# Correlation

__7. Correlation__

Many payoffs that exist today are based upon the performance of multiple assets. When an option derives its value from the price of multiple assets, the relationships between these assets become important. Correlation gives us the strength and direction of a **linear **relationship between different underlyings.

**7.1. Multi-Asset Options**

The creation of such products stemmed from the **concept of diversification**, and there is now a wealth of products structured on multiple underlyings.

Diversification involves combining multiple assets within a portfolio. The central idea is that movements in one asset within a diverse portfolio have less impact on the portfolio and so diversification can lower the exposure to an individual asset.

We will deal with the extremely important **concept of correlation** and its effects because the value of a multi-asset option does not depend only on the underlying asset’s IVs but also on the correlations between these assets. We previously saw that market prices of liquid options can be used to infer the implied volatilities of different individual assets. These **IVs contain additional information about future volatility expectations that is not included in historical volatility**, and in the multi-asset case we would ideally have a similar **implied correlation**, but we do not. The reason is simple: there is no liquid market for such products in the multi-asset case. We must therefore resort to other methods of deciding which correlation to use when pricing multi-asset derivatives. **Correlations change dramatically through time**, which makes the use of realized correlations unreliable and management of correlation risk a difficult task.

Payoffs involving multi-asset options are sensitive to movements in the various underlyings and so the relationships between the underlyings, which are defined using correlations, have an impact on the hedging of any such options. Because such payoffs are non-linear functions in more than one variable, we have **cross-Gamma effects**. The cross-Gamma terms tell us how the **Delta of the option w.r.t. one underlying is affected by a movement in another underlying**, and also depend on how we define the correlations between these underlyings.

**7.2. Correlation: Interpretation**

A statistical correlation will take on values between −1 and +1.

A **negative correlation** indicates that, historically, as one variable has moved up the other has moved down.

A **positive correlation** means that historically both variables have generally moved in the same direction.

The case of **zero correlation** means the two variables move in a generally random manner comparatively.

We must stress that measuring correlation as such gives us information regarding the **linear relationship between two variables**. **Two variables can, for example, have a historical correlation of zero, but not be independent.**

**During a market crash, the realized correlation between various assets could approach 1 and we can definitely witness stocks or indices realizing a correlation of above 90%. **

**7.2.1. Correlation Matrices**

A correlation matrix M_{ρ} is a **square matrix** that describes the correlation among n variables.

This matrix is **symmetric**. Indeed, the correlation between asset i and asset j must be the same as the correlation between asset j and asset i.

The correlation matrix is also **necessarily positive definite**. **If the correlation matrix we assign is not positive definite, then it must be modified to make it positive definite** – see, for example Higham (2002).

Given a basket of n stocks or a basket of n assets S_{1},S_{2},...,S_{n} with respective weights w_{1} , w_{2} , ..., w_{n} , the realized basket correlation is defined simply as the weighted average of the realized correlation matrix between the components, excluding the diagonal of 1’s:

Although we may compute each of these separately, the overall matrix of correlation between the components of the index must be positive definite. For this to have any meaning, the **pairwise correlations must all be computed over the same period**.

To compute the sensitivity of an option to a specific correlation pair, one can bump the correlation between them by 1%, check that the correlation matrix is still valid, and reprice the option to see the difference. If we want to see the effect of an overall move in correlations by 1%, **we will need to bump the entire matrix of correlations to see what the effect would be on the price if the average correlation increases by 1%**. Again, before recomputing the price using the bumped correlation matrix, we should check that this matrix is still a valid correlation matrix.

**7.2.2. Portfolio Variance**

We will now see the implications of correlation on the variance of a portfolio.

As long as the correlation in the above formula is less than 1, holding various assets that are not perfectly correlated in a **portfolio will offer a reduced risk exposure to a specific asset. **

**7.2.3. Implied Correlation**

Although there isn’t an analogy of IV for correlations, we can in practice still define an implied correlation. The usefulness of such implied correlation is subject to debate, but trying to find some method of implying correlations is necessary to say the least.

The market for European options on pairs of underlyings or baskets is not liquid so we cannot extract an implied correlation between the underlyings from these prices.

However, let us take the **case of an index** for which we have both European options on the index itself as well as on each of the underlyings composing the index. Then using market quotes, we can **infer an implied correlation** that is a measure of the dependence between the components of the index.

where n is the number of components, w_{i} is the *i*th component’s weighting in the index, σ_{index} is the IV of the index and σ_{i} is the IV of the ith component of the index.

