Lookback options, also known as Hindsight options are a type of path-dependent option where the payoff is dependent on the maximum or minimum asset price over the life of the option; and this is where the name comes from - the holder of the lookback can 'look back' over time to determine the payoff.

Introduction || Floating Strike || Fixed Strike || Discrete Monitoring || American Lookbacks || Binomial Method || Forward Shooting Grid || Advanced Techniques || Other Known Names & Variants || References || Advanced Readings

Lookback options, also known as Hindsight options are a type of path-dependent option where the payoff is dependent on the maximum or minimum asset price over the life of the option; and this is where the name comes from - the holder of the lookback can 'look back' over time to determine the payoff. They generally come in two distinct forms:

**- Fixed Strike**

This type of lookback option is only settled in cash, and has the strike pretermined at inception and the payoff is the maximum difference between the optimal price and the strike price.

**- Floating Strike**

Introduced in 1979, these can have payoffs which are either cash or asset settled, where the strike is given as the optimal value of the underlying asset. It can be noted that although floating strike options are 'options' per say, they are not actually options as they are always exercised, see Yu, Kwok & Wu (2001).

For both the above types, the respective call and put payoffs are given as:

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.

Relative to vanilla calls and puts, one can consider a long position in a European call option. The call option allows the holder to buy the underlying at a predetermined price. In the case of a lookback option, instead of a predetermined price, the holder of the call can buy the option at either the lowest price over the period (floating), or the difference between the highest observed price and the strike price (fixed). Similarly, we can also apply this to puts.

Heynen & Kat (1994) uses the case of lookback options in order to illustrate a so-called market-entry exit problem.

Consider the example of an investor who anticipates a substantial rise in the market over the next three months and buys a plain European call option on the index, with a strike price equal to the current spot level, say 100. Immediately after the purchase of the option, the index drops unexpectedly by 10%, and then gets in a strong upward trend that last until expiration. The index ends at 120. The option payoff is 20; substantially less than if the option had been bought a few weeks later.

To have gained from these price movements a much better result can be obtained buying a floating strike lookback call, since the lookback option pays off the lowest index value realized. When the index drops, the strike drops with it. The option remains at-the-money until the index begins to rise. In this sense, floating strike lookback options are a useful solution to the market entry problem.

The market exit problem, instead, relies on the use of the fixed strike lookback option. To illustrate, consider the previous example of an investor who, in order to exploit his belief of a rising market, buys an ordinary call option on the index, with strike equal to the current index level of 100. Suppose that, as expected, the index first rises to a level of 125, say; but then, in the last two weeks before expiration, it shows a strong unexpected decline ending at 105. As a result, the option pays off only 5; substantially less than if it had been sold two weeks earlier. A fixed strike lookback call proves to be a better tool to use in this situation, since it pays off on the basis of the highest index value recorded and does not suffer from the plunge just before expiration

**Pricing:**

**Floating Strike Lookback Closed Form Analytical (Goldman, Sosin & Satto):**

Floating strike lookbacks were first introduced in 1979 and can be priced within a Black-Scholes framework by Goldman, Sosin & Satto (1979).

A set of equations given for European floating strike lookback calls and puts are:

Where:

Within the Black-Scholes framework, there is continuous monitoring of the asset price, but in reality, there is likely to be discrete monitoring where the asset price is sampled at particular dates. These are often refered to as "finite sampling lookbacks" and can be priced by making the adjustment given by Broadie, Glasserman & Kou (1998). See also Levy & Mantion (1998) for a slightly different treatment of discrete lookbacks.

**Fixed Strike Lookback Closed Form Analytical (Conze & Vizwanathan):**

As we highlighted in the introduction, fixed strike lookbacks have a strike level which is fixed, and at expiry, the payout is the difference between the highest (for calls) or lowest (for puts) and the fixed strike. A set of analytical formulas can be given as:

When the strike price is greater than the maximum observed price, and

When the strike price is less than or equal to the maximum observed price. Similarly for a put, when the strike is less than the minimum price, the value for a put can be given as:

and where the strike is greater than or equal to the minimum asset price, the put value is:

Where:

Like most other options, the standard European-style lookback has a sibling being the American-style lookback which can be exercised prior to expiry. Continuing from their fixed strike analytical solution, Conze & Vizwanathan (1991), finds that American fixed lookback calls are always equal in value to their European counterparts, whereas American fixed puts show a greater value than their European types. They show a technique which attempts to determine upper bounds of fixed lookback options by making use of a sequence of snell envelopes. Snell envelopes are commonly used for problems involving stochastic convergence for American-style options, but a later paper by Barraquand & Pudet (1994) shows that this convergence is not always effective. Numerous attempts since then have also produced ranging results.

However, Yu, Kwok & Wu (2001) propose a method in which dimensionality can be reduced for floating strike lookbacks, but not for fixed strike lookbacks.

The solution for American floating strike lookbacks can be found by solving the partial differential which governs the options, and this was shown to be similar to the valuation of a plain vanilla American option, with the exception that American floating strikes have a Neumann-type boundary condition and can be solved via simulation techniques.

The technique can be extended to pricing floating strike lookbacks with discrete monitoring.

General pricing of American lookbacks can be undertaken by use of lattice methods, which we look at in subsequent sections.

Kat (1995) introduced the binomial method for pricing lookback options based on an extension of the standard Cox, Ross & Rubinstein methodology. A binomial lattice can be constructed for both floating and fixed strike lookbacks, using the same terms as the standard model:

Babbs (1992) and Cheuk & Vorst (1996) also deal with the pricing of lookback options under a binomial lattice approach.

