Basket & Ranges || Types of Mountain Ranges || Pricing || Closed Form Valuation || Monte Carlo Simulation || Enhanced Monte Carlo Simulation || Hedging || Known Names / Variants || References

 

In 1998, Societe Generale launched a line of products which eventually contributed to SG gaining the award for their equity derivatives house in 2001. We highlight the pricing of these so-called “Mountain Range” options, a series of path-dependent options consisting of a basket of assets, assessing the problems associated in deriving reasonable pricing valuation for these types of options and finally, highlight the troublesome hedging features of mountain ranges.

 

1) Basket and Range Options

Mountain range options are essentially a combination of basket options and range options, two types of exotic derivatives which have particular characteristics which make them useful hedging tools. Broadly speaking, the characteristics include the price of the option being dependent on several underlying assets rather than an individual asset giving way to the ‘basket’ feature and the range attribute is related to the fact that there is a particular period of time when the option is active.

 

2) Types of Mountain Ranges

An overview of several types of mountain range options which have been on offer since these structured equity products were first issued include:

 

a) Altiplano

This entitles the option holder to a large coupon if no stock in a given stock selection reaches a predetermined limit or barrier during a given time period. Otherwise, the option holder receives the payout of a plain vanilla, or sometimes Asian call on the basket. Sometimes considered as a Parisian basket option due to its barrier and Asian characteristics. The payoff is given as:

Where C is the prescribed payout amount if none of the stocks in the basket hits the barrier during the specified time, K is the strike, j represents the jth stock andis a binary variable equal to the condition set for the index value as given as:

Where L is the predetermined limit, represents the start of the limit period and represents the end of the limiting period.

b) Annapurna

Gives the option holder a payoff providing that none of the stocks within the basket falls below a predetermined fraction of the initial value during a period of time.

c) Atlas

A call option which, at maturity, will remove some of the best and some of the worst stocks in the basket.

Where n is the number of stocks, j represents the jth stock given by the iteration counts, and the terms and are constrained by the condition:

+ < n

d) Everest

Gives the option holder a payoff on the worst-performing member of a large basket of stocks at maturity. The main characteristic difference between the Everest and its predecessors is that the Everest is very long term (10-15 years) and the basket contains numerous stocks (usually 10-25 stocks)

Where n is the number of stocks and Si is the i-th stock.

e) Himalayan

Like an Asian option, the Himalaya is a call on the average performance of the best stocks within the basket. Throughout the life of the option, there are particular measurement dates where the best performer within the basket is removed, and this process is continued until all the assets with the exception of 1 have been removed from the basket. The total return on this last stock is taken as the final measure. The payoff is the sum of all the measured returns over the life of the option.

f) Others in the family include Etna options and Kilimanjaro options.

 

3) Pricing

The crucial issue when dealing with mountain range options is that the pricing is structure is primarily dependent on the correlation between the constituent stocks. As the number of stocks in the basket increases, the dimensions of the pricing equation increases to a point where accurate pricing of the option becomes a concern, particularly in larger baskets such as an Everest.

 

To consider this in perspective, we look at a traded Altiplano on 7 blue chip stocks, respectively (ABN Amro, Ahold, Aventis, Dexia, KPN, Nokia and Royal Dutch Petroleum). Estimates of correlation between two assets will inevitably show estimation errors, considering this correlation matrix, we can see that the number of estimation errors are likely to increase as the number of assets increase.

We have a total of 21 unique correlations. If we were faced with an estimation error of 0.5% for 1 set of correlations, the effect of a 7 asset basket would be an estimation error of 10.50%, rendering any option value we find meaningless. Traders often employ estimates of correlation or use historical correlations but that can be meaningless when markets are volatile or when the historical data is inconsistent. This correlation problem renders closed form solutions to be obsolete, unless a small number of stocks exist in the basket and the correlation can be reasonably estimated.

 

With this in mind, throughout the rest of the document we will assume that the correlation values can be reasonable determined through either historical simulation or other applicable method and that the error involved is negligible.

 

A major problem in determining a suitable pricing methodology for this series of options is the difficulty in identifying the smile effect associated with the products (Quessette 2002). The volatility smile becomes distorted when various stocks are removed from the baskets (particularly in the case of Himalayan and Atlas options). When a stock is removed from the basket, the basket volatility will change depending on the individual asset, and it is possible to generalize several forms of mountain range. For further reading, see Quessette (2002).

 

Considering the case of Himalayans, the best performing stock is the one which is removed, and the best performers often possess distinct autocorrelation characteristics with the basket, hence the effective change in volatility for the Himalayan is less than if a stock was removed at random.

 

4) Closed Form Valuation

Closed form solutions would be ineffective in determining pricing of mountain range options as estimating correlations between the underlying assets become problematic in higher-dimensions and hence no analytical closed form solutions exist.

