Chapter 15 Multi-Asset Options

15.1 Introduction

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. These relationships between the underlyings are defined using correlations. Correlation gives us the strength and direction of a linear relationship between different underlyings.

We will start this chapter by a small reminder on the concept of correlation, the different types of correlations, their properties, etc.

15.2 Correlation

15.2.1 Realized/Historical Correlation

If you have never heard about correlation, please read about it first as this sections only serves as a small reminder.

The statistical correlation between two variables is measured as: \(\boxed{\rho_{X,Y} = \frac{\sigma_{X,Y}}{\sigma_{X}\sigma_{Y}}}\)

This statistical correlation, also called realized 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. A correlation of zero means that there is no linear relationship between the two variables but does not imply that these variables are independent. There can be strongly non-linearly correlated! To obtain a more thorough view of the dependence of several variables than a linear relationship between them, one can use copulas.

A couple of practical things can be said about correlation.

First, in practice, one computes the correlation between two financial assets using the historical series of daily log returns. It is very important that the series used to compute the correlations have matching dates. It seems quite obvious but can become problematic when the assets are quoted on different markets.

Then, correlations change dramatically through time. Therefore, the linear relationship described by the historical correlation will not necessarily hold in the future.

Finally, during a market crash, the realized correlation between financial assets tend to increase sharply and we can witness assets realizing a correlation of above 90%.

15.2.2 Correlation Matrices

A correlation matrix \(M_ρ\) is a square matrix that describes the correlation among n variables.

\[M_ρ = \begin{pmatrix} 1 & \rho_{1,2} & ... & \rho_{1,n}\\ \rho_{2,1} & 1 & ... & \rho_{2,n} \\ ... & ... & ... & ... \\ \rho_{n,1} & \rho_{n,2} & ... & 1 \end{pmatrix}\]

For this matrix to have any meaning, the pairwise correlations must all be computed over the same period.

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 is not positive definite, then it must be modified to make it positive definite. How can you achieve this? Well, you can read about the excellent paper of Nicholas Higham to know more about this.

The good news is that when testing the multi-asset options pricer, you will be provided with a quasi-randomly correlation matrix that is already symmetric and positive definite. You can always change it manually and the pricer will let you know if the new matrix does not respect these two constraints.

Given a basket of of n assets \(S_1,S_2,...,S_n\) with respective weights \(w_1 ,w_2,..., w_n\) , the realized basket correlation is defined as the weighted average of the realized correlation matrix between the components, excluding the diagonal of 1’s:

\(\boxed{\rho_{realized} = \frac{\sum_{1 \le i < j \le n} w_i w_j \rho_{i,j}}{\sum_{i < j}^{n} w_i w_j}}\)

15.2.3 Portfolio Variance

The link between volatility and correlation is very important as it will allow us to better understand the cega (sensitivity to the correlation) profile of multi-asset options. So let us now see the implications of correlation on the variance of a portfolio.

\(\boxed{\sigma_{Ptf}^{2} = \sum_{1 \le i \le n} w_i^2 \sigma_i^2 + 2 \sum_{1 \le i < j \le n} w_i w_j \rho_{i,j} \sigma_i \sigma_j}\)

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.

15.2.4 Implied Correlation

The market for European options on baskets of underlyings is not liquid enough to imply a correlation between the underlyings from these prices.

However, if we have both European options on an index and on each of its n underlyings, then we can use market quotes to infer an implied correlation that is a measure of the dependence between the components of the index.

\(\boxed{\rho_{index \; implied} = \frac{\sigma_{index}^{2} - \sum_{i = 1}^n w_i^2 \sigma_{i}^{2}}{2 \sum_{1 \le i < j \le n} w_i w_j \sigma_{i}\sigma_{j}}}\)

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.

The previous formula came from the formula of the variance of a portfolio. This formula assumes that all weights are constant. In the case of an index, this is usually not the case as the weights vary as the components of the index vary. This makes the above formula inexact for an index.

