Monte-Carlo Methods in Financial Modeling

  • Chuanshu JiEmail author
  • Tao Wang
  • Leicheng Yin
Part of the ICSA Book Series in Statistics book series (ICSABSS)


The last decade has witnessed fast growing applications of Monte-Carlo methodology to a wide range of problems in financial economics. This chapter consists of two topics: market microstructure modeling and Monte-Carlo dimension reduction in option pricing. Market microstructure concerns how different trading mechanisms affect asset price formation. It generalizes the classical asset pricing theory under perfect market conditions by incorporating various friction factors, such as asymmetric information shared by different market participants (informed traders, market makers, liquidity traders, et al.), and transaction costs reflected in bid-ask spreads. The complexity of those more realistic dynamic models presents significant challenges to empirical studies for market microstructure. In this work, we consider some extensions of the seminal sequential trade model in Glosten and Milgrom (Journal of Financial Economics, 14(1), 71–100, 1985) and perform Bayesian Markov chain Monte-Carlo (MCMC) inference based on the trade and quote (TAQ) database in Wharton Research Data Services (WRDS). As more and more security derivatives are constructed and traded in financial markets, it becomes crucial to price those derivatives, such as futures and options. There are two popular approaches for derivative pricing: the analytical approach sets the price function as the solution to a PDE with boundary conditions and solves it numerically by finite difference etc.; the probabilistic approach expresses the price of a derivative as the conditional expectation under a risk neutral measure and computes it via numerical integration. Adopting the second approach, we notice the required integration is often performed over a high dimensional state space in which state variables are financial time series. A key observation is for a broad class of stochastic volatility (SV) models, the conditional expectations representing related option prices depend on high-dimensional volatility sample paths through only some 2D or 3D summary statistics whose samples, if generated, would enable us to avoid brute force Monte-Carlo simulation for the underlying volatility sample paths. Although the exact joint distributions of the summary statistics are usually not known, they could be approximated by distribution families such as multivariate Gaussian, gamma mixture of Gaussian, log-normal mixture of Gaussian, etc. Parameters in those families can be specified by calculating the moments and expressing them as functions of parameters in the original SV models. This method improves the computational efficiency dramatically. It is particularly useful when prices of those derivatives need to be calculated repeatedly as a part of Bayesian MCMC calibration for SV models.


Option Price Market Maker Stochastic Volatility Model MCMC Algorithm Market Microstructure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Antonov, A., & Spector, M. (2012). Advanced analytics for the SABR model. SSRN 2026350.Google Scholar
  2. Brooks, S., Gelman, A., Jones, G., & Meng, X.L. (2011). Handbook of markov chain Monte-Carlo. CRC Press.Google Scholar
  3. Das, S. (2005). A learning market maker in the Glosten-Milgrom model. Quantitative Finance, 5(2), 169–180.MathSciNetCrossRefzbMATHGoogle Scholar
  4. Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via EM algorithm (with discussion). Journal of Royal Statistical Society, Series B, 39, 1–38.MathSciNetzbMATHGoogle Scholar
  5. Gelman, A., & Rubin, D. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7(4), 457–472.CrossRefGoogle Scholar
  6. Geweke, J. (1992). Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. Bayesian Statistics, 169–193.Google Scholar
  7. Glosten, L., & Milgrom, P. (1985). Bid, ask and transaction prices in a specialist market with heterogeneously informed traders. Journal of Financial Economics, 14(1), 71–100.CrossRefGoogle Scholar
  8. Hagan, P., Kumar, D., Lesniewski, A., & Woodward, D. (2002). Managing smile risk. The Best of Wilmott, 249–296.Google Scholar
  9. Hagan, P., Lesniewski, A., & Woodward, D. (2005). Probability distribution in the SABR model of stochastic volatility. Large deviations and asymptotic methods in finance (pp. 1–35). Springer.Google Scholar
  10. Harris, L. (2003). Trading and exchanges. Oxford University Press.Google Scholar
  11. Hasbrouck, J. (2009). Trading costs and returns for US equities: estimating effective costs from daily data. Journal of Finance, 64(3), 1–52.CrossRefGoogle Scholar
  12. Meng, X. L., & van Dyk, D. (1997). The EM algorithm: an old folk-song sung to a fast new tune (with discussion). Journal of Royal Statistical Society, Series B, 59, 511–568.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Paulot, L. (2009). Asymptotic implied volatility at the second order with application to the SABR model. SSRN 1413649.Google Scholar
  14. Rebonado, R., McKay, K., & White, R. (2011). The SABR/LIBOR market model: Pricing, calibration and hedging for complex interest-rate derivatives. Wiley.Google Scholar
  15. Robert, C., & Casella, G. (2004). Monte-Carlo statistical methods (2nd ed.). Springer.Google Scholar
  16. Roll, R. (1984). A simple implicit measure of the effective bid-ask spread in an efficient market. Journal of Finance, 39(4), 1127–1139.CrossRefGoogle Scholar
  17. Wang, T. (2014). Empirical analysis of sequential trade models for market microstructure. Ph.D. thesis, University of North Carolina at Chapel Hill.Google Scholar
  18. Yin, L. (2016). Monte-Carlo strategies in option pricing for SABR model. Ph.D. thesis, University of North Carolina at Chapel Hill.Google Scholar

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© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Department of Statistics and Operations ResearchUniversity of North CarolinaChapel HillUSA
  2. 2.Bank of America Merrill LynchNew YorkUSA
  3. 3.Exelon Business Services Company, Enterprise Risk ManagementChicagoUSA

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