Questions tagged [probability]

A probability expresses quantitatively how likely an event is to occur. We often encounter probabilities as conditional probabilities which express how likely an event is to occur in light of certain (given) information.

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39 views

Probability of a certain financial instruments movement

I want to calculate the probability of a certain financial instrument moving above or below a certain threshold within a certain time frame. Let's say up 0.5 % within the next 4 hours. If we assume ...
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88 views

Performance of dollar cost averaging

If we're investing money into a stock $S$ at a continuous rate, $C$, what is the probability distribution of the amount we have invested? For example, modelling a stock as GBM without contributions, $ ...
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Probablity distributions of zero crossings in 1D random-walk

Consider a simple 1D random walk that starts at position zero, and each second changes position by either +1 or -1 with 50-50 probabalities. I know it is proven to cross zero infinitely many times, ...
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111 views

Alternatives to Kelly Criterion

I am preparing for Quantitative Trading interviews and I know that they basically require you to solve problems on the probability of winning in a given game and then they would ask you: How much ...
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A question in information strucutres and probability measures - How are they connected?

Suppose that $\mathcal{I}=(X,\sigma^{\mathcal{X}},\mu)$ is an information strucutre, which is a probability space, where $X=X^1\times X^2$ is the cartesian product of the individual finite sets of ...
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150 views

Requesting for price?

Just for education purpose. Assuming I have some trading ideas that involves the use of OTC derivatives but I may not be able to put them into practice due to regulatory issues and huge minimum ...
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Can I combine the exotics for a payout?

Can I combine a one touch option(barrier lower than current price) and no touch option(barrier higher than current price), so that I get a payout immediately only if the one touch barrier is breached ...
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149 views

What is the probability of touching point A first?

The probability of a stock touching a point A which is below the current spot price is 35%, and the probability of the stock touching a point B which is above the current spot price is 20%. How can I ...
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How does this imply that a Pareto optimum maximizes a weighted average of utility functions?

I'm reading a passage from Asset Pricing and Portfolio Choice Theory by Kerry Back, and I don't understand some of it. I would appreciate any help anyone could provide me. In the passage, Back is ...
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87 views

Compare errors in estimating a probability

Let $X_t$ be a geometric Brownian motion: $dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t$ with $W_t$ a standard Brownian motion. Given the intervals $[t_{j-1}, t_{j}]$ for $j\in {1,...,U,...,N}$, let $M_j$ ...
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108 views

Variance of Random Walk with Drift

For Gaussian random variables $\xi_t$ with mean $\mu_t$ and standard deviation $\sigma$, consider the random walk with initial condition $P_0=100$, such that \begin{equation} P_t=P_{t-1}(1+\xi_t). \...
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140 views

Expected stock price range using implied volatility calculated by Black-Scholes

What's the correct way to calculate the expected stock price range using implied volatility, without the simplifying assumption that the stock price follows a normal distribution?
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Identity of recent books on stock market & risk

Apologies if this seems out of place, but a couple years ago I read several popular books written in the last decade by a single author who was trying to disabuse readers of several fallacies ...
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88 views

Finding Option Probability Density Using Local Volatility from Dupire Model

This question is different than pricing using dupire local volatility model and Is Dupire's local volatility model path independent to recover historical option price? I also asked this on Math ...
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222 views

Trading a Bouncy Stock

I came across the following question and am trying to understand it better. I was hoping you could share your intuitions. For a given stock, you are certain that for the next 100 days, it will move ...
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is the concept of skew observed in fixed odds betting markets?

Bear with me if this sounds a little flippant, but this has got me curious. I know "sports arbitrage" is an active economic activity, although the arbitrage arguments, I think, are not ...
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Statistical significance in the context of financial data?

I understand statistical significance in the general sense: we take a sample from a population and compute some parameter from the sample to infer what is the propulsion parameter to some degree of ...
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178 views

Ito Lemma for Poisson Process

I'm new to stochastic calculus on jump processes and encountered a difficulty. I would appreciate some clarification from the community on the following question. Let $g_t$ be a $\mathcal{F_t}$-...
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305 views

Structuring and Customization

It seems complex derivatives in particular exotic options are not available at any retail broker. Can a regular retail trader get access to these instruments? Maybe through prop firms or banks? ...
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Does time remaining matter in NO Touch-ONE Touch probabilities?