However it still has 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:

\(\boxed{\rho_{index \; implied} = \rho_{index \; realized} + λ \; (1 - \rho_{index \; realized})}\)

If you take two stocks that belong to the same index and for which we have liquid European options on both the index and its components, you can then obtain the value of λ using the above formula and use it alongside the realized correlation of the two stocks to estimate the implied correlation between them.

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. No worries at this stage if you are not yet familiar with the structural correlation position of sell-side desks due to wors-of products. We will discuss them in further details in this chapter and you will quickly get used to it while using the multi-asset options pricer.

As with volatilities, implied correlations will usually be higher than realized correlations (positive λ).

15.2.5 Correlation Skew

Assume we have two assets for which we have liquid vanilla options at different strikes for each of them as well as for the basket. If we imply a correlation at each strike where we used the implied volatilities for the basket and the two constituents, would the implied correlations be the same? Not necessarily so. Plotting implied correlations with respect to the strikes gives a curve known as the correlation skew.

As with implied volatility, implied correlation tends to be higher for lower strike.

Many exotic products have correlation skew exposure in the sense that as the underlying assets move, the correlation sensitivity can vary significantly.

This correlation skew represents an additional risk and you must use a model that knows about it for this risk to be reflected in the product's price.

In the same way, implying correlations may also give rise to a correlation term structure. 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.

15.2.6 Copulas

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.

The importance of copulas in multi-asset derivatives is that they provide a method of expressing joint distributions between assets, allowing for the simulations of these variables, and thus the pricing of multi-asset options.

The traditional correlation does not allow us to specify different behaviours for different parts of the distribution. A copula does precisely this. Modelling with copulas can therefore provide more meaningful hedge ratios.

15.3 Basket Options

15.3.1 Description

A basket option is an option whose payoff depends on the value of a basket of assets.

At maturity, it pays off the greater of zero and the difference between the average return of the n different assets in the basket and the strike price (expressed as a return in this case):

\(\boxed{ \text{Basket Call}_{\text{payoff}} = max \Bigl[0, \sum_{i = 1}^n w_i R_i - K \Bigl]}\)

The typical underlying of a basket option is a basket consisting of several stocks, that represents a certain economy sector, industry or region.

The main advantage of a basket option is that it is cheaper to use a basket option for portfolio insurance than to use the corresponding portfolio of plain vanilla options. Indeed, a basket option takes the imperfect correlation between the assets in the basket into account and moreover the transaction costs are minimized because an investor has to buy just one option instead of several ones. To avoid the problems that could potentially arise from having to deliver multiple underlyings, multi-asset options are generally cash settled.

Unlike a rainbow option in which the weighting at maturity is based on the relative performance of the various assets, the weights of each of the underlyings are known at the outset in a basket option. 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.

Since you can specify the weight of each underlying asset, our multi-asset options pricer is not limited to equi-weighted basket. However, you need to make sure that the weights sum up to 1 for the pricer to work.

Since basket options are vanilla options on a basket. The intuition behind all the greeks discussed in chapter 5 applies. We will therefore focus on a new sensitivity parameter, the Cega.

15.3.2 Cega, sensitivity to correlation

It is easily seen from the portfolio variance formula (\(\sigma_{Ptf}^{2} = \sum_{1 \le i \le n} w_i^2 \sigma_i^2 + 2 \sum_{1 \le i < j \le n} w_i w_j \rho_{i,j} \sigma_i \sigma_j\)) that an increase in correlation implies an increase in the overall basket volatility. Since options have positive vega, the seller of a basket option is thus selling the basket volatility which, in turn, implies that he is selling the correlation between the underlyings. From this formula, you can also imply that the price sensitivity of a basket option to a movement in correlation is not linear.

Using our multi-asset options pricer, you can easily compute the sensitivity of a basket option to a specific correlation pair. To do so, you can simply bump the correlation between the two underlyings by 1% and reprice the basket option to see the difference in prices. You should normally make sure that the correlation matrix is still valid before repricing the basket option but our pricer does it for you ;).

Our pricer allows you to calculate the cega of the basket option. The cega gives you the effect of an overall move in correlations by 1% on the basket option price. It is calculated by bumping the entire matrix of correlations (understand the non diagonal elements of the correlation matrix), making sure the correlation matrix is still valid, and repricing the basket option. You can do it manually and see if you obtain the same result...

