Math in finance or vice versa

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Estimation of Exponential Ornstein-Uhlenbeck process

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In the previous article about process calibration we derived the following updating formula:
P_t - P_{t-1} = \left(\theta(\mu - \ln P_{t-1}) + \frac{1}{2} \sigma^2 \right) P_{t-1} \Delta t + \sigma P_{t-1} \sqrt{\Delta t} W_{t-1}
Now we can rearrange equation’s parts as follows:
\frac{P_t - P_{t-1}}{P_{t-1}} = \left( \theta\mu + \frac{1}{2} \sigma^2 \right) \Delta t - \theta\Delta t \ln P_{t-1} + \sigma \sqrt{\Delta t} W_{t-1}
Thus we can equate it against simple regression formula:
y = a + bx + \epsilon

Therefore, we obtain:
y = \frac{P_t - P_{t-1}}{P_{t-1}}
x = \ln P_{t-1}
b = - \theta\Delta t
a = \left( \theta\mu + \frac{1}{2} \sigma^2 \right) \Delta t
\epsilon = \sigma \sqrt{\Delta t} W_{t-1}
And this gives us the following OLS estimates:
\theta = - \frac{b}{\Delta t}
a = - b \mu + \frac{1}{2} \sigma^2 \Delta t
As it was in the previous articles:
\sigma^2 = \frac{\sigma^2_{\epsilon}}{\Delta t}
a = \frac{1}{2} \sigma^2_{\epsilon} - b \mu
And finally the estimation of \mu is:
\mu = \frac{\frac{1}{2} \sigma^2_{\epsilon} - a}{b}

Thus we obtained the estimation of exponential Ornstein-Uhlenbeck process.


Written by fxpaul

November 20, 2013 at 13:04

Posted in Uncategorized

Recommender systems – Intro

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Recommender System (RS) provides suggestions for items to be of use to a user. As the definition says we should define three pieces, i.e. item, user and useful suggestion.

Item is a general term that describes everything what RS recommends to users. It might be stocks, books, videos or anything else.

User of RS is usually an individual who doesn’t have enough experience or competence to make a decision of item choice. For instance, popular on-line book store Amazon provides recommendations based on what user bought and viewed in past.

To be useful suggestion (or recommendation) should support some decision-making process, like what book to buy, what news to read or what to do in a spare time. In the simplest form, it might be a ranked list of items. This ranking is way to predict what the most suitable items are for user behaviour.

RS should track how users interact with items. For instance, on the book store one might view a book title, look inside of the book and/or buy it. Viewing a title can be considered as an implicit sign of preference. Thus the usual recommender system has to deal with User, Item and User-to-Item actions (transactions).

For RS implementation to be successful it should achieve one or more business goals:

  • Sell more items
  • Sell more diverse items
  • Increase user satisfaction and fidelity
  • Understand user behaviour and habits

And those could be approached by implementation of few tasks:

  • Find some good items: Create a ranked list of items along with predictions of how much user would like them. This is the main task of many RS.
  • Find all good items: Create a complete ranked list of items. Usually it is required when number of items is small and user can benefit from ranking information. Such RS are quite common in financial application. They usually need to examine and to rank all possible scenarios.
  • Annotate items in context: Given an existing context, emphasise items based on long-term user preferences. For instance, such RS might emphasise TV shows in EPG based on previous user behaviour.
  • Recommend a bundle: Suggest a group of items that fits well together. You’ll find such bundles at cable internet providers, travelling agencies etc. For instance, airlines are starting to recommend accommodation and car hire during ticket purchase.
  • Recommend a sequence: Recommend a sequence of items that is pleasing as a whole. For instance, a recommended track of courses at the university might depend not only on chosen major, but also on the absolved courses.
  • Browsing: RS should help the user to browse items that are more likely in the user’s interest in this browsing session.
  • Improve user profile: This task is all-time task of RS. It collects information about user’s actions to provide more personalised recommendations.

That is pretty much what one can expect from such thing as recommender system. In next posts I plan to cover:

  • Overview of basic techniques
  • Clustering
  • Content-based RS
  • Collaborative filtering in RS

Written by fxpaul

November 6, 2013 at 16:11

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Book on Haskell and Financial Mathematics

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Most probably, you’ve noticed the blog was not updated last year. I’ve been writing a book about programming and financial mathematics.

It was not easy for me. When I got a proposal to write a book in this area, I was hesitating if it is something I was able to cope with. For fresh writer it is a daunting task as you should go under the schedule and try to write consistently, every day, at least half page of text. Some parts of book were easy to write as I already wrote about these topics. Some were awful to accomplish as I did not really understand how to explain math and its links to Haskell with a plain and clear language.

I’m quite sure that now I will start again writing this blog, though the new projects are quite far from financial math now but they are still in math and big data projects. Please, also check our new company website to see what is in progress now.

But finally it is out and available in book stores like Amazon, O’Reilly or Safari Book store:

Written by fxpaul

October 30, 2013 at 09:55

Posted in Uncategorized

Mixing up Fx and Stocks

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Anybody can find enormous amount of books about stocks trading but all of Forex books give an expression of spurious market: plain old strategies, positive only examples of profit-loss calculations etc. It’s easy to get impressed as well as to get depressed by results of trading.

I want to explore how we could apply stock strategies to foreign exchange markets. Vast majority of stock strategies should be adopted to turbulent dynamic of Forex markets.

The nearest future is:

  1. Define the common trading task in a precise, mathematical way – this is the starting point.
  2. Monte Carlo methods for optimization and trading experiments.
  3. Simple strategy emulations on model processes

I do hope the list will be continued.

Written by fxpaul

December 27, 2010 at 20:56

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