Apr
5

## Barrier Option Pricing Using Adjusted Transition Probabilities

One big issue of pricing barrier option with Binomial tree or other lattice method is its slow convergence rate, the barrier option value converges very slowly as the number of tree or lattice levels increase, often requiring unattainably large computing times for even a modest accuracy. A typical plot of barrier option binomial tree results against its analytic value looks like

where the pricing performance is in a sawtooth fashion, with severe periodic spikes that move away from the correct result, which is a nightmare for a researcher because adding more steps doesn't necessarily mean to yield a more accurate answer.

The reason for this is that the barrier being used by the tree is generally different from the true barrier value, for example, as demonstrated below, no matter inner barrier or outer barrier is chosen in practice, calculated value will always be smaller or bigger than correct value, where true barrier shall be used.

John Hull presents three approaches for overcoming this problem, namely, positioning nodes on the barriers, adjusting for nodes not lying on barriers, and the adaptive mesh model. Interested readers please refer to chapter 20, from page 467 to 472, the 5th version, Options, futures and other derivatives. Or read another paper in detail "Enhanced Numerical Methods for Options with Barriers" by Emanuel Derman, etc downloadable at http://www.ederman.com/new/docs/gs-numerical_methods.pdf.

The way shared today is distinct from the three approaches, unlike traditional methods to ensure convergence through placing the barrier in close proximity to, or directly onto, the nodes of the tree lattice, this method applies a suitable transition probability adjustment, thereafter called

Please read the paper for detail at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=964623 and check the accompanying C++ codes at http://www.codeproject.com/KB/recipes/Zeppelin_Barrier_Options1.aspx

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*source from paper Enhanced Numerical Methods for Options with Barriers*where the pricing performance is in a sawtooth fashion, with severe periodic spikes that move away from the correct result, which is a nightmare for a researcher because adding more steps doesn't necessarily mean to yield a more accurate answer.

The reason for this is that the barrier being used by the tree is generally different from the true barrier value, for example, as demonstrated below, no matter inner barrier or outer barrier is chosen in practice, calculated value will always be smaller or bigger than correct value, where true barrier shall be used.

John Hull presents three approaches for overcoming this problem, namely, positioning nodes on the barriers, adjusting for nodes not lying on barriers, and the adaptive mesh model. Interested readers please refer to chapter 20, from page 467 to 472, the 5th version, Options, futures and other derivatives. Or read another paper in detail "Enhanced Numerical Methods for Options with Barriers" by Emanuel Derman, etc downloadable at http://www.ederman.com/new/docs/gs-numerical_methods.pdf.

The way shared today is distinct from the three approaches, unlike traditional methods to ensure convergence through placing the barrier in close proximity to, or directly onto, the nodes of the tree lattice, this method applies a suitable transition probability adjustment, thereafter called

**Barrier Option Pricing Using Adjusted Transition Probabilities**, which exhibits increased convergence to the analytical option price,*source from paper Barrier Option Pricing Using Adjusted Transition Probabilities*Please read the paper for detail at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=964623 and check the accompanying C++ codes at http://www.codeproject.com/KB/recipes/Zeppelin_Barrier_Options1.aspx

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期权交易

2010/05/26 17:58 [Add/Edit reply] [Clear reply] [Del comment] [Block]

complicated but interesting. Option trading is an interesting and challenging subject.

aanrechtblad

2015/06/15 08:45 [Add/Edit reply] [Clear reply] [Del comment] [Block]

You can't really say what is beautiful about a place, but the image of the place will remain vividly with you.

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