I asked a question some days back and got an answer which I understand and make sense: Probability of touching short call strike and not touching touching short put strike of a short strangle? However,...
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33 views

How to use CAPM model to calculate expected value of portfolio?

Let's assume that vector $(R_1, R_2, R_3)$ has multivariate normal distribution $N(\mu, \Sigma)$ where $\mu = (2, 6, 4)$ and $$\Sigma^{-1} = \begin{bmatrix} 2 & 2 & 2\\ 2 & 4 & 4 \\ 2 &...
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101 views

Probability of touching short call strike and not touching touching short put strike of a short strangle?

I just came across a blog post. I believe the answer is a correct approximation: http://tastytradenetwork.squarespace.com/tt/blog/probability-of-touching-both-sides I modified the question in the post ...
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105 views

Drift Term in Black-Scholes Model Martingale

How would I prove that a Black-Scholes Model is not a Martingale if it has drift. In many cases it is just stated as a fact (without proof). For instance if Im looking at: $$dS_{t} = \mu S_{t} + \...
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158 views

Determining decomposition long bond yields via Fisher equation and the Expectations Hypothesis 2.0

I've started to get into the weed of UST pricing and was hoping to get some feedback on a "model" I thought about. It is presented in this blog post. https://nonlinearexpectations.blogspot....
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47 views

Risk-Neutral Probability in a Binomial Tree

This question is probably very simple and I'm just missing the easy solution but I'm a bit confused so I thought I might as well try ask here. I've been given this question: When I tried to calculate ...
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Why autocall probabilities are decreasing with time

I am wondering why autocall probabilities decrease with observation dates. Intuitively, I understand that as time goes, if the spot has not breached the barrier, it would need more and more kind of ...
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41 views

Fisher information of an Ornstein-Uhlenbeck process

I would like to compute the Fisher information of an Ornstein-Uhlenbeck process $X_t = Y_t - \beta Z_t$ where $Y_t$ and $Z_t$ are two time-series. My log-likelihood function in this case is: $$\...
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315 views

Probability Distribution at each Simulation Period using Geometric Brownian Motion

I am using the equation $S_t = S_0e^{(\mu-\frac{\sigma^2}{2})t+\sigma\epsilon\sqrt{t}} $ to simulate a financial metric at each $t$, where $t=1$ and $T=5$. Stated in plain English, I am trying to ...
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If arbitrage can happen exactly at one moment, is it really arbitrage?

There are many "interpretations" of what no-arbitrage means in mathematical finance, the most well known is no free lunch with vanishing risk: If $S=\left(S_{t}\right)_{t=0}^{T}$ is a ...
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63 views

Query on Lebesgue Measure

I am reading Steven E. Shreve's book, titled "Stochastic Calculus for Finance II". I have a query w.r.t. an example given in the book which is as follows:-
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64 views

How to prove that the following is still a Brownian motion [closed]

Given a Brownian motion $B_t$ on a filtered probability space, how can I prove that $W_t=B_t+\alpha t$ is still a Brownian motion, with $\alpha \in \mathbb{R}$? Is it always true? Do I need necessarly ...
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79 views

Simulation of Gamma process (distribution of increments)

The gamma process is a Levy process $X$, where $X_t$ has gamma distribution with parameters $at,b>0$ and density $$f\left(x\right)=\frac{b^{at}}{\Gamma\left(at\right)}x^{at-1}e^{-bx}$$ I want to ...
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176 views

Conditional probability of Brownian motion (with drift and scaling) hitting barrier

I am trying to understand the pricing of barrier options, and am considering the Brownian motion $\mathrm{d}X_t=a\mathrm{d}t+b\mathrm{d}W_t$, $a$ and $b$ constant. I am trying to: derive the ...
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150 views

Best way to trade probability density

From the option chain of a security, we can calculate the implied probability density at the maturity $T$ (assume the options are European. Now suppose we have our own view/prediction on the ...
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Testing the fit of an Ornstein-Uhlenbeck process

I would like to check if a time-series follows an Ornstein-Uhlenbeck process defined by an SDE: $$dX_t - \lambda (\mu - X_t) dt = \sigma dW_t$$ where $\lambda > 0$ is the mean-reversion ...
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148 views

Real world probabilities from option implied risk neutral density?

