Lookback Options

Lookback options come up in work on valuing and hedging exotic options. For the buyer, the question is whether the higher premium is worth the payoff. The basics of calls and puts are covered in Option Basics: Calls and Puts.

Two forms that use the highest or lowest price

A lookback option is a path-dependent option, because its payoff depends on prices during the period as well as the price at expiry. Classification of exotic options and the general points on valuation and hedging are covered in Overview of Exotic Options.

In the standard form (floating-strike), a call is the right to buy the underlying asset at the lowest price over the period, and a put is the right to sell it at the highest price¹. The other form (fixed-strike) sets the strike price in advance and compares it with the highest price over the period for a call, and with the lowest price for a put. Let $S_T$ be the price at expiry, $M$ the highest price over the period, $m$ the lowest price, and $K$ the strike price. The payoff at expiry is then as follows. The notation $(x)^+$ means 0 when $x$ is negative.

There are also forms that multiply the highest and lowest prices by a fixed rate, and forms that observe prices over only part of the period up to expiry¹.

The word "lookback" also appears in interest rates, but there it means a calculation convention in compounding in arrears that uses the rate from a number of business days earlier. It is a different meaning from this option.

Comparing payoffs in a hypothetical example

Consider a hypothetical example in which Company B buys a call with one year to expiry from a financial institution. The price on the start date of the contract is 100. The price of the underlying asset is observed on the start date and then four more times, every three months (the last one on the expiry date), and the highest and lowest prices are taken from these five prices.

Path A and Path B both end at a price of 105 at expiry, but they move differently along the way.

Figure description

This figure joins the observed prices of the underlying asset on the observation dates for two paths, shown as the difference from the start-date price of 100. The line at zero is the start-date price, and each point equals the observed price minus 100 (Path A: 100, 92, 85, 110, 105; Path B: 100, 103, 108, 106, 105). The two lines start at the same height and finish at the same height, but their highest and lowest prices along the way differ. Path A falls below the start-date price along the way, and Path B does not. A vanilla option looks only at the price at expiry and cannot tell the two paths apart, while a lookback option tells them apart by the highest and lowest prices along the way.

The strike of the fixed-strike option is 100. For comparison, a vanilla call with a strike of 100 is also shown.

The vanilla call depends only on the price at expiry, so it pays 105 − 100 = 5 on both paths. On Path A the highest price is 110 and the lowest is 85, so the fixed-strike call pays 110 − 100 = 10 and the floating-strike call pays 105 − 85 = 20. On Path B the highest price is 108 and the lowest is 100, the price on the start date, so the fixed-strike call pays 108 − 100 = 8 and the floating-strike call pays 105 − 100 = 5. Puts can be calculated in the same way. With a vanilla put with a strike of 100 shown alongside, the results are as follows.

On Path A the fixed-strike put pays 100 − 85 = 15 and the floating-strike put pays 110 − 105 = 5. On Path B the lowest price is 100, so the fixed-strike put pays 0, and the floating-strike put pays 108 − 105 = 3.

Suppose the vanilla call costs 6 and the floating-strike call costs 12. Interest is ignored. On Path A, Company B gains 8 on the floating-strike call: the payoff of 20 less the premium of 12. The vanilla call loses 5 − 6 = −1. On Path B, the floating-strike call loses 5 − 12 = −7, and the vanilla call again returns −1.

The floating-strike call pays a lot on paths that dip deeply along the way. On paths where the price at expiry is close to the lowest price over the period, the payoff falls short of the premium paid and the buyer loses the difference (on Path B, 5 against 12, or −7).

Why the premium is higher

In the usual form, where the observations include the start date and the price at expiry, the payoff of a lookback option on every path is never smaller than that of a vanilla option with matching terms. The seller, the financial institution, may therefore have to pay more, and the premium is higher.

For a fixed-strike call, the price at expiry $S_T$ is also among the observed prices that make up the highest price $M$, so $M \ge S_T$. Hence $(M - K)^+ \ge (S_T - K)^+$, and on every path the payoff is the same as or larger than that of a vanilla call with the same strike and expiry. For a floating-strike call, the price on the start date $S_0$ is also among the candidates for the lowest price, so $m \le S_0$. Hence $S_T - m$ is at least 0 and at least $S_T - S_0$, so it is never smaller than the payoff of a vanilla call whose strike is the start-date price².

A difference arises on paths where the highest price over the period is above the price at expiry, or the lowest price over the period is below the price on the start date. Path A in the previous section is such a case: the vanilla payoff is 5, while the floating-strike call pays 20. The buyer can take the most favorable price over the period, and in return pays the corresponding premium up front.

Differences from vanilla and barrier options, and points to note

A barrier option adds a condition that reduces the payoff, and often has a lower premium than a vanilla option with the same strike and expiry. A lookback option differs in this respect³.

The buyer pays the premium at the start of the trade and receives the payoff at expiry from the seller, the financial institution. The buyer therefore bears credit risk, the risk that the financial institution cannot pay⁴. The observation dates and the prices used for observation are set in the contract, and the payoff changes with these terms.

Why continuous and discrete observation give different prices

Continuous observation watches the price at all times up to expiry to determine the highest and lowest prices. Discrete observation uses only the prices on observation dates set in the contract. Most path-dependent options traded in the market use discrete observation⁵.

