# Smoothing of the payoff function as a terminal condition for numerical option pricing

I am interested in using a 4th order finite difference method in (underlying asset) space to price a European call basket option. I have developed the solver and everything works as expected, except that no matter how high the order of my numerical method is, the maximum convergence rate is always of the second order due to the discontinuity in the first derivative of the payoff function.

Luckily, there is a smoothing remedy for this, proven around 1970, in a classic paper Smoothing of initial data and rates of convergence for parabolic difference equations by Kreiss et al. This was more recently used in High-Order Compact Schemes for Parabolic Problems with Mixed Derivatives in Multiple Space Dimensions by Düring et al, but without explanation on how the function given by its Fourier transform in Chapter 9 of that paper is inverted and inserted into the smoothing integral.

I would appreciate a lot if someone has an idea about how to perform this 4th order smoothing procedure. Practically, I need to solve the following integral: $$\tilde{u}_0(s)=\frac{1}{h}\int_{-3h}^{3h} \Phi_4\left(\frac{x}{h}\right)u_0(s-x)\text{d}x,$$ where $\Phi_4$ is given by its Fourier transform: $$\hat{\Phi}_4(\omega)=\left(\frac{\sin(\omega/2)}{\omega/2}\right)^4\times\left(1 + \frac{2}{3}\sin^2(\omega/2)\right).$$

Update: I managed to solve the problem using symbolic powers of Mathematica and Matlab. The solution to the inverse Fourier problem is given below as a Matlab code:

f4 = @(x) (1/36)*(1/2)*...
( +56*x.^3.*sign(x) +(x-3).^3.*(-sign(x-3)) +12*(x-2).^3.*sign(x-2) -39*(x-1).^3.*sign(x-1) -39*(x+1).^3.*sign(x+1) +12*(x+2).^3.*sign(x+2) -(x+3).^3.*sign(x+3));


Then one just needs to plug this expression into the integral given in the first equation along with their favorite initial condition and solve it with a method of preference.

• May be the paper "Option valuation using the fast Fourier transform" by Peter Carr and Dilip B. Madan engineering.nyu.edu/files/jcfpub.pdf gives you some idea – Nick Nov 3 '16 at 9:41
• Thank you Nick, though that paper is a well known classic, it does not really attack my problem. – millovanovic Dec 4 '16 at 17:37

## 2 Answers

I just wanted to say that I solved the problem using the symbolic/analytical features of Mathematica and Matlab to perform the inverse Fourier transform and then I used high-order numerical integration to solve the smoothing integral.

• You should add more details to useful as an answer. Otherwise, you should add this to your question or as a comment. – Gordon Dec 4 '16 at 19:37
• @Gordon, it is done. :) – millovanovic Dec 6 '16 at 15:37

Another approach would have been to use some projection as in Pooley and Vetzal Convergence remedies for non-smooth payoffs in option pricing.

In your case, it may be a projection of the initial condition to the RBF space (I have read your paper, and it looks interesting). I wonder a bit how the two approaches compare.