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.

Filter by
Sorted by
Tagged with
1
vote
1answer
116 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?
0
votes
1answer
53 views

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 ...
2
votes
1answer
65 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 ...
3
votes
1answer
205 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 ...
1
vote
0answers
44 views

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 ...
0
votes
1answer
93 views

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 ...
3
votes
1answer
138 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 $\mathcal{F_t}$-...
1
vote
2answers
251 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? ...
2
votes
1answer
78 views

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,...
0
votes
0answers
32 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 &...
2
votes
1answer
92 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 ...
1
vote
2answers
82 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} + \...
1
vote
0answers
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....
0
votes
0answers
45 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 ...
0
votes
0answers
37 views

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 ...
0
votes
0answers
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: $$\...
0
votes
0answers
128 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 ...
4
votes
0answers
84 views

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 ...
0
votes
0answers
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:-
1
vote
1answer
62 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 ...
1
vote
1answer
77 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 ...
3
votes
1answer
165 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 ...
1
vote
1answer
140 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 ...
2
votes
0answers
79 views

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 ...
2
votes
1answer
117 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 ...
0
votes
0answers
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 ...
0
votes
1answer
45 views

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 (...
2
votes
1answer
176 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;...
1
vote
0answers
48 views

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 ...
1
vote
0answers
76 views

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 ...
0
votes
0answers
30 views

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 ...
0
votes
0answers
41 views

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 ...
1
vote
0answers
94 views

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} : ...
0
votes
0answers
53 views

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 &...
0
votes
0answers
71 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 ...
0
votes
1answer
68 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 ...
0
votes
1answer
50 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 ...
0
votes
1answer
46 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 ...
0
votes
1answer
52 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 ...
0
votes
1answer
120 views

Is the portfolio return distribution a weighted combination of individual asset return distributions?

We know that the portfolio expected return is a weighted sum of the individual assets' expected returns (asset means). We also know that the portfolio variance is a weighted combination of the ...
2
votes
0answers
40 views

EMM, Supremum and Expectation

I asked this question on MSE recently. https://math.stackexchange.com/questions/3922347/supremum-and-expectation I want to prove this when $\mathcal{M}$ is a set of equivalent martingale measure. ...
1
vote
1answer
195 views

What's the interpretation of the probability of default implied from CDS spreads?

What's the time horizon of the probability of default implied from a CDS spread? Given CDS = PD*(1-R), if I use a 5yr CDS spread in the formula, is the implied PD the probability that that name ...
2
votes
2answers
333 views

Proving $\mathbb{P}(S_t<0|S_0=s_0)=0$ for Geometric BM

I am trying to prove that for the geometric Brownian motion of a stock $\textrm{d}S_t=\mu S_t\textrm{d}t+\sigma S_t\textrm{d}B_t$ with strictly positive constants $\mu$ and $\sigma$ and and $S_0=s_0&...
2
votes
1answer
166 views

Imperfect Competition among Informed Traders - Back, Chao and Willard

The following assumptions are part of the paper of Back, Chao and Willard and I can not solve for the statistic that is denoted as $\phi$ in the sequel. I would be glad if anyone could help me. Below ...
1
vote
0answers
54 views

Concentration of measure phenomena in financial mathematics

Concentration of measure is a small area of statistics and probability theory that proved inequalities regarding the statistical properties of sets of random variables that exclude one of those random ...
1
vote
1answer
93 views

sub-Gaussian random variables in financial economics

Unlike financial time series that typically possess fat tails, sub-Gaussian random variables have strong decay in the tails of their distribution. do sub-Gaussian random variables or processes appear ...
0
votes
0answers
45 views

Escape Dynamics in financial economics or time series

These slides describe escape dynamics to be a type of, or having some relation to, rare event(s). Black swan events in business cycles was also included under the definition of rare events. My guess ...
3
votes
0answers
165 views

Large deviations theory in finance

In probability theory, the theory of large deviations concerns the asymptotic behavior of remote tails of sequences of probability distributions. A related post says: Large deviations theory is ...
3
votes
0answers
210 views

Does the Shannon entropy of stock returns change over time?

Shannon entropy, $H(X) = -\sum_{i=1}^n p(x) \ln p(x)$ is a probabilistic measure of randomness or disorder within a random variable's probability distribution or histogram. If we take rolling window ...
3
votes
0answers
43 views

Characteristic function of time-changed Levy processes

Let $X_t$ be a Levy process, and $Y_t$ be a subordinator i.e. process with nondecreasing trajectories. I have to find characteristic function of $X_{Y_t}$. I know that I have to calculate: $$E[e^{iuX_{...

1
2 3 4 5
7