84
Other Numerical Methods for European Options
207
If (igreek = 3 And Sgn(iopt
Ł
(S1 - X)) >= 0) Then greek = greek + 1
If (igreek = 5 And Sgn(iopt
Ł
(S1 - X)) >= 0) Then greek = greek + S1
Ł
(Log(S1 / S) - r1)
Next i
If igreek = 1 Then vg = ert
Ł
(greek /S) / nsim
If igreek = 3 Then vg = ert
Ł
X
Ł
tyr
Ł
greek / nsim
If igreek = 5 Then vg = ert
Ł
(greek /sigma) / nsim
QMCOptionGreek135 = vg
The NIOption Value function sets up the share price process and collects the sum for
the numerical integration within a loop. Note how the common components (S and h) are
inserted after the loop:
Function NIOptionValue(iopt,S,X,r, q, tyr,sigma, msd, nint)
’ values optionusing numerical integration
Dim rnmut, sigt, h, sum, zi,payi
Dim i As Integer
rnmut = (r - q- 0.5
Ł
sigmaO2)
Ł
tyr
sigt = sigma
Ł
Sqr(tyr)
h= 2
Ł
msd/ nint
sum = 0
For i = 0 Tonint- 1
zi = -msd + (i + 0.5)
Ł
h
payi = Application.Max(iopt
Ł
(Exp(rnmut + zi
Ł
sigt ) - X/S), 0)
sum = sum + payi
Ł
Application.NormDist(zi, 0, 1, False)
Next i
NIOptionValue = Exp(-rŁ tyr)Ł hŁ SŁ sum
End Function
SUMMARY
In this chapter, we illustrate alternative ways of calculating the expectation of option
value, which underlies the Black–Scholes formula for European options.
Monte Carlo simulation consists of using random numbers to sample from the many
paths a share price might take in the risk-neutral world. An option payoff is calculated
for each path and the arithmetic average of the payoffs, discounted back at the risk-free
rate, used to estimate option value.
When compared with the binomial method of valuation, it produces estimates of option
value with more estimation error. Many more paths must be generated with Monte Carlo
sampling, because unlike the tree, paths do not recombine.
Controlling the random sampling by variance reduction techniques such as using anti-
thetic variables can reduce the estimation error. Quasi-random samples preselect the
sequence of numbers in a deterministic (i.e. non-random) way, eliminating the clustering
of values seen in random numbers. The samples are taken such that they always ‘fill in’
the gaps between existing samples. This means that the error of estimation is proportional
to 1/n rather than 1/
p
n, where n is the number of simulation samples.
Numerical integration is another method that can be adapted to value options. It is
especially useful for options whose payoffs depend on many assets.
REFERENCES
Broadie, M. and P. Glasserman, 1996, “Estimating Security Prices Using Simulation”, Management Science,
42(2), 269–285.
Hull, J. C., 2000, “Options, Futures and Other Derivatives”, Prentice Hall, New Jersey.
Moro, B. 1995, “The Full Monte”, Risk, 8(2), 57–58.