To obtain the **implied correlation over a T-day period**, we must use the **IVs of options with time to maturity T**. In this case we make use of ATM volatilities throughout; however, we discuss below the **correlation skew **that involves implied volatilities of different strikes.

The previous formula came from the formula of the variance of a portfolio. In a portfolio or **basket of stocks** for which we apply this formula, **all weights are assumed to be constant**, whereas in the case of an i**ndex**, the **weights vary as the components of the index vary, thus making this an inexact definition**.

However this does still have some implications and uses. Assume that we have a basket of stocks for which we wish to infer an implied correlation. Assume further that these stocks all belong to the same index.

The idea is to follow a **simple parameterization involving a coefficient λ which relates realized and implied correlations of the index**, and in turn **use this coefficient and also the realized correlations between the index components to infer specific implied correlations. **

**Firstly**, compute the realized correlation of the index, and the implied correlation using formula, then solve for λ in the equation:

Now take two stocks A and B, both of which are in the same index I, for which we have liquid European options on both the index I and its components and, in turn, obtain the value of λ. **With this we can solve for the left-hand side using the realized correlation of A and B on the right-hand side along with the index λ**.

In relevance to pricing, and since this average implied correlation can potentially be hedged, it makes sense that there is some form of implied correlation, and not realized correlation, in the case where there is similar exposure to the correlation pairs between many of the index constituents.

Sell-side desks of multi-asset options will typically be **structurally short correlation**. This is **due to the worst-of feature** in most of the multi asset structured products.

The implied correlations will be higher than the realized correlation levels (assuming the implied λ is positive). In the case where realized correlation is higher than implied, one may want to sell correlation at a level at least equal to the realized correlation. Even in the case where the implied correlation is higher than the realized, the seller of a multi-asset option who is to assume upon the sale a negative position in the correlation, may want to increase the level slightly further. This will depend on three factors:

- The sensitivity of the option to the correlation parameter
- The overall level to which the trader is exposed to the correlations between the assets of the option
- The level to which the trader needs to be aggressive on the trade

**7.2.4. Correlation Skew**

Assume that we have two assets and that we have IV skews for each of them, and also an IV skew for vanilla options on the basket. To have an IV skew for basket options means that, for a fixed maturity, we can find quotes for the prices of basket options with different strikes. If this were the case and we imply a correlation at each strike where we used the IV for the basket and the two constituents taken from each implied skew at this strike, would the implied correlation be the same? Not necessarily so. This curve, when plotted against the strikes used to compute it at each instance is known as a **correlation skew**.

The reason is that **a lower strike holds a higher implied volatility**, but also we expect in this region that **if the index is tanking it means that its components are also tanking and thus their correlation will rise**. Many exotic products have **correlation skew exposure** in the sense that **as the underlying assets move, the correlation sensitivity can vary significantly**.

If we parameterize the skew in the same manner as we did the implied volatility skew, we need the 90% strike, the ATM and the 110% strike options on the index and each component. We can then have a 90–100–110 parameterized correlation skew.

To see the impact of correlation skew on a price one needs to **use a** **model that knows about correlation skew in order that it shows this additional risk**.

On this note, we point out the implying correlations as discussed above may also give rise to a **correlation term structure**. Using index and component option implied volatilities for different maturities may imply different levels or correlations. What is most important is that **whatever correlation we imply, we must use the correct maturities for the relevant implied index and component volatilities**. From a modeling perspective, having a correlation term structure is typically less computationally intense than a correlation skew. To go deeper into the concept of a correlation skew, and have a meaningful method to see this in a model, we will need to look at **copulas [1]**.

**7.3. Basket Options**

A **basket** option is an option whose payoff is contingent on the performance of such a basket. The reference to basket options is where the weights of each of the underlyings is known at the outset.

This is in comparison with what we see later as a distinct set of options called **Rainbows**, where the weighting is specified at maturity and is based on the relative performance of the various assets.

As such, the basket is different from an index in that **the weights in a basket stay the same, whereas in an index they can change as the components of the index move**.

A basket option involves only one transaction to gain exposure to multiple underlyings and thus **lower transaction costs**. It is also because of this multi-asset feature, and the problems that could potentially arise from having to deliver multiple underlyings, that multi-asset options are **generally cash settled**.

An **increase in correlation implies an increase in the overall basket volatility**. Since call options have positive Vega, the seller of the basket call is thus selling the basket volatility which, in turn, implies that the seller is short the correlation between the underlyings. Note the **non-linearity of the basket option’s price sensitivity to a movement in correlation**. If we were to assume that the only impact that correlation has on a basket option is that which it has on the basket volatility, then it is fair to say that the basket call option’s correlation sensitivity is given by

The last term on the right-hand side is positive but is not a linear function in correlation.

