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 )= ∆1∆2 W (θsτ,fcι ,θυτ.fc2 ) Σ Σ e ς'ι ' ς'2 41,j2 —
j2=O j'1=O
— W (θsτ,fcι ,θυτ.fc2 ) ∆1∆2 ς ς e-÷τ∙fc1 <'1 ' —
j2=O jι =O
— W (θβτ,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 (θsτ, 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 ∣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 (θsτ, 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. D1 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, t — t) —
(12)
— e-r(t1-t)E [max {EX (¾ ,¾ ,T — tɪ) , D--1 (s^ ,Vt1 ,T — tx)}].
For a put, EX (st',υt>,T — t') — (X — eSt' )+ and for a call, EX (st',υt>, T — t') — (est' — X)+. X denotes
the strike price. Equation (12) is similar to (7.7) in Epps (2004a).
Next, define:
h- — , n — 1, 2,...
n
11
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