Obviously, your sensitivity to correlation depends on the assets' weight. For example, a basket option on 3 stocks weighted 96%/2%/2% will not have any sensitivity to correlation since it will behave very much like a single-stock option.

15.3.3 Pricing methods / Modelling

In the absence of volatility smile/skew, basket options can be priced using various techniques, including moment-matching or geometric conditioning. The moment-matching methodology consists of modelling the basket as a single log-normal asset so that the classical Black-Scholes formula can be applied. Basically, you find an equivalent log-normal random variable that has the same mean and variance as the basket (whichi is a weighted basket of log-normal random variables).

You can find an analysis of different pricing methods for basket options in this article from Krekel, de Kock, Korn and Man.

In practice, you may want to simply use a Monte Carlo simulation of correlated log-normal random variables. In the case where the basket option has skew dependence, you will need to use a model that knows about skew.

If you have sufficient liquid underlyings for the points to which the basket option has vega exposure, then the calibration of individual loc-vol models to these skews and a simulation of these correlated variables will be enough.

Otherwise, you can handle basket skew by using an index skew as a proxy. You will then have to compare the time series of volatilities 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.

If you are interested about reading a more technical article on pricing basket options with skew, you can read the following article from Dong Qu.

15.4 Rainbow Options

When analyzing basket options, we only spoke about the impact of correlations on the portfolio volatility. This is because basket options are almost not sensitive to the dispersion effect. A higher dispersion does not affect the expected return of the basket. It will not increase the probability of the basket to finish ITM. This is because the weights assigned to the underlying assets do not depend on their respective performance.

For the rest of this chapter, it is quite important to understand how volatility and correlation affect dispersion. If the pairwise correlations between the underlying assets is low, the returns of these underlyings would be quite apart from each other and vice versa. Also, a higher asset volatility leads to asset returns with large deviation from its expected return. Hence, a higher volatility and a lower correlation leads to higher dispersion.

15.4.1 Description

A rainbow is an option on a basket that pays in its most common form, a non-equally weighted average of the assets of the basket according to their performance.

For instance, a rainbow call on 3 assets with weights 50%/30%/20% pays 50% of the best return of the underlying assets at maturity, 30% of the second best return and 20% of the third best.

Rainbow can take various other forms but the combining idea is to have a payoff that is depending on the asset on the assets sorted by their performance at maturity. For instance, worst-of and best-of options can be seen as a particular type of rainbow options and will be describe in the next section.

Note that when pricing rainbow options with our multi-asset options pricer, make sure to set the weights from worst to best. This might not seem very intuitive but it is the way I coded it at the time.

15.4.2 Risk Analysis

As with basket options, the sensitivities to interest rates, dividends and borrowing costs are intuitive. Sensitivity to volatility, correlation and skew are very hard to grasp. It depends on many parameters and the only way I see to determine your position regarding them is to compute the option price using different levels of volatility, skew and correlation.

For instance, the sensitivity to correlation is difficult to predict because there might be two opposite effects. It is the case in the above example. On one hand, increasing correlation would increase the overall basket volatility and will tend to push the option price higher. On the other hand, increasing correlation would decrease the forward price of the basket and will tend to push the option price lower.

Seeing our 50%/30%/20% rainbow option as \(\boxed{0.9*\text{Equiweighted Basket} + 0.2*\text{Best-Of} - 0.1*\text{Worst-Of}}\) should help you understand the second impact.

As with any multi-asset options, the rainbow option will show sensitivity to the implied volatilities of each of the underlyings. The way the vega is spread around the underlying assets depends on the probability of each asset to finish at a particular ranking at maturity (best, second best or third best).

If the weights have values far from each other, the option will behave similarly as a best-of/worst-of and the position regarding volatility and correlation is easy to grasp. Generally, the more dispersed the weights, the more expensive the rainbow option.

15.5 Worst-Of options

We only focus on Worst-Of options as everything that is said about them can be applied to Best-Of options quite easily. You can also price Worst-Of options using our multi-asset options pricer.

