Portal de Periódicos da CAPES

Small-Time Asymptotics for an Uncorrelated Local-Stochastic Volatility Model

2011; Taylor & Francis; Volume: 18; Issue: 6 Linguagem: Inglês

10.1080/1350486x.2011.591159

1466-4313

Martin Forde, Antoine Jacquier,

Stochastic processes and statistical mechanics

Abstract Abstract We add some rigour to the work of Henry-Labordère (2009 Henry-Labordère, P. 2009. Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing, New York, London: Chapman & Hall. [Google Scholar]; Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing (London and New York: Chapman & Hall)), Lewis (2007 Lewis, A. (2007) Geometries and Smile Asymptotics for a Class of Stochastic Volatility Models. http://www.optioncity.net (http://www.optioncity.net) (Accessed: 28 May 2011). [Google Scholar]; Geometries and Smile Asymptotics for a Class of Stochastic Volatility Models. Available at http://www.optioncity.net (accessed 28 May 2011)) and Paulot (2009 Paulot, L. (2009) Asymptotic implied volatility at the second order with application to the SABR model, Working Paper papers.ssrn.com/sol3/papers.cfm?abstract_id=1413649 (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1413649) (Accessed: 11 June 2011). [Google Scholar]; Asymptotic implied volatility at the second order with application to the SABR model, Working Paper, Available at papers.ssrn.com/sol3/papers.cfm?abstract_id=1413649 (accessed 11 June 2011)) on the small-time behaviour of a local-stochastic volatility model with zero correlation at leading order. We do this using the Freidlin—Wentzell (FW) theory of large deviations for stochastic differential equations (SDEs), and then converting to a differential geometry problem of computing the shortest geodesic from a point to a vertical line on a Riemmanian manifold, whose metric is induced by the inverse of the diffusion coefficient. The solution to this variable endpoint problem is obtained using a transversality condition, where the geodesic is perpendicular to the vertical line under the aforementioned metric. We then establish the corresponding small-time asymptotic behaviour for call options using Hölder's inequality, and the implied volatility (using a general result in Roper and Rutkowski (forthcoming Roper, M. and Rutkowski, M. forthcoming. A note on the behaviour of the Black–Scholes implied volatility close to expiry. International Journal of Theoretical and Applied Finance, [Google Scholar], A note on the behavior of the Black–Scholes implied volatility close to expiry, International Journal of Thoretical and Applied Finance). We also derive a series expansion for the implied volatility in the small-maturity limit, in powers of the log-moneyness, and we show how to calibrate such a model to the observed implied volatility smile in the small-maturity limit. Key Words: small-time asymptoticslarge deviationsheat kernel Notes 1 This model is similar to the type of model used in Fouque et al. (2000 Fouque, J. P., Papanicolaou, G. and Sircar, R. K. 2000. Derivatives in Financial Markets with Stochastic Volatility, Cambridge: Cambridge University Press. [Google Scholar]) where the volatility process is bounded away from zero. 2 The existence and uniqueness of a strong solution for the process is standard, and the process can be expressed as a stochastic integral of the process, so is also well defined. is bounded, so the Novikov condition is satisfied and is a martingale. 3 Note that this condition is trivially satisfied if for some , or if α is just a positive constant. 4 Note that the metric associated with the well-known Heston model for unit volatility-of-variance is not complete, because geodesics can escape to the -axis in finite time. Interestingly, Henry-Labordère (2005 Henry-Labordére, P. (2005) A general asymptotic implied volatility for stochastic volatility models,Working Paper http://www.arxiv.org/abs/cond-mat/0504317 (http://www.arxiv.org/abs/cond-mat/0504317) (Accessed: 11 June 2011). [Google Scholar]) and Laurence (2009 Laurence, P. Asymptotics for local volatility and Sabr models. Workshop on Spectral and Cubature Methods in Finance and Econometrics. May2009. Leicester: University of Leicester. [Google Scholar]) point out that this space has infinite curvature at , similar to a black hole in the Schwarzschild metric in general relativity. 5 For the variable endpoint problem that we need to solve, we can restrict attention to geodesics with , because any geodesic starting from with which also passes through the line , , will be longer than the segment of the horizontal line between and , because the two functions and are both strictly decreasing in . 6 See Carmo (1992 Carmo, Do.M. 1992. Riemmanian Geometry, Boston, Basel and Berlin: Birkhäuser. [Crossref] , [Google Scholar]) for a definition of a variation. 7 Note that the Energy functional as we define it here is one-half of the E that is used in Carmo (1992 Carmo, Do.M. 1992. Riemmanian Geometry, Boston, Basel and Berlin: Birkhäuser. [Crossref] , [Google Scholar]). 8 Using the triangle inequality and the Lipschitz property of σ, f and α, we can easily verify that the drift and the diffusion coefficients for this model satisfy the global Lipschitz and linear growth conditions in Theorem 2.9 in Karatzas and Shreve (1991 Karatzas, I. and Shreve, S. 1991. Brownian Motion and Stochastic Calculus, New York: Springer-Verlag. [Crossref] , [Google Scholar]), so a unique strong solution exists.