Идея данной публикации окончательно оформилась после прочтения темы «что это за графики?» форума стокпортала ( http://www.stockportal.ru/forum/index.php?showtopic=10963). Речь пойдет о калибровке торговой стратегии на полностью случайном процессе. Таковым я называю процесс с независимыми приращениями, где приращения берутся из некоего распределения с нулевым средним, то есть каждый следующий член ценового ряда получается из предыдущего по формуле: S(n+1)=S(n)+x(n), где x(n)—случайная величина с плотностью вероятности f(x), причем f(x) не зависит от номера n и имеет нулевое среднее. Примером может служить такой процесс: бросаем монетку, если орел, то S(n+1)=S(n)+1 , если решка, то S(n+1)=S(n)-1. 

Как известно, такие процессы, будучи изображенными в виде графика, чертовски напоминают биржевые котировки (см. например, http://www.stockportal.ru/forum/index.php?showtopic=10963, http://www.russian-trader.ru/forum/viewtopic.php?t=6186&sid=5e6af5d269c40dcaa9723f849360b34d ).

slide1.JPG  
Рис. 1: Реализация процесса
S(n+1)=S(n)+x, где случайная величина x равномерно распределена на отрезке [-1,1]. Случайная величина x сгенерирована функцией RND программы Excel.
  

      

Для таких полностью случайных процессов существует теорема, гласящая, что для них не существует стратегии, дающей прибыль в среднем. Точное доказательство есть, по-моему, в книге Дж. фон Неймана «Теория игр и экономическое поведение», здесь же приведу нестрогое доказательство:

 Доказательство:       Прежде всего, поскольку среднее приращение равно нулю, то есть бессмысленно покупать или продавать навечно (инвестировать), т.к. среднее значение любого члена ряда <S(n)>, будучи суммой n средних приращений, также равно нулю: <S(n)>=<x(1)+x(2)+x(3)+…>=<x(1)>+<x(2)>+<x(3)>+…=n*<x>=n*0=0.  

       Итак, инвестиции бессмысленны, поэтому  рассмотрим спекуляцию по некоторым правилам. Чрезвычайно важно учитывать то, что поскольку приращения независимы, то никакая информация из прошлого никак не влияет на будущие приращения.

      Рассмотрим вначале правила входа. Допустим, наша стратегия велит покупать после трех подряд дней роста (или при цене 100 рублей, или когда быстрая линия MACD пересечет медленную линию или любое другое правило). Поскольку будущие приращения никак не связаны с прошлыми ценами, то с точки зрения прибыльности сделки любой набор правил входа эквивалентен просто случайному входу, поскольку нет никакой связи между будущими приращениями цены и текущими значениями цен и индикаторов.

Итак, вход по любым правилам эквивалентен просто случайному входу.

     Рассмотрим теперь правила выхода. Здесь сложнее дать прозрачное объяснение, приведу лишь доказательства для нескольких типов выхода:

а) Выход по тэйк-профиту. Пусть мы вошли по цене S0. Выйти хотим по цене S1>S0, стоп стоит на цене S2<S0 (хочу отметить, что стоп есть всегда, в предельном случае, когда его не ставят, он находится на нулевой цене). Можно показать (см., например, Д. Письменный, Конспект лекций по теории вероятностей), что вероятность того, что цена дойдет до S1, при этом не коснувшись S2 (мы получим прибыль), равна (S0-S2)/(S1-S2). Вероятность того, что цена дойдет до S2, при этом не коснувшись S1 (мы возьмем стоп-лосс), равна (S1-S0)/(S1-S2). Тогда математическое ожидание прибыли равно (S1-S0)*(S0-S2)/(S1-S2)+(S2-S0)*(S1-S0)/(S1-S2)=0.

