# Risk analysis of Worst-Of and Best-Of options

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In this article we shall consider the risk factors of the Worst-Of and the Best-Of trades. Before we start digging in, it is important to understand the concept of dispersion. Used in the context of multi asset trades, a high (or low) dispersion means that the underlying asset returns are quite different (or similar) from each other. Statistical measures of dispersion include variance, standard deviation, inter quartile range of underlying asset returns.

Figure below shows the effect of dispersion on the returns of the underlying asssets. For higher dispersion, we can expect a higher asset return level from the best performer and a lower asset return level from the worst performer while the basket performance remains unaffected. Similarly a lower dispersion leads to lower expected asset return level from the best performer and a higher expected asset return level for the worst performer.

To analyse the effect of correlations and volatility on the Worst-Of and  best-Of options  it is important to understand how these  factors 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 volatality leads to asset returns with large deviations from it's expected return. Hence, a higher volatility and a lower correlation leads to higher dispersion. Worst-Of and Best-Of trades are good examples of how dispersion dictates some of the risks of multi asset trades. So lets get started!!

## Worst-Of Call

$Payoff\ =\ max\[\ 0,\ min\(R_1(T),R_2(T)....R_n(T)\)\ -\ K\]$

It's straightforward to see that the a Worst-Of call options are cheaper than a basket call options  (see figure above) on the same underlyings. Because they are cheap and offer a large leverage potential, these options are quite popular with the exotic desks.

Interest rates and dividends - Higher the forward prices of the individual underlying stocks, higher will be the price of the call option on the worst performing stocks and vice versa. Since higher interest rates and lower dividends increase the forward prices, a buyer of Worst-Of call is long interest rates and short dividends.

Correlation and Volatility - Higher dispersion would lead to a lower payoff for the call option on the worst performing stock. Since lower correlation would lead to highly dispersed returns of the underlying assets, lower correlations would lead to lower payoffs for Worst-Of call options. Therefore a buyer of Worst-Of call option would be long correlation.

Positions in volatility is however not clear. On one hand increase in volatility increases option prices, on the other hand increase in volatility leads to higher dispersion (as discussed before) which leads to lower payoffs for Worst-Of call options. While a lot of time the dispersion effect dominates, a seller has to be careful about the vega of the trade.

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

## Worst-Of Put

$Payoff\ =\ max\[\ 0,\ K\ -\ min\(R_1(T),R_2(T)....R_n(T)\)\]$

The payoff for a Worst-Of put option is always higher than a payoff for a basket put option on the same underlyings and consequently a Worst-Of put option is costlier than a basket put option on the same underlyings.

Interest rates and dividends - Higher the forward prices of the individual underlying stocks, lower will be the price of the put option on the worst performing stocks and vice versa. Since higher interest rates and lower dividends increase the forward prices, the buyer of worst-Of put is short interest rates and long dividends.

Correlation and Volatility - We should be able to see that a higher dispersion would lead to a higher payoff for the put option on the worst performing stock. Since lower correlation would lead to highly dispersed returns of the underlying assets, lower correlations would lead to higher payoffs for Worst-Of put options. Therefore a buyer of a Worst-Of put option would be short correlation.

We know that higher volatility results in higher put option prices. At the same time, higher volatility leads to higher dispersion which again increases the price of the option. Consequently a buyer of Worst-Of put is long volatilty.

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. As we now know higher volatility leads to higher Worst Of put options. Consequently a buyer of a Worst-Of put is long skew.

## Best-Of Call

$Payoff\ =\ max\[\ 0,\ max\(R_1(T),R_2(T)....R_n(T)\)\ -\ K\]$

Best-Of call options are costlier than a basket call option on the same underlying assets. As a result they are not as popular as the Worst-Of call options.

Interest rates and dividends - Higher the forward prices of the individual underlying stocks, higher will be the price of the call option on the best performing stocks and vice versa. Since higher interest rates and lower dividends increase the forward prices, a buyer of Best-Of call is long interest rates and short dividends.

Correlation and Volatility - Higher dispersion leads to a higher payoff for a Best-Of call option. Since a decrease in correlation leads  to higher dispersion, we conclude that a Best-Of Call is short correlation.

Higher volatility leads to higher option prices and also increases the dispersion. Both cases a higher volatility leads to a higher payoff for a Best-Of call options. We can thus conclude that a buyer of a Best-Of call is long volatility.

Skew - A presence of a skew implies a lower implied volatility on the upside, leading to a lower payoffs for Best-Of call options. Hence a buyer of a Best-Of call is short skew.

## Best-Of Put

$Payoff\ =\ max\[\ 0,\ K\ -\ max\(R_1(T),R_2(T)....R_n(T)\)\]$

Best-Of put options are cheaper than a basket call option on the same underlying assets. Since they offer a higher leverage potential they are quite popular.

Interest rates and dividends - Higher the forward prices of the individual underlying stocks, lower will be the price of the put option on the best performing stocks and vice versa. Since higher interest rates and lower dividends increase the forward prices, a buyer of Best-Of put is short interest rates and long dividends.

Correlation and Volatility - Higher dispersion leads to a lower payoff for a Best-Of put option. Since a decrease in correlation leads  to higher dispersion, we conclude that a buyer of Best-Of put is long correlation.

The effect of volatility is unclear as in Worst-Of call options. One one hand a higher volatility leads to a higher option price, on the ther hand it also increases dispersion which has the oppsosite effect.

Skew - Since the effect of volatility is unclear, the effect of skew is also unclear.

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