Efficient Algorithms for Geometric-Average-Trigger Reset Options

A derivative(or derivative security) is a financial instrument whose value depends on the value of other, more basic underlying variables, such as bonds or stocks. A stock option is a right to buy or sell a stock by a certain date for a certain price. The price in the contract is known as the exercise price or strike price; the date in the contract is known as the expiration date or maturity. Stock options with reset properties are traded actively on many exchanges throughout the world. A reset option is a path-dependent option whose strike price can be reset based on certain criteria. The geometric-average-trigger reset option resets the strike price based on the geometric average of the underlying asset’s price over a certain time period, the so-called monitoring interval. If there are multiple monitoring intervals, multiple resets result. Similar contracts have been traded on exchanges in Asia. For example, Grand Cathay, a securities firm in Taiwan, issued two reset options(Bloomberg 0517TT and 0522TT) in the Taiwan Stock Exchange in 1999. This thesis suggests two numerical approaches for pricing geometric-average-trigger reset options with multiple monitoring intervals. For American-style reset puts, an O(nh)-time algorithm on an n-period binomial lattice is presented, where h is the length (in number of periods) of each monitoring interval. A more efficient O(nhm)-time algorithm, where m denotes the number of monitoring intervals, prices European-style reset options. It is also shown that an American-style reset call will not be exercised early if its underlying asset does not pay dividends. This makes the second approach applicable to American-style reset calls. It can be proved that the price of a geometric-average-trigger reset call is higher than that of an arithmeticone, and vice versa for a put. Monte Carlo simulations suggest that both Europeanstyle geometricand arithmetic-average-trigger reset options have similar values. This suggests that our approaches give very approximate price for the difficult arithmeticaverage-trigger reset options. Experimental data confirm the correctness of the results above. Chapter