Pricing American Options - an Important Fundamental Research in ...

Faculty of Informatics, University of Wollongong 

Pricing American Options - an 

Important Fundamental Research in 

Pricing Financial Derivatives 

Song-Ping Zhu 

School of Mathematics and Applied Statistics 

IWIF-II www.swingtum.com/institute/IWIF PPT 1/68 

Song-Ping Zhu, SMAS, April 26, 2007

Outline 

Outline of the talk 

A Review of Pricing American Options 

An exact and explicit solution for American put 

options 

Examples and Discussions 

Concluding Remarks 


Song-Ping Zhu, SMAS, April 26, 2007 0-1

Background 

What is an option? 

What is a European-style option? 

What is an American-style option? 

What is the optimal exercise price of an American 

option? 

What is financial interpretation of the optimal 

exercise price of an American option? 



Review of Various solution Approaches 

Analytical Approximation Methods 

Numerical Solutions 

Analytical Exact Solutions 




Numerical Approaches: 

Directly solving PDE: 

– Brennan and Schwartz (1977), Wu and Kwok (1997): Finite 

Difference Method (FDM); 

– Allegretto et al. (2001): Finite Element Method (FEM); 

– Forsyth and Vetzal (2002): Finite Volume Method (FVM); 

– Hon and Mao (1997): Radial Basis Function Method (RBF); 

– Benth et al. (2004): Semilinear Approach; 

– Rodolfo (2007): Cubic Splines Approach; 

– Zhu and Zhang (2007b): Predictor-Corrector Scheme. 



A Numerical Example Used in Zhu and Zhang 

(2007b) 

Strike price X = $100, 

Risk-free interest rate r = 0.1, 

Volatility σ = 0.3, 

Time to expiry T = 1 (year). 



A Figure in Zhu and Zhang (2007b) 

100 

95 

O 

Zhu and Zhang’s numerical solution 

Zhu(2006b)’s analytical solution 

Optimal Exercise Price ($) 

90 

85 

80 

75 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 

Time to Expiry (Year) 

A comparison of optimal exercise prices with two different 

approachesIWIF-II www.swingtum.com/institute/IWIF PPT 7/68 


A Figure in Zhu and Zhang (2007b) 

0.8 

N=200 

0.7 

0.6 

M=100 

M=80 

M=60 

CPU Time (second) 

0.5 

0.4 

0.3 

0.2 

0.1 

N=10 

0 

0 1 2 3 4 5 6 7 8 9 

RMSRE (%) 

Accuracy vs. Efficiency 





Risk Neutral Approaches 

– Cox et al. (1979): The Binomial Method 

– Grant et al. (1996): The Monte Carlo simulation method 

– Longstaff and Schwartz (2001): the Least Squares Method 

(Monte Carlo simulation) 





Risk Neutral Approaches 

Under the risk neutral argument, the current value of any 

financial derivative is equal to the expected return at the 

expiry discounted to the present time. 




Zhu and Francis (2004) found an improved algorithm to 

calculate the critical exercise price. 

Table 1. Convergence of the Binomial Method 

Method 

Optimal Exercise Price at Expiry 

Binomial n=10 78.93 



Zhu (2006b) 76.11 

Wu and Kwok (1997) n=100 76.25 




Integral Equation Approach 

Kim (1990) used the integral equation approach to show that 

there is an early-exercise premium associated with the 

American options; 

Jacka (1991) established the equivalence between an optimal 

stopping problem suggested by Karatzas (1988) and the 

integral equation approach; 

Huang et al. (1996) presented a numerical solution to solve 

the integral equation; 

Ju (1998) proposed an approximate method to find the 

solution of the integral equation for the optimal exercise 

boundary; 




Integral Equation Approach 

Jamshidian (1992) also used the integral equation approach to 

derive explicit formulas for American options on coupon bonds 

for the Vasicek (1977) model and the CIR square root model 

(Cox et al. (1985)); 

Zhu and Zhang (2007a) used the integral approach to solve 

the valuation problem of a convertible bond with American 

style conversion. 



A Figure in Zhu and Zhang (2007a) 

Convertible bond price 

200 

180 

160 

140 

120 

100 

80 

60 

40 

20 

new decompostion 

traditional decomposition 

0 

0 50 100 150 200 

Stock price 

A comparison of two decomposition approaches 




Analytical Approximations: 

Geske and Johnson (1984):the compound-option 

approximation method; 

MacMillan (1986) and Barone-Adesi and Whaley (1987): the 

quadratic approximation method; 

Johnson (1983): the interpolation method; 

Carr (1998): the randomization approach; 




Analytical Approximations (cont.): 

Bunch and Johnson (2000): Algebraic Equation Method; 

Zhu (2006a): Laplace Transform based on the 

pseudo-steady-state approximation. 



