Delta and Gamma

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Changes in the value of an underlier are often the primary source of market risk in a derivatives portfolio, so there are two Greek factor sensitivities for measuring such exposure. Delta and gamma represent first- and second-order measures of sensitivity to an underlier.

Consider a hypothetical portfolio whose value depends upon some underlier whose current value is USD 101. Exhibit 1 illustrates how the current market value of a hypothetical portfolio depends upon the current underlier value.

Exhibit 1: The market value in USD millions of a hypothetical portfolio as a function of underlier value in USD. The current underlier value is USD 101. The graph indicates what the portfolio value would be if the underlier had a current value anywhere in the interval USD 99 to USD 103.

Exhibit 1: The market value in USD millions of a hypothetical portfolio as a function of underlier value in USD. The current underlier value is USD 101. The graph indicates what the portfolio value would be if the underlier had a current value anywhere in the interval USD 99 to USD 103.

Exhibit 1 fully describes the relationship between the portfolio’s current market value and the underlier, assuming other market variables are unchanged. With just the two values delta and gamma, we can summarize the information contained in Exhibit 1. Certainly, two numbers cannot describe the wealth of detail contained in a picture, but with delta and gamma we capture the two most important pieces of information in the picture.

Let’s start with delta. The most significant information that Exhibit 1 provides about this particular portfolio is the fact that its value will increase if the underlier increases, and it will decrease if the underlier decreases. This is the information that delta conveys, along with the magnitude of such sensitivity.

If we fit a tangent line to the curve in Exhibit 1 at the underlier’s current value of 101, the slope of that line will capture the direction and magnitude of the portfolio’s sensitivity to the underlier. Delta is the slope of that tangent line.

Exhibit 2: Delta is the slope of the tangent line fit to the portfolio's value function at the current underlier value. In this example, the current underlier value is USD 101, and the slope of the tangent line 700,000.

Exhibit 2: Delta is the slope of the tangent line fit to the portfolio’s value function at the current underlier value. In this example, the current underlier value is USD 101, and the slope of the tangent line 700,000.

For example, in Exhibit 2, the slope of the tangent line is 700,000 (for each unit increase in the underlier, the portfolio’s price appreciates by 700,000). Accordingly, the portfolio’s delta is 700,000.

Fitting tangent lines to functions is the province of calculus, so we turn to calculus for the formal definition of delta. Let p and s be current values for the portfolio and underlier. Delta is the first partial derivative of a portfolio’s value with respect to the value of the underlier:

[1]

This technical definition leads to an approximation for the behavior of a portfolio.

Δp ≈ delta Δs

[2]

Where Δs is a small change in the underlier’s current value, and Δp is the corresponding change in the portfolio’s current value. This is called a delta approximation.

Suppose a portfolio is exposed to IBM stock and has an IBM delta of 1.5MM shares. This means that, for small market moves, the portfolio behaves like a position comprising 1.5MM shares of IBM. It will gain about USD 1.5MM if IBM stock rises USD 1. It will lose about USD 3.0MM if the stock falls USD 2. Enter these numbers into formula [2] and confirm that it is saying the same thing! Note that the portfolio’s exposure could result from direct holdings in IBM stock, a derivatives position with IBM stock as an underlier, or some combination of the two. If it is caused entirely by an outright position in IBM stock, then that position must consist of exactly 1.5 million shares of IBM stock because the delta of one unit of the underlier always equals 1.0.

If the portfolio is exposed to several stocks, then it will have a delta for each. For example, it’s Exxon delta might be –2.5MM shares. This would behave similarly to a short position in 2.5 million shares of Exxon stock. If Exxon stock rose USD 1, the portfolio would lose about USD 2.5MM. If the stock declined USD 0.5, the portfolio would gain about USD 1.25MM.

Now let’s consider gamma. If delta summarizes the most significant piece of information about a portfolio’s sensitivity to an underlier, gamma summarizes the second-most significant piece of information. Delta captured the fact that the graph in Exhibit 1 was upward sloping. It did not capture its downward curvature. Gamma describes curvature.

Exhibit 3 shows the best-fitting parabola for the graph of Exhibit 1:

Exhibit 3: The parabola that best fits the portfolio's value function.

Exhibit 3: The parabola that best fits the portfolio’s value function.

Note that the best-fitting parabola does not exactly overlay the curve in Exhibit 3 because the curve is not itself a parabola. In general, the best-fitting parabola will have the form:

best fitting parabola = c s2 + b s + a

[3]

where ab and c are constants determined to achieve the best fit. Gamma equals 2c. As it turns out, for the best-fitting parabola, the constant b is the portfolio’s delta, and a can be solved for based upon the portfolio’s current market value.

Gamma not only tells us the magnitude of curvature, but its direction as well. Positive gamma corresponds to curvature that opens upward. Negative gamma corresponds to curvature that opens downward.

Exhibit 4: Positive gamma corresponds to curvature that opens upward. Negative gamma corresponds to curvature that opens downward.

Exhibit 4: Positive gamma corresponds to curvature that opens upward. Negative gamma corresponds to curvature that opens downward.

For a formal definition of gamma, we again turn to calculus. Gamma is the second partial derivative of a portfolio’s value p with respect to the value s of the underlier:

[4]

By incorporating gamma, we can improve our approximation [2] for how the portfolio’s value should change in response to small changes in the underlier’s value:

[5]

This is called a delta-gamma approximation.

While sensitivity to the value of an underlier can be one of the most significant determinants of a derivative portfolio’s  market risk, other sensitivities are also important. These include sensitivities to implied volatilities, interest rates, and the passage of time. There are Greek factor sensitivities for each of these, called: vegarho, and theta, respectively.

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