б) Скользящий стоп с фиксированным смещением. Пусть мы вошли в момент k (как показано выше, k может быть случайным, это ни на что не влияет). Выход осуществляется, если цена за один шаг падает на z рублей. Поскольку текущее приращение цен никак не связано с предыдущими ценами, то это уменьшение на z произойдет в абсолютно случайный момент, обозначим его как i. Математическое ожидание прибыли равно <S(k)-S(i)>=<S(k)>-<S(i)>=0-0=0.

в) Выход по указаниям индикаторов, использующих не одну, а несколько прошлых цен. Идеология доказательства аналогична пункту б), используется то, что очередное изменение цен, влияющее на показания индикаторов, никак не связано с предыдущими значениями цен и индикаторов.

Конец доказательства 

В принципе, данная теорема достаточно очевидна и без доказательств, поскольку если приращения независимы, то есть очередное приращение никак не связано с предыдущими, то никакого прогноза будущих  приращений сделать нельзя. Эта  теорема делает тестирование любых стратегий на полностью случайных процессах неплохим инструментом, поскольку позволяет:

1) отсеять системы, заглядывающие в будущее (они будут давать прибыль даже на полностью случайном процессе)

2) независимо оценить свойственную самой системе дисперсию, риски, а затем отделить случайную прибыль от неслучайной при тестировании на реальных котировках.

        Вроде все понятно, но неожиданно появилась интрига. Б. Д. провел тестирование своей стратегии на полностью случайном процессе (как-бы полностью случайном!) и утверждал, что в среднем его система показывает прибыль ( http://www.stockportal.ru/forum/index.php?showtopic=10963&st=50 ). Не верить тестированию Б. Д. нет резона, он человек грамотный. Как же так?  Дело тут вот в чем. Случайный процесс воспроизводился при помощи генератора случайных чисел, встроенного в Excel, по формуле x(n)=2*RND-1, где RND заявлена как случайная величина с равномерным распределением на отрезке [0,1]. Известно, что в компьютере ничего случайного нет, поэтому придуманы алгоритмы, генерирующие так называемые квазислучайные числа. В Excel использован простейший из них, неплохо применимый, скажем, к расчету интегралов методом Монте-Карло. Однако для тестирования хороших биржевых стратегий он плох. Это даже глазом видно, многократно строя подобные графики, замечаешь некоторую однообразность. Таким образом, Б. Д. удалось обнаружить неэффективность генератора случайных чисел Excel. Чтобы избежать этого затруднения, необходимо использовать более изощренные генераторы. Я давно использую так называемый вихрь Мерсенна, он отлично подходит для тестирования стратегий. Для хороших стратегий получаются четкие и приятные ответы—стратегия дает прибыль на реальных котировках и дает 0 на полностью случайном графике.  slide2.JPG 
Рис. 2: Реализация процесса S(n+1)=S(n)+x, где случайная величина x равномерно распределена на отрезке [-1,1]. Случайная величина x сгенерирована при помощи алгоритма Вихрь Мерсенна.
  
Если построить много графиков для экселевского стандартного генератора RND и для генератора вихрь Мерсенна, то видно, что вихрь отличается в среднем от RND. Вихрь дает более изрезанные графики. В результате трендовые стратегии, дающие прибыль а квазислучайном графике RND, не дают прибыль на случайном графике вихрь Мерсенна. Вихрь Мерсенна гораздо более хороший генератор случайных чисел, нежели чем стандартный генератор Excel. 

30 Responses to “Калибровка торговых стратегий на полностью случайных процессах”

  1. Андрей Says:

    Здравствуйте Анатолий, спасибо вам за-то, что пишите очень интересные статьи с доказательствами, поэтому вам можно доверять. Так вот вопрос: у меня МТС созданная в ВЛД 4.4. Подскажите, пожалуйста, как её протестировать в Excel на алгоритме “Вихрь Мерсенна”. Желательнее по доступнее. Заранее спасибо

  2. admin Says:

    Спасибо за высокую оценку :)
    По поводу вашего вопроса:
    У меня тоже стратегии обычно пишутся на велше. Тестирование их на полностью случайном процессе я проводил путем создания файла модельных котировок. То есть вы при помощи Excel создаете txt или dat файл (надо таких файлов родить много, чтобы при тестировании было хорошее усреднение по ансамблю), в котором содержатся ваши модельные котировки. Затем в велше создаете новый datasource с этими файлами и гоняете вашу МТС на этих данных. Теперь, как создать файл при помощи экселя. Я это делаю при помощи VBA. Вам необходимо алгоритмизировать описанный в статье полностью случайный процесс. Для этого нужен генератор случайных чисел. Можно использовать стандартную функцию Excel RND, можно вихрь Мерсенна. Ссылка для скачивания плагина Вихрь Мерсенна для экселя вот: http://www.numtech.com/documents/research/20030106/ntrand-freeware-numeric.php. Вот как-то так вкратце.

  3. admin Says:

    Из моей переписки с Максимом:

    _________________________________________________________________________________

    Анатолий, добрый день!

    Меня зовут Максим, я был до некоторого момента оппонентом Дмитрия (БД) в хорошо знакомой Вам ветке “Что это за графики?”, пока не осознал, что это религиозный спор.

    Наткнулся тут случайно на Ваш анализ этой ситуации: http://stockportal.ru/utkin/?p=113
    Признаюсь, вывод был для меня весьма неожиданный, я искренне считал, что Дмитрий просто накосячил с тестами.
    Из курса о датчиках, прослушанного лет 15 назад на матмехе, я вынес то, что современные датчики для простых (тем более, одномерных) ситуаций вполне хороши. В качестве примера плохого датчика приводилась байка о том, что был захардкожен в 70-х годах в какой-то IBM-вский мейнфрейм датчик, который при моделировании случайных точек в 5-мерном кубе (для примера. размерность не помню на самом деле) давал точки точно принадлежащие 9-ти гиперплоскостям. Пересказ байки весьма приблизительный, как там точно все было - лень искать и не так уж и важно, но подчеркивалось, что тем не менее одномерные и двумерные распределения он давал весьма сносные.
    К чему я это все? К тому, что мне весьма сомнительно, что спустя 30 лет экселевский датчик настолько плох в достаточно простой ситуации. В общем не ради полемики, а чисто ради моего любопытства, не могли бы вы ответить на вопрос, действительно ли Вы сравнивали трендовые стратегии на данных, сгенерированных встроенным датчиком из экселя и Мерсенном? Мне показалось, что из статьи это явным образом не следует.

    Спасибо,
    Максим

    _________________________________________________________________________________

    Здравствуйте, Максим!

    Я прогонял на случайных экселевских и Мерсенновских данных свои трендовые стратегии (ведь стратегий БД я не знаю :) ). Грубо, результаты такие: и там и там в среднем ноль, но разные дисперсии. У Мерсенна дисперсия моих стратегий в среднем больше, чем у экселя.

    Я не могу утверждать, что БД обнаружил неэффективность генератора эксель, но это вполне возможно. Экселевский генератор не нравится мне по следующим причинам:
    1) Глазом видно, что экселевские кривые броуновского движения слишком медленно отходят от нуля, а Мерсенновские–быстрее. То есть они–разные на глаз, что уже странно.
    2) Когда-то давно я строил модельные кривые броуновского движения при помощи монеты–идеального ГСЧ. Мне кажется, что Мерсенн похож на монету гораздо больше, чем эксель.

    Генератор случайных чисел эксель описан здесь http://support.microsoft.com/kb/828795/ru Грубо говоря, это один из первых алгоритмов, ему сто лет. Из моего личного опыта такой генератор весьма неплохо заполняет отрезок 0,1 и поэтому пригоден для расчетов Монте-Карло. Что-же касается корреляций между последовательными значениями–я не уверен, исследований этого я не делал, и вполне возможно, что БД ненароком провел такое исследование :)

    Анатолий

  4. Eldar Says:

    Вихрь Мерсенна не является криптостойким ГПСЧ и не совсем подходит для генерации ПС-чисел для вышеописанных целей. Сам алгоритм достаточно хорош, но…
    Лучше вместо него использовать ISAAC. Данный алгоритм, во первых также быстр, а во вторых - он криптостойкий и генерирует идеальные ПС-числа.

