# 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?

• Is that the VaR of a naked long option, or the VaR of the long option + hedge that you are referring to ? 'Premium of the Option' / 20 seems very low for the daily VaR of a naked position. – Antoine Conze Mar 23 '18 at 12:22
• I think he is referring to a naked position – sen_saven Mar 23 '18 at 12:33
• Quick approximation for an ATM Call: $\text{premium} \approx 0.4 S \sigma \sqrt{\text{maturity}}$ (see quant.stackexchange.com/questions/1150/…), $\Delta \approx 0.5$, and $\text{VaR} \approx 2.3 \Delta S \sigma \sqrt{\text{1 day}}$. So that would give $\text{VaR} \approx \frac{0.5 \times 2.3}{0.4} \sqrt{\frac{\text{1 day}}{\text{maturity}}} \text{premium} \approx 3 \sqrt{\frac{\text{1 day}}{\text{maturity}}} \text{premium}$ e.g. $\text{VaR} \approx \text{premium} / 3$ for a 3 months option – Antoine Conze Mar 23 '18 at 13:58
• thanks for that - quite informative. However, while I agree that this is a much more mathematical approximation than the one mentioned by the risk manager, it still ignores any information regarding the dynamics of the underlying so it doesn't answer the original question looking for references about a 'merge' between the 'premium' methodology and the 'delta/gamma' one. – sen_saven Mar 23 '18 at 14:35
• Personally I think it is a fallacy rather than "a really rough approximation". – Alex C Mar 24 '18 at 17:32

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.

• thanks for your response. Could you please refer me to a paper adding the premium as the portfolio value constraint? – sen_saven Mar 28 '18 at 16:58
• sorry I don't know of any – Antoine Conze Apr 4 '18 at 9:21