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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
    

2 Answers 2

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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
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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

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