Delta-Normal VaR Limits for Stock and Option Portfolios

Summary

The document asks how to calculate a ten-day, 99% Value at Risk for a portfolio that combines shares with short European calls on the same company. It gives the share value, annual volatility, and option delta, but the response cautions that delta alone does not capture the nonlinear payoff of an option. It further argues that adding gamma or higher-order Taylor terms does not necessarily produce a meaningful VaR in this setting.

Instead, the response recommends evaluating portfolio profit and loss across many market-move scenarios. These scenarios can be generated by Monte Carlo methods designed to resemble historical moves, or drawn from actual historical observations. The portfolio should be repriced or otherwise evaluated under each scenario, with shortcuts potentially available to avoid full repricing every time; the tail outcomes then inform VaR. The excerpt does not specify a scenario model, calibration choices, or an implementation, so it provides a high-level method rather than a complete calculation.

Key ideas

• Delta alone does not capture the nonlinear payoff of the short call position.
• The response cautions that adding local Taylor terms may still fail to produce meaningful VaR.
• VaR can be estimated by evaluating portfolio profit and loss across simulated or historical scenarios.
• Scenario analysis should account for the portfolio's stock and option positions together.
• The excerpt leaves model calibration and calculation details unspecified.

Tags

# Delta-normal VaR of portfolio of stock and call option


# Delta-normal VaR of portfolio of stock and call option












I have to calculate the 10-day 99% VaR of a portfolio that consists of a portfolio of 260 stocks of a company $K$ and that is short 500 call (European) options of the same company.

I know that the stocks currently have a value of €73.35, its annual volatility is 17.12% and the call options have a delta is 0.6.

I can compute the VaR of a portfolio of options but I'm a little puzzled how I can do this when we have a portfolio of a stock and an option

## Answer by Dimitri Vulis (score 2)

https://quant.stackexchange.com/a/55323

(I assume that by "260 stocks" you mean 260 shares of same corporation that are also the underlying of the options.)

Since the payoff of the options is non-linear, you can't get a meaningful VaR by multiplying the delta by the volatility of the underlying stock. You can't even get a meaningful VaR by including gamma or higher-order terms of a Taylor expansion.

You need to either use Monte-Carlo to generate lots of posisble market move scenarios that look like historical scenarios, or use lots of actual historical scenarios. You need to estimate the P&L of your portfolio under each scenario (there are shortcuts so you may not have to reprice under every scenario). You need to look at the worst case scenarios.

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