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Electronic 81

American Bermudian option pricing through the optimal exercise boundary approach. (Cont. 2)

American – Bermudian call and put option boundaries

American Bermudian option pricing through the optimal exercise boundary approach. (Cont. 1)

Continue the last post, I present here some numerical results then go to discuss about a parallel approach using grid computing for this algorithm.

* Short description of the algorithm:

[State 1] Create the “Good Lattice Point” dimension

[State 2]

[State 2a]

For to do

[State 2b]

For to do

[State 2c]

For to do

[State 2c-1] Update the vector () by

[State 2c-2] Newton iteration

[State 2c-1.1] Monte Carlo simulation to compute

[State 2c-1.2] Find by the expression =

[State 2c-1.3] Stop when <

end for

[State 2c] Regression where

end for

end for

[State 3] Final Monte Carlo simulation.

NOTE:

[State 2b] We investigated a parallel approach for this state. At each opportunity, we treate each asset independently. (in processing)
[State 2c] This state can be easy parallelized. We distribute the independent GLP to workers then merge the results to a regression process.
[State 3] This state is a standard Monte Carlo simulation and is natural parallel.
* Numerical results

To be continued …

American Bermudian option pricing through the optimal exercise boundary approach.

This approach was proposed by Ibanez and Zapatero (2004)

The main idea of this approach is to determine the optimal exercise boundary before pricing an American option.

Ibanez and Zapatero focused on building a full boundary in a form of a polynomial curve whose dimension depends on the number of underlying assets. However this approach is limited to the monotonic and convex payoff functions. When these properties are not true (e.g. pricing swaptions in a sufficiently general multi-factor term-structure model), the algorithm would have to be revisited. In the first phase, at a given exercise opportunity, the algorithm uses a linear-interpolation or a regression of a quadratic or a cubic function in order to get a parameterization of the boundary. In the second phase, to compute the option price, a MC simulation is run until the price trajectory reaches the dynamic boundary. The main advantages of this method is that it provides a full parameterization of the boundary and the exercise rule.

We consider a Bermudan put option on that stock with maturity at date T and N exercise opportunities at times . The intrinsic value is given by . Then, at any time , the price of the Bermudan option satisfies :

where is the “optimal stopping time” defined as the first time such that , and otherwise. Then the optimal boundart point at time is determined by :

So we have :

(for call)

 

(for call)

 

(for put)

 

(for put)

We discuss here the computation of the boundary from opportunity date m back to m − 1. Notice that at , the boundary is known (e.g. at the optimal boundary point is definitively the strike value). Backward to we compute the boundary point using the previous optimal point. Repeat this process until reaching . Then we have the optimal boundary points for every

Go to the second phase, we use the following equation to pricing the option at 0 :

where

I got a problem that when I pricing a call using the call optimal boundary (the price is not good, it is far way from the reference price) but when I use the put optimal boundary the call price is ok. I really do not understand. Does anyone has experiences on this problem ?

 

 

Note :

American option : quyền lựa chọn kiểu Mỹ. Khác với quyền lựa chọn kiểu Châu Âu, nhà đầu tư chỉ có quyền thực hiện hợp đồng (option contract) vào thời điểm đáo hạn hợp đồng (maturity date), với American option, ta có thể thực hiện contract vào bất cứ lúc nào từ lúc ký contract cho tới khi đáo hạn. Do sự linh hoạt này nên American option được ưa chuộng trong giao dịch. Vậy vấn đề đặt ra là ta nên thực hiện vào lúc nào ? vậy là khái niêm “optimal exercise rule” quy luật tốt ưu để thực hiện hợp đồng (hay optimal exercise boundary : đường biên tối ưu để thực hiện option) ra đời.

To be continued …

Le travail aujourd’hui 30/07/2007

Christianne va partir la semaine prochaine. Elle est gentile dans touts les cas (sweet et sympa). Elle m’aiderait beaucoup, de remplir des dossiers de mission, de faire des visas ou bien des remboursements de l’INRIA … Je me souvien qu’elle viendrait a quel heure toutes les apres midi et qu’elle nous dirait au revoir tous les soirs. Je suis tres desole de ne pas venir ton pot de depart cas je ne sais pas du tout que tu partiras notre equipe. Bon, que tu encore travailles ici jusqu’a Septembre et qu’on peut se recontrer plusieur fois, je veux dire que je te remercie beaucoup pour touts Christianne.

Un mot pour des options Americainne : Comment calculer des Greeks par rapport la frontiere d’exercise optimale ?

Monte Carlo methods in option pricing

I investgate the Monte Carlo method in financial mathematics in order to profit its natural parallel property to gain to the performance on the Grid computing. For instance, I focus on the European, American and their exotic types (i.e. standard, barrier, basket) option pricing.

