Monte-Carlo Methods in Financial Modeling

Abstract

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.