Introduction

In this chapter, we will delve into the Delta-Normal approach, a widely used technique for calculating Value at Risk (VaR) specifically for non-linear derivatives. The Delta-Normal method offers a simplified approximation by assuming that the distribution of portfolio returns is normal and employing the concept of the delta of derivatives. Let’s explore this approach and its application in calculating VaR.

The Delta-Normal approach is a valuable tool for estimating VaR in cases involving non-linear derivatives. This method leverages the delta of derivatives, allowing for a simplified representation of portfolio risk. By understanding the concepts and calculations involved in the Delta-Normal approach, you will be equipped to assess the potential loss associated with non-linear derivatives under different market conditions.

Delta-Normal Approach and Its Explanation

The Delta-Normal approach hinges on the assumption that the distribution of portfolio returns is approximately normal. This simplifies the calculation process while still providing a useful estimate of potential loss. The “delta” of a derivative measures the sensitivity of its value to changes in the underlying asset’s price. In the Delta-Normal method, the VaR calculation involves multiplying the delta of the derivative by the standard deviation of the asset’s return and then by an appropriate confidence level z-score.

Calculating VaR Using Delta-Normal Approach

The formula for calculating VaR using the Delta-Normal approach is as follows:

$$VaR=\Delta \times \sigma \times Z$$

Where:

• $\Delta$ represents the delta of the derivative.
• $\sigma$ is the standard deviation of the asset’s return.
• $Z$ is the $z$-score corresponding to the chosen confidence level (e.g., 1.645 for 95% confidence).

Example: Consider a portfolio that includes call options on a specific stock. The delta of a call option can be less than 1, signifying its non-linear behavior concerning changes in the stock price. Suppose the delta of the call option is 0.7, and the standard deviation of the stock’s return is 0.2. For a 95% confidence level, the $z$-score is 1.645. Using the Delta-Normal formula, we can calculate the VaR as follows:

$$VaR=0.7 \times 0.2 \times 1.645=0.2303$$

Thus, the estimated VaR for the given portfolio is 23.03%.