3.6.3 Analytic Approximations

Basic Prerequisites: 02 · Optimal-Stopping Theory and BSM · Pricing Formulas.

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1. Intuition & Practical Objective

American options have no exact closed form - but they have an exact closed form in one limit and good closed-form approximations everywhere else. This page is the analytic-approximation lookup: the base case (perpetual options, which do solve exactly), the two industry approximations (Barone–Adesi–Whaley, Bjerksund–Stensland), and the transformation that turns every American put into an American call.

The idea behind every approximation is the same as the structure of the problem. The American value is the European value plus a premium for the early-exercise right;

$$V^{\text{Am}}(S)=v^{\text{Eu}}(S)+\text{early-exercise premium}(S).$$

The premium is zero deep out-of-the-money, grows as $S$ moves toward the boundary, and is exactly $(S-X)-v^{\text{Eu}}$ at the boundary. Approximations differ only in how they guess the shape of the premium. Barone–Adesi–Whaley guesses a single power $A_2(S/S^{\ast }{)}^{q_2}$ and solves a one-dimensional root-find for the boundary $S^{\ast }$; Bjerksund–Stensland instead approximates the exercise boundary itself as flat (1993) or two-piece (2002), and integrates the resulting payoff analytically.

Why perpetual matters. With infinite maturity the value is time-homogeneous, the PDE becomes an ODE, and the problem solves exactly - boundary and all. It is both the only true closed form and the $\tau \to \infty$ limit that every finite-maturity boundary tends toward.

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2. Mathematical Ground Truth & Derivations

2.1 Perpetual American put - the canonical solved case (Shreve II §8.3; Björk Prop 21.30)

Infinite horizon, $dS=rSdt+\sigma SdW$ under $\mathbb{Q}$, payoff $g(x)=(K-x{)}^{+}$. The continuation-region ODE $\frac{1}{2}{\sigma }^2{x}^2{v}^{\prime \prime }+rx{v}^{\prime }-rv=0$ has independent solutions $x$ and $x^{-\gamma }$ with

$$\gamma =\frac{2r}{{\sigma }^2}(\text{Shreve Eq. 8.3.14; Bjo¨rk Eq. 21.71}).$$

Boundedness as $x\to \infty$ kills the $x$ solution, leaving $v(x)=B{x}^{-\gamma }$. Value matching $v(L^{\ast })=K-L^{\ast }$ and smooth pasting $v^{\prime }(L^{\ast })=-1$ give the boundary and the value:

$$L^{\ast }=\frac{\gamma K}{1+\gamma }=\frac{2rK}{2r+{\sigma }^2}(\text{Eq. 8.3.12}),v(x)=\{\begin{array}{ll}K-x,& 0\le x\le L^{\ast },\\ (K-L^{\ast }){\left(\frac{x}{L^{\ast }}\right)}^{-\gamma },& x\ge L^{\ast }.\end{array}\text{\hspace{1em}(8.3.13)}$$

Checking smooth pasting directly: the right derivative at $L^{\ast }$ is $-\gamma (K-L^{\ast })/L^{\ast }=-1$ - reproducing $L^{\ast }=2rK/(2r+{\sigma }^2)$ (Shreve Eq. 8.3.14). Note the second derivative jumps at $L^{\ast }$ ($0$ on the left, positive on the right); only $C^1$ smoothness is required.

2.2 Perpetual options with cost-of-carry $b$ (Haug §3.5)

$$c=\frac{({\gamma }_1-1{)}^{{\gamma }_1-1}}{{\gamma }_1^{{\gamma }_1}}{\left(\frac{S}{X}\right)}^{{\gamma }_1}X,{\gamma }_1=\left(\frac{1}{2}-\frac{b}{{\sigma }^2}\right)+\sqrt{{\left(\frac{b}{{\sigma }^2}-\frac{1}{2}\right)}^2+\frac{2r}{{\sigma }^2}}(\text{call, }b<r),$$ $$p=\frac{X}{1-{\gamma }_2}{\left(\frac{{\gamma }_2-1}{{\gamma }_2}\cdot \frac{S}{X}\right)}^{{\gamma }_2},{\gamma }_2=\left(\frac{1}{2}-\frac{b}{{\sigma }^2}\right)-\sqrt{{\left(\frac{b}{{\sigma }^2}-\frac{1}{2}\right)}^2+\frac{2r}{{\sigma }^2}}(\text{put}).$$

2.3 Barone–Adesi–Whaley (1987) (Haug §3.1)

Call ($b<r$; else $C=c_{BSM}$):

$$C(S)=\{\begin{array}{ll}{c}_{BSM}(S)+A_2{\left(\frac{S}{S^{\ast }}\right)}^{q_2},& S<S^{\ast },\\ S-X,& S\ge S^{\ast },\end{array}{A}_2=\frac{S^{\ast }}{q_2}[1-e^{(b-r)T}N(d_1(S^{\ast })\left)\right],$$ $$q_2=\frac{-(N-1)+\sqrt{(N-1{)}^2+4M/K}}{2},M=\frac{2r}{{\sigma }^2},N=\frac{2b}{{\sigma }^2},K=1-e^{-rT}.$$

The critical price $S^{\ast }$ solves $S^{\ast }-X=c_{BSM}(S^{\ast })+(1-e^{(b-r)T}N(d_1(S^{\ast })))S^{\ast }/q_2$ (Newton–Raphson, tolerance $|LHS-RHS|/X<10^{-5}$). The put is symmetric with $q_1=\frac{-(N-1)-\sqrt{(N-1{)}^2+4M/K}}{2}$, $A_1=-\frac{S^{\ast \ast }}{q_1}[1-e^{(b-r)T}N(-d_1(S^{\ast \ast }))]$ and boundary $S^{\ast \ast }<X$.