A. Basso, M. Nardon and P. Pianca’s 2002 working paper 





Zhu and He (2007) identified the missing term in Bunch and 

Johnson’s simple formula and proposed a revised formula: 

where 

g(τ) = ± 

√ 2 ln 

S f (τ) = e −(γ+1)τ−√ 2τg(τ) , (1) 

√ αSf (τ) 

. (2) 

γe −γατ ln 1 1 S f (τ) e− 2 [ ln S f (τ)−ln S f 

√ (τ(1−α))+(γ−1)ατ ] 2 2ατ 



1 

γ=5 

0.9 

0.8 

0.7 

B(τ) 

0.6 

0.5 

0.4 

Carr’s solution 

BJ’s solution 

Zhu & He (2007) 

0.3 

0.2 

0.1 

0 

0 0.02 0.04 0.06 0.08 0.1 

τ 

The comparison of B(τ) from Carr’s solution, Bunch and 

Johnson’s estimation and Zhu and He’s estimation when γ=5. 





Zhu (2006a) also derived a very elegant analytical 

approximation formula for the optimal exercise price of 

American options, Using the Laplace transform approach. 

S f (τ) = 

γ 

1 + γ + e−a2 τ 

∫ ∞ 

e −τρ 

π 0 a 2 + ρ e−f 1(ρ) sin [f 2 (ρ)] dρ, (3) 





where 

[ (√ 

f 1 (ρ) = 1 

a 

b 2 b ln 

2 + ρ 

+ ρ γ 

) 

[ (√ 

f 2 (ρ) = 1 √ a 2 + ρ 

ρ ln 

b 2 + ρ 

γ 

+ √ √ ] 

ρ 

ρ tan −1 ( 

a ) , (4) 

) 

√ ] 

ρ 

− b tan −1 ( 

a ) . (5) 



100 

95 

Analytical Solution 

Numerical Solution of Wu and Kwok (1997) 

Perpetual Optimal Exercise Price 

90 


85 

80 

75 

70 

65 

Perpetual Optimal Exercise Price=$68.97 

60 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 

Time To Expiration (Years) 

Optimal exercise prices for the case in Example 1 




More recently, Zhu (2007) improved the accuracy in Zhu 

(2006a)’s original approximation formula, by changing the 

explicit integral for S f (τ) into an algebraic equation based on 

the proved conjecture Θ(S f (τ), τ) = 0 and the approximating 

formula presented in Zhu (2006a). 




100 

95 

90 

Current Approximation 

Approximation of Zhu (2006a) 



85 

80 

75 

70 

65 

60 

0 2 4 6 8 10 12 

Time To Expiration (Months) 

A comparison of the Critical Exercise Price 




Zhu and Zhang (2006c) extended this approach to the cases 

where there is a continuous dividend payment to the 

underlying asset. 

1 

0.95 

Analytical−Approximation 

Stephest Inversion N=6,8,10 

0.9 

0.85 

0.8 

0.75 

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 


The Critical Asset Prices by Stehfest Method with D 0 = 0 



Main Difficulties in Solving American put option 

problems 

The problem is a moving boundary problem (McKean Jr., 

1965 and Merton, 1973) with the optimal exercise price as the 

moving boundary. This has to be found as part of the solution. 

Singular behavior of the optimal exercise price near expiry. – 

Barles et al. (1995) and Kuske and Keller (1998) showed that 

near expiry, S f (τ) behaves like S f (τ) = 1 − kσ √ τ|lnτ| 



An Exact Solution in Series Expansion Form 

Some Definitions 

“Exact” 

– No approximation is made whatsoever; the differential 

equations, the boundary and initial conditions of the problem 

can all be satisfied to any desired accuracy and the solution is 

infinitely differentiable. 

“Explicit” 

– The solution for the unknown function (or functions) can be 

determined explicitly in terms of all the inputs to the problem. 




Some Definitions (Cont.) 

“closed-form” 

– Gukhal’s definition (2001): “closed-form” solution is the 

one that can be written in terms of a set of standard and 

generally accepted mathematical functions and operations; 






– “An equation is said to be a closed-form solution if it solves 

a given problem in terms of functions and mathematical 

operations from a given generally accepted set. For example, 

an infinite sum would generally not be considered closed-form. 

