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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How to fit KDE from existing probability density function values

I am working with options data, and I am using Breeden-Litzenberger formula to derive the risk-neutral terminal stock price PDF. After applying the formula, here is a scatter plot of strike price vs ...
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Call probability of a callable swap

For one call date, The call probability is just the probability that the swap rate for the remaining life of the swap is below the strike rate. This is easily obtainable in a normal vol model, it is : ...
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Default risk and stock price probability distributions [closed]

First of all, I realise this question might border on `meta-finance', so I'd be totally OK if it gets closed. Having said that, the question itself: Given a stock $S$, in the absence of default it is ...
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What is the meaning of the following mathematical equations? [closed]

Let's say that we have a discrete probability distribution, where $$ x_i $$ represents each of the possible outcomes (discrete set of possible outcomes), and $$ L $$ represents the expected value we ...
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Probability of touching barrier and stochastic interest rates

I am trying to figure out the impact of the stochastic interest rates on the price of barrier option. I was reading the book "FX Barrier Options" by Zareer Dadachanji and in the section ...
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Vol binomial tree

Suppose that we have a stock $X_t$ valued at 100 euros per share. At each time step the price can go up or down 1 euro with prob $1/2$. Assuming that interest rates are $0$ and the volatility of the ...
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calculating probability of a return below a specific value [closed]

assume a probability distribution with a mean of %10 and standard deviation of %1.5. In wanting to solve the probability being lower than %5, the normal distribution is written down and integrated as ...
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Probability distribution function for stock price given many parameters

First of all, I am not in the US market. I am trying to find out if I can do probabilistic analyses of stock price movements using the buy and sell summaries. Kindly let me then explain my problem. My ...
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Convert implied probability into real probability

In this article I have read that: A risk-neutral world is one where all investors are indifferent to risk and don’t require any extra risk premium for the risk they bear. In this world, all assets (...
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Quantile function for fractional Brownian motion (fBm)

If anyone could help me to understand if it is possible calculate the quantile function for fBm? I’ve checked several papers([1],[2],[3]), and although several works stated that it is centralised ...
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If $\Delta \log(V_{t})$ behaves like the increments of fractional Brownian motion, why do we model the rough volatility as follows

From Gatheral's paper, Volatility is rough and empirical evidence, it is clear that $\big\{\log(V_{t+1})-\log(V_{t})\big\}_{t}$ behaves like the increments of fractional Brownian motion $B^{H}$ with ...
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Inconsistency between simulation and the probability of a "stock" hitting take profit before stop loss

Let's assume a stock at time $t$ is worth $X(t)$. If the returns of $X(t)$ are i.i.d. and normally distributed,the probability of $X(t)$ hitting a value $H>X(t)$ before $L<X(t)$ is $\frac{H-X(t)}...
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Are there optimal portfolio theories than instead of the expected value they were based on the Mode of distributions

Are there optimal portfolio theories than instead of the expected value they were based on the Mode of distributions? During my engineer student days I saw the Markowitz theory for portfolio selection ...
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What is the Kurtosis of Returns in Geometric Brownian Motion?

Suppose that $dS_t=S_t(\mu\mathop{dt}+\sigma\mathop{dW_t})$ which has solution $$S_t=S_0\exp\left(t\left(\mu+\frac{\sigma^2}{2}\right)+\sigma W_t\right),$$ such that $W_t$ is a Wiener process, $\mu$ ...
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Term for a class of multi-asset options that determine the joint distribution of a set of assets?

For a single asset we can infer (in theory) the exact distribution of future outcomes via option prices: Either via option butterflies or the Breeden-Litzenberger formula. Is there a name for (one of) ...
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Option implied risk neutral distribution vs BKM risk neutral moments

I am doing some research on the option implied risk neutral distribution and methods calculate it, and so far have come across two ways to do so. The first way is through the Breeden-Litzenberger ...
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How much compensation need to take on risk?

Quant Firm Interview Question We roll three, 8 sided dice. If same face appears 3 times we win 80 dollars. We have a bank of 10,000 dollars. How much are we willing to pay to play? What if we increase ...
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What is the P-probability of an unhedged call-arbitrage to lose money at expiration

Assume that the Risk Neutral Price (under the $\mathbb{Q}$-measure) of an European Call Option with expiration date $T$ has a price of $F(S_0,0)$ at time $t=0$ in the single asset Black-Scholes model ...
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How to compute the combined probability of loss for 2 time series (consisting of historical stock prices)?

May I please ask the community's support with the following problem? I have 2 time series, with approximately 1000 observations each (same number of observations for both). They represent the daily ...
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Analytical evaluation of the following caplet-type product under lognormal assumptions

Let $n \geq 2$, and consider a tenor discretization: $0 = T_{0} < T_{1} < ... < T_{n}$ and associated forward rates evaluated at time $t$, as $L_{i}(t):=L(T_{i},T_{i+1};t)$ for any $i = 0,...,...
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Probability the stock price (following geometric Brownian motion) hits the upper boundary U before there is a retracement from the max by amount R?

I am looking for the probability that the stock price/Geometric Brownian Motion hits the upper boundary U, before there is a retracement (from the maximum price) that exceeds amount R. In other words,...
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Variance of the price from returns variance

Let's say that we have the variance of the daily return at $t_0$: $$\sigma_{r_{t_0}}^2=\text{Var}[r_{t_0}]=\text{Var}[\frac{S_{t_0}-S_{t_0-1}}{S_{t_0-1}}]$$ for price process $S_t$. Is there a way to ...
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Average probability of varying input data

New to the quant finance exchange. I am stuck with a question maybe someone could help me. In the table below I have calculated if price is up/down w.r.t to the open shown in the 2nd column with ...
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There is given a buy limit order. What is a probability that price will drop below this order?

Let's suppose that there is a buy limit order placed at time $t_0$ i.e. $(P,Q,t_0)$. Here $P$ stands for price of this order and $Q$ is the number of shares. Assuming that this order will be filled, ...
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Confidence in Sharpe ratio given performance

Suppose I have a strategy that I believe has a Sharpe ratio of X - not the Sharpe ratio of the backtest (this can be absolutely determined), but the ratio I expect it will actually take on over the ...
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Ito's lemma for option pricing with Levy-alpha stable drift

Consider $$dS=\omega\left(\Lambda-S\right)dt+\sigma_S S dW_t,$$ such that such that $W_t$ is a Wiener process, $\sigma_S$ is constant, $\omega: t\rightarrow\mathbb{R}$ represents anticipated drift and ...
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Why can’t delta’s be used to price double no touch options?

Here is the link to a MATLAB one touch option pricing calculator I used:OT I tried several inputs and I noticed that the one touch option price is approximately twice the delta of an equivalent ...
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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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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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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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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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1 vote
1 answer
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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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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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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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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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1 answer
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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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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 $\mathcal{F_t}$-...
3 votes
2 answers
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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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2 votes
1 answer
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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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2 votes
1 answer
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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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1 vote
2 answers
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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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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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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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