AMERICAN OPTIONS PORTFOLIO VALUE-AT-RISK CALCULATION PROCEDURE



VALUATION METHOD AND ASSUMPTIONS

  1. This method of value-at-risk calculation first values a portfolio of American-style call and put options using risk-neutral binomial tree valuation approach based on current(time-zero) values of underlying share variables.
  2. The underlying share price variables are then projected using their real-world risk-adjusted-discount-rate over one-day using Monte-Carlo simulation to get their real world values after one day.
  3. The portfolio of American-style call and put options is then re-valued after one day using risk-neutral binomial tree valuation approach assuming one day real-world-projected share price variables are realised. To understand the process, suppose for simplicity the portfolio consists of only one American call option that expires in T days from now whose underlying is S. The price of S at time zero is denoted by S(0). First we value the option price at time zero using the binomial tree risk-neutral valuation approach to get its time zero price as C(S(0),T). We then run M real world simulations to get M real world realisations of the share price after one day. We denote one day real world simulation realisation of share price on run i as S(i,1) where i denotes the simulation run index and 1 represents one day. We then value the option price realisation after one day using the binomial risk-neutral valuation approach but now using S(i,1) instead of S(0) and T-1 expiry period instead of T to get the option price at simulation run i as C(S(i,1),T-1).
  4. The volatilities used in valuation of American-style call and put options are implied volatilities determined from actively traded instruments.
  5. The volatility used in one-day share price projection of a particular underlying share variable is the shortest maturity implied volatility of options in that underlying share variable group.
  6. The system also calculates global-fit volatility estimate derived from implied volatilities of options in that underlying share variable group through a simple weighting process. To understand its calculation process, suppose the implied volatilities V(i)'s and maturities T(i)'s of options under a specific underlying share S at times indexed by i are {(V(1),T(1)), (V(2),T(2)),(V(3),T(3))}. The maturity times are indexed in ascending order. The volatility function between time-zero and time T(1) is assumed to be V(1), the volatility function between time T(1) and time T(2) is assumed to be V(2), and the volatility function between time T(2) and time T(3) is assumed to be V(3). We denote our global-fit volatility estimate by G and solve the following equation: T(3)*G^2 = T(1)*V(1)^2 + (T(2) - T(1))*V(2)^2 + (T(3)-T(2))*V(3)^2 . The * symbol represents multiplication and ^ is a power symbol. In situations where the spectrum of available maturities is sufficient, G can be used as constant volatility in projecting the progression of share price S at different time-points up to a period equal to the maturity of the longest maturity option whose underlying is S. A more scientifically appealing way of estimating G is to find the value of G that minimises the sum of squared errors of model prices of actively traded instruments from their market prices.
  7. The shape and level of the yield-curve is assumed to remain unchanged over the one-day period so the same interest rates used on initial portfolio valuation are re-used in its re-valuation after one day.
  8. The known-dividend-yield-factors and present-value-of-discrete-dividend-adjustment are adjusted to take into account the the one-day shift in time-line.
  9. The process is repeated and a distribution of options portfolio values after one day is mapped, sorted and initial value subtracted to get the value-at-risk over one-day period at the required confidence level.
  10. The N day value-at-risk is calculated by multiplying one day value at risk by the square root of N.
  11. Follow the view-page-by-view-page instructions below to calculate value-at-risk.

Financial solutions home page

  1. Log into your account or if you are a new user, register your account and enable two factor authentication.
  2. Click on the [American options portfolio value-at-risk] button, this will take you to the Dynamic-text-blocks parameters page.

Dynamic text-blocks parameters page

  1. Select to enter pairwise-flat or pairwise-varying correlations between underlying share price variables.
  2. Select the number of underlying share price variables in the portfolio. For example, if an American options portfolio consists of 3 long call positions in Microsoft, 2 long put positions in BP Shell, and 1 short call position in BP shell, the number of underlying share variables is entered as 2, ie, Microsoft and BP Shell.
  3. Click the submit button, this will take you to the array populating page.

