Scaling and Aggregating Option Greeks Across a Portfolio

Summary

The document considers how to calculate dollar exposures for a portfolio of equities and vanilla equity options. It asks whether each option Greek can be multiplied by contract quantity and the standard contract multiplier, then summed across positions. A response focuses on gamma, recommending a scaled measure that expresses the change in dollar delta for a percentage move in the underlying. This makes the exposure easier to interpret as the market moves.

The response distinguishes raw Black–Scholes gamma, a second derivative, from a practically scaled gamma exposure. It also advises estimating delta and especially gamma under a sticky-strike volatility assumption, describing a method that adjusts the strike and reprices with the corresponding implied volatility while bumping spot. The material is brief and does not give a complete set of scaling formulas, unit conventions, or treatment for different underlyings and contract specifications. Portfolio sums therefore require consistent definitions and units.

Key ideas

• Multiplying option Greeks by quantity and contract size is a proposed way to express position exposure.
• The response favors gamma expressed as dollar delta change per percentage move for easier interpretation.
• Raw Black–Scholes gamma must be rescaled before it represents that practical exposure measure.
• Sticky-strike implied volatility is suggested when estimating delta and gamma under spot changes.
• Portfolio Greek totals are meaningful only when positions use compatible units and conventions.

Tags

# Portfolio Greek Exposure Equations


# Portfolio Greek Exposure Equations












What are the calculations for calculating greek exposures in a portfolio of equities and equity options? I think I have them but I want to be sure. Are these correct (for vanilla options)?

```
deltaDollars = delta * quantity * 100

gammaDollars = gamma * quantity * 100

vegaDollars = vega * quantity * 100

thetaDollars = theta * quantity * 100

rhoDollars = rho * quantity * 100
```

I think that for calculating exposures for a whole portfolio, I can sum up these values for each position in the whole portfolio. Is this correct?

I am not sure about summing gammaDollars in this way because gamma is a second derivative. If my portfolio has only these positions:

- 23 MSFT options (gamma = 0.1, gammaDollars = $230)

- 29 AAPL options (gamma = 0.2, gammaDollars = $580)

Can I say that my portfolio's gammaDollars are \$810 (\$230 + \$580)? Or can I not add the numbers in this way?

## Answer by Strange (score 1)

https://quant.stackexchange.com/a/4119

I, personally, like to see gamma as change in dollar delta per percent (most systems have it as "GammaP"). This way, it's much easier to think about you delta position as the market is moving around.

The number above is the BS gamma which is an unscaled 2nd derivative of delta. You need to rescale it to get gammap (delta change per percent). In general, it is also a good idea to calculate your delta and especially gamma using sticky strike volatilities - the easiest way to do so is to look at implied vol on the adjusted strikes and reprice your option using that vol and spot bumped up/down by the strike spread.

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