# Webcast: Correlation and Equity Derivatives

**Speaker:** Dr Simon Acomb

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Webinar Agenda

- Market players for equity products on more than one underlying
- Types of equity products seen in the market
- Using multiple equities to enhance upside participation in a structured product
- Using multiple equities to enhance the coupon on a “capital at risk” product
- Correlation and the autocallable market
- Managing correlation risk exposures
- Correlation swaps and offsetting risk in the hedge fund market
- Q&A

Q&A

**1. Could you please comment on using Spearman correlation as opposed to Pearson? Also, how in practice is the implied correlation computed as opposed to the historical correlation?**

A. In the webinar, I was mainly discussing the Pearson (or product moment) correlation coefficient. This is the main type of correlation used when discussing pricing of equity derivative products, as it is the natural way of describing correlation between two stochastic processes. Interestingly, the Spearman correlation coefficient has now become part of regulatory risk calculations and plays a part in the FRTB requirements for testing an Expected Shortfall model. Further details about this can be found in the London Financial Studies course on **''Fundamental Review of the Trading Book''**.

As for how to mark implied correlation, I am tempted to say in the best way you can. For a small number of pairs, options on the sum of 2 assets can be traded. For these pairs the correlation can be implied directly (but do remember that this may be model dependent, as different models for pricing the option on a sum may give different results). For other pairs all that can be measured is the historic correlation. For these pairs we know that the implied correlation will trade at a premium. All we can do is observe the premium on similar pairs and apply the same premium to the pair where only the historic can be calculated.

**2. Why do we have a premium between implied vs historical correlation?**

A. The reason for this is purely supply and demand. Nearly all the products that investment banks sell leave them short correlation, and they find it difficult to buy back the correlation risk. As there isn’t a (liquid) two-way market, the correlations get marked more like an insurance risk – at a level at which investment banks are confident that they will break even or make a profit most of the time. This will push the correlation marks higher, leading to a significant premium between implied and historic correlations.

Just as the premium of implied volatility over historic volatility gives trading opportunities to hedge funds (who received implied and pay historic via variance swap), so the premium of implied correlation over historic also gives rise to opportunities for hedge funds. In the case of correlation, the opportunities are larger due to the bigger premium of implied correlation. However, the ease of extracting value from this premium is more difficult.

**3. I wonder if the concepts and hedging approaches presented by Dr Simon also apply for FX as an underlying asset (i.e. Portfolio composed both FX Fwds and plain vanilla FX Opts)?**

A. The concepts of correlation apply across a range of different asset classes including FX. A typical problem might be to consider the correlation between USDJPY and USDEUR together with the implied volatilities on USDJPY and USDJPY options to imply the volatility on EURJPY options. The key point I tried to make in the webinar is equities are nearly always traded in portfolios and this makes the correlations between each of the equities important when constructing derivatives on these portfolios.

**4. Can you please hint on how Correlation swaps are priced? What kind of mathematics is used? What technique market makers of correlation are used? **

A. Pricing correlation swaps is a difficult topic, but one which is covered briefly in the London Financial Studies course on ** ''Volatility: Trading and Managing Risk''**. The first thought would be that if the correlation between S&P and Nikkei is marked at 0.70 for basket options the correlation swap should have a fair strike of 0.70. Unfortunately, this isn’t true. The number of 0.70 is the implied number you need to combine with volatilities of S&P and Nikkei to get the right covariance when pricing the basket on the sum. The correlation swap divides this covariance by the standard deviation (volatility) of each of the S&P and Nikkei to get a measure of the correlation. In a world where volatilities are stochastic then this means you need a convexity adjustment when pricing a correlation swap. (This is exactly the same when you price an interest rate future, or a volatility swap.) The upshot of this is that to price a correlation swap accurately you need to have a stochastic volatility model for each of the underlyings and then link both of these stochastic volatility models - not an easy thing to do. Of course, if implied correlation is 0.70 and historic is 0.50, and somebody is prepared to pay you 0.65 in exchange for realised correlation it is still a good trade to put on.

**5. There are additional ways to hedge correlation. Assuming equity correlation increases when market goes down, and vols go up, remaining long vega/gamma in the portfolio is one way.**

A. These are indirect hedges, exploiting relationships between financial variables (the fact that when markets crash all correlations go up). It’s not a direct hedge of correlation, but certainly something that will reduce the overall risk on a portfolio. Buying OTM puts would be a good way to put on this sort of macro hedge, or perhaps buying OTM calls on the VIX (options on volatility).

