Choosing Simulation Schemes for Derivative Pricing Models
Summary
The document asks when Monte Carlo discretization is needed in financial modeling, given that geometric Brownian motion can be simulated exactly at selected dates. The responses name the LIBOR market model, Heston stochastic volatility, and local volatility models as examples where simulation generally requires more than exact geometric Brownian motion sampling. They also caution that basic Euler and Milstein schemes can be unsuitable for Heston in particular.
The discussion distinguishes general-purpose discretization methods from schemes designed for specific models. It lists methods such as quadratic-exponential, transformed-volatility, and moment-matching approaches, which are intended to improve convergence or reduce issues such as simulated negative variances. For American option valuation, it mentions a simulation-based least-squares method. The material is a brief overview rather than a comparative study: it gives no implementation details or performance evidence, and method choice depends on the model and pricing objective.
Key ideas
- Geometric Brownian motion can be sampled exactly at a finite set of dates, while many derivative models require approximate simulation.
- The LIBOR market, Heston, and local volatility models are examples where discretization methods may be used.
- Basic Euler and Milstein schemes may perform poorly for Heston dynamics.
- Model-specific schemes can target faster convergence or prevent negative simulated variances.
- Simulation-based regression methods can be applied to American option valuation.
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Full text
# What are some examples of non-solvable SDE where Monte Carlo discretization is necessary # What are some examples of non-solvable SDE where Monte Carlo discretization is necessary Reading Glasserman - "Monte Carlo Methods in Finance" it says in the introduction to Chapter 6 - Discretization Methods, that moste models arising in derivatives pricing can be simulated only approximately. This is in contrast to geometric brownian motion for which it is possible to simulate exactly at a finite set of dates. My interpretation is that it's always suboptimal to use discretization methods when the pricing problem merely involves geometric brownian motion. My question is when does it become necessary to use e.g. Milstein or Euler discretization? What are some common examples? ## Answer by Mark Joshi (score 7, accepted) https://quant.stackexchange.com/a/18951 the LIBOR market model the Heston model -- Euler and Milstein are actually bad for this and much more sophisticated methods are necessary local volatility models ## Answer by user16891 (score 0) https://quant.stackexchange.com/a/18952 Monte Carlo simulation in the context Financial Modeling refers to a set of techniques to generate artificial time series of the stock price,volatility and interest rate and... overtime, from which option prices can be derived. There are several choices available in this regard. The first choice is to apply a standard method such as the Euler, Milstein, or implicit Milstein scheme, as described by Gatheral and Kahl and Jackel, for example. The advantage of these schemes is that they are easy to understand, and their convergence properties are famous. The other choice is to use a method that is better suited, or that is specifically designed for the especial models. These methods include quadratic-exponential scheme of Andersen, the transformed volatility scheme of Zhu, the scheme of Alfonsi, or the moment-matching scheme of Andersen,et al. These schemes are designed to have faster convergence to the true option price, and in some cases, to also avoid the negative variances that can sometimes be generated from standard methods. These and other schemes are reviewed by Van Haastrecht and Pelsser (2010). Also, to valuing American options, you can use the simulation-based algorithm of Longstaff and Schwartz.
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