![]() Here we’ve just used the total probability formula for the probability of B to get above a, representing it as a sum of two conditional probabilities – the probability of B to get above a conditional on the maximum of B to be above a, and conditional on the maximum of B to be below a. To compute this probability let’s consider the probability of the BM itself to be above a at time t: What is the probability that this process reaches certain level a at time t: Having established that simple fact we can now compute several important probabilities.Īs a maximum value of the BM on the interval. So the events and are equally likely, i.e both have probability 1/2. So by symmetry (about a), the path of BM during the time interval is just as likely to lead to as to. But BM, having stationary and independent Gaussian increments, will continue having them after hitting a. Then the value of B(t) is determined by where the BM went in the remaining units of time after hitting a. we know that the BM process hit a at some point before t. Suppose that we define a stopping timeĪs the first (hence the min) time when the Brownian motion (BM) B that starts at 0 hits the value a. The basic intuition behind the reflection principle is very simple. Reflection principle, very simple thing by itself, is used in treating all kinds of exotic options with payoffs depending on the price process reaching certain maximum in a time span, breaking through a barrier and other such things. The correct way of figuring out the expected value of the stopping time is by figuring out its distribution. Strictly speaking this is not a valid derivation because the expectation of Laplace transform can be used as moment generating function only if the moments are finite, and they are not in this case. Remembering that expected value of the Laplace transform of a random variable is the moment generating function of that variable we can compute the expected value of as Strictly speaking we need to consider the expression and take, but we drop the technicality here. If we stop it when W hits a for the first time, that process will also be a martingale according to the stopping theorem. ![]() If we have a Brownian motion, the geometric Brownian motion The theorem basically says that the stopped martingale is also a martingale. We will use the relationship between Brownian motion and geometric Brownian motion (GBM) and optional stopping time theorem. Noting that this expectation is the moment generating function for and as such will give us information about the moments of the distribution. We’ll start by considering the expected value of We are looking for the first passage time – the first time a Brownian motion process that starts from 0 hits the value a: There is a number of question dealing with Brownian motion stopping time (or hitting time or first passage time), this particular one can be formulated as “what is the expectation of ” Another version of the same question – “what is the distribution of the Brownian motion hitting time ” is considered in the next post. If you have a question, but you don’t know an answer, were asked something at an interview that you did not understand, or have an interesting question with an answer that you think will help others, please share it with me. If you wish to refresh your knowledge of a subject category (as opposed to just finding an answer to a specific question), from any post in the category follow the link to the First post and then read the sequence using the Next link in each post. The order of the list corresponds to the order one would follow when studying the subject of the category from square one. Inside each category the entries will be organized as a linked list. ( Reread is a key word here – if you have not red them before this blog will likely not help you).Įach entry will be dedicated to a question from a category, such as calculus, probability, etc. So I decided to create this blog, to help myself, and hopefully, others, to prepare for an interview without having to reread a bunch of textbooks on calculus, linear algebra, stochastic calculus etc. ![]() The questions are from all kinds of topics, and it is my personal experience that people seldom remember everything and often stumble at questions that should not have presented any difficulty for them if they had brushed up the topic before the interview. These are the questions that you can expect to be asked if you interview in a bank, asset manager or a hedge fund for a position of quantitative analyst, quantitative modeler, model validation analyst, quantitative researcher and such. In the process of interviewing people and being interviewed myself for various quantitative financial position I collected a number of “typical” question (many dozens of them, in fact). ![]()
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