Lookback options are a type of exotic option

Exotic option

In finance, an exotic option is a derivative which has features making it more complex than commonly traded products . These products are usually traded over-the-counter , or are embedded in structured notes....

 with path dependency, among many other kind of options

Option style

In finance, the style or family of an option is a general term denoting the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American options. These options - as well as others where the...

. The payoff depends on the optimal (maximum or minimum) underlying asset's price occurring over the life of the option. The option allows the holder to "look back" over time to determine the payoff. There exist two kinds of lookback options: with floating strike and with fixed strike.

Lookback option with floating strike

As the name introduces it, the option's strike price is floating and determined at maturity. The floating strike is the optimal value of the underlying asset's price during the option life. The payoff is the maximum difference between the market asset's price at maturity and the floating strike. For the call, the strike price is fixed at the lowest asset's price of the option's life, and, for the put, it is fixed at the highest asset's price. Note that these options are not really options as they will be always exercised by their holder. In fact, the option is never out-of-the-money, which makes it more expensive than a standard option. The payoff functions are given by, respectively for the lookback call and the lookback put:
where is the maximum asset's price during the life of the option, is the minimum asset's price during the life of the option, and is the underlying asset's price at maturity .

Lookback option with fixed strike

As for the standard European options, the option's strike price is fixed. The difference is that the option is not exercised at the price at maturity: the payoff is the maximum difference between the optimal underlying asset price and the strike. For the call option, the holder choose to exercise at the point when the underlying asset price is at its highest level. For the put option, the holder choose to exercise at the underlying asset's lowest price. The payoff functions are given by, respectively for the lookback call and the lookback put:
where is the maximum asset's price during the life of the option, is the minimum asset's price during the life of the option, and is the strike price.

Arbitrage-free price of lookback options with floating strike

Using the Black–Scholes model, and its notations, we can price the European lookback options with floating strike. The pricing method is much more complicated than for the standard European options, and can be found in Musiela. Assume that there exists a continuously-compounded risk-free interest rate

Risk-free interest rate

Risk-free interest rate is the theoretical rate of return of an investment with no risk of financial loss. The risk-free rate represents the interest that an investor would expect from an absolutely risk-free investment over a given period of time....

  and a constant stock's volatility . Assume that the time to maturity is , and that we will price the option at time , although the life of the option started at time zero. Define . Finally, set that

Then, the price of the lookback call option with floating strike is given by:
where
and where is the standard normal cumulative distribution function

Cumulative distribution function

In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

, .

Similarly, the price of the lookback put option with floating strike is given by: