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- Volatility Derivatives 1
- The world of Structured Products 4
- Library of Structured Products 0
- Table of Contents
- Vanilla Options
- Volatility, Skew and Term Stru
- Option Sensitivies: Greeks
- Option Strategies
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- Dispersion Options
- Barrier Options
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- Autocallable Structures
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- Dispersion Options

# Dispersion Options

__9. Dispersion Options__

**9.1. Rainbow Options**

**9.1.1. Payoff Mechanism**

The rainbow option pays on a return weighted by the performances of the underlying stocks; that is, the weights are agreed in the contract, but the actual payoff at maturity depends on how the assets performed.

Rainbow_{payoff} = Max(0, Return)

Where Return = 50% Best + 30% Second Best + 20% Third Best

**9.1.2. Risk Analysis**

The holder of a rainbow option **has a bullish view**. Therefore, **higher forward** prices increase the rainbow’s price. The seller would be short the forwards, and therefore be short interest rates, long dividends and long borrowing costs.

**The position regarding dispersion depends on the weights of the rainbow option. **

In the above example, it is hard to know and therefore needs to be tested. Typically one bid/asks the levels of the volatility and correlation parameters, and the spread depends on the underlyings and the current state of the market.

The option will show sensitivity to the IVs of each of the underlyings, and the set of European options with the maturity of the rainbow can serve as Vega hedging instruments. The skew position is also dependent on the weights accordingly.

** If the weights were [70%, 20%, 10%], the option’s behaviour is similar to a best-of call option**, then the trader selling this option would be short dispersion, which means short volatility and long correlation. If the rainbow’s weights are dispersed, the option’s price is higher.

**9.2. Individually Capped Basket Call (ICBC)**

**9.2.1. Payoff Mechanism**

This product is based on a basket of stocks. For instance, let’s take a 3-year maturity individually capped basket call based on a basket of N stocks. The holder of the option receives an annual coupon, Coupon(i) (floored at 0%) of value of the arithmetic average of the capped returns.

**9.2.2. Risk Analysis**

The buyer of an ICBC has a **bullish view** on the underlying stocks. He is therefore **long the forward prices**. The seller would then be short the stock’s forwards and therefore be short interest rates, long dividends and long borrowing costs.

The **holder of an ICBC is short dispersion**. If dispersion is high, this means that volatility is high and correlation is low. If we take the case of low-correlated stocks, we get a lot of positive returns and a lot of negative returns. If at the same time, volatility is high, this means that we would get returns far from their expected value, and thus more extreme values. When averaging the returns to determine the option’s payoff, the **positive large values are capped but the negative values are not floored**. The **downside effect is then more important than the upside effect**. Therefore, the potential payoff is lower when dispersion is higher. In other words, the seller of this option is long dispersion; which means he would be long volatility and short correlation.

**The seller of an ICBC is short skew**.

**The lower the number of stocks composing the basket, the lower the downside effect with respect to the upside effect, and consequently, the higher the ICBC price would be.**

Do you think a capped basket call option is cheaper or more expensive than the ICBC you were about to offer? Moreover, do you believe the risks associated with hedging a capped basket call are similar to those associated with the ICBC?

**The price of the ICBC is always cheaper than the price of the capped basket call because its payoff is lower. **

The slight modification has a big effect on the risks associated with hedging a position in this option.

Obviously, the holder of the capped basket call is **still long the forward**.

Now, the analysis starts to be more interesting when we are talking about the impacts of volatility and correlation. Recall the dispersion effect on the ICBC: the dispersion was coming from the idea that the downside effect was more important than the upside effect when simulating paths. But in the case of a capped basket call, there are no caps on the individual returns; the cap is global, and so **the downside paths do not gain relative importance**. Thus **there is no dispersion effect** on the capped basket call. This is an intuitive way of understanding that **a capped basket call has risks similar to a simple basket call**.

We are just applying a global cap to the payoff of a basket call. Therefore, a **higher volatility increases the capped basket call price** since it increases the potential payoff of a basket call. Moreover, a **higher correlation increases the volatility of the basket and then increases its payoff**.

When a trader sells an ICBC, they are buying the volatilities and selling the correlations between the underlyings. When selling a capped basket call, they are no longer dealing with dispersion, and the risks involved are slightly simpler, but it is still necessary to know that they are selling the volatilities and correlations. **The capped basket call has Greeks that resemble the call spread**, for which we know **the Vega can become negative depending on the paths of the underlying basket and position of the basket forward.**

**9.3. Outperformance Options**

**9.3.1. Payoff Mechanism**

The outperformance option, also referred to as a spread option, is typically European style, and has a payoff based on the positive return of an asset S_{1} over another asset S_{2}. At maturity, the outperformance option holder receives a payoff given by

When the payoff takes the form described above, the outperformance option still makes a positive payout even if both underlying assets decrease in value, as long as S_{1} has a negative performance lower in absolute value than the negative performance of S_{2}. Ideally, the holder of an outperformance option would rather S_{1} increase and S_{2} decrease, and the payoff would then be higher.

**9.3.2. Risk Analysis**

The **seller of this option is essentially short the forward price of S _{1} and long the forward price of S_{2}**. The seller will be long the dividends of S

_{1}because he will have to go long an amount of S

_{1}to hedge the risk to this underlying, and short the dividends of S

_{2}.

The **seller of this option is obviously short dispersion**, which means that he is long correlation between the indices and short their volatility.

As is the case for best-of options, outperformance options are another way for an exotic trader to balance his global position in dispersion. Selling such options involves selling volatility, but more importantly, involves buying correlation. This sets it out from the majority of multi-asset options in which the seller is short correlation.

Extending the payoff of the outperformance option, the two assets whose performances are being compared could be baskets of assets B_{1} and B_{2}. The seller of the option will still be long the correlations between the two baskets, i.e. long the pairwise correlations between any one element of B1 and one element of B2, but will be short the correlations within each basket, i.e. the pairwise correlations between any two elements of B1 or any two elements of B2. The sub-basket correlation positions can be seen by recalling the formula for the volatility of a basket, it involves the covariances and thus the correlations, and so an increase in the correlations within a basket raises the overall basket volatility. The seller of the option is short the volatility of each basket and is thus short these correlations.