Pathwise method for digital option

This forum is to discuss the book "the concepts and practice of mathematical finance" by Mark Joshi.

Pathwise method for digital option

Postby kelvin_dc200015b » Thu Mar 06, 2014 9:54 pm

Hi Mark,

'This is sometimes called the pathwise method. The main difficulty
with this method is how to interpret f'(ST) when f is discontinuous. Jump discontinuities
will give rise to delta functions in the derivative. We can therefore write
f = g + h, with g continuous and h piecewise constant. Then g' is well-behaved,
and its contribution to the Delta can now be evaluated by Monte Carlo. The derivative
of h is a sum of Delta functions so the integral can be computed analytically
as a finite sum and we are done.'

Can you explain how the last sentence can be applied to a simple example of a digital call? (though such simple product is not evaluated through MC in practice). I know we can use smoothing technique to approximate the digital payoff using cumulative normal distribution. Appreciated if you could tell me there is a way to compute it analytically.

Many thanks.

regards,
Kelvin
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Re: Pathwise method for digital option

Postby mj » Fri Mar 14, 2014 2:23 am

essentially you just observe that the derivative of the Heaviside is
\delta(S_T-K)
so that when you integrate against you just pick up the rest of the integrand times the density at K.

see this paper

http://www.quarchome.org/nthdefaultjoshikainth.pdf
mj
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