conclusion is that the two parts of PAjoin smoothly without a kink or discontinuity. This is known as the “smoothness condition”.
S0*
S0 PA
X - S
0
f (S0 , T)
Figure 6.2 American put option price
(v) This section might have increased the reader’s understanding of the nature of the Black Scholes equation and its application to American options; but unfortunately there is no tidy, analytical solution of the equation available for American options, analogous to the Black Scholes model for European options.
Very general solutions are available using numerical techniques (the binomial method) and will be explored in detail in later chapters. These methods are so widely understood and accessible that they have really swept the board as the main tools for evaluating a wide range of American-style options. The one initial drawback was that they took a lot of computing power, but this resource has become cheaper, numerical methods have eclipsed a number of formerly useful and ingenious closed form solutions of the Black Scholes equation which were devised for special cases of American options. However, it is worth briefly looking at a couple of these methods, if only to illustrate the theory developed in the last few pages.
6.2 BARONE-ADESI AND WHALEY APPROXIMATION
(i) This method can be applied to continuous dividend puts and calls. We will restrict our analysis to put options where price divergence between European and American options is greater, but the analysis for calls is exactly analogous (Barone-Adesi and Whaley, 1987).
The price of an American put option can be writtenPA=PE+ϕ, wherePEis the price of the European option andϕ is a premium for the possibility of early exercise. This method seeks a way of calculating ϕ; the Black Scholes model is used to calculate the values ofPE.
PAis a solution to the Black Scholes equation in the regionS0∗<S0, i.e. above the exer- cise boundary. Therefore in this region,ϕ is also a solution of the Black Scholes equation.
Rearranging from the normal order a little gives
1
2S02σ2∂2ϕ
∂S02 +(r−q)S0
∂ϕ
∂S0 −rϕ− ∂ϕ
∂T =0 (6.3)
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6 American Options
(ii) Consider the evolution ofϕover time which is illustrated in Figure 6.3. The key properties of this graph are as follows:
S0 2* PA
S PA
PE
PE S0 1*
t1
t2 j = PA - PE
t1 < t2
Figure 6.3 American put options
(A) The quantityϕis defined only in the regionS0∗<S0. (B) S0∗is a function ofr, q, Tandσ; it decreases asTincreases.
(C) AsS0 → ∞we expectϕ→0 since it is unlikely that the stock price will reach theS0∗ where early exercise occurs.
(D) IfS0is small (but nevertheless aboveS0∗),ϕwill approach its asymptotic value (X−S0)− (Xe−r T −S0e−q T)≈X(1−e−r T) for small dividend yieldq.
(E) IfT →0 we must haveϕ→0 since the early exercise possibility ceases to have any meaning.
(iii) Define a new variablev=ϕ/(1−e−r T); differentiating with respect toTgives
−rϕ−∂ϕ
∂T = − rϕ
(1−e−r T)−(1−e−r T)∂v
∂T Substituting this back into equation (6.3) gives
1
2S02σ2∂2ϕ
∂S02 +(r−q)S0∂ϕ
∂S0
− rϕ
(1−e−r T) −(1−e−r T)∂v
∂T =0 (6.4)
Consider now the last term in this equation:
r
From (C) above, asS0→ ∞,ϕ →0 and thereforev→0; therefore the last term in equa- tion (6.4) goes to zero.r
From (D), whenS0→0, ϕ, approachesX(1−e−r T); thereforevapproaches a constant and the last term in equation (6.4) goes to zero.r
From (E), asT →0, the expression in brackets in the last term of equation (6.4) goes to zero.In each of these limits of the variablesS0andT, the last term of equation (6.4) may be set to
6.2 BARONE-ADESI AND WHALEY APPROXIMATION
zero. The Barone-Adesi and Whaley (BAW) method assumes that this last term in the equation mayalwaysbe set equal to zero.
(iv) The BAW equation can now be written as S02d2ϕ
dS02 +bS0dϕ dS0
+cϕ=0 where b=(r−q)
1
2σ2 ; c= −r
1
2σ2(1−e−r T) This is a standard differential equation known as the Cauchy linear differential equation. Its general solution is
ϕ=ASγ1+B Sγ2 whereAandBdepend on the boundary conditions and
γ1= 12{−(b−1)+
(b−1)2−4c}; γ2=12{−(b−1)−
(b−1)2−4c} (v)cis always negative, so thatγ1andγ2 must be real; furthermore,γ1 must be positive andγ2
must be negative. But ifγ1is positive, then the boundary condition limS0→∞ PA→0 means that we must haveA=0. We are then left with the following two-part solution:
PA=
X−S0 S0<S0∗
PE+ϕ=PE+B S0γ2 S0∗<S0
These two complementary solutions forPAmust be equal at the pointS0=S0∗. Furthermore, the smoothness condition of Section 6.1(iv) means that the slopes of the two functions must be the same at this point. This leads to the conditions
X−S0∗=PE∗+B S0∗γ2
∗E+Bγ2S0∗γ2−1= −1
or equivalently
X−S0∗=PE∗+ S0∗ γ2
{1+ ∗E} B= − 1+ ∗E
γ2S0∗γ2−1
wherePE∗and ∗Eare Black Scholes values calculated atS0=S∗0. The value ofS0∗ must be calculated numerically from the implicit equation which is the first on the right-hand side above. The easiest way to do this is to use the Black Scholes formulas forPE∗and ∗Eand use a “goal seek” function on a spread sheet. Finally, the formula for the price of an American put above the exercise boundary is
PA=PE−1+ ∗E γ2
S0∗ S0
S0∗ γ2
(vi) This model is quite ingenious, but the real test is how accurately it prices an option. Table 6.1 gives the prices of put options with X =100,r =10%,q =1%, σ =25% for a range of values ofS0 andT. BN is the value of an American put calculated with the binomial model using a large number of steps and may be taken as the “right answer”. BS is the Black Scholes price of a European put.
The results are fairly good overall, and are in line with the nature of the approximation made: whenTis either large or small, subsection (iii) shows that the last term in equation (6.4)
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6 American Options
Table 6.1 Comparison of models
American Put European Put
S0 BAW BN BS
T =1 month 90 10.00 10.00 9.52
95 5.58 5.61 5.42
100 2.56 2.56 2.51
105 0.94 0.94 0.92
110 0.27 0.27 0.26
S0∗=90.31 115 0.06 0.06 0.06
T =10 years 70 30.00 30.00 5.33
80 20.50 20.56 4.23
90 14.34 14.35 3.40
100 10.42 10.38 2.75
110 7.81 7.71 2.25
S0∗=75.40 120 6.00 5.87 1.86
T =3 years 70 30.00 30.00 15.49
80 20.17 20.21 11.30
90 13.35 13.33 8.18
100 9.08 8.96 5.88
110 6.29 6.10 4.22
130 3.14 2.91 2.18
150 1.63 1.45 1.13
S0∗=77.40 200 0.37 0.27 0.23
may be dropped. For intermediate values ofT(we have taken 3 years), this assumption is less justified and the results show that the errors are greater.