Conze & Vizwanathan (1991)'s American extension to fixed strike lookbacks makes use of snell envelopes which was shown to lack effective convergence by Barraquand & Pudet (1994). In their paper, B&P proposed the use of a *forward shooting grid* (FSG) method which aims to cope with the problem associated with the degenerate advection-diffusion partial differential equation which governs the lookback, equations which generally require the use of character-based finite element methods. The method allows us to solve the dual-dimensionality problem, but this can be reduced to a single state variable as the dimensionality problem is homogenous in S.

The forward shooting grid method provides an exact numerical solution for American lookbacks (both fixed and floating) when the asset price follows a binomial process, and gives us a solution to the non-holonomic stochastic control problem associated with the degenerative diffusion PDE.

The forward shooting grid is, in effect a lattice method, and we can generalise this model further as an extension of the Cox, Ross & Rubinstein (1979) binomial tree method, and results show that the FSG method produces results which have an error to the order of 0.001 (B&P - 1994), making this method arguably as effective as Monte Carlo simulation and superior to the closed form approach provided by (C&V - 1991), particularly for American type options.

Pricing methods for lookbacks continue to be proposed in academia. Here, we highlight several recent techniques which have been used within research and trading to provide valuation for these exotics.

Bermin (1998) brings together several essays linking the pricing of lookbacks with a path of mathematics known as Malliavin calculus. By introducing a Malliavin derivative of the stochastic variables of a lookback and the use of the Clark-Ocone formula, Bermin shows that a closed form analytical framework can be applied to the pricing and hedging of lookbacks, as well as extending this to partial lookbacks, look-barriers as well as extreme spread options. The details of Malliavin calculus is not the scope of this text and we recommend Bermin's text for interested readers.

Alpha-Quantile Options

Asian Lookbacks

Hindsight Options

Lookback Spread Options

Mocatta Options

Partial Time Lookbacks

Quanto Lookback

**Additional/Useful List of resources**

**Papers:**

**Babbs, S., **"*Binomial Valuation of Lookback Options*", Journal of Economic Dynamics & Control, 24, pp. 1499-1525 (1992, 2000)**Barraquand, J., & Pudet, T., **"

*Pricing of American Path-Dependent Contingent Claims*", Journal of Mathematical Finance, 6, 1, pp. 17-51 (1994)

**"**

Bermin, H.,

Bermin, H.,

*Essays on Lookback & Barrier Options - A Malliavin Calculus Approach*", Working Paper (1998)

Black, F. & Scholes, M.

Black, F. & Scholes, M.

*"The Pricing of Options & Corporate Liabilities",*The Journal of Political Economy (May '73)

Cheuk, T., Vorst, T.,

Cheuk, T., Vorst, T.,

*"Currency Lookback Options & Observation Frequency: A Binomial Approach",*Journal of International Money & Finance, 16, 2, pp 173-187, 1997

**"**

Conze, A., & Viswanathan, R.,

Conze, A., & Viswanathan, R.,

*Path Dependent Options: The Case of Lookback Options*", Journal of Finance, S. 1893-1907 (1991)

Cox, J., Ross, S., & Rubinstein M.,

Cox, J., Ross, S., & Rubinstein M.,

*“Option Pricing: A Simplified Approach."*Journal of Financial Economics, 7. (Sept '79).

**Goldman, B., Sosin, H., Gatto, M. A.,**"

*Path Dependent Options: Buy at the Low, Sell at the High*", Journal of Finance, S.1111-1127 (1979)

**Heynen, R., & Kat, H.**

*"Lookback options with Discrete and Partial Monitoring of the Underlying Price",*Applied Mathematical Finance, Vol 2, pp273-283. (1995)

**Hakala, J., & Wystup, W.,**

*"Foreign Exchange Risk - Models, Instruments and Strategies",*Risk Publications, 2002

**Hull, J.**

*, "Options, Futures & Other Derivatives"*, 5th Edition 2002 - Chapter 12

**Kat, H. M.,**"

*Pricing Lookback Options Using Binomial Trees: An Evaluation*", Journal of Financial Engineering, 4, 375-397 (1995)

**Kat, H. M.,**"

*Selective Memory*", Risk, 7, 11, (Nov 1994)

**Levy, E., & Mantion, F.,**"

*Approximate Valuation of Discrete Lookback & Barrier Options*" Working Paper, 1998

**Merton, R.,**

*"Theory of Rational Option Pricing"*, Bell Journal of Economics & Management (June '73)

**Vorst, A., Cheuk, T.,**"

*Lookback Options and the Observation frequency: A Binomial Approach*", Working Paper, University of Rotterdam (1994)

**Yu, H., Kwok, Y.K., Wu, L.,**"

*Early Exercise Policies of American Floating Strike & Fixed Strike Lookback Options*", Nonlinear Analysis (2001)

**Bermin, H., **"*Essays on Lookback & Barrier Options - A Malliavin Calculus Approach*", Lund University Publication (1998)**Lai, T. L., Lim, T. W.,** "*Efficient Valuation of American Floating-Strike Lookback Options Using a Decomposition Technique*", National University of Singapore + Stanford Universty Joint Working Paper, (Aug '01) **Xu, C., & Kwok, Y.K.,** "*Integral Price Formulas for Lookback Options*" Hong Kong University Working Paper, 2003