2.4 Bjerksund–Stensland 1993 (Haug §3.2)

Model the boundary as a flat trigger $I$ and integrate the payoff analytically:

$$C=\alpha S^{\beta }-\alpha \varphi (S,T,\beta ,I,I)+\varphi (S,T,1,I,I)-\varphi (S,T,1,X,I)-X\varphi (S,T,0,I,I)+X\varphi (S,T,0,X,I),$$ $$\alpha =(I-X)I^{-\beta },\beta =\left(\frac{1}{2}-\frac{b}{{\sigma }^2}\right)+\sqrt{{\left(\frac{b}{{\sigma }^2}-\frac{1}{2}\right)}^2+\frac{2r}{{\sigma }^2}},\varphi (S,T,\gamma ,H,I)=e^{\lambda }{S}^{\gamma }\left[N(d)-{\left(\frac{I}{S}\right)}^{\kappa }N\left(d-\frac{2\ln (I/S)}{\sigma \sqrt{T}}\right)\right],$$ $$\lambda =(-r+\gamma b+\frac{1}{2}\gamma (\gamma -1){\sigma }^2)T,d=-\frac{\ln (S/H)+(b+(\gamma -\frac{1}{2}){\sigma }^2)T}{\sigma \sqrt{T}},\kappa =\frac{2b}{{\sigma }^2}+2\gamma -1.$$

Trigger $I=B_0+(B_{\infty }-B_0)(1-e^{h(T)})$, $B_{\infty }=\frac{\beta }{\beta -1}X$, $B_0=\max (X,\frac{r}{r-b}X)$, $h(T)=-(bT+2\sigma \sqrt{T})\frac{B_0}{B_{\infty }-B_0}$; if $S\ge I$, exercise ⇒ $S-X$. The 2002 refinement uses a two-piece boundary (needs the bivariate normal $M(\cdot ,\cdot ,\rho )$) and is more accurate for long maturities.

2.5 The American put-call transformation (Haug §3.4)

$$P(S,X,T,r,b,\sigma )=C(X,S,T,r-b,-b,\sigma ).$$

Every American-put result - perpetual, BAW, BS-1993 - is reused for calls and vice versa, which is why Haug only tabulates one side.

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3. Computational Implementation - the approximations, benchmarked

Perpetual closed forms, BAW and BS-1993, each checked against its Haug cell and against a high-resolution CRR tree. Stdlib only.

(The BAW call values differ from Haug’s printed cells by $2\times 10^{-4}$ - his Newton tolerance, not the formula. BS-1993 reproduces Haug’s $5.2704$ exactly.) BAW’s error is tens of basis points against a 4000-step tree - good for desk use, too coarse for risk sensitivities near the boundary.

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4. Failure Modes & First-Principles Breakdowns

1. BAW is not uniformly accurate. Error grows with maturity - $+0.45\%$ at $T=0.5$ drifting to $+2\%$ at $T=3$ (see 05 · Failure Modes). It is a short-maturity tool; long-dated Americans should use BS-2002 or a tree.
2. The critical price must actually be found. BAW’s $S^{\ast }$ is a Newton root of a transcendental equation; bad seeds or loose tolerance move the answer by more than the approximation error. Use robust bracketing and a tight tolerance.
3. Wrong branch when $b\ge r$. For a call with $b\ge r$ (no dividend / futures/forward-type carry) BAW is the European formula - assembling the premium branch there double-counts value.
4. Applying the put formula where the transformation belongs. Haug tabulates calls; puts come via $P(S,X,T,r,b,\sigma )=C(X,S,T,r-b,-b,\sigma )$. Mixing signs ($r-b$ vs $b-r$) is the classic transcription error, and it silently shifts the boundary.
5. Perpetual formulas misused for long-but-finite maturity. The perpetual put ignores the approach to maturity, where the boundary collapses to $K$; it overstates the value of a $T=5$ option unless the extra early-exercise value is genuinely there.

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5. References

• Haug, The Complete Guide to Option Pricing Formulas
• Shreve, Stochastic Calculus for Finance II
• Björk, Arbitrage Theory in Continuous Time
• Barone-Adesi, G. & Whaley, R. (1987), Efficient analytic approximation of American option values, JF 42(2). Bjerksund, P. & Stensland, G. (1993, 2002), NHH working papers.

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6. Connected Graph Bridges

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• Forward: 04 · Free Boundary · 05 · Failure Modes
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