Lookback options:
Lookback options are path dependent options. The payoff from a Lookback call (put) depends on the exercise price being set to the minimum (maximum) asset price achieved during the life of the option. Thus, a Lookback call (put) allows the purchaser to buy (sell) the asset at its minimum (maximum) price.
Lookback option functions:
Extreme Spread 
The time to maturity of an extreme spread option is divided into two periods: one period starting at time zero and ending at some arbitrary date, and another starting at that arbitrary date and ending at the expiration date. A payoff at maturity of an extreme spread call (put) option equals the positive part of the difference between the maximum (minimum) value of the underlying asset of the second (first) period and the maximum (minimum) value of the underlying asset of the first (second) period.[1] The payoff at expiration of a reverse extreme spread call (put) option equals the positive part of the difference between the minimum (maximum) of the underlying asset of the second (first) period and the minimum (maximum) value of the underlying asset of the first (second) period. Extreme spread options can be priced analytically using a model introduced by Bermin (1996). The FinOptions function ExtremeSpread can be used to evaluate European extreme spread options. 


Floating Strike Lookback 
The Lookback call (put) option gives the holder the right to buy (sell) an asset at its lowest (highest) price observed during the life of the option. This observed price is applied as the strike price. The payout for a call option is essentially the asset price minus the minimum spot price observed during the life of the option. The payout for a put option is essentially the maximum spot price observed during the life of the option minus the asset price. Therefore, a floating strike Lookback option is always in the money and should always be exercised. Floating strike options can be priced analytically using a model introduced by Goldman, Sosin, and Gatto (1979). Monte Carlo simulation is used for the numerical calculation of a European style floating strike options. FinOptions has two functions to evaluate European floating strike Lookback options: Lookback and LookbackMC. The latter using Monte Carlo simulation to evaluate the theoretical value. 


Fixed Strike Lookback 
For a fixed strike Lookback option, the strike price is known in advance. The call option payoff is given by the difference between the maximum observed price of the underlying asset during the life of the option and the fixed strike price. The put option payoff is given by the difference between the fixed strike price and the minimum observed price of the underlying asset during the life of the option. A fixed strike Lookback call (put) option payoff is equal to that of a standard plain call (put) option when the final asset price is the maximum (minimum) observed value during the options life. Fixed strike Lookback options can be priced analytically using a model introduced by Conze and Viswanathan (1991). The FinOptions function LookbackFixed can be used to evaluate European fixed strike Lookback options. 


Partialtime Fixed Strike 
For a partialtime fixed strike Lookback option, the Lookback period starts at a predetermined date after the initialization date of the option. The partialtime fixed strike Lookback call option payoff is given by the difference between the maximum observed price of the underlying asset during the Lookback period and the fixed strike price. The partialtime fixed strike Lookback put option payoff is given by the difference between the fixed strike price and the minimum observed price of the underlying asset during the Lookback period. The partialtime fixed strike Lookback option is cheaper than a similar standard fixed strike Lookback option. Partialtime fixed strike Lookback options can be priced analytically using a model introduced by Heynen and Kat (1994). The FinOptions function LookbackPFixed can be used to evaluate European partialtime floating strike Lookback options. 


Partialtime Floating Strike 
For a partialtime floating strike Lookback option, the Lookback period starts at time zero and ends at an arbitrary date before expiration. Except for the partial Lookback period, the option is similar to a floating strike Lookback option. The partialtime floating strike Lookback option is cheaper than a similar standard floating strike Lookback option. Partialtime floating strike Lookback options can be priced analytically using a model introduced by Heynen and Kat (1994). The FinOptions function LookbackPFloat can be used to evaluate European partialtime floating strike Lookback options. 
The ExtremeSpread function calculates the theoretical price, sensitivities, the implied volatility, and the implied strike value of a European extreme spread option using Bermin’s model.
The Lookback function calculates the theoretical price, sensitivities, the implied volatility, and the implied strike value of a European floating strike Lookback option using Goldman, Sosin, and Gatto’s model.
The LookbackFixed function calculates the theoretical price, sensitivities, the implied volatility, and the implied strike value of a European fixed strike Lookback option using Conze and Viswanathan’s model.
The LookbackPFixed function calculates the theoretical price, sensitivities, the implied volatility, and the implied strike value of a European partialtime fixed strike Lookback option using Heynen and Kat’s model.
The LookbackPFloat function calculates the theoretical price, sensitivities, the implied volatility, and the implied strike value of a European partialtime floating strike Lookback option using Heynen and Kat’s model.
The LookbackMC function calculates the theoretical price of a European floating strike Lookback option using either an Antithetic or Control Variate Monte Carlo technique.
References
[1] Haug E.G., The complete guide to option pricing formulas, 1998, McGrawHill