Binomial trees as the number of time steps is increased (or equivalently as the time step tends to 0), converge to the exact value for an option. So why do people use FDM for pricing options (for example an American Put), if Binomial Trees give already accurate results and converges quickly?
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1$\begingroup$ The size of the binomial tree also doubles with each additionnal step. $\endgroup$– LlianeFeb 27, 2019 at 9:06
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2$\begingroup$ Not if using a recombining tree (CRR model) $\endgroup$– Antoine ConzeFeb 27, 2019 at 17:33
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2 Answers
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Actually recombining binomial trees are only a particular case of an explicit FDM scheme. But they have obvious limitations, the foremost being that they cannot accomodate local volatilities. Also 1/2 explicit 1/2 implicit FDM schemes (Crank-Nicolson) have faster convergence with respect to the size of the time step. And FDM schemes can accomodate all sorts of boundary conditions including Dirichlet which is necessary to accurately price barrier options.
answered Feb 27, 2019 at 17:41
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$\begingroup$ Why can't one accomodate local vols into a binomial tree scehme? For example, changing th U parameter as a function of the time step and spot $\endgroup$– RodrigoJan 27 at 12:24
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Our company chose to use FDM for calculating American Options. According to colleagues I talked with, binomial trees are efficient and accurate When there are a small number of option values. But it has a couple of weaknesses:
(1) Binomial tree models are generally inefficient when cash dividends should be taken into consideration;
(2) Compared with FDM, binomial trees are less efficient for multiple options valuations;
(3) Additionally, binomial trees are inefficient in valuing American options compared with European options.
Please correct me if I'm wrong.
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$\begingroup$ I disagree with point (3), in the sense that trees and more general methods both price european exercise as a special case of bermudan/american exercise. $\endgroup$– Brian BDec 28, 2020 at 16:49