Other methods exist whereby the **basket is modelled as a single log-normal asset** so that the Black–Scholes formula can be applied. This breaks down to finding the equivalent mean and variance, and thus involves moment matching. One can ask: given a set of variables all of which are log-normal and for which we know the mean and variance, can we find an equivalent log-normal random variable that has the same mean and variance as the weighted basket of these log-normals? In Brigo et al. (2004) the authors use a **moment-matching method** to give a closed formula equivalent log-normal process for the basket.

In practice, we may want to simply apply a **simulation-based pricing method**. Once the volatilities and correlations are specified, basket options can then be priced using **Monte Carlo simulation correlated log-normal random variables**. In the case where there is skew dependence, for example an OTM basket call option, skew models will be needed. The seller of an OTM basket call option is short the individual OTM implied volatilities of the underlying assets, and as skew increases these values go down, thus the seller of the OTM basket call option is long the individual skews.

**One methodology to handle basket skew is to use an index skew as a proxy for the basket skew**. One can compare the time series of the volatility of the basket to that of the index to decide the level at which to buy/sell volatility if the basket option’s Vega is to be hedged with options on the index. This becomes necessary when dealing with baskets of underlyings for which we do not have liquid individual underlying OTM European options data but still need to price skew correctly. In the case where one has **sufficient liquid individual underlying OTM option quotes for the points to which the basket option has Vega exposure, then the calibration of individual local-volatility models to these skews, and a simulation of these correlated variables, will suffice.**

**7.4. Quantity Adjusting Options: “Quantos”**

**7.4.1. Quanto Payoffs**

A quanto option is an option denominated in a currency other than the currency in which the underlying is traded. Cashflows are computed from the underlying in one currency but the payoff is made in another. The idea behind the quanto is that it handles the risk to foreign exchange rates which are found in foreign derivatives.

In a European payoff, for example, the strike price is set in the currency of the underlying.

where FX(0) is the exchange rate at time 0, defined as the domestic currency per one unit of the foreign currency. This is fixed in the above payoff. This option gives the buyer exposure to the upside in the index above the specified strike, but without the payout having any exposure to changes in the USD (in which the underlying is traded in this case) and GBP (in which the payout is being computed) exchange rate. The payoff can be modified to include the exchange rate at maturity, FX(T), however the option will no longer provide protection against the FX risk.

**7.4.2. Quanto Correlation and Quanto Option Pricing **

Let r_{stock} denote the RFR of the currency in which the underlying is traded, and let q denote the dividend yield and σ_{S }its volatility. Denote also by σ_{FX} the volatility of the exchange rate. If we make BS assumptions and also assume a log-normal process for the FX process, then analytical pricing solutions for quanto European options exist. The result is the same as a BS formula for the non-quanto case, using the RFR r_{stock }and dividend yield q, plus what is known as a **quanto adjustment** which accounts for the quanto effect. The adjustment is added to the dividend yield and is given by

where ρ_{quanto} is known as the quanto correlation and is the correlation between the underlying equity and the FX rate. Let’s be clear on the FX rate and quanto, going back to the example of the call option on the S&P 500 index: when denominated in GBP, the quanto correlation is the correlation between the USD–GBP exchange rate and not the GBP–USD exchange rate. Note that σ_{FX} is the volatility of the FX rate and will be the same for USD–GBP and GBP–USD.

Like many equity–equity correlations, it is hard to correctly obtain an implied quanto correlation from market data. In the general case where we cannot imply and hedge the quanto correlation risk, the seller of the quanto option will have to resort to looking at the realized correlation and taking a margin. When computing such a correlation from two time series, we do as before and use data of the log-returns for the asset and the log of the FX rate, not the price and exchange rates themselves.

**7.4.3. Hedging the Quanto Risk **

Firstly, we think about the effect the quanto adjustment has on the forward. As it appears above, applied to the dividend but with a negative sign, it impacts the forward in the opposite way from dividends. An increase in the quanto correlation, the FX volatility or the volatility of the underlying will have the same effect as a decrease in dividends. Therefore, increasing the quanto correlation and the FX volatility increases the forward.

One thing to note is that the volatility of the underlying appears in the adjustment. The seller of the put option is short the volatility of the underlying, the quanto effect here has the opposite effect. Generally speaking, the quanto effect will be secondary and the seller of the quanto put will still be overall short the volatility of the underlying.

Assume that a trader sells the above call option on the S&P 500 denominated in GBP. Then to hedge, the seller will need to buy Delta of the underlying, which involves selling GBP and buying USD. The seller of the quanto call is thus short the quanto correlation.

