Delta Exposure

Delta exposure, sometimes referred to as dollar delta or delta adjusted exposure, measures the first order price sensitivity of an option or portfolio to changes in the price of an underlying security. For example, if the delta exposure of a portfolio is \$200 million and the underlying security undergoes a 1% return, then the portfolio should increase in value by approximately \$2 million = \$200 million x 1%.

Delta exposure can be used to measure the sensitivity of a portfolio with or without options. If the portfolio does contain options then the delta exposure will be most accurate for small change in the value of the underlying.

For an option, the delta exposure is equal to the delta of the option multiplied by the price of the underlying security. Mathematically, for an option with value V, Δ = ∂V/∂S and underlying price S, the delta exposure is ∆\$ = ∆S. The change in the value of the option due to a small change in the underlying is approximately ∆dS:

dV ≈ ΔdS

Multiplying and dividing the right-hand side of the equation by S, we have:

In other words, to a first order approximation, the change in the value of the option is equal to the dollar delta multiplied by the percentage change in the value of the underlying security.

Because the delta of an equity is equal to 1, the delta exposure of an equity is equal its market value.The delta exposure of a portfolio is equal to the sum of the delta exposures of all of the securities in the portfolio.

As an example, if a portfolio contains 10 call options on IBM, each with a Δ = 0.50, and IBM is trading at \$100, then the delta exposure of each option is \$50 = \$100 x 0.50 and the delta exposure of all 10 options is \$500. If IBM increases in value by 1%, we would expect the 10 options to increase in value by approximately \$5. If the portfolio also contained 2 shares of IBM stock, then the delta of the entire portfolio would be \$700 = 10 x 0.50 x \$100 + 2 x \$100.

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