# 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?

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

• Hi Mark, except for Euler and Milstein, what others would you recommend for Heston model? Thanks. Jul 20 '15 at 12:56
• QE is popular but bad for Greeks and long steps. our scheme is good for both but a lot more complicated: ssrn.com/abstract=1617187 Jul 20 '15 at 21:11

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.

• +1 for links but did not actually answer the question.
– UmaN
Jul 20 '15 at 10:40
• I think that, I have exactly mentioned advantages and deficiencies of Euler and Milstein Scheme.
– user16891
Jul 20 '15 at 11:06
• but that wasn't the question.
– AFK
Jul 25 '15 at 2:52
• I think that we can use Euler and Mistein schemes for approximation every things but these Methods is not convergent in especial case.
– user16891
Jul 25 '15 at 10:36