# Lookback option: Wikis

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# Encyclopedia

The Lookback options are a type of exotic options with path dependency, among many other kind of options. 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 fix 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 there 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:

$LC_{float}=\max(S_T - S_{min},0) = S_T - S_{min}, ~~\text{and} ~~ LP_{float}=\max(S_{max} - S_T,0) = S_{max} - S_T,$

where Smax is the maximum asset's price during the life of the option, Smin is the minimum asset's price during the life of the option, and ST is the underlying asset's price at maturity T.

## 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:

$LC_{fix}=\max(S_{max}-K,0), ~~ \text{and} ~~ LP_{fix}=\max(K-S_{min},0),$

where Smax is the maximum asset's price during the life of the option, Smin is the minimum asset's price during the life of the option, and K 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 r > 0 and a constant stock's volatility σ > 0. Assume that the time to maturity is T > 0, and that we will price the option at time t < T, although the life of the option started at time zero. Define τ = Tt. Finally, set that

$M = \max_{0\leq u \leq t} S_u, ~~m= \min_{0\leq u \leq t} S_u \text{ and }S_t = S.$

Then, the price of the Lookback call option with floating strike is given by:

$LC_t = S\Phi(a_1(S,m)) - me^{-r\tau}\Phi(a_2(S,m)) - \frac{S\sigma^2}{2r} ( \Phi(-a_1(S,m)) - e^{-r\tau}(m/S)^{\frac{2r}{\sigma^{2}}}\Phi(-a_3(S,m))),$

where

$a_1(S,H) = \frac{\ln(S/H) + (r+\frac12\sigma^2)\tau}{\sigma\sqrt{\tau}}$
$a_2(S,H) = \frac{\ln(S/H) + (r-\frac12\sigma^2)\tau}{\sigma\sqrt{\tau}} = a_1(S,H) - \sigma\sqrt{\tau}$
$a_3(S,H) = \frac{\ln(S/H) - (r-\frac12\sigma^2)\tau}{\sigma\sqrt{\tau}} = a_1(S,H) - \frac{2r\sqrt{\tau}}{\sigma},\text{ with }H>0, S>0,$

and where Φ is the standard normal cumulative distribution function, $\Phi(a) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{a} e^{-\frac{x^2}{2}}\, dx$.

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

$LP_t = -S\Phi(-a_1(S,M)) + Me^{-r\tau}\Phi(-a_2(S,M)) + \frac{S\sigma^2}{2r} ( \Phi(a_1(S,M)) - e^{-r\tau}(M/S)^{\frac{2r}{\sigma^{2}}}\Phi(a_3(S,M))).$

## Arbitrage free price of Lookback options with fix strike

Using the Black-Scholes Model, and its notations, we can also price the European Lookback options with fix strike. Assuming the same as for the Lookback option with floating strike and using the same notations, the price at time t<T of the European Lookback call option with fix strike is given by the following statement.

If M < K,

$LC_t^K = S\Phi(a_1(S,K)) - Ke^{-r\tau}\Phi(a_2(S,K)) + \frac{S\sigma^2}{2r} ( \Phi(a_1(S,K)) - e^{-r\tau}(K/S)^{\frac{2r}{\sigma^{2}}}\Phi(a_3(S,K))),$

and if M > K,

$LC_t^K = (M-K)e^{-r\tau} + S\Phi(a_1(S,M)) - Me^{-r\tau}\Phi(a_2(S,M)) + \frac{S\sigma^2}{2r} ( \Phi(a_1(S,M)) - e^{-r\tau}(M/S)^{\frac{2r}{\sigma^{2}}}\Phi(a_3(S,M))).$

The fact that there is two different prices depending on whether M is greater than K or not, can be explained by the following. If M > K, then the event {Smax > K} is always true. This implies that the expected payoff function at time t is then SmaxK instead of max(SmaxK,0).

Similarly, the price at time t<T of the European Lookback put option with fix strike is given by the following statement.

If m > K,

$LP_t^K = -S\Phi(-a_1(S,K)) + Ke^{-r\tau}\Phi(-a_2(S,K)) - \frac{S\sigma^2}{2r} ( \Phi(-a_1(S,K)) - e^{-r\tau}(K/S)^{\frac{2r}{\sigma^{2}}}\Phi(-a_3(S,K))),$

and if m < K,

$LP_t^K = (K-m)e^{-r\tau} - S\Phi(-a_1(S,m)) + me^{-r\tau}\Phi(-a_2(S,m)) - \frac{S\sigma^2}{2r} ( \Phi(-a_1(S,m)) - e^{-r\tau}(m/S)^{\frac{2r}{\sigma^{2}}}\Phi(-a_3(S,m))).$

## References

1. Musiela, Marek; Marek Rutkowski (2004). Martingale methods in Financial Modelling. Springer. pp. 202–206.
2. Conze; Viswanathan (1991). "Path dependent options:the case of lookback options". J. Finance 46: 1893–1907.