Abstract

In this paper, we prove a large deviations principle for the class of multidimensional affine stochastic volatility models considered in [C. Gourieroux and R. Sufana (2010), J. Bus. Econom. Statist., 28, pp. 438--451], where the volatility matrix is modeled by a Wishart process. This class extends the very popular Heston model to the multivariate setting, thus allowing us to model the joint behavior of a basket of stocks or several exchange rates. We then use the large deviations principle to obtain an asymptotic approximation for the implied volatilities of various options and to develop an asymptotically optimal importance sampling algorithm, to reduce the number of simulations when using Monte Carlo methods for price derivatives. The theory is illustrated with foreign exchange option pricing examples.

Keywords

  1. large deviations
  2. Wishart process
  3. importance sampling
  4. basket options
  5. implied volatility

MSC codes

  1. 60F10
  2. 91G20
  3. 91G60

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References

1.
A. Ahdida and A. Alfonsi (2013), Exact and high-order discretization schemes for Wishart processes and their affine extensions, Ann. Appl. Probab., 23, pp. 1025--1073.
2.
A. Alfonsi (2015), Affine diffusions and related processes: Simulation, theory and applications, Bocconi and Springer Ser. 6, Springer, Cham, Switzerland.
3.
A. Alfonsi, A. Kebaier, and C. Rey (2016), Maximum likelihood estimation for Wishart processes, Stochastic Process. Appl., 126, pp. 3243--3282.
4.
A. Benabid, H. Bensusan, and N. El Karoui (2008), Wishart Stochastic Volatility: Asymptotic Smile and Numerical Framework, preprint, hal-00458014v2.
5.
M. F. Bru (1991), Wishart processes, J. Theoret. Probab., 4, pp. 725--751.
6.
I. J. Clark (2011), Foreign Exchange Option Pricing: A Practitioner's Guide, Wiley, Chichester, England.
7.
C. Cuchiero, D. Filipović, E. Mayerhofer, and J. Teichmann (2011), Affine processes on positive semidefinite matrices, Ann. Appl. Probab., 21, pp. 397--463.
8.
J. Da Fonseca, M. Grasselli, and C. Tebaldi (2007), Option pricing when correlations are stochastic: An analytical framework, Rev. Deriv. Res., 10, pp. 151--180.
9.
J. Da Fonseca, M. Grasselli, and C. Tebaldi (2008), A multifactor volatility Heston model, Quant. Finance, 8, pp. 591--604. \iffalse
10.
D. Morris, T. Mitchell, and D. Ylvisaker (1993), Bayesian design and analysis of computer experiments: Use of derivatives in surface prediction, Technometrics, 35, pp. 243--255. \fi
11.
A. Dembo and O. Zeitouni (1998), Large Deviations Techniques and Applications, 2nd ed., Appl. Math. 38, Springer, New York.
12.
M. Forde and A. Jacquier (2011), The large-maturity smile for the Heston model, Finance Stoch., 15, pp. 755--780.
13.
A. Genin and P. Tankov (2019), Optimal importance sampling for Lévy processes, Stochastic Process. Appl., to appear.
14.
C. Gourieroux (2006), Continuous time Wishart process for stochastic risk, Econometric Rev., 25, pp. 177--217.
15.
C. Gourieroux and R. Sufana (2010), Derivative pricing with Wishart multivariate stochastic volatility, J. Bus. Econom. Statist., 28, pp. 438--451.
16.
P. Guasoni and S. Robertson (2008), Optimal importance sampling with explicit formulas in continuous time, Finance Stoch., 12, pp. 1--19.
17.
S. Heston (1993), A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6, pp. 327--343.
18.
A. Jacquier and M. Keller-Ressel (2018), Implied volatility in strict local martingale models, SIAM J. Financial Math., 9, pp. 171--189.
19.
A. Jacquier, M. Keller-Ressel, and A. Mijatović (2013), Large deviations and stochastic volatility with jumps: Asymptotic implied volatility for affine models, Stochastics, 85, pp. 321--345.
20.
H. Pham (2007), Some applications and methods of large deviations in finance and insurance, in Paris-Princeton Lectures on Mathematical Finance 2004, Springer, Berlin, pp. 191--244.
21.
S. Robertson (2010), Sample path large deviations and optimal importance sampling for stochastic volatility models, Stochastic Process. Appl., 120, pp. 66--83.
22.
R. T. Rockafellar (1970), Convex Analysis, Princeton University Press, Princeton, NJ.
23.
M. R. Tehranchi (2009), Asymptotics of implied volatility far from maturity, J. Appl. Probab., 46, pp. 629--650.

Information & Authors

Information

Published In

cover image SIAM Journal on Financial Mathematics
SIAM Journal on Financial Mathematics
Pages: 942 - 976
ISSN (online): 1945-497X

History

Submitted: 29 June 2018
Accepted: 16 September 2019
Published online: 17 December 2019

Keywords

  1. large deviations
  2. Wishart process
  3. importance sampling
  4. basket options
  5. implied volatility

MSC codes

  1. 60F10
  2. 91G20
  3. 91G60

Authors

Affiliations

Funding Information

Chaire Risques Financiers

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