Apr 18

VaR of binary option portfolio (Par 4 difficulty)

by David Harper, CFA, FRM, CIPM

FRM | Risk |

Sudoku

VaR is just a quantile

For FRM candidates, one thing I'd like to impart with yesterday's and today's "pop quiz" is that VaR is pretty harmless. VaR is just a quantile.  While we are shown three approaches (parametric, historical sim and Monte Carlo; plus hybrids) and complexity ensues in the methodology, the one thing they all have in common is this quantile: the expected loss (over a horizon) with some probability.

In Gujarati (Chapter 2), we are shown the cumulative distribution function (CDF). If we assume a standard normal random variable, the CDF = F(X) = P(X <= x). As for example, P(X <= 1.645) = 95% or, its equivalent for our purposes, P (X <= -1.654) = 5%. That means, "95% of time, the standard normal random variable will fall less than 1.645 standard deviations away from the mean."

Once you get that, you will see that all parametric VaRs are inverse CDFs. To use Excel's function terminology, if our assumption is normality (note: convenient but empirically incorrect!), then 95% VaR = NORMSINV(95%) = 1.65 standard deviations. So, we only need to "un-standardize" and multiply by volatility: 95% VaR = (1.65)(volatility%) and relative dollar VaR = (1.65)(volatility)(portfolio value). We could do the same with any parametric distribution; e.g., lognormal VaR is based on =LOGINV(confidence, mean, volatility).

Question (adapted from 2006 practice exam)

Assume you own a portfolio that is short two binary call options (i.e., you have written two call options. Your counterparty is long.) Both options have the same payoff: 10% probability of no exercise (i.e., the options will expire out of the money) and 90% probability of exercise with a $100 loss to you. The options are uncorrelated. Assume a time horizon of one period.

  • What is the portfolio's 90% value at risk (VaR)?
  • Bonus: the example happens to expose a failure in VaR. Wilmott's Intro to VaR chapter cites four criteria for a proper risk measure, which does the example fail?

 

(do not peek until you've tried!)

 

 

 

(do not peek until you've tried!)

Answer:

Please see the EditGrid spreadsheet below. (It's calculator mode, it can be changed without altering the original!)

The PDF consists of only three joint probabilities: P (no exercises), P(one exercise) and P(both exercised). The joint P(no exercises) = (10%)(10%) = 1%. The joint P (both exercised) = (90%)(90%) = 81%. That's actually all we need to do. We can see the 90% VaR will fall in between these outcomes, where one option is exercised. Therefore, the 90% VaR = $100 (or -$100).

Note at the bottom of the sheet, I use =CRITBINOM() function with is an inverse CDF.

Bonus: it fails sub-additivity which renders VaR incoherent! Why? Because the 90% VaR of each individual (option) instrument is 0, but the 90% VaR of the portfolio = $100. That's anti-diversification!

 

Spreadsheet:

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