15.5.1 Worst-Of Put

As its name indicates, a worst-of put is a put option on the worst-performing underlying asset among a basket. It is obviously more expensive than a vanilla put option on the same basket.

\(\boxed{\text{WO Put}_\text{Payoff} = max[0, K - min(S_1(T),S_2(T),...,S_n(T))]}\)

As previously mentioned, the sensitivities to interest rates, dividends and borrowing costs are intuitive.

15.5.1.1 Sensitivity to underlyings' price

A WO put option is still a put option and therefore the intuition behind the sign of the delta is straightforward.

The cross-gamma effect between the underlyings is more interesting. As an underlying asset declines and takes the role of the Worst-Of, you will expect its delta to increase in absolute value and, at the same time, the delta on the other underlyings to decrease in absolute value. That is what we call a cross-gamma effect. A movement on a underlying asset will typically have an impact on the delta of the other underlyings. The more probability an underlying asset has to finish as the Worst-Of, the more the delta will be allocated to this underlying asset. This cross-gamma sensitivity is more pronounced when the forward price of the underlying assets are close to each other (it is not clear which asset will be the worst-of). More generally, the more probability an underlying asset has to finish as the Worst-Of, the more the greeks will be allocated to this underlying asset. So the option will also show higher vega to the volatilities of the underlyings that perform the worst for example.

15.5.1.2 Sensitivity to volatility

We already know that the holder of a vanilla put option is long volatility. At the same time, higher volatility would lead to higher dispersion which again increases the price of the option. Consequently the buyer of WO put option is long volatility.

15.5.1.3 Sensitivity to correlation

A higher dispersion would lead to a higher payoff. Since lower correlation would lead to more dispersed returns, lower correlation would lead to higher payoff for worst-of put options. Therefore the buyer of a WO put option would be short correlation.

Just remind yourself that the holder of a worst-of/best-of is long dispersion and therefore short correlation and long volatility.

15.5.1.4 Sensitivity to volatility skew

Skew results in a return distribution that are negatively skewed with higher probability of downward movements leading to higher implied volatilities on the downside. Since Worst-Of put options pay on the downside, an increase in skew will raise their prices.

15.5.1.5 Sensitivity to correlation skew

Let us assume we are long a WO put option and therefore short the correlation between the underlying assets.

If the market goes down, the WO put becomes more ITM and hence we will be shorter correlation at a time when correlation tends to spike up.

If the market goes up, the correlation tends to go down but the WO put will be OTM and our sensitivity to correlation decreases.

In the first case, the skew hurts us and, in the second case, it does not benefit us. As a consequence, a WO put option is cheaper when the correlation skew is taken into account.

15.5.2 Worst-Of Call

As its name indicates, a worst-of call is a call option on the worst-performing underlying asset among a basket. It is obviously less expensive than a vanilla call option on the same basket.

\(\boxed{\text{WO Call}_\text{Payoff} = max[0, min(S_1(T),S_2(T),...,S_n(T)) - K]}\)

15.5.2.1 Sensitivities

You should be able by now to deduce most of the greeks and their behaviour for a Worst-Of call.

I will just emphasize the effects of volatility and correlation as the position in terms of dispersion is different as for a WO put option. Indeed, a WO call price decreases as dispersion goes up. Based on dispersion, you could think that the holder of a WO call is short volatility and long correlation. However, the position in volatility is not this obvious as volatility now has two opposite effects on the WO call price. On the one hand, volatility increases the expected payoff of the call (typicall volatility effect that you see on a vanilla option). On the other hand, it increases dispersion which lowers the forward level of the worst performing underlying and thus decreases the expected payoff of the WO call.

Most of the time, the dispersion effect is dominant but that is not always the case (for example in a high correlation environment). It is also quite common to see a trader that is selling the WO call being short volatility on one underlying, usually the one with the highest volatility, and long volatility on the other underlyings.

Since the position in volatility is not clear, we dont know whether the WO call option holder would benefit or lose due to volatility skew. Hence skew dependance would again depend on the actual trade parameters.

As previously discussed with WO put options, you can also expect many cross-risks with WO call options.