  5. admin Says:

    Я не специалист, поэтому поправьте, если что. Насколько я понимаю, криптостойкий генератор псевдослучайных чисел (ГПСЧ)–это генератор, образующий последовательность чисел, удовлетворяющую неким придуманным стандартам. То есть само понятие криптостойкости–условное. Но, естественно, криптостойкий генератор лучше некриптостойкого в том смысле, что у криптостойкого труднее выявить псевдослучайность.

    Генератор ISAAC изучу. Спасибо большое за ссылку. Если честно, я про него просто не знал :)

    Теперь общие слова. Данная заметка написана, в основном, для любителей искать в броуновском движении скрытые мотивы и смыслы. Знаете, есть такой класс людей–конспирологи. Ищут “хозяина тренда”, “кукловода”, и.т.д. А для того, чтобы визуально представить себе броуновское движение, понять его основные законы (в нем есть тренды, дисперсия пропорциональна времени, среднее равно нулю, и.т.д.), подойдет любой ГСЧ, даже эксельный :)

  6. Eldar Says:

    Если на основе сгенерированной последовательности у генератра можно вычислить его первоначальную энтропию (состояние), то значит можно узнать какую будущую последовательность генератор будет генерировать. У криптостойких генераторов это сделать невозможно (по крайней мере очень сложно - на сегодняшний день с сегодняшней вычислительной техникой). Касательно алгоритма ISAAC - на него уже производилась атака Мариной Пудовкиной - Вы можете изучить ее научный доклад. Он лежит на сайте автора авлгоритма. Но если честно, то я думаю что вролне сгодится и Вихрь Мерсенна, но точно не экселевский генератор :) Кстати у Вихря есть более улучшенный последователь - WELL (Well equidistributed long-period linear).

  7. admin Says:

    Ясно. Спасибо за квалифицированный комментарий! ISAAC и WELL обязательно поизучаю–хорошие ГСЧ дело нужное в хозяйстве :)

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    When you’re correct you can not become too revolutionary;
    when you’re drastically wrong, you can not end up being as well conventional.
    -Dr. Martin Luther Master, Jr.
    Practically nothing in all the planet will be much more hazardous as compared to genuine lack of edcuation and careful silliness.
    -Dr. Martin Luther Full, Jr.
    Show people, usually, and also tyranny along with oppressions regarding body and mind may fade away like mood in the birth of morning.
    -Thomas Jefferson
    The person whom says almost nothing is best prepared compared to gentleman who reads it will always be classifieds.
    -Thomas Jefferson
    The may with the men and women is the solely legitimate groundwork connected with just about any authorities, and to guard their free appearance ought to be each of our very first object.
    -Thomas Jefferson
    In case Now i’m completely wrong, say to myself I am just inappropriate.
    Never consult me personally about project.
    -Dr. Gregory Property
    Think outside the box, after that rape the particular pack, shit within the package along with
    mild the item on fire, subsequently rape the idea yet again.

    -Anonymous
    Of all awful adult males faith based undesirable males
    include the most severe.
    -C. Utes. Lewis
    We hold it that a little rebellion, on occasion, is an excellent matter, and as important from the political entire world as hard thunder storms
    within the bodily.
    -Thomas Jefferson
    People who would likely quit necessary liberty to get
    a little momentary basic safety, should have none liberty neither safe practices.

