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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Lévy alpha-stable distribution and modelling of stock prices.
Since Mandelbrot, Fama and others have performed seminal work on the topic, it has been suspected that stock price fluctuations can be more appropriately modeled using Lévy alpha-stable distrbutions ...
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How useful is Markov chain Monte Carlo for quantitative finance?
Naively, it seems that Bayesian modeling, structural models particularly, would be quite useful in finance because of their ability to incorporate market idiosyncrasies and produce accurate ...
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Probability of touching
For a vanilla option, I know that the probability of the option expiring in the money is simply the delta of the option... but how would I calculate the probability, without doing monte carlo, of the ...
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Random matrix theory (RMT) in finance
The new kid on the block in finance seems to be random matrix theory. Although RMT as a theory is not so new (about 50 years) and was first used in quantum mechanics it being used in finance is a ...
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How are distributions for tail risk measures estimated in practice?
Let's say you want to calculate a VaR for a portfolio of 1000 stocks. You're really only interested in the left tail, so do you use the whole set of returns to estimate mean, variance, skew, and shape ...
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$\mathbb{P}$ vs $\mathbb{Q}$ Probabilities - Transitioning Between Measures
I'd like this question to definitively guide a practitioner to using both $\mathbb{P}$ vs $\mathbb{Q}$ probabilities in trading and research.
Let's take only one fact as given: if I have a risk-...
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How do you distinguish "significant" moves from noise?
How do you distinguish between losses that are within the normal range for day-to-day shifts and situations with a real potential for loss? The specific application I have in mind is pattern ...
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How to estimate the probability of drawdown / ruin?
A fairly naive approach to estimate the probability of drawdown / ruin is to calculate the probabilities of all the permutations of your sample returns, keeping track of those that hit your drawdown / ...
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Bayes' rule for conditional expectations (Proof review)
The Baye's rule for conditional expectations states
$$ E^Q[X|\mathcal{F}]E^P[f|\mathcal{F}]=E^P[Xf|\mathcal{F}] $$
With $f=dQ/dP$ - thus being the Radon-Nikodyn derivative and $X$ being
...
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How to calculate stock move probability based on option implied volatility and time to expiration? (Monte Carlo simulation)
I am looking for one line formula ideally in Excel to calculate stock move probability based on option implied volatility and time to expiration?
I have already found a few complex samples which took ...
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How does left tail risk differ from right tail risk?
How does left tail risk differ from right tail risk? In what context would an analyst use these metrics?
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open problems in mathematical finance
What are open problems in mathematical finance that use fundamental concepts of mathematics (functional analysis, geometry and topology, algebra and number theory etc.) and not data-driven.
I have ...
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How to estimate probability of default from bond prices?
How do you use bond prices/yields to infer probabilities of default? I would think of it as follows:
Create a relationship between default free (e.g., Germany) and defaultable (e.g., Greece) bond ...
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Quantum Mechanics and Economics... What
I was reading this paper: Financial Turbulence, Business Cycles and Intrinsic Time in an Artificial Economy.
The author has the model presented here: Quantum Evolutionary Financial Economics
But I am ...
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Probability density function of simple equation, compound Poisson noise
I would like to find the probability density function (at stationarity) of the random variable $X_t$, where:
\begin{equation*}
dX_t = -aX_t dt + d N_t,
\end{equation*}
$a$ is a constant and $N_t$ is a ...
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Do people use unbounded interest rate models, and what alternatives exist?
A simple interest rate model in discrete time is the autoregressive model,
$$
I_{n+1} = \alpha I_n+w_n
$$
where $\alpha\in [0,1)$ and $w_n\geq 0$ are i.i.d. random variables. When working with ruin ...
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What are some examples of Compound Poisson processes in insurance?
I'm writing the Bachelor thesis but I need some information. I need to find some practical examples and applications of the Compound Poisson Process in insurance. Does anyone have any good examples?