I would cite a section which describes about the Monte Carlo method in finance from the wikipedia :

In the field of financial mathematics, many problems, for instance the problem of finding the arbitrage-free value of a particular derivative, boil down to the computation of a particular integral. In many cases these integrals can be valued analytically, and in still more cases they can be valued using numerical integration , or computed using a partial differential equation (PDE), for example Black-Scholes. However when the number of dimensions (or degrees of freedom) in the problem is large, PDE’s and numerical integrals become intractable, and in these cases Monte Carlo methods often give better results. For large dimensional integrals, Monte Carlo methods converge to the solution more quickly than numerical integration methods, require less memory and are easier to program. The advantage Monte Carlo methods offer increases as the dimensions of the problem increase. Monte Carlo methods were first introduced to finance in 1977 by Phelim P. Boyle in his seminal paper “Options: A Monte Carlo Approach” in the Journal of Financial Economics. This article discusses typical financial problems in which Monte Carlo methods are used. It also touches on the use of so-called “pseudo-random” methods such as the use of Sobol sequences.

So the Monte Carlo methods are ideally suited to evaluating difficult integrals.

In finance the underlying random variables (such as an underlying stock price) often follow a path of Brownian motion process. Through the Black – Scholes model, a stock price is :

Where is the stock price at a given time , is the spot price at time 0, is the interest rate, is the volatility rate and is a standard Brownian motion in one dimension. Notice that is the solution of the following equation:

By discretizing this equation, we can simulate the trajectory of the underlying asset price

With is the Gaussian random numbers.

We can compute the option price through the expectation of the discounted payoff (under the risk neutral probability)

Option price Call =

Option price Put =

In the other hand, to approaches these expectations above by using Monte Carlo methods, we simulate a large number random simulations discretized in time of option price at T, then do an avarage like (in case of a Put):

With is the number of Monte Carlo simulations.

It is important to calculate the convergence rate and the confident interval of this method

Follow the law of large numbers we have :

Follow the central limit theorem we have :

convergence rate is

95% of confident interval is

Next post : Pricing the European option using Monte Carlo methods.

 

Grid computing

Grid computing is a phase in distributed computing (or parallel computing).

In distributed computing (or parallel computing) you have some concurrency such as super computing, cluster computing, grid computing. (ref. wikipedia)
There are many ways to distinguish the conceptions above but I like an easily way that bases on the property of the components.

Homogeneous components:

• Super computing : Bases on the super computers. A super computer is a very powerful, massively parallel computer. It is applied to the study the very intensive computational problem in the biometric, geometric sciences … (i.e. IBM Blue Gene series). In finance, the IBM Deutschland research center has built a platform bases on Cell Broadband Engines and provided a specific SDK for numerical applications.
• Cluster computing : We do the cluster computing on the cluster computers. The cluster computers is a group of computers closed together. The computers are usually commonly, they are connected by a high speed local network (geographically limit).

Advantages

• High speed communication
• Same system operation
• High trust, high

Disadvantages

• Expensive
• Hard to enlarge the system

Heterogeneous components:

• Grid computing : bases on one or many groups of computers which are very different and may be do not fully trust each other, or which are geographically dispersed. They are connected through Internet by a conventional network interface.

Advantages

• Open resources
• Open standards
• Cheap
• Easy to enlarge the system

Disadvantages (also as challenges)

• Hard system configuration
• Load balancing
• Fault tolerant
• Synchronization

Then how to apply Grid computing in finance (particularly in my case is pricing the derivative productions) ? A very good question. I will try to figure it out

PicsouGrid

I am a PhD candidate in Computer Science, not in Applied Mathematics. My PhD thesis aims to explore the derivatives pricing algorithms using Monte Carlo methods then to highlight their potential of parallel techniques applied on Grid infrastructure. My objective is designing a grid-based software platform (bases on ProActive, a grid middleware for parallel, distributed and multi-threaded computing), able to run financial risk evaluations too .

Our platform, we call it PicsouGrid (if you read the Donald duck comic, you have to know uncle Scrooge McDuck the richest man in the world and Picsou is his French name). More information about PicsouGrid is in the link below.

http://www.spmetric.com/wordpress/2006/12/21/picsougrid-vd/

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Daily working 26/07/2007

Today afternoon, I’ve just had a talk with Michael Mascagni about the last forgettable conference at Reading. This year, the MCM2007 was not itself. We did not understand what was happened during this conference. It had a very poor technical program. The only thing that made me satisfy is that I could meet the most of key persons in Monte Carlo and Quasi Monte Carlo method field. And because there were not many people so I could talk with them easily.

For the next MCM conference in 2009, they would organize it in Zurich, Switzerland due to the big financial support from the local organisers. Micheal said that it will not be the same as this year and I hope so. He also suggested me something about the grid based applications which are familiar with my PhD thesis. He is very kind, funny and in some cases is talkative (positive senses of course) . A collaboration with him is not bad, right ? who know.

Return to my PicsouGrid architecture, it is not perfect yet. I have 2 parallel algorithms and I had a problem in synchronization the distributed tasks. I have to finish it as soon as possible to continue my numerical experiments. Another meeting will be held next Thursday. To help me Arnold suggested me to use the method setScatterGroup(). I probably do not like to create many objects at the beginning but it seems there is no way rather than it.

Fine, I spent a couple of hours for these posts. It helps me out of stress.