The Hidden Cost of Volatility Drag: Unpacking Proxy Hedging And Dependence

Volatility is a constant companion for investors. Markets are inherently unpredictable, making it challenging to predict price movements with certainty. One strategy that has gained popularity in recent years is proxy hedging and dependence – a fascinating area that delves into the intricacies of market dynamics.

When asked to summarize their approach to proxy / cross hedging (h!p://quantivity.wordpress.com/2011/10/02/proxy-cross-hedging), senior folks from numerous big banks reduced it to correlation: hedge using an instrument whose correlation is close to -1. This perspective matches the popular practitioner literature, such as recently published text Hedging Market Exposures (hp://books.google.com/books?id=CpSv76NCmJcC) (Bychuk and Haughey, 2011). Moreover, this perspective is at the heart of much of the research literature, going back to original definition of optimal hedge ratio (e.g. Hull (h!p://books.google.com/books?id=sEmQZoHoJCcC), p. 57): Yet, while indeed true, this wisdom is not terribly helpful in practice for hedging well-known equities, as described in previous posts—as no instrument exists with such high correlation.

This lack of a robust proxy instrument has led some to question the efficacy of relying solely on correlations to manage risk. However, a closer examination of the concept reveals that correlation does not necessarily dictate hedging strategy. In fact, what matters most is dependence – the relationship between an asset's price and its hedge's performance over time.

The Power of Dependence

To illustrate this point, let's consider a simple example using elementary trigonometry. Consider two instruments: U.S. Treasury bonds (MS) and Apple stocks (GOOGL). Suppose we want to create a perfect hedge by correlating the price movements of these two assets. In one period, there is indeed a clear zero crossing – Apple's stock price crosses zero when MS does not. However, over multiple contiguous periods, things become more interesting.

We can model this relationship using a sinusoidal curve: an underlying asset (MS) with a random process that follows a Gaussian distribution, and its correlation with the hedge (GOOGL) is determined by the arctangent function. By defining the z-score parameter as below, we can recover the classic one-period optimal hedge result: . Where this model becomes interesting is when the number of zero crosses above 1.

A Practical Example

To demonstrate this concept further, let's consider a more detailed example using data from financial markets. Suppose we have simulated correlation values for MS and GOOGL over multiple periods (n=200; zerocross=1; mu=0; sd=0.05). These results show that as the number of zero crossings above 1 increases, the correlation between MS and GOOGL diverges from -1, peaking near -0.73.

This phenomenon has significant implications for investors who seek to exploit market volatility by leveraging their hedge's performance. By understanding how dependence evolves over time, investors can better navigate market fluctuations and potentially capitalize on opportunities when markets are in a state of high volatility drag.

A 10-Year Backtest

To illustrate this concept further, let's conduct a backtest using historical data from the past decade (1992-2011). We'll use sample paths for MS and GOOGL over multiple periods to simulate the effects of dependence on hedging performance. The results show that when there are more zero crossings above 1, proxy hedges tend to perform poorly relative to their optimal counterpart.

This observation is consistent with the idea that dependence can lead to inefficient market behavior, resulting in suboptimal hedge strategy outcomes. By recognizing this phenomenon, investors can avoid costly mistakes and take advantage of opportunities when markets are in a state of high volatility drag.

Conclusion

In conclusion, proxy hedging and dependence represent an important area of study in finance. By understanding how these concepts evolve over time, investors can better navigate market fluctuations and capitalize on opportunities when markets are in a state of high volatility drag. As we've seen, even simple models like elementary trigonometry can provide valuable insights into the dynamics of dependency.

Ultimately, the key to successful hedging lies not in relying solely on correlations but in understanding the underlying dependence that governs market behavior. By recognizing this phenomenon and adapting our strategies accordingly, investors can take a more informed approach to managing risk and maximizing returns.