Lookback options.

The payoff from lookback options depend on the maximum or minimum of the underlying achieved through the period.

For this particular option an analytical solution has been found, due to Goldman et al. (1979), which is implemented below.

// file exotics_lookback_call.cc // author: Bernt A Oedegaard #include <cmath> #include "normdist.h" double exotics_lookback_european_call(double S, double Smin, double r, double q, double sigma, double time){ if (r==q) return 0; double sigma_sqr=sigma*sigma; double time_sqrt = sqrt(time); double a1 = (log(S/Smin) + (r-q+sigma_sqr/2.0)*time)/(sigma*time_sqrt); double a2 = a1-sigma*time_sqrt; double a3 = (log(S/Smin) + (-r+q+sigma_sqr/2.0)*time)/(sigma*time_sqrt); double Y1 = 2.0 * (r-q-sigma_sqr/2.0)*log(S/Smin)/sigma_sqr; return S * exp(-q*time)*N(a1) - S * exp(-q*time)*(sigma_sqr/(2.0*(r-q)))*N(-a1) - Smin * exp(-r*time)*(N(a2)-(sigma_sqr/(2*(r-q)))*exp(Y1)*N(-a3)); };

An often used technique to estimate prices of various exotics is simulation, see the chapter on general simulations.