**6. Reverse convertible WOP shows correlation skew as well that needs to be addressed with long skew position (vega) as well.**

A. Indeed correlation skew is an important issue that needs to be addressed, but first of all we need to define terms. I generally find that if you ask 2 traders you get 3 definitions of correlation skew. My starting point is to relate correlation skew to volatility skew. Volatility skew says that you need a different volatility when using the Black-Scholes model to price European options depending on the strike of the option. Similarly, I would define correlation skew to be that you need different correlations when pricing basket options using a local volatility model depending on the strike of the option. Of course this is model dependent. If you use a different model for pricing basket options (say an SLV model) then you will have a different concept of correlation smile.

**7. Can you help define variance swaps and comment on hedging techniques for variance swaps?**

A. This is covered in the London Financial Studies course on ** ''Volatility: Trading and Managing Risk''**. You may also be interested in the London Financial Studies webinar ** “Volatility Trading - Does a variance swap have a delta?”**

**8. You’ve mentioned copulas to model more complex dependencies. Are ML based approaches an alternative to capture non-linear dependencies? Or more generally, are ML based approaches (deep hedging) possible alternative hedging approaches?**

A. Deep hedging and ML approaches are not my specialism. However, I have tried investigating using different copulae to price index options. The argument goes as follows. For the EuroStoxx50 you can trade options on all 50 of the underlyings. Can I take the volatilities smiles of each of the 50 underlyings and use these to imply a distribution of each of the underlyings. (This is easier than you might think.) If I can do this, can I then use a copula to price the option on the basket of the 50 underlyings which is the EuroStoxx50 index. If the copula choice is a good one, then the basket options should be in agreement with the index options on the exchange no matter what the strike. (For the fixed income readers, this is a similar sort of problem of calibrating a BGM model to caplets and then trying to accurately price swaptions from it.) So far, I haven’t found a great copula for doing this, and the Gaussian copula certainly won’t do this. I suppose that this just goes to demonstrate the importance of the concept of a correlation smile.

**9. What's the "best" calculation method to calculate the historical correlation? You used yearly roll method, monthly roll method? Does it depend on the type of underlying?**

A. I’m not sure I can say for certain which is the best method, but I can give you some pointers in the right direction. The problem with correlation is that you need a large amount of data to get a statistically significant estimate. (If you are interested, confidence intervals can be estimated by the use of something called a Fisher statistic.) If you are only looking at equities in **one** time zone, then I would use daily data to get as large a sample size as possible. The problem comes when having to deal with equities in different time zones, as daily data then becomes asynchronous. In this case you really have to use weekly data and a longer rolling window.

**10. The infamous correlation swap SX5E v/s EURKRW is -40% on CorrelSwap market but -80% on quato option. So is it different correlations for different products?**

A. There are multiple issues going on here.

a) Correlation swaps don’t trade at the correlation used to mark basket (or in this case quanto) options. This is due to a convexity adjustment (see my answer to question 4).

b) Quanto options are only very weakly dependent on correlation. Correlation has to change a great deal to have an impact on price. Is the a supply and demand problem here. Is everybody one way on one of these products. Is it possible to cross the bid/offer spread between these two products to put on an arbitrage, or is the difference in correlation marks within the B/O on these two products?

c) Correlation is definitely model dependent. Different models for quanto options will give different definitions of an implied correlation.

**11. When one prices a correlation product, is it more usual to use a flat correlation or a pairwise correlation?**

A. I always use a pairwise correlation.

**12. What do you think of Sebastien Bossu’s local correlation model?**

A. It's not something I have studied in great detail. My initial thoughts are to compare this to the pros and cons of a local volatility model. A local volatility model is a simple way to calibrate to all European options (regardless of strike and maturity) in a single process. Similarly, a local correlation model will do the same thing for basket options and so remove some of the problems associated with correlation smile. However, local volatility models are not without problems:

a) You need a large amount of data which isn’t always present, and it needs to be arbitrage free. (This is actually the most difficult part of building a local volatility model.)

b) The dynamics of the model are not well adjusted to market realities. (This is the reason many people in the FX world reject local volatility models).

I suspect that these issues are amplified with a local correlation model. In particular, I would imagine that the calibration with scant data and always ensuring that the model is arbitrage free is especially difficult.

**13. In terms of market liquidities, which is higher? Variance swaps correlation synthetic trades or direct correlation swaps?**

A. Variance swaps are generally more liquid.

**14. How do you mark correlation on far otm options?**

A. This is truly a difficult subject. A starting point has to be how are correlations marked when you consider an index option as a basket on all of its underlyings. This should give you some idea of a “correlation smile”, or how correlation change with strike. A nightmare product would be something like an option on the “Ethical EuroStoxx 50”, which a client defines as the EuroStoxx with all the defence and oil stocks removed. Pricing this as a basket you would definitely want to ensure that the correlation smile was compatible with the information you have on the pricing of EuroStoxx 50 options.

**Thank you to those attendees who submitted their questions.**

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