Estimating One-Day VaR for Long and Short European Calls
Summary
The example estimates one-day Value-at-Risk for a large position in at-the-money European call options on a non-dividend-paying stock. It uses a first-order approximation: convert annualized volatility to a daily standard deviation, take the adverse return quantile for the stated confidence level, and apply the option position’s approximate delta to translate the stock move into a portfolio loss. The example treats the calls as roughly equivalent to holding a fraction of the underlying shares.
This delta-only calculation gives an approximate loss for the long position. The answer explains that gamma refines the estimate: long calls have positive gamma, which cushions losses on a downward move, so the long-position VaR is somewhat smaller in magnitude than the linear estimate. For a short position, delta exposure is similar at the outset, but negative gamma worsens the adverse effect. The calculation is a back-of-the-envelope approximation, not a full option repricing or a general VaR model; its result depends on the stated assumptions and omits other risks.
Key ideas
- A delta approximation translates an underlying stock return into an approximate option portfolio loss.
- Annualized volatility must be scaled to a daily volatility for a one-day risk estimate.
- Positive gamma in a long call position can reduce losses relative to a delta-only estimate.
- Short option gamma can increase losses compared with the linear delta approximation.
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# How do I compute Value at Risk of a European call option?
# How do I compute Value at Risk of a European call option?
Consider a European call option on a non-dividend paying stock, where the option has strike K = 100 and expiry T = 0.25, i.e. the option expires 3 months from now. The option is on a single share. The current price of the stock is 100, the riskfree interest rate is zero, and the option premium (i.e. price) is equal to the price given by the Black-Scholes formula using a volatility of 30% (where this is quoting volatility on an annualized basis). The expected return on the stock is 10% annually.
Let Π denote a portfolio consisting of a long position of 100, 000 of these options. At a confidence level of 95%, what is the 1-day Value-at-Risk of the portfolio Π?
What changes it instead of a long position I take it short?
## Answer by ZRH (score 2)
https://quant.stackexchange.com/a/43847
Quick back-of-the-envelope calculation would be to say that the downmove which will not be exceeded in 95% of cases, is $N^{-1}(0.05)=-1.64$ of a daily return standard deviation $\sigma_{daily}=30\%/\sqrt{365}=1.57\%$. So that means a downmove of $-1.64*1.57\%=-2.58\%$. Your option is ATM, so ca. 50% delta or the equivalent of 50'000 shs. 50'000 shs at price 100 correspond to 5'000'000 invested, a $-2.58\%$ move on this is ca. -129'000. For a more accurate value, you would have to include gamma, which on your option position is long, hence VaR will turn out slightly smaller than the -129'000 done with this simple analysis.
On a short position, the analysis remains the same for delta, gamma however will be systematically to your disadvantage, making VaR<-129'000.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.