The work of Breeden and Litzenberger-formula (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2642349) gives us a risk neutral probability distribution of a stock price, depending on the option ...
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86 views

Are these two expectations the same?

I'm studying Markov Processes and Ito diffusion, I'm just at the beginning but I can't understand the different formulation of the expectation formulated in two different books. I'm talking about ...
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What day of a week should we pick something to happen to minimize it happening on the fourth business day of the month?

This is an extension of problem 3.16 in Mark Joshi's book. My answer is to avoid Thursday, and all other weekdays are equally good. The probability that the fourth business day is Thursday is 3/7 (...
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233 views

Optimal Strategy in 3 Dice Game

In a recent interview I received the following question (an optimisation/strategy game)...which left me a bit stumped. The rules of play, you start with 0 points, then: Roll three fair six-sided dice;...
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Why are prediction markets based on logarithms when a linear solution can suffice?

For example, take a binary outcome; A coin toss, heads or tails. If heads, then those that picked heads receive \$1 and tails receive \$0. To quote the prices for each bet Hanson's LMSR uses ...
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Risk neutral probabilities in binomial option pricing with discrete dividends — whose argument is correct?

In trying to discover more about pricing American options with dividend payouts, I found the the post linked here. I notice two disagreeing answers when it comes to determining the replicating ...
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Value the claim $(X-K)1_{X>K}1_{L<Y<U}$

Consider two correlated assets $X$ and $Y$ with marginals $f_X$ and $f_Y$ and linear correlation coefficient $\rho$. Assume a Gaussian copula, $C_{X,Y}(x,y,\rho)$, can approximate the joint CDF well ...
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Weighted and Probability Graph

I have a simple markov chain with A, B and C states. For each state I have a probability and beyond that, a value. So, for each state transition I have two informations: the probability of the ...
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Interpretation of Value at Risk

Let $X$ be a Loss random variable (Positive values of X represents Losses) and let $p \in (0,1)$. I know that the Value at Risk at level $p$ of $X$ is defined as: $$VaR_p(X) = inf{\{x \in \mathbb{R} : ...
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Given the density function of $S^{1}$ in one-period model, find the risk-neutral measure

Consider the one period market model $\left(\overline{\pi},\overline{S}\right)$ consisting of a risk-free asset $\left(\pi^{0},S^{0}\right)=(1,1+r)$ and a risky $\left(\pi^{1},S^{1}\right)$ Let $ r &...
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72 views

Martingale stochastic processes

Does anyone know how to do this question? A player whose initial holding is $N$ bets 1 on each game of a series of independent identical parts. He loses his bet whether he loses or he wins but, if he ...
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69 views

Physical Probability Measure vs. Risk Free Probability Measure (State Contigent Claims)

currently I am working on a problem regarding state contingent claims. I have 5 securities (one of the security is a risk-free security) and in the next period, these securities will end up in one of ...
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59 views

Expected Loss on a Portfolio, which contains an asset and a default protection contract, due to credit defaults

A portfolio consists of one (long) 100 million asset and a default protection contract on this asset. The probability of default over the next year is 10% for the asset, 20% for the counterparty that ...
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51 views

Calculation Expecting Credit Loss from a Portfolio

I have the following question: An investor holds a portfolio of 50 million dollars. This portfolio consists of 'A' rated bonds (30 million dollars) and 'BBB' rated bonds (20 million dollars). Assume ...
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67 views

Calculating the cumulative probability of default from recovery rate, yield and coupon rate

I have the following details: A 10-year U.S.Treasury strip has a yield of 6% and a 10-year zero issued by XYZ Inc, rated A by S&P and Moody's, has 7% (semi-annual compounding). Assuming a recovery ...

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