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Fixed income modeling
I am currently working on my research paper and trying to explain a two-dimensional variable: volume and instrument of corporate debt financing.
Independent variables that I believe must be included ...
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Distribution of Geometric Brownian Motion
Please let me know where I have been mistaken!
Let the SDE satisfied by the GBM $S(t)$ be
$$
\frac{dS(t)}{S(t)} = \mu dt + \sigma dW(t).
$$
Then, the underlying BM $X(t)$ will satisfy
$$
dX(t) = \...
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How do I estimate the joint probability of stock B moving, if stock A moves?
I have two stocks, A and B, that are correlated in some way.
If I know (hypothetically) that stock A has a 60% chance of rising tomorrow, and I know the joint probability between stocks A and B, how ...
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Arbitragefree Pricing: Q vs. P
I read that the Fundamental Theorem of Asset Pricing states, that a market is arbitrage-free if and only if there exists an equivalent martingale measure Q, under which the discounted asset price ...
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If stock A has a 60% chance of rising, and stocks A and B have 80% correlation, what is the chance of stock B rising?
As in the subject, I'm interested in a math puzzle of sorts:
If stock A has a 60% chance of rising, and stocks A and B have an 80% correlation, what is the chance of stock B rising?
Would it be ...
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Distribution of hitting time of the integrated CIR process
If an increasing process $X_t$ has a known Laplace transform $\mathbb{E} e^{-s X_t} = m_t(s)$, define its hitting time $\tau$ to some level $B$ to be
$$
\tau = \inf\{ u > 0 : X_u \geq B \}.
$$
Can ...
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Strategies for Liar's Poker
This question is only tangentially related to quantitative finance. Scott Patterson's book The Quants describes how a quant at Kidder Peabody figured out a strategy to playing Liar's Poker in the late ...
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t-statistics for the mean return, using Newey-West standard errors
I have seen that in several papers, where the aim was to evaluate the performance of a certain investment strategy, they use t-statistics to test for significance in the results. However, this seems a ...
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Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative
The problem:
Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in ...
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What distribution should I apply to estimate the likelihood of extreme returns?
Say I have a limited sample, a month of daily returns, and I want to estimate the 99.5th percentile of the distribution of absolute daily returns.
Because the estimate will require extrapolation, I ...
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Transition densities in the Heston model
Knowing the Characteristic function $\Phi_{T,t} = \mathbb{E} [ e^{i u S_T} | S_t, V_t]$ (or equivalently, the Laplace transform) of an affine process, it's possible to know the distribution of the ...
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2-state HMM / ARMA process?
I have issues with this problem:
Let $\{X_t, t\in \Bbb N\}$ be a 2-state stationary Markov chain, with transition $M$ (and $M(1,2)\neq 0 \neq M(2,1)$), let $\{W_t, t\in \Bbb N\}$ be a strong Gaussian ...
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Probability Puzzle from a Quant Interview
An urn contains 20 balls colored each of the 7 colors of the rainbow (140 total balls). We select balls one-by-one without replacement. Given that in the first 70 draws we selected 5 more red balls ...
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Simulating the joint dynamics of a stock and an option
I want to know the joint dynamics of a stock and it's option for a finite number of moments between now and $T$ the expiration date of the option for a number of possible paths.
Let $r_{\mathrm{s}}$ ...
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Prove $E_{\mathbb Q}[X_t | \mathscr F_u] = X_u$ given $Y_t$ is a martingale
Edit years later: No idea why I'm upvoted. I actually am not sure how I'm correct. But maybe I haven't forgotten conditional expectation as much as I thought I have.
We are given a filtered ...
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Calculate the expectation of a shift CDF
Suppose $X$ is a normal random variable with mean 0, and variance $\sigma^2$. $F(x)$ is the CDF(cumulative distribution function) of a standard normal random variable(mean 0 and variable 1), how to ...
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Difference betweem martingale property and adapted filteration
What is the difference between a random process that is adapted to a filteration and one that had the martingale property. It seems the two notions are quite similar and would be helpful to construct ...