 

5) Monte Carlo Simulation

By far, the most effective known method for pricing mountain range options is use of Monte Carlo Simulation using a stochastic volatility model and a correlation matrix linking the constituent assets as well as the assets to volatility.

 

We consider the Hull-White (1987) stochastic volatility (SV) model to be more accurate compared to other SV models such as Heston (1993) for one major reason. W hen Heston's SV model is used in the case of basket options, the consideration of correlation becomes problematic and is difficult to apply in a valuation context.

 

The Hull-White model is more suitable as there is no correlation between the volatility processes and the assets’ one and it can be adjusted to fit the case of correlated assets as shown by Lewis (2002).

 

Continuing with our example of Himalayan options, the number of points along the Monte Carlo path to be simulating is not computationally intensive and in the case of 10 assets, only 20 random number samples per path are required as noted by Overhaus (2002). Furthermore, as assets are removed, the number of random number samples per path can be reduced to arise at an ever faster convergence.

 

In order to price mountain range options, we can undertake the following steps:
1) Determine a suitable correlation matrix for the stocks considered.
2) Set up a simulation for the paths.
3) Generate random normal variables with mean of 0 and unit variance.
4) Standardise the variables.
5) Apply Cholesky decomposition – but noting that some correlation values cannot be used as they are not positive.
6) For each path, loop the simulations from 0 to n and 0 to i, where n is the number of stocks and i represent the number of time periods.

 

With mountain ranges which have a greater number of dimensions such as Everest options, the difficulties in rapid and efficient convergence of simulation techniques becomes noticeable. Although variance reduction techniques by means of antithetic variables (Newton 1997) is not difficult to implement, the computational effectiveness does not necessarily improve and other methods such as combining stratification sampling with importance sampling vastly improves the effectiveness of pricing higher order problems, such as that considered by Glasserman, Heidelberger, Shahabuddin (1999).

 

Note: This family of options is highly sensitive to volatility skew, particularly near sampling dates, but as the skew decays with respect to time, we can determine more accurate pricing of the option using volatility skew closer to the maturity date. see Buhler (2002)

 

For baskets which have strong positive correlation (i.e. near +1), there is a greater probability that the stocks being removed have shown similar performances over time and this factor can increase the accuracy of pricing the options on a large basket of options. Conversely, a basket of assets which have highly mixed correlations spanning from -1 to +1, is likely have varying performances. We suggest the addition of an error term equal to an estimated spread to the price of the option based on a distribution of correlations. In reality, the correlation structure is likely to be mixed and will rarely tend towards either end of the spectrum, in which case, Monte Carlo simulation and also the consideration of a large spread should be applied.

 

6) Enhanced Monte Carlo Simulation (Duan)

A more accurate approach to pricing of these exotics is that of Enhanced MCS, which is in effect an extension of the path-dependent simulation method given by Broadie & Glasserman (1996). Duan (2003) proposes the general use of the EMCS method for pricing of several types of path-dependent options and it is claimed that the simulation errors given by the standard Monte Carlo pricing can be reduced due to shortening of the contingent claim, albeit of minute proportions.

 

In the case of mountain range options, the correction is likely to have a larger effect, as the number of time periods being simulated can be reduced by 1 and this will likely reduce additional errors. Illustrations of the error reduction will be shown here in due course.

 

Hedging

Other Known Names / Variants

Altiplano
Basket Options
Etna
Everest
Flexi-Range Options
Himalaya
Kilimanjaro
Mountain Range Options
Range Options

 

Additional/Useful List of resources

Papers:

Broadie, M., Glasserman, P., "Monte Carlo Methods for Security Pricing", Journal of Economic Dynamics & Control, 21, pp. 1267-1321 (1996)
Buhler, H., "Applying Stochastic Volatility Models for Pricing & Hedging Derivatives", Deutche Bank Global Quantitative Research Presentation, (Dec 02)
Cheng, Y., Hong, R., Keller, A., "Effective Dimension of Option Pricing Problems", City University (Hong Kong) Presentation (Nov 2002)
Duan, J.C., "An Enhanced Path-Derivative Monte-Carlo Method for Computing Option Greeks" University of Toronto Working Paper, (Mar '03)
Levy, G., “Multi-Asset Derivative Pricing Using Quasi-Random Numbers & Monte Carlo Simulation”, Numerical Algorithms Group, Sep-Oct 2002
Lewis, A., "The Mixing Approach to Stochastic Volatility & Jump Models", Wilmott Mag (Mar '02)
Macaskill, J., "Managing the Downturn", Thomson IF Review, 2001
Overhaus, M., "Himalaya Options", Deutsche Bank Masterclass, Risk.net, (March 2002)
Quessette, R., "New Products, New Risks", Deutsche Bank Masterclass, Risk.net (March 2002)