Definitions and Calculation of Delta and Gamma in Structured Option Products

Structured option products are customized derivatives based on traditional options, and their pricing and risk management are relatively complex. For these products, sensitivity indicators (the "Greeks") remain essential tools for risk measurement, with Delta (the sensitivity to changes in the underlying asset's price) and Gamma (the sensitivity of Delta's change) being the most crucial. The following is a summary of their definitions and calculation methods.

1. Definition and Calculation of Delta

1.1 Traditional Definition

Traditionally, Delta is defined as the first derivative of the option value $V$ with respect to the underlying asset price $S$:

$$\Delta = \frac{dV}{dS}\,.$$

This value describes the absolute change in the option's price (in option price units) for a unit change in the underlying asset's price (for example, 1 currency unit).

1.2 Conversion to a Percentage Delta

In practical risk management, it is often more intuitive to convert Delta into a measure that indicates the relative (percentage) change in the option’s value when the underlying asset's price changes by a specific percentage, commonly called delta(%). The calculation proceeds as follows:

1. Suppose the current underlying asset price is $S$. A 1% change corresponds to a price change of

   $$\Delta S_{1\%} = 0.01 \times S\,.$$

2. According to the traditional definition of Delta, the approximate change in the option’s price is

   $$\Delta V \approx \Delta \times \Delta S_{1\%} = \Delta \times (0.01 \times S)\,.$$

3. If $V$ is the current option price, the relative (percentage) change in the option's value is then

   $$\text{delta}(\%) = \frac{\Delta V}{V} \times 100\,\%.$$

4. Rearranging the formula gives the following

   $$\text{delta}(\%) \approx \frac{\Delta \times S}{V}\,.$$

Note:
The formula $\text{delta}(\%) = \frac{\Delta \times S}{V}$ yields a number that represents the percentage part. For example, a result of 3 means 3% (and not 0.03). If a decimal representation is needed, the result should be divided by 100.

1.3 Consideration of Trade Direction in Structured Products

In structured option products, numerical methods (such as finite difference methods) are typically used to calculate the option’s theoretical value. During this process, the trade direction (e.g., the option premium being negative for a purchase) is not incorporated into the theoretical valuation.

• Pricing Stage: The theoretical value of the option is calculated without embedding any directional information. Consequently, the computed Delta and Gamma provide absolute sensitivity measures.
• Risk Management Stage: The final sign of the sensitivity indicators is adjusted based on the actual trading position (whether it’s a long or short position). For instance, in forex options, a call may be positive while a put is negative. In structured products, the sign adjustment is made after obtaining the sensitivity values.

2. Definition and Calculation of Gamma

2.1 Traditional Definition

Gamma is the second-order sensitivity measure in option pricing and is defined as the second derivative of the option value $V$ with respect to the underlying asset price $S$:

$$\Gamma = \frac{d^2V}{dS^2}\,.$$

Its unit is typically expressed in option price units per square unit of the underlying asset’s price (for example, currency unit per (currency unit)$^2$).

2.2 Conversion to a Percentage Gamma

Similar to Delta, Gamma can also be converted into a measure that represents the relative (percentage) effect on the option's value due to a 1% change in the underlying asset's price—this is often denoted as Gamma(%). The derivation is as follows:

1. Consider a small change in the underlying asset with $dS = S\,\epsilon$, where $\epsilon$ represents the relative change (with $\epsilon = 0.01$ corresponding to a 1% change). Using a Taylor expansion, the approximate second-order change in the option’s value is

   $$dV \approx \frac{1}{2} \Gamma \,(dS)^2 = \frac{1}{2} \Gamma \, (S \epsilon)^2 = \frac{1}{2} \Gamma \, S^2 \epsilon^2\,.$$

2. The relative change in the option’s price, expressed as a percentage, is

   $$\frac{dV}{V} \approx \frac{1}{2} \frac{\Gamma S^2}{V} \epsilon^2\,.$$

3. Hence, Gamma(%) is defined as

   $$\boxed{\Gamma(\%) = \frac{\Gamma \, S^2}{V}\,.}$$

Explanation:
Gamma(%) represents the relative effect on the option’s price (due to second-order sensitivity) when the underlying asset experiences a 1% change. The result is expressed as the numeric part of the percentage (e.g., a Gamma(%) of 2 means 2%, not 0.02).

3. Summary

For structured option products, it is critical to separate the absolute theoretical valuation from the directional factors tied to trading positions:

• Pricing Stage:
  Utilize numerical methods (e.g., finite difference) to calculate the theoretical value of the option, along with Delta $\left(\Delta = \frac{dV}{dS}\right)$ and Gamma $\left(\Gamma = \frac{d^2V}{dS^2}\right)$. At this stage, the computed values are absolute and do not include any information about the position’s direction.

• Conversion to Percentage Measures:
  Based on the current underlying asset price $S$ and the option price $V$, the sensitivity measures are converted into percentage terms:

  • Delta(%):

    $$\text{delta}(\%) = \frac{\Delta \times S}{V}\,.$$

  • Gamma(%):

    $$\Gamma(\%) = \frac{\Gamma \, S^2}{V}\,.$$

• Risk Management and Hedging:
  Finally, during risk management or hedging operations, the computed sensitivity measures are adjusted (by assigning appropriate signs) according to the actual trading direction—for example, long positions may be positive, while short positions may be negative.