If only observation dates are looked at, highs and lows reached between observation dates do not enter the payoff. As a result, the highest price under discrete observation is at most the highest price under continuous observation, and the lowest price is at least the lowest price under continuous observation. Suppose that on Path A in the previous section, the price fell to 80 between the observations after 6 months and after 9 months. Under continuous observation the lowest price is 80, and the floating-strike call pays 105 − 80 = 25. Under discrete observation, which looks only at the observation dates, the lowest price stays at 85, and the payoff is 20. With other terms the same, the payoff under discrete observation is no larger than under continuous observation, and the premium is smaller too².

The frequency of observation affects the price of barrier options in the same way⁶. Increasing the number of observations brings the price closer to continuous observation, but the difference is not necessarily small. In a numerical example for a barrier option, a large difference between the discrete and continuous prices remains even with daily observation (50 observations)⁷. Because the observation dates and the prices used are set in the contract, valuation is done with the observations as the contract specifies.

Whether an analytical solution exists, and valuation methods

For continuous-observation lookback options, there are price formulas under the assumptions of the Black-Scholes model. Goldman, Sosin, and Gatto derived the formula for the floating-strike form in 1979⁸, and Conze and Viswanathan (1991) and others also gave formulas for continuous observation⁹.

For discrete observation, there is no closed-form formula except one that uses a normal distribution function whose dimension equals the number of observations. Even that formula is not easy to compute when the number of observations exceeds 5⁹. Methods that approximate the discrete price by adding a correction to the continuous-observation formula (continuity corrections) have therefore been studied. For lookback options, the correction is given by Broadie, Glasserman, and Kou⁷.

In numerical calculation, the highest or lowest price is carried as a state variable, because a lookback option can be valued by tracking the current price and the highest or lowest price (or both)¹⁰. In binomial trees, Hull and White kept separate states at each node, while Cheuk and Vorst devised a tree that merges the states at each node into one⁸. A plain Monte Carlo method or binomial tree can take a long time to reach good accuracy⁹.

Once the assumptions of the Black-Scholes model are dropped, valuation usually relies on the time-consuming Monte Carlo method or on numerical solution of partial differential equations. In some models with jumps, the price can be written as a Laplace transform¹¹.

The seller's hedge

The sensitivities that the seller, the financial institution, uses for hedging (such as delta) come from the valuation model. The Black-Scholes model has a price formula. For some models with jumps, formulas that give the delta and gamma of a fixed-strike option in the form of Laplace transforms have been presented¹². Because the valuation includes the current price and the highest or lowest price, the sensitivities also depend on them¹⁰.

Notes

1. Steven Kou, "Discrete Barrier and Lookback Options" (2008), Section 1.1 (page 344). It gives the floating-strike and fixed-strike payoffs, with the notation changed in the text. The forms that multiply the prices by a rate and the forms that observe only part of the period are in the same section. ↩ ↩²

2. A comparison derived in the text from the payoff formulas in footnote kou-11; the source has no statement of it. It uses a contract whose observations include the expiry and start-date prices, and the fact that discrete observation uses only some of the points in time used for continuous observation. ↩ ↩²

3. Board of Governors of the Federal Reserve System (FRB), "Trading and Capital-Markets Activities Manual," Section 4330.1 "Options," "Exotic Options" (Page 2; PDF page 40). The passage on the lower premium of barrier options. ↩

4. The same FRB manual, Section 4330.1 "Options," "Risks," "Credit Risk" (Page 7; PDF page 45; April 2003 edition). The passage stating that credit risk arises when a financial institution buys an option; the explanation is for over-the-counter trades. ↩

5. Kou, same paper, Section 1 "Introduction" (page 343). The passage stating that most path-dependent options traded in the market use discrete observation. ↩

6. The same FRB manual (February 1998), Section 4330.1 "Options," "Exotic Options" (Page 2; PDF page 40). The passage stating that the value of lookback and barrier options can depend on the frequency of observation. ↩

7. Kou, same paper, Chapter 4 "Continuity corrections," Section 4.1 (pages 355–356) and Table 1. The barrier option numerical example, and the statement that Broadie et al. (1999) gave the correction for lookbacks. ↩ ↩²

8. Karl Grosse-Erdmann and Fabien Heuwelyckx, "The pricing of lookback options and binomial approximation" (arXiv:1502.02819, 2015), Section 1 (pages 1–2). The passage explaining the formula of Goldman et al. and the trees of Hull and White and of Cheuk and Vorst. ↩ ↩²

9. Kou, same paper, Section 1.1 (page 345). The passage on the sources of the continuous-observation formulas, the absence of a closed form for discrete observation, and the computing time of a plain Monte Carlo method or binomial tree. ↩ ↩² ↩³

10. Mark Broadie and Jérôme Detemple, "Option Pricing: Valuation Models and Applications" (Management Science, 2004), Section 5.1.1 (page 1166). The passage stating that tracking the highest or lowest price is enough for valuation. ↩ ↩²

11. Markus Hofer and Philipp Mayer, "Pricing and hedging of lookback options in hyper-exponential jump diffusion models," Abstract (page 1) and Section 1 (page 2). The passage on valuation methods in more general models and on the Laplace-transform form of the price. ↩

12. Section 3.2 of the same paper (pages 8–11). Delta and gamma are in Theorem 3.2 and the sensitivity to maturity in Theorem 3.3; the text mentions only delta and gamma. ↩