Abstract
Replication of an option is used to realize the same payoff that could have been realized by holding a position in the option itself when the target option is not available. In order to ensure that a portfolio’s market value does not decrease beyond a certain level, it is necessary to insure it by hedging with options. The project discusses the basic replication strategy, followed by its application on S&P 500 index futures September 2009 contract. The Greeks (delta, gamma, rho and kappa) of the option are used, in this replication strategy, to replicate the payoff. Sources of Data: The historical prices, for June and September 2009 contracts on S&P 500 index futures and the options on these contracts, are retrieved from Bloomberg.com and Reuters.com. The project does not list the data obtained from those sources because of their license restrictions. The Eurodollar historical prices data from November 2008 to September 2009 are obtained from the website of Federal Reserve Statistical Release, H.15 which is released everyday (unless holiday) at 2:30pm. The yield curves historical data from June 2000 to October 2009 are retrieved from the website of United States –Department of the Treasury, in the Interest Rates Statistics section. For analysis purposes, during the project, information is obtained from websites of Yahoo! Finance and CME Group/Chicago Board of Trade. Conclusions Reached: The results of the project lead to a conclusion that an option pay-off can be replicated by hedging the Greeks (delta, gamma, rho and kappa). The basic strategy of replication, which provides the hedge to the delta of an option, shows that investing a proportion of total investment, equal to the delta of the option, in the underlying asset of an option, will provide the same pay-off that could be realized by holding 100% position in the option itself. The hedging for gamma can be executed by investing a portion in another option with a higher gamma or a more liquid option. This proportion should be a fraction needed to neutralize the gamma of the target option payoff curve. Investing in more than one option, with shorter maturity period or higher gamma as compared to the target option, will reduce the gamma for replication of the target option. The importance of rho hedging is realized when there is a change in carrying cost; in this project, it is the interest rate. Whenever there is a high change in price along with the change in the interest rate, the replication is adjusted for risk if it is hedged by a proper interest rate addressing instrument like Eurodollar futures or other interest rate futures. After empirical analysis, it is found that replication by hedging for delta, gamma and rho, all at a time, provide a better replication to that the delta hedge or delta-gamma hedge by itself. The project shows that replication of September 2009 contract for S&P 500 index futures. The net replication error from a delta-gamma neutral hedge is less as compared with the replication error from delta-neutral hedge. The project calculates the replicating error difference between delta-gamma neutral hedge and delta-neutral hedge. Delta-neutral hedge was $0.6187, $0.8773, and $0.1937 for September’09; call option with strike price of $1025 on January 15, 2009, January 27, 2009, and April 9, 2009, respectively. The calculations also provide an explanation for theta and rho exposures hedge in the replication.