Analytic Approximation of American Options, G. Barone-Adesi and R. E. Whaley (1987)

On this page, we discuss an analytic approximation of American options. If both, the European and the American options, follow the Black-Scholes PDE, it is then that the early exercise premium of the option as well. For an American Put option, the early exercise premium is defined as $$ \epsilon (S,t)=P(S,t)-p(S,t), $$ where \(P\) is the American option value and \(p\) is the European option value. The partial differential equation of the early exercise premium is therefore $$ - r \epsilon + \frac{\partial \epsilon}{\partial t} + r S \frac{\partial \epsilon}{\partial t} + \frac{1}{2} S_t^2 \sigma^2 \frac{\partial^2 \epsilon}{\partial S^2}= 0. $$ The early exercise premium is written as \(\epsilon(S,K)=K(t)~f(S,K)\), with \(K(t)=1-e^{-rt}\) and under the assumption that$$ (1-K)\frac{2r}{\sigma^2} \frac{\partial f}{\partial K} \approx0. $$ For options with very short (long) expiry time, this assumption is reasonable, since as \(t\) approaches 0 \((\infty)\), \(\frac{\partial f}{\partial K}\) approaches 0 ( \(K\) approaches 1), and the term, \((1-K)\frac{2r}{\sigma^2} \frac{\partial f}{\partial K}\), can be ignored. From now on, we denote the quantity \(M:=\frac{2r}{\sigma^2}\) to get the following approximation to the PDE of the early exercise premium. $$ S^2 \frac{\partial^2 f}{\partial S^2} + M ~S \frac{\partial f}{\partial S} - M~ f=0. $$ The previous equation is a second order ordinary differential equation with two linearly independent solutions of the form \(aS^q\). The general solution for \(f\) is therefore $$ f(S)=aS^{q_a}+bS^{q_b}, $$ with$$ q_a= \displaystyle{\frac{\left(-(M-1)-\sqrt{(M-1)^2+4M/K} \right)}{2}}; q_b= \displaystyle{\frac{\left(-(M-1)+\sqrt{(M-1)^2+4M/K} \right)}{2}}. $$ Since \(q_a\)<0 and \(q_b>0\), \(b\) is set to zero, because if \(b\) is a non-zero quantity, then the solution violates the boundary condition of \( \lim_{S \to \infty} P(S,t)=0\) . The solution is therefore simplified to $$ P(S,t)=p(S,t)+a~K(t)~S^{q_a}, $$ where the value of \(a\) depends on the value of \(S^{\ast}\) and it is not of our interest. The price of an American option is a boundary problem. Below the critical value \( S^{\ast}\), the American Put option is equal to its exercisable value of \(E-S\). While above \( S^{\ast}\), the value of the American option \(P(S,t)\) satisfies the above solution.

To find the critical price \( S^{\ast}\), we need to equate the exercisable value of \(E-S\) to the value of \(P(S,t)\), when \(S=S^{\ast}\). That is
$$ E-S^{\ast}=P(S^{\ast},t)=p(S^{\ast},t)+a~K(t)~S^{\ast^{q_a}}, $$ and the slope of the exercisable value of the put option, -1, is set equal to the slope of \(P(S^{\ast},t)\), that is $$ -1=-N \left[-d_1(S^{\ast}) \right]+a~K(t)~q_a~S^{q_a-1}, $$ where \(-N \left[-d_1(S^{\ast}) \right]\) is the partial derivative \( \frac{\partial p}{\partial S^{\ast}}\), and where \(d_1(S)=\left(log(S/E)+(r+\frac{1}{2} \sigma^2)T \right)/(\sigma~\sqrt(T)) \).

We then solve two equations with two unknowns, \(a\) and \( S^{\ast} \), that shoud be solved numerically using an iterative method. The American put option value is then simplified to \begin{align*} \begin{array}{lll} P(S,t)&=p(S,t)+A(S/S^{\ast})^{q_a}, ~~~ &\text{when}~~ S > S^{\ast}\\ P(S,t)&=E-S, ~~~ & \text{when} ~~S \leq S^{\ast}, \end{array} \end{align*} where \(A=-\left( S^{\ast}/q_a \right) \times \left(1-N \left[-d1(S^{\ast}) \right] \right) \).

A numerical algorithm for finding \( S^{\ast} \)
Initialize the values \(LS=0\), \(RS=2E\) and \( S^{\ast}=E/4\).

• Compute the following, while \( |\frac{LS-RS}{E}| < 10^{-5} \),
  1. \( d_1:=d_1(S^{\ast}) \).
     
  2. \(LS:= E-S^{\ast} \).
     
  3. \(RS:=p(S^{\ast},t)- \left( 1-N[-d_1] \right) S^{\ast}/q_a \).
     
  4. \(b:= N[-d_1] \left(\frac{1}{q_a}-1 \right) - \frac{1}{q_a}\left(1+\phi[-d_1]/(r \sqrt{t}) \right) \), where \(\phi \) is the probability density function of a standard normal distribution, and \(b\) is derived as the slope of \(RS\) at \(S^{\ast}\).
     
  5. \(S_{new}^{\ast}= \left( E-RS+b S^{\ast} \right)/(1+b) \).
     
  6. \(S^{\ast}=S_{new}^{\ast}\).

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References

1. BARONE-ADESI, GIOVANNI, and Robert E. Whaley. Efficient analytic approximation of American option values. The Journal of Finance 42, no. 2 (1987): 301-320.