Publisher Name: Springer, London
Print ISBN: 978-1-4471-6505-7
Online ISBN: 978-1-4471-6506-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)