Aug 06

Value at Risk (VaR): 2007 FRM. Part 5 - Structured Monte Carlo

by David Harper, CFA, FRM, CIPM

FRM | Risk |

smc_intro

Learning Outcome

  • LO 7.12: Describe the structured Monte Carlo approach to measuring VAR, and identify the advantages and disadvantages of the SMC approach.

Monte Carlo Simulation with multiple risk factors

We previously introduced Monte Carlo simulation (MCS) with the classic example: we simulated a single stock price, as it moves forward in time, which showed randomness according to geometric Brownian motion (GBM). In that example, we modeled a single risk factor, the randomized volatility of a stock.

Linda Allen's text, Understanding Market, Credit, and Operational Risk is used for the 2007 FRM to explain the structured Monte Carlo simulation. Here, the structured Monte Carlo refers to an approach for dealing with multiple risk factors. Of course, a single risk factor is rarely practical. Realistic problems (portfolios) have multiple risk factors.

The example: the VaR of a straddle

In the EditGrid spreadsheet below, I illustrate numerically Linda Allen's straddle example. The idea is simply to estimate the VaR of a straddle. To straddle is to go long/short both a call and a put on the same asset with the same strike and expiration (technically, below is a top straddle because we are buying both; the bottom straddle, where we sell both to hopefully earn income on a languishing non-moving stock is extremely risky).

So note below the stock-based assumptions: initial stock price, expected return and volatility. Then, to the right of that, the call/put assumptions.

Below this are the two VaR outcomes, which are based on the one hundred MCS trials. Note that each trial is simply a future stock price (one period assumption). This produces a call and put payoff. The net payoff is the gain, if any, on the call or the put minus (-) the premiums paid. (You can see how the straddle works here: it hopes to profit by large moves in either direction, such that the gain on the large move outweighs the premiums).

The resulting "Monte Carlo VaR" is then simply the bottom percentile of the sorted list; i.e., the 95th and 99th percentile-ranked outcomes.

The point of the exercise is to show you that we are simulating the underlying asset (the stock price) but the VaR is estimated based on the derivative's (straddle) payoff.

Structured Monte Carlo

Alas, the VaR of the straddle is not really structured Monte Carlo. Structured Monte Carlo is when we model additional risk factors (e.g., more stocks in the portfolio) by taking the additional step of incorporating the correlation among the additional risk factors. (If we just model a vector of independent risk factors and aggregate them into a portfolio, we are unrealistic in ignoring their correlations).

So Linda Allen's structured Monte Carlo extension looks like this:

smc

And you see it looks just like our single-factor MCS except that it substitutes a matrix (A') for the single-factor volatility and a vector (Z) for the standard normal variable. (the matrix contains correlated variables by virtue of a Cholesky factorization).

In a nutshell, the structured Monte Carlo refers to Monte Carlo with multiple risk factors, but where we replace a matrix of uncorrelated variables with a matrix of correlated variables.

Takeaways

In regards to the learning outcome, the key advantage of structured Monte Carlo is that it generates correlated scenarios.

The disadvantages include:

  • Though less a concern for those with powerful tools, we don't overcome the "curse of dimensionality." At a minimum, we need perform calculations on the order of the product of [number of risk factors][number of trials][number of path-steps within each trial].
  • Simulations are not necessarily predictive of the future. Importantly, adding more trials does not necessarily improve the simulation.
  • Related, and perhaps another form of model risk, the whole thing is built on the robustness of the embedded covariance matrix (i.e., which gives the covariances between factors). If that not applicable going forward and/or under stressful conditions, the simulation is flawed.
EditGrid Spreadsheet by bt/frm2007.

Comments

  1. John C. Vesey 06 Aug 2007

    As usual, David, you are doing an outstanding analytical presentation. Even an old man, my age, can understand the subject after reading your dissertatioin.

    I hope all is well with you.

    JCV

  2. mikey10011 19 Oct 2008

    On what page in Linda Allen’s book can I find the straddle example?

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