Introduction To Outperformance Options
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Derivatives

Introduction To Outperformance Options

Outperformance options, also known as exchange or spread options are exotic derivatives that allow an investor to gain leveraged exposure to the percentage price performance of one security or index over another.

Outperformance options, also known as exchange or spread options are exotic derivatives that allow an investor to gain leveraged exposure to the percentage price performance of one security or index over another. In general they are traded on an over-the-counter basis. OTC outperformance options are typically cash settled and European-style exercise. In this Learning Curve, we will discuss options on the outperformance spread between global stock index pairs such as the Nikkei 225 index and the Standard & Poor's 500 index.

The payoff at expiry of a call option on the percentage outperformance of over is:

 

 

 

 

As an illustration, we plot the rolling percentage outperformance of the Nikkei 225 index in local currency over the S&P 500 index in local currency on a rolling basis:

 

 

 

 

 

 

 

 

Motivation for Outperformance Options

Some investors have more confidence in their ability to predict the relative performance between global stock indices than the absolute levels. Such investors may prefer to buy a Nikkei 225 over S&P 500 outperformance option than a plain vanilla call on the Nikkei 225 index.

We can compare the relative prices of outperformance options and vanilla options. We find that the difference is strongly influenced by the implied correlation input for the outperformance option. In general, if the forward factors for the two indices are equal and the volatilities of the two underlying indices are equal then an outperformance option will be cheaper than a plain vanilla call on either of the two indices when the implied correlation is greater than +0.5.

 

 

 

This means that for highly correlated indices such as the S&P 500 index and the OEX 100 indices an outperformance option will usually be cheaper than a vanilla call on the S&P 500 index.

Another motivation for buying outperformance options can be the relatively low cost in relation to the magnitude of the possible upside; the reward can be significantly higher than the risk. As an example, we note that on Nov. 24, a one-year Nikkei 225 index over S&P 500 index outperformance-option quanto USD was quoted indicatively at 4.2% bid, 5.0% offered. On that date the Nikkei 225 had outperformed the S&P 500 index by 28.5% over the previous year. The option premium was therefore relatively low in relation to the magnitude of the performance, offering an attractive reward/risk ratio for buyers. Buying this option on Nov. 24 would have been profitable; from the trade date to Dec. 30 the Nikkei 225 index outperformed the S&P 500 by 10.7%.

It should be noted that in an outperformance option, the initial levels of the two indices are just as important in determining the payoff as the final levels. The timing, therefore, of any purchase or sale is an important consideration.

 

Correlation Exposure

An outperformance option buyer will be short the implied correlation between the log returns of the two underlyings over the life of the option. The correlation implied by the price of an outperformance option may be interpreted as what the options market is implying will be realized over the life of the option between the natural log returns of the two underlying indices. Therefore, when calculating the historical correlation between the underlying indices, it is important to calculate the correlation using log returns and not price levels.

 

Outperformance Option Pricing

The price of an outperformance option will primarily be a function of:

a) Forward factor differential

The forward price for an index is a function of the interest rate and the value of the expected dividends to expiry:

 

 

 

 

 

 

An outperformance option value will be strongly influenced by the difference between the "forward factors":

 

 

 

 

The following table summarizes the forwards for the Nikkei 225 index over the S&P 500 index as of 11-Nov-05, detailing the contribution from interest rates and dividend yield:

 

 

 

 

From the above table we can see that the forward factor for the Nikkei 225 is 3.7% below that for the S&P 500 index. This makes an outperformance option on the Nikkei 225 over S&P 500 index cheaper than would have been the case if the interest rates and dividend yields were the same for both indices. By multiplying by the deltas for the two indice, we can estimate that the forward factor differential significantly lowers the outperformance option price by about 1.5% of notional.

b) Volatility

For most values of implied correlation, an increase in implied volatility for either of the underlyings of an outperformance option will lead to an increase in the value of the outperformance option. The volatility of the spread between A and B increases as the volatility of A increases and therefore the option value increases.

However, this is not always the case. Lower volatility can sometimes lead to higher outperformance option prices. This is usually only observed when the implied correlation is positive and high, or alternatively if the volatility of one of the underlying indices is low in relation to the other underlying index volatility.

This can be explained as follows. A drop in the volatility in one of the two underlyings may increase the value of the outperformance option because that underlying can no longer "keep pace" with the other underlying. The volatility of the spread increases as a result.

c) Correlation

Another factor influencing the pricing is the correlation between the underlying indices (specifically the correlation between the log returns). Using a 1-factor approximation for the spread, the following equation can be used to estimate the sensitivity of the price of an outperformance option to changes in the implied correlation. The spread volatility below may be input to a modified version of the Black-Scholes equation (Margrabe) to assess the sensitivity:

 

 

 

 

 

 

 

 

 

The historic or realized correlation between indices may be used as a guide to the correlation used to price an outperformance option. Buying an outperformance option means that the investor is short the implied correlation between the performance of the two indices. The following graph illustrates the impact of varying implied correlation for an outperformance option where the implied volatilities and the forward factors for the two indices are equal:

 

 

 

 

 

 

 

 

d) Quanto FX hedging adjustment

When specifying an outperformance option, a currency must be chosen for the option. If the currency in which the outperformance option is paid for is different from the currency of one or more of the underlying indices, then the option is said to be quantoed into the option premium currency.

The outperformance option price will need to be adjusted with a quanto FX adjustment to take into account the possible future profit or loss for the provider from the combined effect of movements in the indices and the currencies.

 

Pricing: 2-Factor Models

Margrabe (1978) introduced a pricing model for valuing European exchange options under a Black-Scholes framework. However, this model does not provide all the risk exposures required to adequately manage or hedge an outperformance option. Furthermore, it can only be used to price an outperformance option that has zero strike. Many market participants therefore employ a true "2-factor model". Such a pricing model is readily available from quantitative software vendors and typically involves numerical integration of a one dimensional integral. Alternative models work using Monte Carlo simulation. The output from a 2-factor model will include two deltas, two gammas, a cross-gamma term, two vegas and a correlation sensitivity.

 

This week's Learning Curve was written Anjam Ahmad, v.p. and equity derivative strategist at Citigroupin London.

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