In this course, we provide a methodology for calculating the Value at Risk for options and futures. The methodology that we employ uses a Monte Carlo Simulator to first generate a terminal price series. Then we calculate the related payoffs and option price series. We use the option price series to determine the return series. And use the returns to calculate the volatility and VaR calculations.

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

## a. VaR for Options – method 1

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

In this step of the Value at Risk for options process, we construct a Monte Carlo simulator to determine the terminal price of the underlying. As we are interested in the daily prices of the options, the interval or time step length should be for a day. In our illustration, we assume that the option contract will expire after 10 days. Hence, we use ten intermediate steps to simulate the development of prices of the underlying security for this period.

Simulate the prices using the Black Scholes’ Terminal Price formula:

**S _{t}=S_{0}*exp[(r-q-0.5sigma^^{2})t+sqroot(sigma)*z_{t}]**

- Where S
_{0}is the spot price at time zero, - r is the risk free rate
- q is the convenience yield
- sigma 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. Obtain z_{t}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 for options and futures, 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, repeat this process 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 dataset 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 a long position in these contracts are calculated as follows:

- Futures payoff = Terminal Price – Strike
- Call option payoff = Maximum of (Terminal Price –Strike, 0)
- Put option payoff = Maximum of (0, Strike-Terminal Price)

The figure below illustrates a subset of futures payoffs:

*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

Using the payoffs, 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 calculate 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. The figure below illustrates the 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 we have the prices, 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. The figure below illustrates returns for a subset of the futures, call option and put option contracts:

*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 for Options**

Next, we calculate the VaR for options using the techniques in our Calculating Value at Risk course. In particular, we use the Simple Moving Average (SMA) Variance Covariance (VCV) Approach and the Historical Simulation Approach.

For our illustration, we calculate the 10-day holding period Value at Risk for options and futures at different confidence levels, using the VCV approach. The results are as follows:

A graphical representation of the results for VaR is given below:

The figure below shows the 10-day holding period Value at Risk for options and futures. It uses the Historical simulation approach at the 95% confidence level:

## b. Alternate VaR method for FX Forwards: Delta VaR

If you need to calculate VaR for foreign exchange forward contracts there is a shorter, alternative approach. It combines the underlying currency pair VaR estimate with the delta estimate for the forward contract.

To factor in the impact of the interest rates differential between the foreign and domestic risk free rates, the forward exchange rates risk factor is considered. The VaR for the forward contract will approximately equal this factor’s VaR times the sensitivity of the forward’s price to fluctuations in the underlying factor. The sensitivity is measured as the forwards delta[1]. In particular, the VaR of the forward position will be:

**VaR _{forward position} = Delta*VaR _{forward exchange rates}**.

Where Delta = e^{-rfT}

r_{f} is the foreign risk free rate as of the report date

T is the days to maturity (DTM) (=midpoint of DTM bucket, *see below*), expressed in years

### i. Data requirements

- FX Forward Exchange Rates history for the lookback period
- Foreign risk free rate for the report date for each currency where a position is present

### ii. Guide to calculating VaR for forwards & swaps

The steps for calculating VaR for forwards and swaps are given below.

#### Preliminary Steps

**Step 1**: Identify the currencies (Foreign currency (FCY) & domestic currency (DCY)) for each forward deal. Treat the near and far legs of a swap deal as two separate forward deals.

**Step 2**: Identify the long and short positions for each forward deal.

**Step 3**: Calculate the Days to Maturity (**DTM**) for each position and allocate pre-specified standardized DTM buckets to each position. We have used the following DTM buckets with the midpoint for each bucket specified below. This midpoint will be used to select the relevant Forward Exchange Rate buckets to use:

DTM Bucket | Midpoint (days) |

1 -15 days | 8 |

16 -30 days | 23 |

1-3 months | 60 |

3-6 months | 135 |

6-9 months | 225 |

9-12 months | 318 |

greater than 1 year | 366 |

#### Gross and Net Positions

**Step 4**: Sum all the long positions by currency and DTM bucket. Sum all the short positions by currency and DTM bucket.

**Step 5**: Calculate the Gross position by currency and DTM bucket. This is the sum of the absolute value of the long and absolute value of short positions.

**Step 6**: Calculate a Net position by currency and DTM bucket. This is the sum of the long and short positions for the bucket.

**Step 7**: Using the interpolated[2] forward foreign exchange rate for the calculation date calculate the MTM of the position on a Gross and Net basis (MTM) (Gross), i.e.:

- MTM (Gross) = Gross Position*FX Forward Exchange Rate* Delta;
- MTM (Net) = Net Position * FX Forward Exchange Rate * Delta

#### Vol & VaR

**Step 8**: Calculate the holding volatility & VaR% for 1 unit of an FX forward exchange rate in the given currency:

- Obtain the FX forward exchange rates for the specified lookback period
- Calculate the return series for these rates
- Determine the daily volatility for the returns & the holding volatility based on the selected holding period
- Calculate the holding VaR % based on the selected confidence level

**Step 9**: Multiply the holding VaR % with the MTM (Gross) & MTM (Net) amounts respectively to determine the Holding VaR (Gross) & Holding VaR (Net) amounts for each currency & DTM bucket.

#### Portfolio VaR

**Step 10**: Calculate portfolio VaR (Total (Gross) & Total (Net)) across all instruments & currencies

- Calculate weights for each currency and DTM bucket using the absolute value of the MTM (Gross) & MTM (Net) respectively
- Using the return series of the for each currency and DTM bucket, and the weights determine a weighted average return series for the portfolio
- Determine the daily volatility for the returns & the holding volatility based on the selected holding period
- Calculate the holding VaR based on the selected confidence level
- Multiply the resulting portfolio VaR%s with the MTM (Gross) and MTM(Net) total amounts to determine Holding VaR (Gross) & Holding VaR (Net) for the portfolio.

[1] Understanding market, credit and operational risk – The value at risk approach –Linda Allen, et al.

[2] Interpolated based on the relevant midpoint of DTM bucket