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# Lookback Option – Definition

### Lookback Option Definition

A lookback option is a type of option that gives the holder the opportunity of knowing its history when deciding on the appropriate time to exercise it. This option has a benefit of reducing the uncertainties arising when timing the market entry. It belongs to a category of exotic option possessing path dependency. Exotic options are classified as more complex than the vanilla options and are traded over the counter.

### A Little More on What is Lookback Option

The strike price of the lookback option is floating, and it’s determined at maturity. This strike price is derived from the optimal value of the price of the underlying asset during the lifespan of the option. It allows the holder to review the historical prices of the underlying asset over the life of the option.

These types of options are unlisted and therefore do not trade on the formal exchanges. Upon their execution, the holder is awarded a cash settlement equivalent to the profits that would have been realized in the buying or selling of the underlying asset. They are two types of lookback options which are the fixed strike price and the floating strike price, and they are both present for the call and put options.

A strike price is that at which exercising a derivative contract can be done and it mostly relates to stock and index options. The strike price in call options is that at which the buyer can purchase the security until the date of expiry. On the other hand, for put options, the strike price is best known as the price at which the option buyer can sell the shares.

### Examples of Lookback Options

Assume that an option has a three-month contract and it trades at \$100 at the beginning and end of the contract. At a certain point in the options period, it reaches a high price of \$120 and a low price of \$80.

For a fixed strike lookback option, the best price reached is \$120 while the strike price is \$100. Based on these, the holder’s profit can be calculated as follows: \$120 – \$100 = \$20.

For the floating strike option, the stock matures at \$100 which is also the strike price. Since the lowest price is \$80, the holder’s profit will, therefore, be \$100 – \$80 = \$20

The reason these profits are similar is that during the option’s lifespan, the stock shifted both higher and lower by the same amount.

### Academic Research on Lookback Option

Robust hedging of the lookback option, Hobson, D. G. (1998). Finance and Stochastics, 2(4), 329-347. This article’s primary objective is finding the bounds on exotic derivative’s prices in terms of the market prices of call options. It holds a particular emphasis on the lookback option.

Lookback option valuation: A simplified approach, Choi, S., & Jameson, M. (2003). The Journal of Derivatives, 11(2), 53-64. This paper presents a technique developed by Choi and Jameson that can be used to value a lookback option without needing to have an auxiliary variable.

Lookback option pricing using the Fourier transform B-spline method, Haslip, G. G., & Kaishev, V. K. (2014). Quantitative Finance, 14(5), 789-803. This study formulates a new closed-form formula that proves to be efficient in the approximation of discrete option’s lookback prices that have underlying asset prices driven by process of an exponential semimartingale.

Pricing lookback option under CEV process [J], Chi, X. I. E. (2001). Journal of Systems Engineering, 4, 008. This paper develops an evaluation model on the basis of the binomial approximation created by Nelson and Ramaswamy and the method of applying it when faced with a problem of pricing European lookback options.

The British lookback option with fixed strike, Kitapbayev, Y. (2015). Applied Mathematical Finance, 22(3), 238-260. This article gives a general view of the work involving the British Russian option and presents a financial analysis of lookback options with a non-zero strike.

American lookback option with fixed strike price—2-D parabolic variational inequality, Chen, X., Yi, F., & Wang, L. (2011). Journal of Differential Equations, 251(11), 3063-3089. This paper examines a 2-dimensional parabolic variational inequality with a financial background and develops a basic formula to get the existence and uniqueness of the problem.

On the lookback option with fixed strike, Kitapbayev, Y. (2013). This article attempts to examine the payoff mechanism based on the fixed price lookback option which protects the holder against unfavorable stock prices scenarios.

A kind European lookback option pricing model in mixed fractional Brownian motion environment [J], Zhao-qiang, Y. A. N. G. (2012). Journal of Shandong University (Natural Science), 9, 020. This paper presents a study of an exotic European lookback option through the Ito formula for mixed fractional Brownian motion.

Some lookback option pricing problems, Guo, X. (2002). In Recent Developments In Mathematical Finance (pp. 39-48). This article presents several problems associated with the pricing of lookback options and proposes various solutions to curb or minimize them.

Pricing Futures Contract Considering Reserve Margin Based on Lookback Option Model [J], HUANG, W., & LIU, H. L. (2009). Journal of Shanghai Jiaotong University, 4, 007. This paper uses a lookback option model to try and price future contracts while still putting considerations into the reserve margin

Efficient valuation and exercise boundary of American fractional lookback option in a mixed jump-diffusion model, Yang, Z. (2017). International Journal of Financial Engineering, 4(02n03), 1750033. This study develops an effective method to be used in pricing American fractional lookback options in situations where their prices adhere to a mixed jump diffusion fraction Brownian motion.

Study on Pricing European Lookback Option with Transaction Cost under the CEV Process, Li-wei, W. J. W. W. (2008). Application of Statistics and Management, 3, 026. This paper uses the CEV process to conduct pricing on European lookback options that have transaction costs.

On the lookback option with fixed strike, Kitapbayev, Y. (2014). Stochastics An International Journal of Probability and Stochastic Processes, 86(3), 510-526. This paper examines a fixed strike lookback option in a fixed horizon using the solution to the optimal stopping problem for a three-dimensional Markov process.