Calculating Value at Risk using Monte Carlo Simulation (Futures, options &Equity) Group members Najat Mohammed James Okemwa Mohamed Osman

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Introduction

Step 1:Construct a Monte Carlo Simulator for prices of the underlying

Step 2: Expand the Monte Carlo Simulator In order to calculate the Value at Risk (VaR) measure we require a series of returns which in turn requires time-series price data. To simulate this particular environment we assume that we have a series of similar option contracts that commence and expire on a one-day roll-forward basis. We assume that time to maturity for our portfolio is one month(30 days). Suppose that an option commences at time 0 and expires at time 30. The next commences a time 1 and expires at time 31, the next at time 2 and expires at time 32 and so on. Based on this premise we will obtain a time series of daily terminal prices. In our illustration we have repeated this process in order to generate time-series data for terminal prices for a period of 180 days(half a year) as shown here.

Step 3: Run scenarios Step 2 above generates a 180-day terminal price series under a single scenario. The process now needs to be repeated several times (in our illustration we have used 1000 simulation runs). After running 1000 scenarios take a simple average across all the 180 days for each data point.

Step 4: Calculate the intrinsic value or payoffs then calculate the discounted value of the payoffs. This two steps combined give us the prices for our instruments(futures, options and equity) Payoff for a long futures = Terminal Price – Strike Price=Payoff*e -rT Where r is the risk free rate and T is the time to maturity of the option, future or equity i.e. 30 days.

Step 5: Calculate the return series Now that we have the derivatives average price series we will determine the return series by taking the natural logarithm of successive prices. The return on Date 3 for a futures contract will therefore be ln((0.37)/(0.32)) =12%.

Step 6: Calculate the VaR measure In this step we calculate the Value at Risk Using the Historical method where we refer to the simulated daily return processes as our historical Returns. As such VaR is calculated by computing a percentile over the range of the return data set under consideration. This is easily calculated in Excel by the function =PERCENTILE (array, (100-X)%) Where X is the confidence interval. In our illustration, the 1day Value at Risk at different confidence levels, using the Historical returns is calculated as shown below. These are the 30-day holding VaR’s for the various instruments. To interpret this table for example it tells us that we are 95% confident that if we held put options our loss would not exceed 8%. You can also tell that it is risky to invest in futures contract since the loss percentages are quite high. To calculate the N-day VaR we use the formula, N-day VaR 99% =1-day VaR 99% * √ N

Value at Risk, Histograms and risk management in Excel Finally we seek to answer three important questions a) How bad can things get when they really get bad? b) What is the most that you can lose on a really bad day? c) What is the worst that can happen? We can summarize this in one sentence as; What is the worst that can happen and over what period and with what odds?

Probabilities of losses occuring For instance there is a 0.56% chance that the worst scenario (23.43% loss) would happen

CONCLUSION Putting all this information together we can say that; if one should invest 100 SEK in long call options then the maximum they can lose is SEK, in any given day with a probability of 0.56%. Value at Risk is a widely used risk measure of the risk of loss on a specific portfolio of financial assets. For a given portfolio, probability and time horizon, Value at Risk is a threshold value such that the probability that the market to market loss on the portfolio over the given time horizon exceeds this value in a given probability level. Value at Risk and volatility are the most commonly used risk measurements. Value at Risk is easy to calculate and can be used in many fields.

“Always borrow money from a pessimist. He won’t expect it back”-Oscar Wilde You have been a nice audience Thank you!