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Carr-Madan proved that there is a simple relation between call-prices and the characteristic function of the underlying model.

See Equation 5 and 6 in their original paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.348.4044&rep=rep1&type=pdf.

For many models, we do have characteristic functions. (Levy models, Heston models, and many other stoch-vol models).

So the above integral can readily be evaluated in practically no time using Gaussian quadrature or Trapezoid rule!

So the question I ask is ... why is there continued focus on numerical European option pricing in the literature?

Is the problem not solved?

Or to rephrase my question: when does above approach "not work"? Or not work "fast enough"/"accurately enough"?

I mean, you can go to arxiv.org and find tons of bizarre and convoluted methods to price Europeans and I am just left here standing thinking "uhh ... ok ... but why not just use Carr-Madan formula???".

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Firstly, there is nothing wrong with the fast Fourier transform approach from Carr and Madan (1999). However, there is a whole range of reasons why there is research about other numerical approaches.

  • You may have a model where you do not know the characteristic function (e.g. local volatility). Then, the Carr Madan method does not apply at all and you got to have fast alternatives for computing prices and Greeks.
  • You may want to price path-dependent options where Carr Madan does not apply either and you use European-style options as an example.
  • You may be able to find an even better method which is simply faster than Carr Madan. Crisostomo (2018) argues that Carr Madan is not the fastest possible method and highlights the importance of strike price vectorisation. Similarly, the COS approach is celebrated for its speed.
  • You have the issue of finding a suitable (optimal) damping factor $\alpha$, which requires some optimisation in the beginning. There's a paper from Lord and Kahl (2007) which discusses how to find an optimal $\alpha$. Furthermore, the Carr Madan method struggles with highly OTM options.

That having said, Carr Madan is a simple, popular and powerful method for pricing European-style options, particularly for Levy models and stochastic volatility models. Some well-known extensions are the OTM approach presented in the same paper (Carr and Madan (1999)) and using (for example) the Black-Scholes option price as a control variate. But that does not mean one cannot seek to improve it. Finally, sometimes research is even about trying out things that do not work and confirming that already known algorithms/approaches are indeed optimal. But you only find such results, if you're curious and try out new things.

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The problem is not solved. The assumptions necessary for Ito calculus to work are very strong and the normal and log-normal were chosen because they had known solutions, not because anyone thought they were true.

One of the assumptions in the calculus is that the parameters are known. So, if you had the autoregressive equation $x_{t+1}=\beta{x}_t+\epsilon_{t+1}$ and the parameters are known, then it is a little painful because of the skill level required but it is still a tractable problem.

If $\beta$ is not known and $\beta>1$, then there is a proof that no meaningful solution to the problem exists that also remains inside the axioms being used. There is a conflict between the math and economics. Of course, if $\beta\le{1}$ then no one would invest.

I have proposed a new calculus to resolve this problem. You can find a link to it at https://www.datasciencecentral.com/profiles/blogs/a-generalized-stochastic-calculus.

I dropped the assumption that the parameters are known. It solves a decision-theoretic problem that has always been present because I require the solution is a sufficient statistic that first-order stochastically dominates alternative solutions and minimizes the loss from an unfortunate sample. The problem is that if there is a requirement that $\theta$ is known and it is not knowable, then you cannot make a good decision on a price.

On the other hand, if $\hat{\theta}$ is a sufficient statistic for the decision, then knowledge of $\theta$ is irrelevant. It is removed from the problem. Unfortunately, it seems I cannot get it published so I cannot get the options pricing model published.

You can also watch this video to see part of the problem https://youtu.be/R3fcVUBgIZw.

Sorry, the entire math is way to long to post here.

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