Editing Lookback option (section)

==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''.<ref>
{{Cite book
  | last = Musiela
  | first = Mark
  | last2 = Rutkowski
  | first2 = Marek
  | title = Martingale Methods in Financial Modelling
  | publisher = Springer
  | date = November 25, 2004
  | isbn = 978-3-540-20966-9
}}</ref> Assume that there exists a continuously-compounded [[risk-free interest rate]] <math>r>0</math> and a constant stock's volatility <math>\sigma >0</math>. Assume that the time to maturity is <math>T>0</math>, and that we will price the option at time <math>t<T</math>, although the life of the option started at time zero. Define <math>\tau=T-t</math>. Finally, set that 
:<math> M = \max_{0\leq u \leq t} S_u, ~~m= \min_{0\leq u \leq t} S_u  \text{ and }S_t = S.</math>

Then, the price of the lookback call option with floating strike is given by:
:<math> 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))),</math>
where 
:<math> a_1(S,H) = \frac{\ln(S/H) + (r+\frac12\sigma^2)\tau}{\sigma\sqrt{\tau}}</math>
:<math> a_2(S,H) = \frac{\ln(S/H) + (r-\frac12\sigma^2)\tau}{\sigma\sqrt{\tau}} = a_1(S,H) - \sigma\sqrt{\tau}</math>
:<math> 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,</math>
and where <math>\Phi</math> is the [[standard normal]] [[cumulative distribution function]], <math>\Phi(a) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{a} e^{-\frac{x^2}{2}}\, dx</math>.

Similarly, the price of the lookback put option with floating strike is given by:
:<math> 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))).</math>