Dear Wise minds of Wilmotts.

I've been working for a small investment firm for the past 3 months.
My current boss gets a data set from morgan stanley every few months with simulations of inflation rates and average wage inflation.
He wants me to consider possible frameworks for similar simulations on bond prices, stock prices, option prices and entire portfolios. In which the previous assets are combined in various ways, say perhaps 40% bonds, 50% stocks, 10% options.

In this my question is two-fold, I assume, there are methods for simulation, which would be sufficiently accurate for the vast majority of investment firms.
I'd like to ask if I've understood correctly, that monte carlo simulations and Geometric brownian motions are amongst these and perhaps the one's you'd recommend?
Beyond this I've been trying to read up on more advanced methods, in the hopes of finding simulation methods with more accuracy for large portfolios.
My understanding is, that Stochastic portfolio theory could be such a method, can anyone confirm or deny this proposition?

Best regards CtoL (conventional take-off and landing?)

As either Neils Bohr, or maybe it was Yogi Berra, said, “Prediction is very difficult, especially if it's about the future!” 

I doubt there's much consensus about how to do what your boss wants, as it's quite vague. And, even it wasn't, there are a zillion ways to proceed.  

Here's what I like to do. For stocks and bonds, there's high quality historical total returns in the US going back to 1926. So, I would pick a couple large chunks, say 1926 to date, and post WWII to date. Create the historical returns for whatever mixture you want. These sets of returns are called the empirical distributions. For simulations, just make random repetitive draws from these empirical distributions and chain together the returns for whatever time horizons you want. This is a simple Monte Carlo procedure. Develop whatever statistics you want as well as their standard errors.  

For options or portfolios with options, I wouldn't presume to offer any advice without knowing what kind of options and the purpose of the simulation.  

That’s probably a very crude estimation? Could he use ARIMAX with some relevant external regressors (such as the IR trends from his boss - but would be relevant to know how those were modelled)?
I’d pack it all in some LSTM

Alan and Katastrofa, those are both really good advice.

It's starting to seem to me, like my boss wants something he can also understand. So ARMA might be the upper limit, as to how advanced it can be. I think the procedure described by Alan could be easy to explain and showcase to my boss. Which at the moment will be the most important, since I don't work in a quant fund yet.
I think Alan's advice might be down that alley. 

I have been adviced to try LSTM and ARIMAX before, so it could be worth writing up the code and procedures for the future when I will be working in a quant fund.

How do the both of you feel about combining the methods? I've been speculating about the following idea.

If I were to run 100 simulations with ARMA, geometric brownian motion and LSTM and than finding the mean path for all. Would this get me closer to the actual mean of say a stock price movement. Or would it just be gibberish?
My expectation is that it would simply return gibberish. But perhaps the two of you have other opinions or some nice arguments like now?

I don't know about what you are suggesting. 

All I can speak to is the procedure I suggested. Its main weakness is that it fails to capture volatility clustering. If you want to also capture that in a simulation, instead of IID (daily) draws from an empirical distribution, you can try making "block draws" --  say N days at a time. This is not going to change any simulated stock means/growth rates but it should produce some more realistic 2nd order behavior, like the auto-covariance of squared returns. 

Both procedures (my original suggestion and my block-draw suggestion) will lead to simulated stock return distributions with 'wide tails', which is a desirable realistic feature. I would have some doubts that anything using GBM, which you mention, will have this property.  

Volatility clustering, rare events - and also correlations are potentially important missing effects?

I personally like the model ensemble idea you had - averaging several models, each having its own strengths and weaknesses, can be a powerful approach.i

Volatility clustering, rare events - and also correlations are potentially important missing effects?

Sure. Why do people do simulations? It is often for asset allocation/risk-management -- to see if you can stomach some worse case scenarios. Sometimes it is for option/derivative valuation.

Rare events.
The IID draws from an empirical distribution that contains the Black Monday crash will not only pick up the occasional market crash in the simulation, but the possibility of a crash followed the next day by another crash, etc.  Of course, the probabilities, while greater than zero,  are quite small (although orders of magnitude greater than under GBM with typical vols).

Volatility clustering/correlations.
This is important for option/derivative valuation. Without it, typically simulated option "skews" decay too rapidly relative to what is observed. However, the simple IID model or even the block-draw model is not going to generate option prices that match the market, so that's why I didn't really discuss options.

The general rule is you'd like the simulated series to behave like the empirical series.

Alan, is it common/correct to perform some common factor analysis for the assets in the portfolio and simulate them instead? (I’m thinking PCA for example.)

Also, is it actually possible to build a robust portfolio model without including any exogenous factors?

Hope CtL will forgive me using his thread for asking my own questions!

This is very interesting things.So please do go on.

I might be misunderstanding, but isn't rare events like volatility clustering and crashes usually portrayed by jump processes?
I do really like the idea of using empirical distributions to portray both.
But I worry that trying to justify the use of long term empirical distributions with crashes and various volatility clusters can be difficult to do.
I guess I am asking for advice on how to convince my boss, who has his own opinions of what the probability of a large loss is, that the use of empirical distributions over longer time horizons and with possible outliers in his opinion, is correct.

Note to self: Heavy tailed distributions, because of possible large positive or negative returns

@kat, 
Sure you could first do a cross-sectional analysis of portfolio returns to identify explanatory factors, and then simulate those. In the original post, I interpreted "40% bonds, 50% stocks" to refer to broad-market indexes. But if the portfolios have active bets on factors beyond the "broad-market factor", then simulating the factors makes sense. 

I'm not sure about exogenous factors. I suspect it depends on the purpose of the simulation. If it was a stress test for a bank over the next year or so, explicitly including "recession scenarios", for example, seems sensible. But, for a long-term asset allocation exercise for a pension fund, as another example, the historical empirical distributions already reflect multiple recessions.   

@CtL,
 If you're drawing from a daily empirical return distribution, every draw is a jump! The probability of a Black Monday crash in the simulation is 1/N, where N is the number of days in the empirical distribution, assuming it includes Oct. 19, 1987. 

If instead, you're simulating a parameterized continuous-time process, then yes, you'll need a jump component to see crashes -- as a diffusion is typically too benign. The empirical distribution simulation is easy, although simulating something like Merton's jump-diffusion process is pretty simple also. Like I said, there are a zillion ways to proceed. A GARCH process with (log-return) innovations drawn from a non-normal distribution (say with exponential, rather than Gaussian, tails) is also nice.