# Mountain range (options)

Mountain ranges are exotic options originally marketed by Société Générale in 1998. The options combine the characteristics of basket options and range options by basing the value of the option on several underlying assets, and by setting a time frame for the option.

The mountain range options are further subdivided into further types, depending on the specific terms of the options. Examples include:

• Altiplano - in which a vanilla option is combined with a compensatory coupon payment if the underlying security never reaches its strike price during a given period.
• Annapurna - in which the option holder is rewarded if all securities in the basket never fall below a certain price during the relevant time period
• Atlas - in which the best and worst-performing securities are removed from the basket prior to execution of the option
• Everest - a long-term option in which the option holder gets a payoff based on the worst-performing securities in the basket
• Himalayan - based on the performance of the best asset in the portfolio

Most mountain ranges cannot be priced using closed form formulae, and are instead valued through the use of Monte Carlo simulation methods.

## Everest Options

Although Mount Everest is the highest point on earth, the Everest option payoff is on the worst performer in a basket of 10-25 stocks, with 10-15 year maturity. (Richard Quessette 2002). Given n stocks, $S_1, S_2,..., S_n$ in a basket, the payoff for an Everest option is: $\min_{i=1...n}(\frac{S_i^T}{S_i^0}).$

## Atlas Options

Atlas was a Titan who supported the Earth on his back. The Atlas option is a call on the mean (or average) of a basket of stocks, with some of the best and worst performers removed. (Quessette 2002). Given n stocks $S_1, S_2,..., S_n$ in a basket, define:

$R_{(1)}^t=\min{\{\frac{S_1^t}{S_1^0},\frac{S_2^t}{S_2^0},...,\frac{S_n^t}{S_1^n}\}},$
$R_{(n)}^t=\max{\{\frac{S_1^t}{S_1^0},\frac{S_2^t}{S_2^0},...,\frac{S_n^t}{S_n^0}\}},$

where $R_{(i)}^t$ is the i-th smallest return, so that:

$R_{(1)}^t \leq R_{(2)}^t \leq \dots \leq R_{(i)}^t \leq \dots \leq R_{(n)}^t.$

The Atlas removes a fixed number ($n_1$) of stocks from the minimum ordering of the basket and a fixed number ($n_2$) of stocks from the maximum ordering of the basket. In a basket of n stocks, notice that ($n_1+n_2 < n$), to leave at least one stock in the basket on which to compute the option payoff. With a strike price $K$, the payoff for the Atlas option is:

$\sum_{j=1+n_1}^{n-n_2}{\frac{R_{(j)}^T}{n-(n_1+n_2)}-K})^{+}.$

## Himalayan Options

A Himalayan option with notional $N$, and maturity $T$ starts with a basket of $m$ equities. The terms of the contract will specify $m$ payoff times: $t_0 = 0 < t_1 < t_2 < \dots < t_m = T$. At payoff time $t_i, \ i=1:m$, the percentage returns since inception of all equities currently in the basket are computed, and the equity with the largest return is noted; denote this equity by $S_{k_i}, \ 1\leq k_i \leq m$. The derivative then makes the payoff: $N \max \left(\frac{S_{k_i, t_i} - S_{k_i, t_0}}{S_{k_i, t_0}}, \ 0\right)$, and $S_{k_i}$ is removed from the basket. The procedure is repeated until maturity, at which time the final payoff occurs and the basket is emptied.