In general, exotic desks tend to be long delta and therefore one would expect an exotics desk to be structurally short the quanto correlation between various underlyings and the relevant currencies. Using realized correlation plus a margin is in some ways the best one can do to price this quanto risk; however, the fact that it cannot be hedged in the market means that the seller will have to essentially sit on this risk.

**7.5. Trading Correlation**

Here we discuss two of three possible **correlation trading strategies**. Traditionally one makes use of European options on the index and its components and can trade these against each other in the form of straddles. A more specific and pure correlation trade is the correlation swap. A third method involves trading variance swaps (or Gamma swaps), again on the index versus the components to get a cleaner exposure than the straddle version. This method will be considered after our discussion of variance swaps.

**7.5.1. Straddles: Index VS Constituents**

Consider a trade where we go **long straddles on an index** and **short straddles on each of the individual components**. In this case the holder of this position is long the average correlation of the index and not the individual pairwise correlations.

The weights in such a strategy must be specified for the component straddles according to the weights of the index, and will need to be readjusted if the weights change. **Straddles are used because a Delta-hedged straddle can provide exposure to volatility, although trading straddles does not give a pure exposure to volatility**.

The idea is that **by gaining exposure to just the volatility of the index and those of the components, the spread will leave us with an exposure to correlation**.

The **variance swap, or Gamma swap, provides a purer exposure to volatility**, and thus trading spreads between the variance swaps of an index versus those of the components is a more transparent method for trading the average correlation in an index.

**7.5.2. Correlation Swaps**

The correlation swap is an OTC product typically of medium-term maturity between 1 and 3 years. It allows the investor to obtain a **pure exposure to the average correlation among a basket of underlyings**.

The correlation swap consists of a fixed leg and a floating leg with payments made on the basis of a prespecified notional that we denote N_{corr}.

The **fixed leg** of the swap pays this notional times the strike ρ_{strike}.

The **floating leg** pays the annualized realized correlation between the underlying assets of the swap, thus the need for price data for each underlying.

At expiry, the payoff of the correlation swap is given by the difference in percentage points times the notional. For the payer of the fixed leg this is

If the correlation swap is written on a basket of underlyings then the floating leg is the average correlation. Each pairwise correlation is computed using the log daily returns of each underlying. An investor who is short the swap, meaning one who pays the floating level, makes money if the correlation realized is lower that the specified strike level.

Sell-side desks will be **structurally short correlation on a book level** because of the sale of multi-asset options, the majority of which set the seller of the option short the correlation.

Although spread positions in straddles allow one to hedge the average correlation of a basket or index, the risk to pairwise correlations remains, and this can potentially be very large for certain underlyings. The correlation swap provides a method for the sell side to buy back some of the correlation they have sold, providing a counterparty for such a swap can be found.

The problem with **correlation swaps** is that they **cannot be replicated or priced in a simple and arbitrage-free manner**. The strike of the correlation swap would thus generally be some estimation of future realized correlation.

Bossu (2005) shows that the fair strike of the correlation swap on the realized correlation of the components of an index is in fact related to the implied correlation of the components. These problems have left the **correlation swap market relatively illiquid**.

Assuming that one were able to trade a correlation swap on the underlyings to which a book is most exposed, this is **not the absolute solution**. On day 1 of selling an option the trader can know the correlation sensitivity of the option. However, this **correlation sensitivity changes over time**.

We saw that for a basket call option, its correlation sensitivity is directly proportional to the vega sensitivity. The vega of a call option is obviously sensitive to movements in the underlying and is a function of moneyness. We can therefore expect the **correlation sensitivity of the option to change as the underlyings move**.

Thus hedging such correlation risk using a correlation swap – which obviously has a fixed notional – is not the absolute answer to the correlation problem. Although the **correlation swap doesn’t provide a complete hedge**, it can prove valuable on a book level to at least partially hedge the correlation risk to specific pairs or baskets to which the trader has large short exposures.

Generally, the **correlation sensitivity of a multi-asset option will move as the volatilities of the underlying assets change**. One way to incorporate this is to consider covariance swaps, defined analogously, but involving both the correlation between assets and also their volatilities. However, these will again suffer from the same liquidity problems owing to the lack of a correct replication methodology.

[1] A copula is a multivariate probability distribution for which the marginal probability distribution of each variable is uniform. Copulas are used to describe the dependence between random variables.

Sklar's theorem states that any multivariate joint distribution can be written in terms of univariate marginal distribution functions and a copula which describes the dependence structure between the variables.