# Consistency of economic scenarios in nested stochastics simulation

I am interested in references on research regarding the consistency of economic scenarios in nested stochastics for risk measurement.

Background:

1. Pricing by Monte-Carlo: For pricing complex derivatives Monte-Carlo simulation is a well established method. It works by generating iid samples of the underlying risk-factors, evaluating and discounting the pay-off in each sample path and finally calculating the average over all samples. The distribution used for sampling is a "risk-neutral" distribution. The process of deriving and calibrating those distributions is technically difficult but well understood on a conceptual level.

2. Risk Measurement There are several approaches to measuring the risk of such a position. One approach, which is employed in an insurance and banking regulation context, works by estimating the distribution of position prices after a prescribed period (such as one year for insurance) and applying a risk measure such as Value at Risk to the resulting p&l, which is just the difference in value between t=1 and t=0, say.Of course the change in the underlying risk factors over the measurement period is modeled by a distribution. This distribution is different from the distribution for valuation under 1. on the interval [0;1] and is called the real-world distribution of market states.

3. Nested Stochastics for risk measurement: Creating sample paths for risk measurement using Monte-Carlo simulation, now works in a two-step process. i) Generate a sample of market states at t=1 according to the/a real world distribution (these are called outer nodes or outer samples) ii) For each of those samples of market states, generate risk neutral samples of paths from t=1 onwards which are consistent with the market state. (called inner samples) The actual distribution of prices at t=1 is then calculated by applying the Monte-Carlo valuation as described in 1. for each set of inner samples created in ii).

Question of consistency: There is one straightforward question of market consistency at each outer node, i.e. how to ensure that risk neutral scenarios are consistent with the market states of the outer node. But except for practical issues, I think this is just the same question as the calibration at t=0, i.e. how to ensure pricing consistent with current market parameters.

But I see another issue with consistency, which is harder for me to understand. Already at t=0 risk neutral and real world measures are constrained by well defined consistency conditions. They are equivalent measures connected by a change of measure involving the current price of risk.For nested stochastics, you need to have some sort of consistency between the t=0 risk neutral measure and the sample of measures, each being contingent on the market states at t=1. Already in plain-vanilla processes, there are straightforward consistency requirements, such as expectation at t=0 being the integral of expectations at t=1. But the case here is more difficult, due to the mix of measures over the first and subsequent periods.

So here is my question: Do you know of any research such as textbooks or papers explaining or discussing these questions? Note that I am currently neither concerned with the formidable practical questions of the approach nor with potential alternatives or any simplifications to it.

Any hints are highly appreciated

• hope someone else can help you with that. Seems we don't get much further due to lost-in-translation. Cheers Jan 30, 2013 at 11:10
• Sorry, could you please state the consistency part more precisely, to better agree on terms and the problem? Anyway I do not see well what's the issue: at the different RW scenarios in t=1 you calibrate and price with a corresponding RNt1_i measure, which sure will all be somehow related to RNt0 by construction, but why care about that? Jan 31, 2013 at 14:00
• The question is exactly how this “related somehow by construction” can be achieved. Look at a 2period option in a Black-Scholes Model. Black-Scholes assumes deterministic implied vola. So it makes no sense within this model to apply an implied vola distribution after one period to assess the risk of this option. The price at t=0 will be inconsistent with all prices at t=1 by construction.
– g g
Feb 2, 2013 at 12:08
• "Why care": This is more difficult to comment on. Some people think logical consistency is an end in itself, some are more pragmatic But I think that a logically inconsistent risk measuring approach is inherently dangerous. At least as long as you do not understand the nature and scope of those limitations.
– g g
Feb 2, 2013 at 12:15
• Only once one understands what is going, one might start to ask commercial questions. Imagine an institution managing vega-risk in such a dont-care fashion with a model where implied vola is assumed constant. Can you make money off of them? Or at least offer them vega-protection cheaper by having a better(?) model?
– g g
Feb 2, 2013 at 12:25