Risk Latte - Quanto Asian Rainbow Call Option

Quanto Asian Rainbow Call Option

Team Latte
Apr 28, 2005


We can suggest a really “funky” exotic option that a bank can sell to a hedge fund or any other big institutional client that wants an exposure to the upside of a weighted basket of equity indices. This is not an original idea and it has been inspired by a few recent structures that we have seen in the market.

The option is a Quanto Call option with an embedded Rainbow as well as Asian feature.

Option Type : Quanto Asian Rainbow Call Option
Underlying Indices : Oil Futures, Nikkei 225, FTSE and S&P500
Strike Price : 35%
Initial Reference P0i : The Initial Reference of each individual Index (i-th index) means the closing price of the i-th Underlying Index on the Initial Reference Setting Date
Initial Reference Setting Date & Maturity : 4 January, 2005 and 31 December 2013
Final Reference Pi : The arithmetic mean of the closing prices of the i-th underlying asset on 31 March 2005 and thereafter on 30 June, 30 September and 31 December of each calendar year up to and including the Expiration Date. For each i-th Underlying Index, there are a total of 32 observations.

Mathematically defined as:



Where Pij is the price of the i-th index in the asset basket on the j-th fixing date. If the price of the relevant asset is lower than the corresponding start price then the price will be replaced by the start price in order to calculate the average.


Return :
Performance Measurement : Return Ri of the Asset;
Option Payoff at Expiry : Max [(40% Best Performing Asset + 30% Second Best Performing Asset + 20% Third Best Performing Asset + 10% Fourth Best Performing Asset) , 0]

Back of the Envelope Pricing:

Since the option has both a Rainbow as well as an Asian feature this option can be easily priced using a Monte Carlo Simulation. Trying to find a closed form solution to this option will not only be tedious but may also break down at certain points.

We have priced the above option using a Monte Carlo simulation with the following volatility and correlation numbers, which are rather arbitrary (and chosen only as a very rough estimate) and are assumed to be constant throughout the life of the option:

Volatility

Oil Futures : 18%
Nikkei 225: 18%
FTSE : 15%
S & P 500 : 12%

Correlation Matrix:

oil nikkei FTSE S&P500
oil 1.00 0.34 0.25 0.45
nikkei 0.34 1.00 0.75 0.67
FTSE 0.25 0.75 1.00 0.80
S&P500 0.45 0.67 0.80 1.00

We used a Monte Carlo simulator working on a Cholesky factorization schedule and we ran 1000, 2000 and 5,000 simulation runs on an Excel/VBA engine. The price(s) of the option that we got was:

1,000 Runs : 3.41%
2,000 Runs : 3.20%
5,000 Runs : 3.11%

Advanced Pricing:

For a detailed and advanced pricing of the option we have used a C/C++ engine. And given the fact that both volatilities and correlations can become unstable Cholesky factorization may not always work. Therefore, we have designed the Monte Carlo simulator to run using an Eigensystem (eigenvalues and eigenvectors).

The following points need to addressed very well in an advanced pricing model:

  • Since the option is a quanto option all the price changes (returns) of each asset need to be transformed into the price changes (returns) in the base currency of the option seller/buyer; for this all returns need to be transformed into the quato return using the volatility of the asset, volatility of the FX rate (e.g. for Nikkei225 the vol of USD/JPY if USD is the base currency) and the correlation between the FX rate and the asset;

  • A volatility term structure should be used to simulate the asset price path over the years;

  • Plugging implied volatilities into the Monte Carlo simulator is not going to give the right price of the option and we have used a GARCH model for volatility and correlation to be input in the simulator;

  • Volatilities and correlations need to be stressed adequately to check the explosion or the implosion of the price.

Any comments and queries can be sent through our web-based form.

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