TY - JOUR
T1 - A general valuation framework for SABR and stochastic local volatility models
AU - Cui, Zhenyu
AU - Kirkby, J. Lars
AU - Nguyen, Duy
N1 - Publisher Copyright:
© 2018 Society for Industrial and Applied Mathematics.
PY - 2018
Y1 - 2018
N2 - In this paper, we propose a general framework for the valuation of options in stochastic local volatility (SLV) models with a general correlation structure, which includes the stochastic alpha beta rho (SABR) model and the quadratic SLV model as special cases. Standard stochastic volatility models, such as Heston, Hull–White, Scott, Stein–Stein, a-Hypergeometric, 3/2, 4/2, mean-reverting, and Jacobi stochastic volatility models, also fall within this general framework. We propose a novel double-layer continuous-time Markov chain (CTMC) approximation respectively for the variance process and the underlying asset price process. The resulting regime-switching CTMC is further reduced to a single CTMC on an enlarged state space. Closed-form matrix expressions for European options are derived. We also propose a recursive risk-neutral valuation technique for pricing discretely monitored path-dependent options and use it to price Bermudan and barrier options. In addition, we provide single Laplace transform formulae for arithmetic Asian options as well as occupation time derivatives. Numerical examples demonstrate the accuracy and efficiency of the method using several popular SLV models, and reference prices are provided for SABR, Heston-SABR, quadratic SLV, and the Jacobi model.
AB - In this paper, we propose a general framework for the valuation of options in stochastic local volatility (SLV) models with a general correlation structure, which includes the stochastic alpha beta rho (SABR) model and the quadratic SLV model as special cases. Standard stochastic volatility models, such as Heston, Hull–White, Scott, Stein–Stein, a-Hypergeometric, 3/2, 4/2, mean-reverting, and Jacobi stochastic volatility models, also fall within this general framework. We propose a novel double-layer continuous-time Markov chain (CTMC) approximation respectively for the variance process and the underlying asset price process. The resulting regime-switching CTMC is further reduced to a single CTMC on an enlarged state space. Closed-form matrix expressions for European options are derived. We also propose a recursive risk-neutral valuation technique for pricing discretely monitored path-dependent options and use it to price Bermudan and barrier options. In addition, we provide single Laplace transform formulae for arithmetic Asian options as well as occupation time derivatives. Numerical examples demonstrate the accuracy and efficiency of the method using several popular SLV models, and reference prices are provided for SABR, Heston-SABR, quadratic SLV, and the Jacobi model.
KW - American options
KW - Asian options
KW - Barrier options
KW - CTMC
KW - Continuous-time Markov chains
KW - Exotic options
KW - Occupation time derivatives
KW - Option pricing
KW - Quadratic local volatility
KW - SABR
KW - Stochastic local volatility
UR - http://www.scopus.com/inward/record.url?scp=85049723045&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85049723045&partnerID=8YFLogxK
U2 - 10.1137/16M1106572
DO - 10.1137/16M1106572
M3 - Article
AN - SCOPUS:85049723045
VL - 9
SP - 520
EP - 563
JO - SIAM Journal on Financial Mathematics
JF - SIAM Journal on Financial Mathematics
IS - 2
ER -