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.

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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
so that when you integrate against you just pick up the rest of the integrand times the density at K.

see this paper
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