Array populating page

  1. Identifier indices used in this section start identification counting from 0 to make them consistent with programming of array indices.
  2. Enter name of indicated underlying share price variable.
  3. Enter number of discrete dividends before expiry of longest maturity option (enter zero if none) of indicated underlying share variable. Discrete dividend is such that, if the underlying share price value just before dividend payment is S, and the value of discrete dividend is D, then the value of the share price just after discrete dividend payment is S-D.
  4. Enter the number of known dividend yields before expiry of longest maturity option (enter zero if none) of indicated underlying share variable. Known dividend yield is such that, if the underlying share price value just before dividend payment is S, and the value of known dividend yield is q, then the value of the share price just after dividend payment is S*(1-q).
  5. Enter the total number of different option groups whose underlying is the indicated share variable. The rules for subdividing options into different groups are: Options in the same group must have the same maturity period, must be of the same type, must belong to the same underlying, must be of the same position(ie all long or all short), and must have the same strike price. For example, suppose an American options portfolio consists of 3 long call positions in Microsoft, 2 long put positions in BP Shell, and 1 short call position in BP shell. The number of option groups under Microsoft can be 1 with number of positions equalling 3 if all the call options are of the same maturity and strike price. The number of options groups under BP shell cannot be less than two. It can be 2 if the put options are of the same maturity and strike price. Repetition of the same option group is allowed. For example, one can enter the number of option groups under Microsoft as 3 each with number of positions equalling 1 even if they are of the same maturity and have the same strike price. The number of positions will be entered in the American options portfolio value-at-risk calculation parameters page.
  6. Enter pairwise correlations between indicated underlying share price variables.
  7. Click the submit button, this will take you to the American options portfolio value-at-risk calculation parameters page.

American options portfolio value-at-risk calculation parameters page

  1. Select the currency of valuation from the drop-down list.
  2. Enter the options portfolio name.
  3. Enter the projection period for value at risk calculation in days.
  4. Enter the continuously compounded zero-rate for one-day share price projection period.
  5. Enter the continuously compounded expected return on market portfolio.
  6. Enter the confidence-level for value at risk calculation as a percentage.
  7. Enter the initial prices of indicated share price variables.
  8. Enter the continously compounded dividend yields of indicated share price variables. The continously compounded dividend yields together with discrete dividends, and known dividend yields, enable any dividend payment patterns to be accomodated in the pricing model.
  9. Enter the market betas of indicated share price variables.
  10. Enter the maturities of indicated options in years.
  11. Enter the continuously compounded zero rates for the indicated options expiry periods.
  12. Enter the strike prices of indicated options.
  13. Enter the annualised implied volatilities of indicated options.
  14. Enter allocated expenses and tax to be added to the indicated pure option prices.
  15. Enter profit fraction to be loaded to the calculated sum of indicated pure option price and dealing expenses and tax. If the profit-loading is x and the sum of pure option price plus expenses and tax is P, then the final price will be calculated as (1 + x)*P. It is important to note that valuation is done from a dealer/broker's perspective. The profit-charge is charged by the dealer/broker.
  16. Select indicated option type.
  17. Select indicated option position.
  18. Enter the number of indicated option positions. For example, suppose an American options portfolio consists of 3 long call positions in Microsoft, 2 long put positions in BP Shell, and 1 short call position in BP shell. If the call options under Microsoft are of the same maturity, same strike price and are classified under one group, the number of positions is entered as 3. Similarly if the long BP Shell put options are of the same maturity, same strike price and are classified under one group, the number of positions is entered as 2. As for the short BP call option, it will be classified in its own group and the number of position is entered as 1. It is important to note that no negative sign is entered to indicate the short position. This system will identify the short position selection and internally inserts the negative sign.
  19. Enter discrete dividend values and their timing for indicated underlying share price variables.
  20. Enter known dividend yield values and their timing for indicated underlying share price variables.
  21. Click the submit button and this will take you to the output display page.

Output display page

  1. You can view pricing model output together with input parameters you entered.
  2. At the bottom of the page you can click on the button to create database record for the current model output. This will take you to the create database record page where you should click on the create button to create the database record.
  3. After clicking on the create button to create the database record, you will be taken to the database records view page, where you can scroll vertically or horizontally to view database records including the one you just created.
  4. You can filter database records according to the currency of valuation using the filter box. You can click on the details link on the extreme right of a particular database record, this will take you to the particular database record details page.
  5. To view the one-day options portfolio gain frequency distribution graph, click on the graph link in the output display page.