Modeling Value-at-Risk for Long Options with Payoff Bounds

Summary

The document considers how to estimate daily value-at-risk for a long option and asks whether the estimate can reflect the fact that the position cannot lose more than its premium. It contrasts a rough premium-based rule with delta-gamma VaR, which approximates portfolio changes using the underlying’s price movement and option sensitivities.

The answer says a semi-analytical approach based on a quadratic portfolio return distribution may not readily impose a bound on portfolio value. Monte Carlo simulation can instead evaluate the portfolio’s bounded payoff directly, so it can represent the zero-value floor of a long call. The discussion is conceptual and offers no empirical comparison or implementation details; the appropriate VaR estimate still depends on the chosen risk horizon, market assumptions, and simulation design.

Key ideas

• A long option’s loss is bounded by the premium paid.
• Delta-gamma VaR may not naturally capture a hard floor on portfolio value.
• Monte Carlo simulation can apply the option’s bounded value function to simulated market outcomes.
• The document does not establish a specific VaR model or compare estimates empirically.

Tags

# VaR of long options


# VaR of long options












I just had a chat with a risk manager who thinks that the daily VaR of a long option with a maturity under three months should be 'Premium of the Option' / 20 (assuming twenty days in a month)

Obviously, this looks like a really rough approximation and there are a few approaches which look much more scientific - this link contains one of them . However, I think that there is some value in considering the fact that the VaR cannot be more than the premium for a long option position - is there any model/framework which takes this 'cap' into account?

## Answer by Antoine Conze (score 0, accepted)

https://quant.stackexchange.com/a/38968

If I understand correctly from your comments, your question could be rephrased as "is it possible to incorporate known bounds on the portfolio value in the delta gamma VaR approach" ? (e.g. for a long call the value cannot go below zero). I don't think the semi-analytical approach, which is based on an FFT applied to the quadratic portfolio return moment generating function to obtain the PDF - see for instance http://www.financerisks.com/filedati/WP/paper/RM%20FOR%20FINANCIAL%20INSTITUTIONS.pdf, can do that. However Monte Carlo methods can easily accommodate bounds on the portfolio value function.

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.