Class defining an analytical approximation for American option prices as derived by Bjerksund and Stensland (2002).
The price of a call is given by: $$ \begin{align*} C = &\alpha_2 S^\beta - \alpha_2 \phi(S, t_1, \beta, I_2, I_2)\\ & + \phi(S, t_1, 1, I_2, I_2) - \phi(S, t_1, 1, I_1, I_2)\\ & - K\phi(S, t_1, 0, I_2, I_2) + K\phi(S, t_1, 0, I_1, I_2)\\ & + \alpha_1 \phi(S, t_1, \beta, I_1, I_2) - \alpha_1 \psi(S, T, \beta, I_1, I_2, I_2, t_1)\\ & + \psi(S, T, 1, I_1, I2, I_1, t_1) - \psi(S, T, 1, K, I_2, I_1, t_1)\\ & - K\psi(S, T, 0, I_1, I_2, I_1, t_1) + K\psi(S, T, 0, K, I_2, I_1, t_1) \end{align*} $$ where $$ \begin{align*} t_1 &= \frac{(\sqrt{5} - 1)T}{2}\\ I_1 &= B_0 + (B_\infty - B_0)(1 - e^{h_1})\\ I_2 &= B_0 + (B_\infty - B_0)(1 - e^{h_2})\\ B_0 &= \frac{\beta K}{\beta - 1}\\ B_\infty &= \max\left(K, \frac{rK}{r-b}\right)\\ h_1 &= \frac{(bt_1 + 2\sigma\sqrt{t_1})K^2}{B_0(B_0 - B_\infty)}\\ h_2 &= \frac{(bT + 2\sigma\sqrt{T})K^2}{B_0(B_0 - B_\infty)}\\ \alpha_1 &= (I_1 - K)I_1^{-\beta}\\ \alpha_2 &= (I_2 - K)I_2^{-\beta}\\ \beta &= \frac{1}{2} - \frac{b}{\sigma^2} + \sqrt{\left(\frac{b}{\sigma^2} - \frac{1}{2}\right)^2 + \frac{2r}{\sigma^2}} \end{align*} $$ The function $\phi(S, T, \gamma, H, I)$ is defined as $$ \begin{align*} \phi(S, T, \gamma, H, I) &= e^\lambda S^\gamma\left[N(-d_1) - \left(\frac{I}{S}\right)^\kappa N(-d_2)\right]\\ d_1 &= \frac{\ln(\frac{S}{H}) + (b + (\gamma - \frac{1}{2})\sigma^2)T}{\sigma\sqrt{T}}\\ d_2 &= \frac{\ln(\frac{I^2}{SH}) + (b + (\gamma - \frac{1}{2})\sigma^2)T}{\sigma\sqrt{T}}\\ \lambda &= -r + \gamma b + \frac{\gamma(\gamma - 1)\sigma^2}{2}\\ \kappa &= \frac{2b}{\sigma^2} + 2\gamma + 1 \end{align*} $$ and the function $\psi(S, T, \gamma, H, I_2, I_1, t_1)$ is defined as $$ \begin{align*} \psi(S, T, \gamma, H, I_2, I_1, t_1) = &e^{\lambda T} S^\gamma\left[M(d_1, e_1, \rho) -\left(\frac{I_2}{S}\right)^\kappa M(d_2, e_2, \rho) - \left(\frac{I_1}{S}\right)^\kappa M(d_3, e_3, -\rho) +\left(\frac{I_1}{I_2}\right)^\kappa M(d_4, e_4, \rho)\right] \end{align*} $$ where $$ \begin{align*} d_1 &= -\frac{\ln(\frac{S}{I_1}) + (b + (\gamma - \frac{1}{2})\sigma^2)t_1}{\sigma\sqrt{t_1}}\\ d_2 &= -\frac{\ln(\frac{I_2^2}{SI_1}) + (b + (\gamma - \frac{1}{2})\sigma^2)t_1}{\sigma\sqrt{t_1}}\\ d_3 &= -\frac{\ln(\frac{S}{I_1}) - (b + (\gamma - \frac{1}{2})\sigma^2)t_1}{\sigma\sqrt{t_1}}\\ d_4 &= -\frac{\ln(\frac{I_2^2}{SI_1}) - (b + (\gamma - \frac{1}{2})\sigma^2)t_1}{\sigma\sqrt{t_1}}\\ e_1 &= -\frac{\ln(\frac{S}{H}) + (b + (\gamma - \frac{1}{2})\sigma^2)T}{\sigma\sqrt{T}}\\ e_2 &= -\frac{\ln(\frac{I_1^2}{SH}) + (b + (\gamma - \frac{1}{2})\sigma^2)T}{\sigma\sqrt{T}}\\ e_3 &= -\frac{\ln(\frac{I_2^2}{SH}) + (b + (\gamma - \frac{1}{2})\sigma^2)T}{\sigma\sqrt{T}}\\ e_4 &= -\frac{\ln(\frac{SI_1^2}{HI_2^2}) + (b + (\gamma - \frac{1}{2})\sigma^2)T}{\sigma\sqrt{T}} \end{align*} $$ and $\rho = \sqrt{\frac{t_1}{T}}$ and $M(\cdot, \cdot, \cdot)$ is the CDF of the bivariate normal distribution (see {@link com.opengamma.analytics.math.statistics.distribution.BivariateNormalDistribution}). The price of puts is calculated using the Bjerksund-Stensland put-call transformation $p(S, K, T, r, b, \sigma) = c(K, S, T, r - b, -b, \sigma)$.