Calculating Value at Risk for Options and Futures

In this course we provide a methodology for calculating the Value at Risk (VaR) measure for futures and options. The methodology that we have employed uses a Monte Carlo Simulator to first generate terminal prices series, then calculates the related payoffs and prices series. The prices series is used to determine the return series which is used in the volatility and VaR calculations.

As a pre-requisite to this course the user may like to review the following two courses:

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

The first step of the process involves the construction of a Monte Carlo simulator for determining the terminal price of the underlying. As we are interested in daily prices of the options, the interval or time step length should be for a day. In our illustration we have assumed that the option contract will expire after 10 days so we have used ten intermediate steps to simulate the development of prices of the underlying security for this period.

The simulated prices are generated based on the Black Schole’s Terminal Price formula:

S_t=S₀*exp[(r-q-0.5?²)t+??tz_t]

Where S₀ is the spot price at time zero,

r is the risk free rate

q is the convenience yield

? is the annualized volatility in the commodity’s price

t is the duration since time zero, and

z_t is a random sample from a normal distribution with mean zero and standard deviation of 1. z_t was obtained in these models by normally scaling the random numbers generated using Excel’s RAND() function, i.e. NORMINV(RAND()).

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. Suppose that for the original option the commencement was at time 0 and the expiry was at time step 10, the next option will commence at time 1 and expire at time step 11, the next will start at time 2 and expire at time step 12, 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 365 days.

Step 3: Run scenarios

Step 2 above generates a 365-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) in order to generate a data set of times series data with the aid of EXCEL’s Data Table functionality. Once this process has been completed an average terminal price time-series will be calculated, by taking a simple average of the terminal prices at each future date across all the simulated runs. The figure below shows this process for our example.

The Average Terminal Price for Date 1 is the average of all the terminal prices generated for this date across the 1000 simulated runs. The Average Terminal Price for Date 363 is the average of all the terminal prices generated for this date across the 1000 simulated runs.

Step 4: Calculate the intrinsic value or payoffs

Individual payoffs at each data point

For each data point given in the terminal price data set mentioned in Step 3 above we now have to calculate the payoffs or intrinsic values of the derivatives contract. In our illustration, we have assumed that we have a futures contract, a European call option and a European put option all having a strike or exercise price of 1300. The payoffs for these contracts are calculated as follows:

Payoff for a long futures = Terminal Price – Strike

Payoff for a long call option = Maximum of (Terminal Price –Strike, 0)

Payoff for the long put option = Maximum of (0, Strike-Terminal Price)

This is illustrated for a subset of futures payoffs below:

For example, for scenario 3 (third data row) on date 2 (second data column) the Terminal price is 1333.04. The strike price as mentioned earlier is 1300. The futures payoff therefore works out to Terminal Price – Strike Price = 1333.04 – 1300 = 33.04.

Average payoff time-series

Once all the payoffs have been calculated we determine the average payoff time series by taking a simple average of the payoffs at each future date across all the simulated runs.

Step 5: Calculate discount values of payoffs, i.e. prices

Individual prices at each data point

For each data point given in the terminal price data set mentioned in Step 3 above for which we have determined the payoffs or intrinsic values of the derivatives contract as mentioned in Step 4 above, we will now calculated their discounted values as follows:

Payoff * e^-rT

Where r is the risk free rate and

T is the tenor of the option, i.e. 10 days.

The discounted values derived are the values/ prices of the futures contract and the call and put options respectively. This is illustrated for a subset of futures prices below:

For example, for scenario 3 (third data row) on date 2 (second data column) the payoff is 33.04. The risk free rate is 0.15% and as mentioned earlier the tenor of the contract is 10 days. The futures price therefore works out to Payoff * e^-rT =33.04*exp (-0.15%*(10/365))=33.03.

Average price time-series

Once all the prices have been calculated we determine the average price time series by taking a simple average of the prices at each future date across all the simulated runs.

Step 6: 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. This is illustrated for a subset of the futures, call option and put option contracts below:

The average prices of a call on Date 1 and 2 are 12.31 and 12.65 respectively. The return on Date 2 will therefore be ln(12.65/12.31) =2.71%.

Step 7: Calculate the VaR measure

Next we have calculated the VaR measure using the techniques outline in our course Calculating Value at Risk, in particular we have used the Simple Moving Average (SMA) Variance Covariance (VCV) Approach and the Historical Simulation Approach.

For our illustration, the 10-day holding period VaR at different confidence levels, using the VCV approach has been calculated as follows: