Theory - 2023.2 English

Vitis Libraries

Release Date
2023.2 English

Because the American option can be exercised anytime prior to the stock’s expire date, exercise price at each time step is required in the process of valuing the optimal exercise. More precisely, the pricing process is as follows:

  1. Generate independent stock paths
  2. Start at maturity \(T\) and calculate the exercise price.
  3. Calculate the exercise price for previous time steps \(T-1\), compare it with the last exercise price, select the larger one.
  4. Keep rolling back to previous time steps until \(t\) is \(0\) and obtain the max exercise price \(V_t\) as the optimal exercise price.

In the process above, for each time step \(t\), conditional expectation \(E_t[Y_t|S_t]\) of the payoff is computed according to Least-Squares Monte Carlo (LSMC) approach, proposed by Longstaff and Schwartz. Mathematically, these conditional expectations can be expressed as:

\[E_t[Y_t|S_t] = \sum_{i=0}^{n}a_iB_i(S_t)\]

where \(t\) is the time prior to maturity, \(B_i(S_t)\) is the basis function of the values of actual stock prices \(S_t\) at time \(t\). The constant coefficients of basis functions are written as \(a_i\) , and acts as weights to the basis functions. The discounted subsequent realized cash flows from continuation called \(Y_t\).

Here we employed polynomial function with weights as the basis functions, so the conditional expectations can be re-written as:

\[E_t[Y_t|S_t] = a + bS_t + cS_t^2\]

where 3 functions: 1, \(S_t\) and \(S_t^2\) are employed. The constant coefficients \(a\), \(b\) and \(c\) are the weights.

By adding immediate exercise value \(E_t(S_t)\) to the Equation above, and exchanging the side of elements, the equation is changed to

(3)\[a + bS_t + cS_t^2 + dE_t(S_t) = E_t[Y_t|S_t]\]

\(a\), \(b\), \(c\), \(d\) are constant coefficients. These coefficients need to be known while calculating the optimal exercise price in mcSimulation model. More details of American-style optimal algorithm used in our library refer to “da Silva, J. N., & Fernández, L. A Monte Carlo Approach to Price American-Bermudan-Style Derivatives.”


Theoretically, the basis functions \(B_i(S_t)\) can be any complex functions and the number of basis function may be any number. The American Option implemented in our library employs three polynomial functions, namely, 1, \(S_t\) and \(S_t^2\), which proved by Longstaff and Schwartz that works well, and is a typical setup in real implementations.