# How to price barrier options (binomial tree)

What is the easiest way to price single barrier options using binomial tree? I found This method. Is this method good or maybe should I use another one? Does this price converge to price from BS model?

## 1 Answer

This may not be answering your question - but it is worth noting that valuing barrier options on a binomial / trinomial tree is at best problematic. It is difficult to enforce the boundary conditions because nodes will not typically sit on the barrier itself, necessitating some kind of probability-weighted interpolation - which is unlikely to be numerically stable. Greeks will be even worse. Strongly suggest you look at PDE approaches which are the industry-standard for Barrier options. Google "barrier options PDE" - there's a tonne of literature out there.

• Thanks! Can you say sth about this method which I posted in link? It looks simple, i.e. it is just such a very simple scheme which in some way gives a fair price fot this type of option? Commented Nov 30, 2020 at 19:51
• The method does not recognise or address the numerical issues of pricing barrier options. This is a serious gap. There are no numerical results which makes me more suspicious. Setting option values to zero in the knocked-out regions is very likely to introduce numerical instability and poor convergence. To mitigate this, try probability weighting, e.g. for up and out, multiply the option price at the node immediately below the barrier by the prob (p) of breaching the barrier over the next timestep. Similar for above the barrier. Trees are a very poor choice for discountinuous payoffs. Commented Nov 30, 2020 at 21:32
• Rephrased a little bit differently: Yes, you could use a binomial tree for barrier option pricing, but you will have to use a very unwieldy number of steps in your tree. In my experience the challenge with barrier options is their $\Delta t \to dt$ behaviour: You need extremely small time steps to get towards reasonable prices when compared to quasi closed form PDE solutions. Thus, we see another challenge: You will also have problems when using numerical PDE schemes: Explicit schemes need a humongous amount of steps; implicit schemes introduce errors when pricing path-dependency. Commented Dec 1, 2020 at 8:53
• I have 2 questions. First: which method is the most commonly used for barrier options? Monte carlo simulations or PDE approach? Second : I have question about binomial tree. if we consider only 5 steps, we gets 32 different possible payoffs (the option price tree is not recombining). So if I want to price option by binomial tree I need calculate how much these payoffs are and then go back up the tree to time 0 in the usual way. But I have no idea how to calculate these payoffs at maturity that is, how to distinguish all these trajectories (I work in VBA) Commented Dec 2, 2020 at 20:33
• Do you know maybe Boyle, Lau (1994) method for pricing barrier options? They use $F\left(m\right)=\frac{m^{2}\sigma^{2}T}{\ln\left(\frac{S}{H}\right)^{2}}$ to find optimal number of time steps but I hae problem with understadning why they use formula like this Commented Dec 23, 2020 at 14:58