Asymptotic computation of Greeks under a stochastic volatility model

Park, Sang-Hyeon;Lee, Kiseop

  • 투고 : 2015.08.16
  • 심사 : 2015.12.05
  • 발행 : 2016.01.31


We study asymptotic expansion formulae for numerical computation of Greeks (i.e. sensitivity) in finance. Our approach is based on the integration-by-parts formula of the Malliavin calculus. We propose asymptotic expansion of Greeks for a stochastic volatility model using the Greeks formula of the Black-Scholes model. A singular perturbation method is applied to derive asymptotic Greeks formulae. We also provide numerical simulation of our method and compare it to the Monte Carlo finite difference approach.


computation of Greeks;asymptotics;stochastic volatility;singular perturbation;Malliavin calculus


  1. Fouque JP, Papanicolaou G, and Sircar R (2000). Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press.
  2. Fournie E, Lasry JM, Lebuchoux J, and Lions PL (2001). Applications of Malliavin calculus to Monte Carlo methods in finance II, Finance Stochastics, 5, 201-236
  3. Fournie E, Lasry JM, Lebuchoux J, Lions PL, and Touzi N (1999). Applications of Malliavin calculus to Monte Carlo methods in finance, Finance Stochastics, 3, 391-412.
  4. Fouque JP, Papanicolaou G, and Sircar R (2003). Singular perturbations in option pricing, SIAM Journal on Applied Mathematics, 63, 1648-1665.
  5. Fouque JP, Papanicolaou G, Sircar R and Solna K (2011). Multiscale Stochastic Volatility for Equity, Interest-Rate and Credit Derivatives, Cambridge University Press.
  6. Glasserman P and Yao DD (1992). Some guideline and guarantees for common random number, Management Science, 38, 884-908.
  7. Glynn PW (1989). Optimization of stochastic system via simulation, In Proceedings of the 1989 Winter simulation Conference. San Diego: Society for Computer Simulation, 90-105.
  8. Heston S (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6, 327-343.
  9. Hull J and White A (1987). The pricing of options on assets with stochastic volatility, The Journal of Finance, 42, 281-300.
  10. Takahashi A and Yamada T (2012). An asymptotic expansion with push-down of Malliavin weights, SIAM Journal on Financial Mathematics, 3, 95-136.
  11. Takahashi A and Yamada T (2014). On error estimates for asymptotic expansions with Malliavin weights: application to stochastic volatility model, Mathematics of Operations Research, 513-541.