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.