# Valuation and Risk

The number of assets in Mountain Range options generally ranges from a low of 4-5 to a high of about 20.

Owing to their complex payoff and path-dependency, idiosyncratic characteristics of each asset need to be taken into account.

Hence one cannot assume homogeneity of assets for either small or large baskets, making any closed-form approximation intractable.

As a result, Mountain Range options, even the non-path-dependent Atlas options, are calculated using MC simulation.

MC methods, especially for high-dimensional payoffs with large number of assets and time points, are slow to converge, and usually one or more variance-reduction techniques are employed. This problem is exacerbated further when calculating first and second order Greeks.

The other challenge posed by these options is the correlation. Even in the simple lognormal model, the sheer size of the correlation matrix can become a challenge. Since for n assets, there can be n (n-1)/2 distinct correlations.

Moreover, it is not clear how one can obtain the pairwise correlations themselves.

If, theoretically speaking, there existed n(n-1)/2 traded spread options on each pair, their implied correlations could be used with the spread options as hedges. However, it is unlikely that every pair of assets in a basket would have a traded spread option. Even if they did, their sheer number would make transaction costs prohibitive.

Hence historical correlations are more often used, even though they are hard to hedge and can change with macro- and microeconomic shifts.

When all assets belong to the same sector, a single correlation number is commonly used.

The high amount of asset interdependence makes cross-gammas important, adding further to the hedging complexity.

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