Implications for the Valuations and Risk Management of Correlation-Dependent Derivatives
The benefits of diversification in equity portfolio construction have long been extolled in both academic works and among industry practitioners. As any investment management professional will be acutely aware, however, these benefits tend to disappear at the exact times when they are needed the most: during periods of severe stress and market dislocation. There is clear empirical evidence that asset correlations tend to increase sharply during periods of crisis. In this short paper we share some of our experience in how such shifts in correlation regimes can pose challenges for the valuation and risk management of correlation-dependent derivatives.
The Quantitative Risk and Valuations Advisory team at BDO UK LLP was recently engaged to independently value a set of barrier options written on a basket of liquid blue-chip equities as part of an audit of an independent financial institution operating in the brokerage space. The payoffs of these options were structured as a fixed percentage of the notional, payable only if each equity in the basket exceeds its own barrier level at each payment date. The payoffs are then further ratcheted up by the passage of time and decreased by any coupons already paid at earlier payment dates, thus introducing path dependence. Each option was written on a basket of six well-known, liquid blue-chip equities.
The valuation of this derivative requires simulating the joint evolution of all equities in the basket. While the individual evolution of each equity can be modelled and calibrated in a straightforward way due to the availability of options traded in sufficient volume for a range of strike prices and maturity dates, the values of the correlation model parameters are somewhat less clear cut. This is problematic since payoffs defined as above tend to be most sensitive with respect to the correlation parameters.
In picking the right values for the correlation parameters in the absence of traded instruments to which correlations can be explicitly calibrated, a number of questions need to be asked:
- What should be the length of the look-back window used in the estimation of the correlations?
- Which past interval of equity returns should be used for the estimation of the correlations?
- What frequency of returns should be used for the estimation? (E.g. daily, weekly, monthly etc.)
Because of the non-stationary nature of the correlation processes, different answers to the questions above will yield very different numerical values for the parameters. The answer to the first question is perhaps the most obvious: the length of the look-back period should match the length of the interval between payment dates. For example, for semi-annual payment dates six-month periods should be used.
The answer to the second question is less obvious. Once the look-back length has been determined, the most recent set of six-month returns could be used, or the six-month returns starting one year in the past and so on. The theoretically correct answer would be to use an historical period where economic conditions most closely resembled the ones expected to occur in the future. This guideline is not particularly helpful from a practical standpoint, however, and therefore some sort of average rolling correlations is usually recommended.
And the answer to the third is not immediately obvious either. Some rules of thumb that can be used are the frequency of portfolio rebalancing and the trade-off between large sample size and noise. For daily portfolio rebalancing, correlations estimated from daily returns appear to be a logical choice. Daily returns would provide the largest sample size over the look-back interval but at the expense of noisier observations.
To illustrate the variability in possible correlation parameter values and the effect of general market conditions on correlations, we provide two graphs showing the evolution of pair-wise equity rolling correlations through time.
Figure 1: Six-Month Rolling Pair-Wise Correlations for a Basket of 6 Equities¹
Source: Capital IQ and BDO UK research.
Figure 1 above clearly shows two regimes in the evolution of pair-wise correlations. The first regime lasts until approximately February 2020 and is characterised by lower levels of correlations. The second regime starts after February 2020 and is caused by market stress related to the fallout from the global COVID-19 outbreak. We note that in this regime not only are correlation levels higher, but their behaviour is qualitatively different. In the first regime the correlations evolve more or less randomly with respect to each other. This is contrasted with the crisis regime in which the correlation processes themselves are correlated regardless of industry and exposure to different sectors of the economy.
Despite the different sectors for these six equities, during the crisis regime only the correlation of Barrick Gold Corporation with Facebook and Costco Wholesale Corporation remained flat or upward-sloping. All other correlation processes regardless of sector or their usual pre-crisis behaviour declined in unison as they started to revert to their pre-crisis levels. This illustrates how strong the effects of the regime-switching behaviour are: they can entirely supersede the effects of any sector-specific differences.
By September 2020 this regime shift had started to dissipate and correlations had started reverting back to their pre-crisis levels. The situation is markedly different for longer look-back intervals, however.
Figure 2: One-Year Rolling Pair-Wise Correlations for a Basket of 6 Equities²
Source: Capital IQ and BDO UK research
Figure 2 demonstrates that the same regime shift behaviour is observed for longer look-back intervals, but its effects are even more pronounced than for shorter look-back periods. The correlation processes in the crisis regime are markedly higher, but also less volatile and more correlated with respect to each other. Crucially, however, this effect is sticky and takes much longer to dissipate than when shorter look-backs are used.
The implications of this regime-switching behaviour for the risk management of correlation-dependent derivatives can be far reaching.
- Firstly, the correlation risk associated with a sudden spike in correlations is not easy to offset, since correlation processes behave uniformly regardless of sector or market factor exposure. They also become much more correlated with each other, which additionally impacts the valuation of derivatives.
- Secondly, this example clearly illustrates the drawbacks of traditional sensitivity and risk measures, which rely mostly on smooth changes in the risk factors. Risk is calculated on the basis of small changes in the current values of the risk factors. However, this example illustrates that the main risk facing financial institutions writing correlation-dependent derivatives is that of a sudden, discontinuous jump to persistently higher correlations. That is why we are of the opinion that when quantifying risks and judging the effectiveness of correlation hedges, it is important to stress derivative portfolios using regime-switching correlations models, perhaps implemented in the form of Hidden Markov Models.
- Thirdly, this example also says something about the appropriateness of standard practices which apply risk measures uniformly across the entire portfolio. Clearly, a basket option structured on the basis of six-month payment periods is exposed to correlation risks, which are different in nature to the ones that an option written on the basis of annual payment is exposed to. While it may be tempting to simplify and apply risk measures on a risk-factor level, it is important to keep in mind that parts of the derivative portfolio may behave quite differently purely in the basis of differences in payment structure even if they are exposed to the same risk factors as the rest of the portfolio.
Derivatives which depend on equity correlations can be difficult to value due to the inherent issues with estimating the values of the correlation parameters. These values usually need to be estimated empirically and give rise to substantial model risk. Furthermore, correlations tend to exhibit a highly regime-dependent behaviour. Equity correlations tend to exhibit not only sudden jumps to persistently higher values, but also qualitative changes in behaviour, such as less volatility and higher correlations among the correlation processes with respect to each other. Crucially, the strength of these effects varies with different choices for the length of the look-back interval used in estimation. As a consequence, different parts of a derivative portfolio may react differently to changing market conditions, even if they are exposed to the same broad risk factors. Care should therefore be taken not to underestimate the amount of correlation risk inherent in a derivative portfolio.
If you would like to find out more information, please get in touch with Simon Greaves or Martin Anastasov.
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1. The equities in the basket are: United Health Group (UH), Micron Technology (MT), General Electric (GE), Costco Wholesale Corporation (CO), Facebook (FB), and Barrick Gold Corporation (BG).
2. The equities comprising this basket are: International Business Machines (IBM), McDonald’s (MCD), Philip Morris International (PM), Qualcomm (QC), AT&T (ATT), and Walmart (WM).