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



Then:


Q (ʃ5 Cι,<2)

P 5 c1,c2)


β - P7iC i


^r -' - 1

Л----7=---

Beτ√A + 1

τ (r - δ - — "ʌ 1 + -^ τβ + τ√"7^ + 2 In ——+-—

V 7 J     72 L            Be^ + 1

Caveat. This solution is valid for (ζ 1,C2) = (O, O),Given the trial solution for Ψ and the dehnition
of a ch.f., it must be the case that q (τ
; O, O) = p (τ;O, O) = O, Vτ. I check that the above expressions
for q (τ
; ζι,C2) and p (τ; ζ 1,C2) do not contradict the uniform continuity property of Ψ, that is, whether
lim(c,G)'(o,o)' q( τ; ci,c2) = iim(c,G)'(o,o)' p( τ; Cι,C2) = o :

Iim
(ζ12)'(0,0)'


q(τ ; c 1 ,c 2)


1

72


'    Γ2eτ^2 - lim1∕B (C 1,C2)

β - V β —/=-----------

eτ√^2 + lim1∕B (C 1,C2).


-2 [β - β] = o,
72


lim
(GX2)'(0,0)'


p(τ ; c 1 ,c 2)


ɪ τβ + τ√β2 + 2ln X±lim1/B (C 1,C2)

7 L                  eτβ2 + lim1∕B (C 1,C2).

72 [τβ + τβ - 2 ln eτ"] = O,


because lim(ζ12)'(o,o)' B (C 1, C2)


= , if β > O (it is straightforward to extend the proof


to the subcase β = O^.


Case 2. 7 = O, β > O.


q (τ ; c 1,c 2)   =

- 2ξ [e βτ (c 1+ iC 1+ 2 '/C2) - c 1 - iC 1],
2β

p (τ; c 1, c2)  = τ (r - £)iC 1 - 2⅛ [2 (e βτ - 1)iC2+ τ'β 1+ τc2] -

2β


α (e ^τ - 1) r 9       -l

-      2.,.2       [C ? + ≈C l].

2β


Case 3. 7 = β = O.


q (τ; c 1,c2) = -2 [C1 + iC 1] + iC2,

p (τ; c 1,c2) = τ [r - β iC 1 - ɪ [τ (c 1 + iC 1) - 4iC2] .

BlackScholes obtains with 7 = β = α = O and p (τ; C1, C2) = τ [r - £] iC 1.




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