    -Benjamin Franklin
    Loyalty to be able to petrified viewpoint by no means broke some sort of
    string or maybe separated the human being spirit.
    -Mark Twain
    So long as the costa rica government is perceived as being employed by the
    advantage of the youngsters, folks will probably enjoyably put up with nearly every curtailment of liberty in addition to
    nearly every starvation.
    -Adolf Hitler
    It is the stupidest young children who will be essentially the most
    idiotic plus the stupidest grown-ups that are by far the
    most grown-up.
    -C. Ohydrates. Lewis, This Gold Easy chair
    If you chat reality, have one particular foot or so within the stirrup.

    -Turkish proverb
    The 1st Change offers assembled the wall in between religious organization in addition to condition.
    In which wall membrane has to be maintained high and also impregnable.
    We’re able to not necessarily approve your merest breach.
    -Justice Hugo Black color
    Meaning indignation: envy using a halo.
    -H. Grams. Bore holes
    We have to esteem the opposite fellow’s religion, although only inside the perception in order to the extent that we regard his theory that
    their better half will be beautiful along with his children clever.

    -H. D. Mencken
    No one is actually far more actually week, far more
    entirely a unwilling recipient, as compared to this individual who is
    going to neither decide on nor transform nor get away his protectors.

    -John Holt
    Many excellent realities begin because blasphemies.

    -George Bernard Shaw
    Any terrorist is usually somebody who has some sort of blast yet can?
    t have the funds for a great atmosphere pressure.
    -William Blum
    Disobedience, inside the little brown eyes of whoever has examine record,
    can be guy’s initial virtue. It is through disobedience in which development has become manufactured, by way of disobedience along with by way of rebellion.
    -Oscar Wilde
    Madness inside people is actually anything rare; however in groups, celebrations, nations around the world, along with epochs it’s the principle.

    -Nietzsche
    Disobedience may be the legitimate footing connected with liberty.

    The actual obedient has to be slaves.
    -Henry Brian Thoreau
    We have to dilemma the story reasoning of needing the all-knowing
    all-powerful God, exactly who creates defective people, after which blames these individuals regarding his or her own problems.

    -Gene Roddenberry
    The problem using the world is that this dumb usually are cocksure plus the intelligent are
    generally brimming with hesitation.
    -Bertrand Russell
    Patriotism is the best sentence that country is usually superior to all
    countries because you were given birth to from it.

    -George Bernard Shaw
    I doubt the individuals exactly who know so very well
    what exactly Goodness wishes those to do simply because I detect it usually
    coincides because of their unique dreams.
    -Susan T. Anthony
    Every state realizing it’s got the one true religion as well as the just satisfied program regarding govt, every single despising all of the people, each a good rear end and not suspecting the item.
    -Mark Twain
    Whenever fantastic improvements happen in history, as soon as excellent guidelines are participating, usually the majority are drastically wrong.
    -Eugene Versus. Debs
    Us citizens employed to roar such as elephants with regard to freedom; today we bleat such as sheep with regard to safety measures.
    -Norman Vincent Peale
    Every person’s concerned about stopping terrorism. Properly, you will find there’s not that hard technique; cease playing the idea.
    -Noam Chomsky
    Democracy is actually four baby wolves plus a lamb voting of what to get for meal.
    -Ambrose Bierce
    The history associated with liberty is usually a history of amount of resistance. The history associated with freedom is really a historical past involving disadvantages associated with governmental power, certainly not your increase from it.
    -Woodrow Wilson
    We suspend the particular petty thieves along with employ the fantastic ones to help public workplace.
    -Aesop
    Within is important associated with conscience, the law connected with bulk doesn’t
    have spot.
    -Mohandas Gandhi
    Of which which is often damaged with the reality need to
    be.
    -P. D. Hodgell
    The issues which exist these days is not resolved simply by the
    level of thinking that created these people.

    -Albert Einstein
    Innocence is lack of education inside a frilly white gown
    -Julio Gutierrez
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    Just about any trick can make a new concept, in addition to each fool will probably head the item.

    -Henry Mark Thoreau
    Logical arguments tend not to usually work with religious individuals; otherwise, there could well be no spiritual individuals.

