Calculating Portfolio Greeks for Options and Underlying Shares
Summary
The document explains how to aggregate the five standard option Greeks when a portfolio also holds shares of the underlying. Portfolio sensitivity is calculated by differentiating the combined position value: option exposures are multiplied by their quantities, while the share position contributes delta equal to its share count and contributes zero gamma. In the Black–Scholes setup described, shares also contribute no theta, vega, or rho, so those exposures come from the options.
The example combines two option positions with a share holding and shows how the share terms enter each derivative. The method assumes the options and shares refer to the same underlying, allowing a common underlying-price change in the delta and gamma calculations. The answer cautions that options on different underlyings require separate sensitivities; their price moves cannot automatically be treated as the same change. The explanation is specific to the stated model and portfolio setup, rather than a full treatment of dividends, financing, or other market conventions.
Key ideas
- Portfolio Greeks can be found by differentiating the value of the combined positions.
- Shares contribute one unit of delta per share and zero gamma.
- In the stated Black–Scholes setup, shares contribute zero theta, vega, and rho.
- Aggregating delta and gamma this way assumes a shared underlying price movement.
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Full text
# Equity option portfolio greeks with underlying
# Equity option portfolio greeks with underlying
I'm curious about how to construct the five basic greeks for an equity option portfolio when there are shares of the underlying in the portfolio.
For example, a portfolio of 100 call options and 100 put options has a portfolio delta of `100 * call_delta + 100 * put_delta` (assuming the 100 calls are the same and 100 puts are the same). A portfolio of 100 short call options and 100 short put options has a portfolio gamma of `-100 * call_gamma - 100 * put_gamma` (again assuming the 100 calls are the same and 100 puts are the same).
What about a portfolio of 100 call options and 100 put options and 10 shares of the underlying? How would one include for the other greeks (gamma, theta, rho, vega)?
## Answer by strimp099 (score 3, accepted)
https://quant.stackexchange.com/a/4423
Ok so for completeness, assuming Black-Scholes and an example portfolio of 100 long $C_1$, 100 long $C_2$ (both on the same underlying), and 10 long shares of the same underlying, $S$.
Portfolio delta:
$$\frac{\partial}{\partial S} (100C_1 + 100C_2 + 10S) = 100\frac{\partial C_1}{\partial S} + 100\frac{\partial C_2}{\partial S} + 10\frac{\partial S}{\partial S}$$
Where $10\frac{\partial S}{\partial S}$ term is 10.
Portfolio gamma:
$$\frac{\partial^2}{\partial S^{2}} (100C_1 + 100C_2 + 10S) = 100\frac{\partial^2 C_1}{\partial S^2} + 100\frac{\partial^2 C_2}{\partial S^2} + 10\frac{\partial^2 S}{\partial S^2}$$
Where $10\frac{\partial^2 S}{\partial S^2}$ term is 0.
Portfolio theta:
$$-\frac{\partial}{\partial \tau} (100C_1 + 100C_2 + 10S) = -100\frac{\partial C_1}{\partial \tau} - 100\frac{\partial C_2}{\partial \tau} - 10\frac{\partial S}{\partial \tau}$$
Where $10\frac{\partial S}{\partial \tau}$ term is 0.
Portfolio vega:
$$\frac{\partial}{\partial \sigma} (100C_1 + 100C_2 + 10S) = 100\frac{\partial C_1}{\partial \sigma} + 100\frac{\partial C_2}{\partial \sigma} + 10\frac{\partial S}{\partial \sigma}$$
The $10\frac{\partial S}{\partial \sigma}$ term is 0.
Portfolio rho:
$$\frac{\partial}{\partial r} (100C_1 + 100C_2 + 10S) = 100\frac{\partial C_1}{\partial r} + 100\frac{\partial C_2}{\partial r} + 10\frac{\partial S}{\partial r}$$
The $10\frac{\partial S}{\partial r}$ term is 0.
Note this assumes the options are on the same underlying. This is important because the partials assume a small (or at least constant) change in the underlying across the portfolio. If $C_1$ and $C_2$ were on different underlyings, we cannot necessarily assume that a small change in the underlying of $C_1$ will be the same small change in the underlying of $C_2$
## Answer by mynegation (score 0)
https://quant.stackexchange.com/a/4411
Delta is a derivative of the price with respect of the price of underlying, so for the unit stock position delta is 1 and gamma is obviously 0. As for theta, rho and vega of the stock position, they do not make sense, at least not in the Black-Scholes setting they don't. You would not be able to hedge say vega or theta risk with stock positions.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.