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Quoting Hull's book:

When gamma is positive, theta tends to be negative. The portfolio declines in value if there is no change in S, but increases in value if there is a large positive or negative change in S. When gamma is negative, theta tends to be positive and the reverse is true: the portfolio increases in value if there is no change in S but decreases in value if there is a large positive or negative change in S. As the absolute value of gamma increases, the sensitivity of the value of the portfolio to S increases.

So there is a clear opposite sign correlation but I don't understand why if gamma is negative then theta tends to be positive and the portfolio increases in value if there is no change in S?

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    $\begingroup$ Gamma is a desirable feature of options, it affords the holder of the option some protection against unfavorable price moves. Theta is a disadvantage of options, it causes their value to decay over time. It seems intuitive that these have to balance each other, since there is no free lunch, or you can't have a benefit without a cost. $\endgroup$
    – nbbo2
    Jun 25, 2017 at 21:03
  • $\begingroup$ Theta is not always negative? $\endgroup$
    – Carlo
    Jun 25, 2017 at 21:22
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    $\begingroup$ Not always. Theta is always negative when $r=q=0$ but for big interest rates or dividend rates it is not guaranteed. $\endgroup$
    – Alex C
    Jun 25, 2017 at 22:23
  • $\begingroup$ Positive Theta means that, for example, an option increases to the diminishing of residual life. Negative Gamma means that the price decreases as the underlying price varies. How are these two concepts correlated? $\endgroup$
    – Carlo
    Jun 25, 2017 at 23:07
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    $\begingroup$ Let's look at it from pt of view of option holder. Negative theta (the normal case) means the option value decreases as time advances. Positive gamma (the normal case) means the Delta decreases as the underlying $S$ decreases, providing a "cushion" (protection) against a further fall. $\endgroup$
    – Alex C
    Jun 26, 2017 at 0:28

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I think I've found the answer to my question (I'm waiting for confirmation from you in the comments)

The intuitive difference in this negative sign correlation depends on the position taken on options in the portfolio:

  • Gamma is always positive when you buy an option (Theta acts negatively when buying options);
  • Gamma is always negative when selling an option (Theta acts positively in case of sale).
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    $\begingroup$ This is correct $\endgroup$
    – zer0hedge
    Jul 5, 2017 at 10:17
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    $\begingroup$ Depends on what you call $\theta$ I Guess. Is ot the partial derivative or the carry cost which should offset the convexity? The partial derivative is not always positive. $\endgroup$
    – Quantuple
    Jul 6, 2017 at 10:29
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I think this is very well explained (with almost no maths) in the first chapter of Lorenzo Bergomi's book "Stochastic Volatility Modeling" (sample available here for download). Note that he explains it for a delta-hedged portfolio, which is not exactly your question but I think it can help anyways (and too long so that I can post it as a comment).

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  • $\begingroup$ But, as @Tom Au also answered: Theta is a "greek"that represents time decay. All other things equal, the longer the time elapsed before the maturity date, the less the value of the option. That is, theta is negative over time. $\endgroup$
    – Carlo
    Jun 26, 2017 at 16:12
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    $\begingroup$ That is not true in general. Think of a call with dividends for instance. $\theta$ simply figures a financing cost. $\endgroup$
    – Quantuple
    Jun 26, 2017 at 16:16
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Theta is a "greek"that represents time decay. All other things equal, the longer the time elapsed before the maturity date, the less the value of the option. That is, theta is negative over time.

Gamma refers to the "second derivative" of the price of the underlying security. (The option captures the "delta," or the first derivative). Because it is a second derivative, gamma is positive when the price of the underlying security moves towards the strike price of the option, and negative when it moves away. So depending on the price movements, gamma could be either positive or negative, while theta is negative, and the two could thus be positively or negatively correlated.

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  • $\begingroup$ I agree with your answer, but reading Hull: 'When gamma is positive, theta tends to be negative. The portfolio declines in value if there is no change in S, but increases in value if there is a large positive or negative change in S. When gamma is negative, theta tends to be positive and the reverse is true: the portfolio increases in value if there is no change in S but decreases in value if there is a large positive or negative change in S. As the absolute value of gamma increases, the sensitivity of the value of the portfolio to S increases' So there is a clear opposite sign correlation $\endgroup$
    – Carlo
    Jun 26, 2017 at 8:25
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    $\begingroup$ Important: when Hull talks about "the portfolio" he means the delta hedged portfolio, not just the option itself. Let us be clear which we are talking about, the option or the option plus the delta hedge. Otherwise it is confusing. $\endgroup$
    – nbbo2
    Jun 26, 2017 at 12:15
  • $\begingroup$ The forced correlation in the opposite sign between Gamma and Theta depends on whether we are talking about delta hedged portfolio or just the option itself? $\endgroup$
    – Carlo
    Jun 26, 2017 at 16:06
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Gamma is the second derivative of price (the first is delta, there are third and fourth and on-up derivatives that are largely not useful. I refer to these as color and temperature). Since your portfolio delta measures your directional assumption, the gamma measures the propensity to a price movement Practically this is only useful when looking at near expiration term, at the money options.

Theta will be positive if you sell options (theta decay - you sell high and buy back low). Buying options (do not do this unless you have to) gives negative theta.

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  • $\begingroup$ When gamma is negative, theta tends to be positive: the portfolio increases in value if there is no change in S but decreases in value if there is a large positive or negative change in S. Why? $\endgroup$
    – Carlo
    Jul 2, 2017 at 9:32
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The gamma/theta concept is simple.

Let's say there's an asset with a price of 100 dollars. I say to you: "I'm going to give this cool portfolio, where if the price goes from 100 dollars to 102, you will make 4 dollars. But if the price goes from 100 dollars to 98, you will also make 4 dollars. Will you take this portfolio?"

I hope your answer is: heck yeah !!! Will I give it to you? Yes... but at a price. The portfolio I'm giving you has a positive gamma. But it's NOT free. It costs money (that's theta).

So if you graph the portfolio PnL, it will look like a U shape. In mathematics, it has a positive 2nd derivative = convex. The more convex it is, the better its profile is (for the owner), but the more expensive it will be.

If the portfolio's profile was inverted, we say it's concave (or having a negative 2nd derivative). This portfolio sucks. It loses money on an up move, and loses money on a down move. You'd better have someone pay you money to lay this risk on you.

Being long options means you have a positive convexity (long gamma). If you make one hedge (equal to delta) at the time of buying the option (sell the underlying asset for a call or buy the underlying asset for a put), you will create a U-shaped portfolio. Now if the underlying moves up or down, you make $$$.

The options (ATM and around ATM) get steeper and steeper (more U-shaped) as time moves. Therefore, their gamma gets bigger. Not only do they lose more value, but they lose value (theta) at a faster rate is you get closer to expiration.

I hope this helps.

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