## Monte Carlo VaR

Given the limitations of the Analytic risk model we can look at other approaches to Value at risk. A very natural approach to consider is using a Monte Carlo VaR Simulation. Monte Carlo simulation will be able to cope with stochastic jumps and complicated payoff functions much better than an analytical framework. Any non-linear payoff especially from a portfolio of options would be better served by a Monte Carlo simulation based model for VaR purposes. Options portfolios can sometimes have big P&L swings from small moves in the underlying. Analytical VaR models are modeled so that the maximum loss occurs when the underlying moves by a large amount. This would be quite dangerous for options strategies like straddles and strangles positions where the largest loss will usually happen when there is no market volatility.

The idea behind a Monte Carlo simulation is quite simple, we specify the random process and then simulate various different scenarios. Each scenario will simulate a P&L outcome from which we can build a distribution (options will be revalued at each simulation to get a proper price distribution). Once we have a distribution we can very simply read off the appropriate confidence level to get the VaR.

### Monte Carlo VaR Methodology

Let us examine more closely how a Monte Carlo VaR Simulation works. We have a stock price S, which follows a geometric Brownian motion.

where μ is expected rate of return and σ is the volatility of the asset. The term dW represents a Weiner process, shows by :

So our initial equation becomes :

So what we are saying in the above equation is that the stock price returns are normally distributed with a mean of μdt and a standard deviation of σ√dt. So in a sense the same criticisms that applied to analytic VaR for assuming normality of distribution will also apply here.

The Steps for a monte Carlo VaR simulation are as follows :

– We know the length of time we want to simulate the price over. The first step is to divide this time period T up into a large number of smaller increments N. That is to say our above equation is dt = T / N. We then have a starting value (usually the current market price) S(0) which is then incremented randomly until we get to the end of the time period, that is to say the total number of N increments.

– This is then repeated again and again so we have a sufficient number of simulations.

– The idea behind the Monte Carlo simulation is to estimate a large number of terminal values for the stock price. Once we have this , we can then plot it in a histogram (or in an ordered list) and simply read off a the right confidence interval to get the VaR. It would generally be very useful to carry out this analysis in a program such as excel. Excel Tip : Use the in built function percentile to quickly get your answer, it takes the form Percentile (Simulated Price Array, Confidence Interval)

We have an excel spreadsheet which shows how to implement a Monte Carlo for a single asset. For multiple assets this is of course more complicated and will be covered later on.

### Applications of the Monte Carlo VaR

– Risk factors that show fat / heavy tails or are subject to jumps. There could also be a mix of different risk factors – normal markets along with credit risk factors.

– An option portfolio with non-linear payoffs. This is the most common use case as Monte Carlo can deal with the non-linear payoffs far better than the analytical risk models.

– Complex instruments like mortgage backed securities, credit derivatives.

### Pros and Cons of Monte Carlo VaR

Pros

– Ability to model a wide range of risk factors, with different market dynamics.

– Ability to deal with non-linear portfolios and path dependent options

– Capture any risk that arises from smaller market moves, while at the same time also capturing risk of much larger market moves.

– Can model difficult scenarios – scenarios which may apply an unpredictable shock – weather or political risk.

Cons

– The oft cited drawback of Monte Carlo is that it is difficult to implement and time consuming in terms of computer power. This can also mean that practically speaking risk reports are more time consuming to run and deadlines more difficult to achieve potentially.

– Monte Carlo approaches also tend to be quite black box from the point of view of users. The assumptions and the working are difficult to fathom except for the people who have actually built the system.

– There is still the necessity for an underlying assumption to be made about the distributions of price returns. This is considered an inherent weakness as most models assume Normal distribution which is far from the case in reality in the markets.

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