    -Dr. Gregory House
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    -C. Ersus. Lewis
    In the event that Christ had been the following, there exists the one thing however definitely not possibly be:
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    It can be much better to recognize your universe
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    -Carl Sagan
    In case all of us are unable to believe pertaining to yourself, in the event we live resistant in order to issue guru, and then we’ve been merely putty inside the fingers of the people within strength. However, if the people are informed as well as style their very own viewpoints, and then those inside energy work for us all.
    -Carl Sagan
    With or maybe without religious beliefs, you would include excellent persons accomplishing nutrients and bad persons accomplishing unpleasant issues. Except for excellent visitors to do unpleasant points, in which requires religion.
    -Steven Weinberg
    In no way credit for you to malice that that is sufficiently explained by simply absurdity.
    -Hanlon’s blade
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    -Grey’s Rules
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    -Elbert Hubbard
    I acquire requested if i intellect as soon as someone melts away my personal music. I’m simply flattered that people wish to.

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    Folks ought not promise allegiance for the federal government; the costa rica government should pledge allegiance
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    -Michael Lind
    Arrange the directly to consider, with regard to
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    -Hypatia regarding Alexandria
    Wizard’s First Tip: Individuals are ridiculous. They will imagine a lay sometimes since they are interested to be true as well as because they are afraid it can be genuine.
    -Terry Goodkind
    Wizard’s Next Tip: The maximum cause harm to could result from the top goals.

    -Terry Goodkind
    Wizard’s 3rd Rule: Passion regulations explanation, Regarding superior or maybe pertaining to worse.
    -Terry Goodkind
    Wizard’s Last Concept: There’s wonder in honest forgiveness; from the forgiveness a person supply, although in addition within the forgiveness you will get.
    -Terry Goodkind
    Wizard’s Fifth Rule: Intellect what folks carry out, besides precisely what that they say,
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    -Terry Goodkind
    Wizard’s 6th Concept: Really the only sovereign it is possible to make it possible for to concept you can be reason.
    -Terry Goodkind
    Wizard’s Seventh Guideline: Life would be the foreseeable future, not way
    back when.
    -Terry Goodkind
    Wizard’s 9th Rule: Ought to have success.
    -Terry Goodkind
    Wizard’s 9th Concept: Some sort of contradiction cannot occur in reality.
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    -Terry Goodkind
    Wizard’s Tenth Tip: Willfully transforming in addition to the the reality is treason to help a person’s
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    -Terry Goodkind
    Wizard’s Eleventh Guideline: You are able to ruin those that talk the facts, however you cannot eliminate the reality per se.
    -Terry Goodkind
    I am moving through just how far better instruct each of our learners for you to precisely how effortlessly you can take action.
    -Tim Callahan
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    Every single creation imagines by itself to get a lot more intelligent compared to the one which gone prior to the idea, along with better compared to one which comes after the idea.
    -George Orwell
    Lets get rid of the word ‘childish’ mainly because grownups typically trigger the greatest problems.
    -Adora Svitak
    For just a short although very funny time period [Dance Dance Revolution] was your next big ‘Threat In your Young children.
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    -Cracked. com
    Childhood is usually wasted for the young : Guru can be wasted for the old
    -Jackie Ferro
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    Struggle can be prosperous older men defending their home by
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    -George Carlin
    As soon as one individual has a new misconception, it is called madness.

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    -Robert Michael. Pirsig
    Very first, whilst you were asleep yesterday evening,
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    Subsequent, almost all of that you do not supply a shit.
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    -Pastor Tony adamowicz Campolo
    Never become a spectator of unfairness or even battiness.
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    benefit.
    -Christopher Hitchens
    Suspicion empathy; desire dignity on your own among others.
    You shouldn’t be scared to be imagined haughty or even egoistic.
    -Christopher Hitchens
    Get the risk associated with pondering yourself. Additional contentment, simple fact, attractiveness, and also information arrive to your account because of this.
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