# 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)?

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You're going to kick yourself when you learn the answer.... –  Brian B Oct 23 '12 at 21:16
–  Alexey Kalmykov Oct 23 '12 at 21:33

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 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 10

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 $C_2$

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Well, $\frac{\partial^2 S}{\partial S^2 }$ is zero while $\frac{\partial S}{\partial t }$ is nonzero in the case of a dividend stream or other cashflow.... –  Brian B Oct 26 '12 at 17:52
Knew my calculus was rusty... Can you give me some further insight into how $\frac{\partial S}{\partial t}$ can become nonzero? –  strimp099 Oct 26 '12 at 20:02

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.

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Close, but you should fix your answer because $\theta$ is generically nonzero for positions in the underlying. –  Brian B Oct 24 '12 at 14:49
Fixed. Thank you, Brian. What I meant is that you cannot use underlying positions to hedge theta. –  mynegation Oct 24 '12 at 15:56
I'm rusty on my calculus (a while since Brian B's class :) ), am I on target? $$\frac{\partial}{\partial S} (100C_1 + 100C_2 + 10S) = 100\frac{\partial C_1}{\partial S} + 100\frac{\partial C_2}{\partial S} + 10\frac{\partial S}{\partial S}$$ Where of course $10\frac{\partial S}{\partial S}$ gives us 10. In the case of the other greeks, vega for example: $$\frac{\partial}{\partial \sigma} (100C_1 + 100C_2 + 10S) = 100\frac{\partial C_1}{\partial \sigma} + 100\frac{\partial C_2}{\partial \sigma} + 10\frac{\partial S}{\partial \sigma}$$ The $10\frac{\partial S}{\partial \sigma}$ term is 0. –  strimp099 Oct 24 '12 at 17:45
That's the right idea Strimp. @mynegation, your answer is still not quite correct. Strimp shows above that vega is zero rather than nonsensical. –  Brian B Oct 24 '12 at 20:44