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KMV-Merton Probabilties of Default vs Moody's EDF
Moody's used to publish probability of default estimates from their Moody's EDF model, but they have temporarily discontinued it. I understand that the Moody's EDF model is closely based on the Merton ...
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Normally Distributed Returns Become Leptokurtic Due to Compounding
I was running a bunch of simple simulations in excel the other day in excel. Using the NORM.INV(RAND(),0,1) to simulate daily stock returns I noticed that the more compounded the returns, ie, the more ...
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Heuristics for calculating theoretical probabilities of being ITM at time T for listed options
I'm looking for a heuristic way to calculate the probabilities of being in the money at expiry for non-defined risk options combinations (listed options).
I use delta as a proxy for this probability ...
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What is the connection between the risk neutral implied density and the real world density?
I understand that we can use option prices to imply volatilities and ultimately to imply a risk neutral density. I also understand that this implied density is not the same as the "real world density"....
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How do I calculate probability distribution of stock prices given option prices?
I'd like to calculate a probability distribution for prices given the option prices for that stock? Any ideas how to do this?
My desire is to do this daily and then see how the price PD changes over ...
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How to fit probability density function from sample moments?
If I have calculated the sample mean, variance, skew and kurtosis of a set of data, how would I go about fitting a probability distribution to match these moments (i.e. choosing a probability ...
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Modeling Call Price w.r.t. Strike w Models that Capture Vol Smile
I am trying to model $C(K)$, the price of the call $C$ as a function of strike $K$. Because this is tied to Prob ITM - and in fact the probability density function of that particular expiration (https:...
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How can we have negative probabilities in finance? Can we have negative payments in bonds? If not, how else can we have negative probabilities?
In Half of a Coin: Negative Probabilities, the author mentions bond duration.
Suppose we have payments at times $t = 1,2,...,n$ denoted respectively by $R_1, R_2, ..., R_n$ and the discount factor is ...
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on "recovering probability distributions from option prices" - how to subtract influence of stochastic volatility?
This is based on a 1995 paper by Rubinstein/Jackwerth by the above title where the authors produces a distribution of stock prices inferred from option prices. But their approach only produces a joint ...
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How do you synthesize a probability density function (pdf) from equally weighted price data?
What I'm working with:
I have a collection of prices that has very few to no repeating values (depending on the look back period) ie each price value is unique, some prices are clustered and some can ...
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Can the concept of negative probabilities be used to price a call option?
Edit: I'm a dumbass. The thing below is supposed to be just the motivation of asking. I want to ask for below and in general, hehe.
Assume that we have a general one-period market model consisting of ...
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Definition of orthogonality and independence for a stochastic processes
Somehow I can't find the explicit definition of when two processes are supposed to be orthogonal or independent anywhere. I think orthogonality and independence should mean the same thing in this ...
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pricing of heat rate-linked derivative
It's a simplified model.
Suppose $U_t$ is a random variables subject to Lognormal($x_1$, $z_1^2$)distribution. $V_t$ is a random variables subject to Lognormal($x_2$, $z_2^2$)distribution. Suppose ...
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Do futures follow physical or risk-neutral distributions
I've spent a while looking for an answer to this question and while I feel it is a simple question I have not found an answer.
I know prices of option contracts follow an implied, risk-neutral ...
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What is the distribution of Brownian Bridge over a given time interval?
I know from Karatzas & Shreve (1991) that a Brownian Bridge $B(t)$ from $a$ to $b$ on time interval $[0,T]$ satisfies:
$$B(t)=a(1-t/T) + b*t/T + [W(t) - W(T)*t/T]$$
where $W(t)$ is a standard ...
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Probability in different measures
I'm having some troubles understanding a problem.
The problem:
"Show how a measure change can be used to estimate the probability for $Y > 100$ when $Y \sim \mathcal{N}(0, 1)$.
The book I'm using ...
