A financial institution owns a portfolio of options on the...

A financial institution owns a portfolio of options on the U.S. dollar-sterling exchange rate. The delta of the portfolio is 56.0 . The current exchange rate is 1.5000 . Derive an approximate linear relationship between the change in the portfolio value and the percentage change in the exchange rate. If the daily volatility of the exchange rate is $0.7 \%$, estimate the 10 -day $99 \% \mathrm{VaR}$.

Key Concepts

Time Horizon Scaling

Time horizon scaling, often implemented via the square root of time rule, is used to adjust volatility estimates from a one-day period to a longer period, such as 10 days. This method assumes that the underlying asset’s returns are independently and identically distributed over time, allowing volatility (and thus VaR) to be scaled by the square root of the number of days in the period.

Volatility

Volatility represents the standard deviation of the percentage changes in the underlying exchange rate over a specified period. It is a crucial component in risk management as it quantifies the normal variability or uncertainty associated with the asset’s price movements, which is then used to compute the potential extreme variations in portfolio value.

Value at Risk (VaR)

Value at Risk (VaR) is a statistical measure that estimates the maximum potential loss of a portfolio over a given time period and confidence level under normal market conditions. In this case, the 10-day 99% VaR is estimated by scaling the daily volatility over 10 days and applying the corresponding critical value from the normal distribution to assess the level of risk that the portfolio might experience.

Linear Approximation

The linear approximation uses the delta of the options portfolio to estimate the change in portfolio value as a product of delta and the change in the underlying asset's value. This approximation assumes that over a small change, the relationship between the portfolio’s value and the underlying asset’s price is nearly linear, allowing us to relate percentage changes in the exchange rate to changes in the portfolio value.

Delta

Delta is a measure of the sensitivity of the value of a portfolio of options to changes in the price of the underlying asset. In this context, it quantifies how much the portfolio’s value is expected to change for a small absolute change in the currency exchange rate. Essentially, delta provides a first?order linear approximation for the option's price movement relative to its underlying asset.

Exchange Rate Percentage Change

Relating the change in the portfolio value to the percentage change in the exchange rate involves comparing the absolute change in exchange rate (derived from the current rate and its percentage change) with the delta-adjusted change in value. This process emphasizes how relative changes (i.e., percentage changes) in the underlying variable are converted into absolute changes in portfolio value through the sensitivity metric (delta).

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