compilers presently fail to allocate arrays that require memory in excess of 2GB. It can be shown that with
double precision number storage, the largest Fourier coefficient matrix one can use is of size 2N1 x N2 :
V1 . N2 = 226.
FFT routines were compiled and run on the Birch Linux cluster at UVA. Each node of the cluster is a
double-Pentium IV 2.4GHz system with 2GB RAM. FFT program requires memory storage of slightly above
1GB, therefore, there is no need to use virtual memory and the transformation itself takes approximately 1
minute. Extra 2 minutes are needed to form a 226-element Fourier coefficient matrix and to write out the
results: one round of FFT takes slightly more than 3 minutes in total. For comparison, one round of FFT
takes 6-8 minutes on the Aspen Linux cluster and over 25 minutes on the Unixlab cluster.
Each (2k∣ + 1, k2 + 1)-indexed element of the inverse Fourier coefficient matrix10 (normalized and smoothed
by (2π) 2 ∆∣Δ2H' (θsτ fcι, θvτ fc2 )) is f (sτ,k1 ,vτ,k2 ), the value of the p.d.f. corresponding to one particular
point in (sτ,vτ)' grid.
There is no need to save and use the whole 226-element inverse coefficient matrix. The p.d.f. converges
to 0 long before sτ and vτ attain their maximum grid values and long before sτ goes all the way down to
the minimum grid value. For practical purposes of option pricing, it is more than sufficient to extract the
part of the matrix that covers: 5-times up and down movement of the underlying’s price Sτ = esτ from
St and 4-times up and down movement of √Fτ from y"vj. Resulting grids of (sτ,vτ)' on average contain
800 ∙ 120 elements. This extracted matrix needs to be processed to zero out very small negative values at
some levels of sτ far from st, which infrequently occur because (ζ 1,ζ2)' grid is bounded and imperfectly
fine. Such negative values are always small in absolute value (on average, < —10 6) and negligibly small if
compared to the values of the p.d.f. around the peak (on order of +102). I also verified that the imaginary
parts of the inverse Fourier coefficients were close to zero (the imaginary part of the integrand function must
integrate out to zero by the inversion theorem).
As an illustration, consider Figures 3 and 4. In these Figures, I plot f (sτ,υτ) with parameters as of
June 30 for two different τ’s: a “small” one, which corresponds to ∣ of the time span (in years) between
June 30 and 3rd Friday of July, and a “large” one, which stands for ∣ of the time till 3rd Friday of December.
These p.d.f.’s are, in fact, used in pricing “Bermudan” options expiring in July and December that offer an
opportunity of an early exercise half-way to expiration. The p.d.f. is a unimodal function with a peak at
(sτ,υτ) = (st,υt). It has a long “tail” in υτ direction. It is also evident that for smaller τ, the function is
“concentrated” around the peak; for larger τ, it is more “diffuse” . This result has an intuitive explanation:
one should be more uncertain about relatively distant future.
10Note that fcχ = 0, ..., Nt — 1, fc2 = 0, ∙∙∙, N — 1 and recall that real parts of the coefficients are stored in odd-numbered rows
and imaginary parts are in even-numbered rows.
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