Pricing American-style Derivatives under the Heston Model Dynamics: A Fast Fourier Transformation in the Geske–Johnson Scheme



to notice is that the weighting function W (θsτ, θυτ ) must be real and nonnegative.


Then, at syjfe1 jv-ι δ1i , uτ,fe2 jv2δ22 :


-    / {3tm ,υτ,k2 )=12 W (θsτ,fcι υτ.fc2 ) Σ Σ e         ς'ι '     ς'2 41,j2

j2=O j'1=O

— W (θsτ,fcι υτ.fc2 ) 12 ς ς eτfc1 <'1 '                 —

j2=O jι =O

— W (θβτ,kl υτ.fc2 ) ι2e-(τ∙^1   +     2) ς ς           ɪ+ k2 )ψjι,j2 .

j2=O j1=O


It remains to propose a kernel function and derive W sτ, θυτ ).

My suggestion is to use a simple kernel function of the form:


K (x,y) —


_______________(1-M)2(1-H)2_______________

2y2 + 2(1-y)2 + (1-)2y2 + (1-)2(1-y)2 ,

0,


if both |ж| ≤ 1 and y1 .
elsewhere.


Properties of K (x, y) are discussed in the Appendix. It turns out that the weighting function does not
have a closed-form expression. However, W
sτ, θυτ ) can be approximated with arbitrary precision.


5. Geske-Johnson Scheme with Richardson Extrapolation


Consider a sequence of “Bermudan”-style derivative securities,
{D- (st,vt,T t)}=1, where each D- can be exercised just at times tj t + j(T- t), j — 1,...,n, prior
to expiration at T. An American-style option is the limit of the sequence. D
1 is the value of a corresponding
European-style derivative.

If EX (st' ,vt' ,T t') is the exercise value of the security at t', then, D-,s obey the following recursion:


d- {st∙, Vt, tt) —


(12)


— e-r(t1-t)E [max {EX,T tɪ) , D--1 (s^ ,Vt1 ,T tx)}].


For a put, EX (st't>,T t') — (X eSt' )+ and for a call, EX (st't>, T t') — (est' X)+. X denotes
the strike price. Equation (12) is similar to (7.7) in Epps (2004a).

Next, define:

h- —        , n1, 2,...

n


11




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