However, the choice of what to call closed-form and what 

not is rather arbitrary since a new ”closed-form” 

function could simply be defined in terms of the infinite 

sum.”; (cf. mathworld.wolfram.com) 






– On the Wikipedia, Zhu’s solution has been declared as a 

semi-closed-form solution. 

(http://en.wikipedia.org/wiki/American option) 




The PDE system for pricing American put options 

determining the optimal exercise boundary 

⎧ 

⎪⎨ 

⎪⎩ 

∂V 

∂t + 1 2 σ2 S 2 ∂2 V ∂V 

+ rS 

∂S2 ∂S 

V (S f (t), t) = X − S f (t), 

∂V 

∂S (S f (t), t) = −1, 

lim V (S, t) = 0, 

S→∞ 

V (S, T ) = max{X − S, 0}. 

− rV = 0, 




In the above PDE system, 

– X is the strike price of the option; 

– r is the risk-free interest rate; 

– σ is the volatility of the underlying asset price. 

In the current solution, r and σ are assumed to be 

constant. 




The normalized PDE system becomes 

⎧ 

⎪⎨ 

⎪⎩ 

− ∂V 

∂τ + S2 ∂2 V ∂V 

+ γS − γV = 0, 

∂S2 ∂S 

V (S f (τ), τ) = 1 − S f (τ), 

∂V 

∂S (S f (τ), τ) = −1, 

lim V (S, τ) = 0, 

S→∞ 

V (S, 0) = max{1 − S, 0}, 

in which γ ≡ 2r can be viewed as an interest rate relative to the 

σ2 volatility of the asset price. 




We begin with introducing the Landau transform 

x = ln 

S 

S f (τ) . 

To make the current approach more general for 

other derivatives of American-style exercise, a new 

expansion is currently being worked, in which the 

Landau transform is no longer needed. 




The PDE system under the Landau transform becomes 

⎧ 

⎪⎨ 

∗ 

∂V 

∂τ − ∂2 V 

− (γ − 1)∂V 

∂x2 ∂x + γV = 1 

S f (τ) 

V (x, 0) = 0, 

V (0, τ) = 1 − S f (τ), 

dS f 

dτ 

∂V 

∂x , 

⎪⎩ 

∂V 

∂x (0, τ) = −S f (τ), 

lim V (x, τ) = 0. 

x→∞ 




The homotopy-analysis method 

Ortega and Rheinboldt (1970): Solving nonlinear algebraic 

equations; 

Liao (1997) and Liao and Zhu (1999): Solving heat transfer 

problems; 

Liao and Zhu (1996) and Liao and Campo (2002): Solving 

fluid flow problems. 

The essential concept of the method is to construct a 

continuous “homotopic deformation” through a series 

expansion of the unknown function. 




Let’s now construct two new unknown functions ¯V (x, τ, p) and 

¯S f (τ, p) that satisfy the following differential system, 

⎧ 

⎪⎨ 

⎪⎩ 

(1 − p)L[ ¯V (x, τ, p) − ¯V 0 (x, τ)] = −p{A[ ¯V (x, t, p), ¯S f (τ, p)]}, 

¯V (x, 0, p) = (1 − p) ¯V 0 (x, 0), 

¯V (0, τ, p) + ¯S f (τ, p) = 1, 

∂ ¯V 

[ 

∂x (0, τ, p) + ¯S f (τ, p) = (1 − p) 1 + ∂ ¯V ] 

0 

∂x (0, τ) − ¯V 0 (0, τ) , 

lim ¯V (x, τ, p) = 0, 

x→∞ 




where L is a differential operator defined as 

L = ∂ 

∂τ − ∂2 

∂ 

− (γ − 1) 

∂x2 ∂x + γ, 

and A is a functional defined as 

A[ ¯V (x, τ, p), ¯S f (τ, p)] = L( ¯V ) − 

1 ∂ ¯S f ¯V 

(τ, p)∂ (x, τ, p). 

