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also commonly referred to as the time decay of the option. Theta is mathematically represented as follows:

      (1.17)

      where V is the price of the option (a call or a put) and t is time. The sign of theta depends on the type of position and is opposite gamma:

      ● Long call and long put: .

      ● Short call and short put: .

      In other words, the time decay of the extrinsic value decreases the value of the long position and increases the value of the short position. For instance, a long call with a theta of –5 per one lot is expected to decline in value by $5 per day. This makes intuitive sense because the holders of the contract take gradual losses as their asset depreciates with time, a result of the value of the option converging to its intrinsic value as uncertainty dissipates. Because the extrinsic value of a contract decreases with time, the short side of the position profits with time and experiences positive time decay. The magnitude of theta is highest for ATM options and lower for ITM and OTM positions, all else constant.

      There is a trade‐off between the gamma and theta of a position. For instance, a long call with the benefit of a large, positive gamma will also be subjected to a large amount of negative time decay. Consider these examples:

      ● Position 1: A 45 DTE, 16Δ call with a strike price of $50 is trading on a $45 underlying. The long position has a gamma of 5.4 and a theta of –1.3.

      ● Position 2: A 45 DTE, 44Δ call with a strike price of $50 is trading on a $49 underlying. The long position has a gamma of 7.9 and a theta of –2.2.

      Compared to the first position, the second position has more gamma exposure, meaning that the contract delta (and the contract price) is more sensitive to changes in the underlying price and is more likely to move ITM. However, this position also comes with more theta decay, meaning that the extrinsic value also decreases more rapidly with time.

      To conclude this discussion of the Black‐Scholes model and its risk measures, note that the outputs of all options pricing models should be taken with a grain of salt. Pricing models are founded on simplified assumptions of real financial markets. Those assumptions tend to become less representative in highly volatile market conditions when potential profits and losses become much larger. The assumptions and Greeks of the Black‐Scholes model can be used to form reasonable expectations around risk and return in most market conditions, but it's also important to supplement that framework with model‐free statistics.

      Covariance and Correlation

      Up until now we have discussed trading with respect to a single position, but quantifying the relationships between multiple positions is equally important. Covariance quantifies how two signals move relative to their means with respect to one another. It is an effective way to measure the variability between two variables. For one signal, X, with observations and mean , and another, Y, with observations and mean the covariance between the two signals is given by the following:

      (1.18)

      Represented in terms of random variables X and Y, this is equivalent to the following:15

      (1.19)

      Simplified, covariance quantifies the tendency of the linear relationship between two variables:

      ● A positive covariance indicates that the high values of one signal coincide with the high values of the other and likewise for the low values of each signal.

      ● A negative covariance indicates that the high values of one signal coincide with the low values of the other and vice versa.

      ● A covariance of zero indicates that no linear trend was observed between the two variables.

      Covariance can be best understood with a graphical example. Consider the following ETFs with daily returns shown in the following figures: SPY (S&P 500), QQQ (Nasdaq 100), and GLD (Gold), TLT (20+ Year Treasury Bonds).

Figure 1.9 (a) QQQ returns versus SPY returns. The covariance between these assets is 1.25, indicating that these instruments tend to move similarly. (b) TLT returns versus SPY returns. The covariance between these assets is –0.48, indicating that they tend to move inversely of one another. (c) GLD returns versus SPY returns. The covariance between these assets is 0.02, indicating that there is not a strong linear relationship between these variables.

      Covariance measures the direction of the linear relationship between two variables, but it does not give a clear notion of the strength of that relationship. Because the covariance between two variables is specific to the scale of those variables, the covariances between two sets of pairs are not comparable. Correlation, however, is a normalized covariance that indicates the direction and strength of the linear relationship, and it is also invariant to scale. For signals with standard deviations and covariance , the correlation coefficient ρ (rho) is given by the following:

      (1.20)

The correlation coefficient ranges from –1 to 1, with 1 corresponding to a perfect positive linear relationship, –1 corresponding to a perfect negative linear relationship, and 0 corresponding to no measured linear relationship. Revisiting the example pairs shown in Figure 1.9, the strength of the linear relationship in each case can now be evaluated and compared.

      For Figure 1.9(a), QQQ returns versus SPY returns, the correlation between these assets is 0.88, indicating a strong, positive linear relationship. For Скачать книгу