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Limits of Delta-Only VaR for Currency Option Portfolios

Article Quant Q&A · Author: May
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Summary

The document considers a ten-day, 99 percent value-at-risk estimate for a portfolio of options on the USD/GBP exchange rate. It presents a delta-equivalent position and a daily-volatility-based calculation that scales daily risk by the square root of the horizon, then applies a normal quantile. The example also raises a unit conversion question, but does not establish a corrected numerical VaR.

The substantive answer cautions that delta-only risk can be inadequate for options because their value responds nonlinearly to the underlying exchange rate. Gamma can make a Taylor approximation inaccurate for a large move, and implied-volatility changes can affect value through vega. A delta-only estimate is more appropriate for a cash foreign-currency position or options far enough in or out of the money that these effects are immaterial. The document does not provide an option revaluation procedure or a complete VaR model.

Key ideas

  • Square-root-of-time scaling is used in the example to extend daily risk to a multi-day horizon.
  • A normal quantile is applied to translate volatility into a confidence-level risk estimate.
  • Delta alone may poorly estimate option losses for a large underlying move because option value is nonlinear.
  • Gamma and changes in implied volatility can materially affect option portfolio VaR.
  • Delta-only risk is more suitable for cash currency exposure or options with negligible nonlinear effects.

Tags

ForexOptionsRisk managementVolatility
Full text
# 10-day VaR for a portfolio


# 10-day VaR for a portfolio












So, Bank ANZ owns a portfolio of options on the USD/GBP exchange rate. The delta equivalent position of the portfolio is GBP 56.00. The current exchange rate is 1.5, with a daily volatility of 0.7 percent. Using the given information and assuming that changes in portfolio value are normally distributed, the 99 percent /10-day VaR for this portfolio is:

I have noted that: Daily VaR = Daily Volatility * Delta Equivalent Position * Exchange Rate)

How do I use this to calculate the 10-day VaR?

## Answer by Dimitri Vulis (score 1)

https://quant.stackexchange.com/a/59964

In general, the P&L of options is non-linear with respect to the underlying. Unless the options are very far in or out of the money, the delta alone does not accurately tell you what the value of the option will be if the underlying exchange rate moves 2.33 standard deviations.

Even if you were told the gamma, estimating the change in value of the options using delta and gamma (Taylor expansion) for a 2.33 sd move would be very inaccurate.

Also, unless again the options are very far in or out of the money, they have material vega. Your VaR should include the possible change in option value because implied volatility changes.

Your delta-only approach would be OK if instead of options you just had foreign currency as cash.

## Answer by May (score 0)

https://quant.stackexchange.com/a/59963

I multiplied the delta equivalent by the daily volatility by the current exchange rate: 56 x 1.5 x 0.7 = 58.8

I then multiplied this by the square root of 10 to get the 10-day VaR = 185.942.

I then multiplied this answer by 2.33 (99 percent confidence interval) to get 433.245.

I assume we then divide this by 100 to get the final answer of 4.33?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.