¯S f (τ, p) ∂τ ∂x 




With p = 0, we have, from the differential system * 

⎧ 

L[ ¯V (x, τ, 0)] = L[ ¯V 0 (x, τ)], 

¯V (x, 0, 0) = ¯V 0 (x, 0), 

⎪⎨ 

¯V (0, τ, 0) + ¯S f (τ, 0) = 1, 

⎪⎩ 

∂ ¯V 

∂x (0, τ, 0) + ¯S f (τ, 0) = 1 + ∂ ¯V 0 

∂x (0, τ) − ¯V 0 (0, τ), 

lim ¯V (x, τ, 0) = 0. 

x→∞ 




Clearly, the solution of this differential system is 

⎧ 

⎨ ¯V (x, τ, 0) = ¯V 0 (x, τ), 

⎩ 

¯S f (τ, 0) = 1 − ¯V 0 (0, τ) = ¯S 0 (τ), 

so long as the initial guess ¯V 0 (x, τ) satisfies the condition 

lim ¯V 0 (x, τ) = 0. 

x→∞ 




When p = 1, the differential system * becomes 

⎧ 

L[ ¯V (x, τ, 1)] = 

1 ∂ ¯S f ¯V 

(τ, 1)∂ (x, τ, 1), 

¯S f (τ, 1) ∂τ ∂x 

⎪⎨ 

¯V (x, 0, 1) = 0, 

¯V (0, τ, 1) = 1 − ¯S f (τ, 1), 

⎪⎩ 

∂ ¯V 

∂x (0, τ, 1) = − ¯S f (τ, 1), 

lim ¯V (x, τ, 1) = 0. 

x→∞ 




Comparing this PDE and the original one we were trying to solve, 

it is obvious that the solution we seek is nothing but 

⎧ 

⎨ V (x, τ) = ¯V (x, τ, 1), 

⎩ 

S f (τ) = ¯S f (τ, 1). 




To find the values of ¯V (x, τ, 1) and ¯S f (τ, 1), we can now expand 

the functions ¯V (x, τ, p) and ¯S f (τ, p) as a Taylor’s series expansion 

of p 

¯V (x, τ, p) = 

¯S f (τ, p) = 

∞∑ 

m=0 

∞∑ 

m=0 

¯V m (x, τ) 

p m , 

m! 

¯S m (τ) 

p m , 

m! 




where ¯V m is the mth-order partial derivative of ¯V (x, τ, p) with 

respect to p and then evaluated at p = 0, 

¯V m (x, τ) = ∂m 

∂p ¯V m (x, τ, p) 

∣ , 

p=0 

and ¯S m is the mth-order partial derivative of ¯S(τ, p) with respect 

to p and then evaluated at p = 0, 

¯S m (τ) = ∂m 

∂p ¯S m f (τ, p) 

∣ . 

p=0 




To find all the coefficients in the above Taylor’s expansions, we 

need to derive a set of governing partial differential equations and 

appropriate boundary and initial conditions for the unknown 

functions ¯V m (x, τ) and ¯S m (τ). They can be derived from 

differentiating each equation in the differential system * with 

respect to p and then setting p equal to zero. 




The 1st order differential system: 

⎧ 

L[ ¯V 1 (x, τ)] = −L[ ¯V 0 (x, τ)] + A ′ (x, τ, 0), 

¯V 1 (x, 0) = − ¯V 0 (x, 0), 

⎪⎨ 

¯V 1 (0, τ) + ¯S 1 (τ) = 0, 

⎪⎩ 

∂ ¯V 1 

∂x (0, τ) + ¯S 1 (τ) = ¯V 0 (0, τ) − ∂ ¯V 0 

(0, τ) − 1, 

∂x 

lim ¯V 1 (x, τ) = 0. 

x→∞ 




The nth order differential system: 

⎧ 

L[ ¯V n (x, τ)] = n ∂n−1 A ′ 

∂ n−1 , 

p ∣ 

p=0 

⎪⎨ 

¯V n (x, 0) = 0, 

¯V n (0, τ) + ¯S n (τ) = 0, if n ≥ 2, 

where 

⎪⎩ 

∂ ¯V n 

∂x (0, τ) + ¯S n (τ) = 0, 

lim ¯V n (x, τ) = 0, 

x→∞ 

A ′ (x, τ, p) = 

1 ∂ ¯S f ¯V 

(τ, p)∂ (x, τ, p). 

¯S f (τ, p) ∂τ ∂x 




After eliminating ¯S n (τ) from the two boundary conditions at x = 0 

in above two equations, we can rewrite them in a general form 

⎧ 

L[ ¯V n (x, τ)] = f n (x, τ), 

⎪⎨ 

¯V n (x, 0) = ψ n (x), 

⎪⎩ 

∂ ¯V n 

∂x (0, τ) − ¯V n (0, τ) = φ n (τ), 

¯V n (∞, τ) = 0, 

with f n (x, τ), ψ n (x) and φ n (τ) being expressed respectively as 




⎧ 

⎪⎨ 

f n (x, τ) = 

⎪⎩ 

ψ n (x) = 

−L[ ¯V 0 (x, τ)] + A ′ (x, τ, 0), if n = 1, 

n ∂n−1 A ′ 

∂p n−1 ∣ ∣∣∣∣p=0 

, if n ≥ 2, 

{ − ¯V0 (x, 0), if n = 1, 

0, if n ≥ 2, 

⎧ 

⎨ 

φ n (τ) = 

¯V 0 (0, τ) − ∂ ¯V 0 

(0, τ) − 1, if n = 1, 

⎩ 

∂x 

0, if n ≥ 2. 



