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 (θ_sτ, θ_υτ ) must be real and nonnegative.

Then, at sy_jfe₁ — _jv-_{ι δ}¹_i , uτ,fe₂ — jv₂δ²₂ ^:

-    / {_3tm ,_υτ,_k2 )= ∆₁∆₂ W (θ_sτ,_fcι ,θυτ._fc2 ) Σ Σ e         ^ς'ι '     ^ς'2 41,j2 —

j2=O j'1=O

— W (θ_sτ,_fcι ,θυτ._fc2 ) ∆1∆₂ ^{ς ς} _e-÷^τ∙^fc1 <'1 '                 —

j2=O jι =O

— W (θ_βτ,_kl ,θυτ._fc2 ) ∆ι∆₂e-≈(≡^τ∙^^{1   +}     “^{2) ς ς}           ɪ^{+ k2} ⅛ )ψjι,j2 .

j2=O j1=O

It remains to propose a kernel function and derive W (θ_sτ, θ_υτ ).

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 ∣y∣ ≤ 1 .
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 (θ_sτ, θ_υτ ) can be approximated with arbitrary precision.

5. Geske-Johnson Scheme with Richardson Extrapolation

Consider a sequence of “Bermudan”-style derivative securities,
{D_- (s_t,v_t,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. D₁ is the value of a corresponding
European-style derivative.

If EX (s_t' ,v_t' ,T — t') is the exercise value of the security at t', then, D_-^,s obey the following recursion:

^d- {^st∙, Vt^{, t — t) —}

(12)

— e^-r(t1-t)E [max {EX (¾ ,¾ ,T — tɪ) , D--1 (s^ ,Vt1 ,T — t_x)}].

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

Next, define:

^h- —        ^{, n} — ^{1, 2,}...

n

11