# A forward Monte Carlo method for American Options Pricing

I am trying to implement the forward Monte Carlo algorithm from the paper "A Forward Monte Carlo Method for American Options Pricing" by Daniel Wei-Chung Miao and Yung-Hsin Lee. I am a little bit confused by the following notation:

Under the Pseudo-Critical Prices section, the authors state:

First, consider an American call option. According to Barone-Adesi and Whaley (1987) (BAW), the optimal exercise boundary $S_c^{*}$ for the call option should solve the nonlinear equation at any time $t\in[0,T]$

\begin{equation} \label{eq:2} S_c^{*} = \frac{Q_2(C_e(S_c^{*}) + K)}{Q_2 - (1-C_e^{'}(S_c^{*}))} \end{equation}

where $C_e(S)$ is the European call option price calculated by the Black-Scholes (1973) formula, $K$ is the strike price, together with the notation $Q_2 = \frac{-(n-1)+\sqrt{(n-1)^2 + 4m/k}}{2} > 0$ in which $m = \frac{2r}{\sigma^2}$, $n = \frac{2(r-q)}{\sigma^2}$, and $k = 1 - e^{-r(T-t)}$. Note that this study used more streamlined notations such as $S_e^{*} = S_c^{*}(t)$, $C_e(S) = C_e(S,t)$ when some dependent parameters are not stressed.

Replacing the critical price $S_c^{*}$ on the right-hand side of the equation above with the current stock price $S$ yields a new but closely related function $f_c(\cdot)$. $$\hat{S_c} = f_c(S) = \frac{Q_2(C_e(S) + K)}{Q_2 - (1-C_e^{'}(S))}$$ where $\hat{S_c}$ represents the pseudo-critical price.

In the algorithm I need to compute $\hat{S_c}$ but what I am not understanding is it seems that $Q_2$ will just equal some number but then the formula for $\hat{S_c}$ has $Q_2$ acting as some function which does not make sense since it is just a number. Any suggestions or comments on the matter are greatly appreciated.

## 1 Answer

To simply answer this question the author is just multiplying the numbers.

• Thanks for contributing back by answering your question! – LocalVolatility Mar 23 '17 at 23:51