The Solution for the Option Price 

¯V n (x, τ) = e − 1 2 (γ−1)x− 1 4 (γ+1)2 τ 

{ 

∫ ∞ [ 

·e (γ−1)√ τξ−ξ 2 dξ + 

x 

2 √ τ 

+e − 1 2 (γ−1)x ψ n (2 √ ] 

τ ξ − x) 

{ ∫ 

√1 

π 

e 1 2 (γ−1)x 

(γ−1)x+ 

(γ+1)2 

4 τ 

x 

2 √ τ 

− 

x 

2 √ τ 

e 1 2 (γ−1)x ψ n (2 √ τ ξ + x) 

} 

e (γ−1)√ τξ−ξ 2 dξ 

−(γ + 1) √ τ e − 1 2 

∫ ∞ 

· ψ n (2 √ τξ − x)e 2γ√ τ ξ · erfc(ξ + 

x 

2 √ τ 

− 2 √ π 

e (γ+1)2 

4 τ 

[ 

·e − (γ+1)(x+η) 

4ξ 

∫ ∞ 

0 

∫ ∞ 

e − (γ+1) η 2 

] 2−ξ 

2 

dξdη 

x+η 

2 √ τ 


ψ n (2 √ τ ξ + x) 

(γ + 1) √ τ)dξ 

2 

( 

) 

(x + η)2 

φ n τ − 

4ξ 2 


The Solution for the Option Price 

∫ τ 

+ 

0 

⎧ 

⎨ 

⎩ 

[ 

e (γ+1)2 η ∫ 

4 

√ e 1 2 (γ−1)x 

π 

e (γ−1)√ τ−η ξ−ξ 2 dξ 

∫ ∞ 

x 

2 √ τ−η 

x 

2 √ τ−η 

− 

x 

2 √ τ−η 

[ 

e 1 2 (γ−1)x f n (2 √ τ − η ξ + x, η) 

f n (2 √ τ − η ξ + x, η) 

+ 

] 

+e − 1 2 (γ−1)x f n (2 √ ] 

τ − η ξ − x, η) e (γ−1)√ τ−η ξ−ξ 2 dξ 

−(γ + 1) √ τ − ηe − 1 (γ+1)2 

(γ−1)x+ τ 

2 4 

· e 2γ√ τ−η ξ erfc(ξ + (γ+1) 

2 

∫ ∞ 

x 

2 √ τ−η 

√ τ − η)dξ 

} 

dη 

f n (2 √ τ − η ξ − x, η) 

} 

, 




Initial Guess 

Choosing the corresponding European option value as the initial 

guess yields the following three apparent merits: 

The boundary condition at x = ∞ is automatically 

satisfied 

f 1 (x, τ) is further simplified because the first term 

on the righthand side vanishes 

ψ 1 (x) vanishes as well, so the integral involving ψ n 

is entirely eliminated 




Optimal Exercise Price 

S f (τ) = 2 √ π 

e − 1 2 (γ−1)x− 1 4 (γ+1)2 τ 

∫ ∞ 

η 

2∫ √ τ τ 

+ 

− 

0 

( ) 

φ n τ − η2 

4ξ 

[e 2 ∫ ∞ 

(γ+1)2 η 4 

∞∑ 1 

{e (γ+1)2 τ 4 

n! 

] 2−ξ 

2 

dξdη 

[ n=0 

e − (γ+1)η 

4ξ 

√ π 

2 (γ + 1)√ τ − ηe (γ+1)2 

4 τ 

· e 2γ√ τ−η ξ erfc(ξ + (γ+1) 

2 

0 

∫ ∞ 

0 

e − (γ+1) 

2 η 

f n (2 √ τ − η ξ, η)e (γ−1)√ τ−η ξ−ξ 2 dξ 

∫ ∞ 

0 

f n (2 √ τ − η ξ, η) 

} 

√ τ − η)dξ 

] 

dη 




The convergence criterion is 

Convergence of the series 

lim ( m 

m→∞ m + 1 )| ¯V m+1 

|p < 1, 

¯V m 

for p ∈ [0, 1], according to the d’Alembert’s ratio test. 

