Timeline for Option pricing using characteristic function
Current License: CC BY-SA 4.0
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| Nov 10, 2021 at 15:30 | comment | added | Stéphane | As a last tip, note that we typically do not model the term structure of interest rates when pricing options, especially at shorter horizons. However, you can respect it by matching contracts with a constant interest rate taken from fixed income instruments of a maturity that is closest to that of your option contract. That means you can make groups of option contracts on the basis of (days, maturity) pairs and price all strikes simultaneously if you code it smartly. Also, the integrands in Fourier pricing tend to converge fast to zero, so consider a log grid for numerical integration. | |
| Nov 10, 2021 at 15:25 | comment | added | Stéphane | For the comment about Carr and Madan's approach, all Fourier-based pricing is based on the knowledge of a characteristic function (conditional on current price and volatility, for the log asset price). If your model does not allow you to compute it in closed form, you then turn towards Monte Carlo methods. If you do that, remember to use the correction proposed by Duan and Simonato (1998). A problem with naive simulations is that the martingale property being only approximately enforced can lead to option prices that violate rational pricing bounds in simulation which pollutes your estimates. | |
| Nov 10, 2021 at 15:18 | comment | added | Stéphane | That book does not cover this topic. You'd have to read the actual papers. I'd suggest reading Christoffersen, Dorion, Jacobs and Wang (2010) because they have many models in one place and the paper is very pedagogical. Note that you can also risk-neutralize with quadratic kernel (you have a risk premium and a variance premium directly built into the stochastic discount factor) -- the theory is in Christoffersen, Elkamhi, Feunou and Jacons (2010) and an application can be found in Christoffersen, Jacobs and Heston (2013). | |
| Nov 10, 2021 at 15:10 | comment | added | Stéphane | Affine refers to the conditional volatility entering the expected return linearly inside the exponential -- i.e., it's a GARCH-in-mean type of model. That's necessary in discrete time models to obtained a closed-form expression for the characteristic function. Otherwise, you have to price using Monte Carlo methods. | |
| Nov 9, 2021 at 14:12 | comment | added | Trettman | I'm also thinking about other methods, for example in the presented in "Option valuation using the fast Fourier transform" by Peter Carr and Dilip B. Madan. In section 3, they introduce a method for calculation of option prices that I believe would work for any model? Atleast, I don't see any reason for why it would not. | |
| Nov 9, 2021 at 13:57 | comment | added | Trettman | Great answer! Thanks for your response! Does Derivative Analytics with Python explain the difference between affine and non-affine GARCH models and why this approach works on one but not the other, or do you have some other resource for this? | |
| Oct 19, 2021 at 4:01 | history | answered | Stéphane | CC BY-SA 4.0 |