Lookback Options

A lookback option, also known as a hindsight option is a path dependent option where the payoff is determined by the maximum or minimum asset price achieved during the entire life of the option. Essentially, at expiration, the holder can "look back" over the life of the option and exercise based upon the optimal asset price achieved during that period. Lookbacks can be structured as puts or calls and come in two basic forms:

• A fixed strike lookback option is cash settled and has a strike set in advance. It is exercised based upon the optimal asset price achieved during the life of the option. In the case of a call, this is the highest asset price achieved, so the call has a payoff equal to the greater of: zero or the difference between that highest value and the fixed strike. In the case of a put, the optimal value is the lowest asset price achieved, and the payoff is the greater of: zero or the difference between the strike and that lowest value.[1]
(1)
$$c_{fixed} = max(0,S_{max} - X)$$
(2)
$$p_{fixed} = max(0,X - S_{min})$$

[2]

• A floating strike lookback option, first introduced in 1979, can have payoffs settled in either cash or assets. It settles based upon a strike that is set equal to the optimal value achieved by the underlying asset over the life of the option. In the case of a call, that optimal value is the lowest value achieved by the underlying asset, so the call has a payoff equal to the difference between the value of the asset at expiration and the lowest value achieved by the asset over the life of the option. In the case of a put, the payoff is the difference between the highest value achieved by the underlying asset and the value of the asset at expiration.[1]
(3)
$$c_{float} = max(0,S - S_{min})$$
(4)
$$p_{float} = max(0,S_{max} - S)$$

[2]

It can be noted that although floating strike options are 'options' per say, they are not actually options as they are always exercised.[2]

An attractive benefit of lookback options is that they are never out-of-the-money, but the result is that lookbacks are often more expensive than similar vanilla style options. [2]

Lookback options have obvious appeal, but they are expensive. Their structure doesn't mimic typical business liabilities, so they are largely a speculative device.

Bibliography
1. "Lookback Option." Riskglossary.com. 30 Sept. 2008 <http://www.riskglossary.com/link/lookback_option.htm>.
2. "Lookback Options." Global Derivatives. 30 Sept. 2008 <http://www.global-derivatives.com/index.php?id=32&option=com_content&task=view>.
page revision: 8, last edited: 28 Mar 2010 02:25