This is thus equivalent to 

lim | ¯V m+1 

| < 1. 

m→∞ ¯V m 

Numerical evidence will be provided to support the satisfaction of 

this convergence criterion. 



Examples and Discussion 

Verification Through Examples: 




Example 1: 

Use the same example used in Wu and Kwok (1997) and Carr and 

Faguet (1994): 




Time to expiry T = 1 (year). 

In terms of the dimensionless variables, the two parameters 

involved are γ = 2.2222 and τ exp = 0.045. 




100 

95 

90 




85 

80 

75 

70 

65 

60 

0 2 4 6 8 10 12 

Time To Expiration (Months) 

Optimal exercise prices for the case in Example 1 




1 x 10−5 

0 

−1 

Dimensionless V n 

value 

| 

−2 

−3 

−4 

n=24 

n=25 

n=26 

n=27 

n=28 

n=29 

−5 

−6 

−7 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 

Logarithm of the dimensionless stock price 

Dimensionless option prices with six different total summation 

numbers inIWIF-II Example 1www.swingtum.com/institute/IWIF PPT 58/68 



100 

90 

80 

70 

τ=1.000 Year 

τ=0.738 Years 



Option Price ($) 

60 

50 

40 

30 

20 

10 

0 

0 20 40 60 80 100 120 140 160 180 200 

Stock Price ($) 

Option IWIF-II prices at www.swingtum.com/institute/IWIF different times to expiration inPPT Example 59/68 1 



Example 2: 

This is the same example used in Bunch and Johnson (2000) (For 

this example, Bunch and Johnson have compared theirs results 

with those generated by the binomial method) 




Time to expiration T = 1 (year), 


involved are 

Relative risk-free interest rate γ = 1.084, 

Dimensionless time to expiration τ exp = 0.045. 




40 

38 

The number of terms summed up in the series expansion = 29 

The number of terms summed up in the series expansion = 30 

36 


34 

32 

30 

28 

26 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 


Convergence IWIF-II of www.swingtum.com/institute/IWIF the optimal exercise price Example PPT 61/68 2 



40 

38 

Eq. (23) in Bunch and Johnson (2000) 

CRR(800) in Cox. et. al. (1979) 


36 


34 

32 

30 

28 

26 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 


Optimal IWIF-IIexercise www.swingtum.com/institute/IWIF prices for the case ExamplePPT 2 62/68 



40 

35 

30 


25 

20 

15 

τ=1.000 Year 




10 

5 

0 

0 10 20 30 40 50 60 70 80 90 100 


Option IWIF-II prices at www.swingtum.com/institute/IWIF different times to expiration inPPT Example 63/68 2 



1 

0.9 

0.8 

Dimensionless Option Price 

0.7 

0.6 

0.5 

0.4 

0.3 

γ=2.2222 

γ=1.084 

0.2 

0.1 

0 

0 0.5 1 1.5 2 2.5 

Dimensionless Stock Price 

A comparison IWIF-II www.swingtum.com/institute/IWIF of dimensionless option price withPPT different 64/68 γ values 



Example 3: 

This is the same example used in Bunch and Johnson (2000), 

except with a very long life time. 




Time to expiration T = 40 (years), 


involved are 

Relative risk-free interest rate γ = 0.5, 

Dimensionless time to expiration τ exp = 0.4. 




100 

90 

Optimal Exercise Price Calculated From the Analytical Solution 

Perpetual Optimal Exercise Price 

80 


70 

60 

50 

40 

Perpetual Optimal Exercise Price=$33.33 

30 

0 5 10 15 20 25 30 35 40 


Optimal IWIF-IIexercise www.swingtum.com/institute/IWIF price for the case Example PPT 3 66/68 



100 

90 

80 

70 






60 

50 

40 

30 

20 

10 

0 

0 50 100 150 


Option IWIF-II prices at www.swingtum.com/institute/IWIF different time to expiration in PPT Example 67/68 3 


Conclusions 

Conclusions 

A review is given to the research area of pricing 

American options; 

Pros and Cons of each approach are summarized; 

With the homotopy-analysis method, an exact and 

explicit solution (in a Taylor series expansion form) 

of the well-known Black-Scholes equation is 

obtained for the first time;

Pricing American Options - an Important Fundamental Research in ...

Create successful ePaper yourself

Delete template?

Save as template?