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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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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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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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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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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Understanding the solution of this integral
The following integral represents an expected value of a geometric brownian motion for $S_T>K$ (i.e. part of the Black-Scholes call option price):
$$\int_{z^*} (S_te^{\mu\tau-\frac{1}{2}\sigma^2\...
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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? ...
user57027
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Probability of exercise in the Black-Scholes Model
What's the intuition behind the fact that the limit of $\mathcal{N}(d_2)$, i.e. the (risk-neutral) probability of exercise, in the Black-Scholes Model tends to $0$ when the volatility tends to ...
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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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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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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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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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Pricing when arbitrage is possible through Negative Probabilities or something else
Also now asked about here: Is it fair in an introductory stochastic calculus/derivatives pricing class to ask for the price when absence of arbitrage is violated?
Assume that we have a general one-...
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Dumb question: is risk-neutral pricing taking conditional expectation?
Dumb question: is risk-neutral pricing taking conditional expectation? $\tag{1}$
In trying to recall intuition for risk-neutral pricing, I think I read that we should price derivatives risk-neutrally ...
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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 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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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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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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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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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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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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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 ...
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probability question about brownian motion
Assume $W_{t}$ is a standard Brownian Motion, calculate the the probability that $W_{t}*W_{2t}$ is negative, i.e., $P(W_{t}*W_{2t}<0)$. I find it tricky to calculate the probability.Thank you.
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Joint probability distribution only measures product sets?
According to these notes (top of p 133), "We say that random variables $X_1, X_2, \ldots X_n : \Omega \to \mathbb{R}$ are jointly continuous if there is a joint probability density function $p(x_1, ...
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Option and probability of finishing in the money?
This seems to be another easy question but I am a bit confused. I know delta is a proxy for an option finishing ITM. Delta also happens to be N(d1) in the BSM pricing model. N(d1) usually is pretty ...
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What are $d_1$ and $d_2$ for Laplace?
What are the formulae for d1 & d2 using a Laplace distribution?
user3232
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What exactly is/How exactly do we interpret the binomial model's Radon-Nikodym derivative?
Related: Dumb question: is risk-neutral pricing taking conditional expectation?
Maybe there's not quite an interpretation given Lewis' triviality result if $E^Q[X]$ is a real world conditional ...
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How to compute the conditional probability for a geometric Brownian process?
Somewhat embarrassingly I'm stuck with something very elementary.
I want to find the conditional probability of a stock movement (GBM):
$$\mathbb{P} \big( S_t \geq b \vert S_s \leq b) $$
for $ t &...
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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 ...
user56826
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Calculating probability of options with normal/lognormal distribution: does time make a difference?
I'm trying to calculate the probability of a calendar spread resulting in a profit at expiration, when estimating it is modeled as a lognormal distribution, by getting:
...
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Quantile normal and lognormal
Let's assume we have a normal distribution $X\sim \mathcal{N}(\mu,\sigma^2)$. In a normal distribution the quantile can be calculated as follows:
\begin{equation}
\Phi_X ^{-1}(p)=\mu +\sigma {\sqrt {...
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Confidence Intervals of Stock Following a Geometric Brownian Motion
In preparation for my Options, Future's and Risk Management examination next week, I have been presented with a series of questions and their answers. Unfortunately, my lecturer, one of the less ...
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Prove uniqueness, and prove $Y_t$ is a martingale by considering $dZ_t$ and $dL_t$
Suppose we are given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t \in [0,T]}, \mathbb{P})$, where $\{\mathscr{F}_t\}_{t \in [0,T]}$ is the filtration generated by standard $...
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Large deviations theory and extreme value theory
I'll enter into details of both, sooner or later, but for the moment I'm concerned about the differences (and relationships, if any) between these two theories. Can someone give me a brief, but still ...
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probability that the stock price is below the strike price
How can I prove that under the risk-neutral probability:
$\mathbb{P}[S_{t}<K]=-\frac{\partial{C}}{\partial{K}}(K,T)$
where
$S_{t}$ is the stock price, K is the strike price, C is the call ...
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Effects of random-generator-choice on derivative's price
There is a plethora of pseudo-random-generators out there. Some of them are definetly better and some of them severily underperform.
My standard tool is Mersenne Twister - when I need to generate ...
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How to test signifcance of a sharpe ratio
Let say you have measured a Sharpe Ratio of $S^*$. What is the simplest way (ie no fancy distributions) to do a hypothesis that this is different from $0$?
So $H_0: \text{ The sharpe ratio is equal ...
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Joint distribution from expectations
Given two random variables $X$ and $Y$ and let $K$ be a constant value. Assume the expectation $\mathbb{E}[X(Y-K)^{+}]$ is given for all possible values of $K\geq 0$. Is there a way to derive the ...
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Interpreting Autocorrelation as probability
I was recently asked:
Given a random time series of 1s and -1s. Eg of a sample = [1, 1, 1, -1, -1, 1, -1,..]. The autocorrelation of this series is Z. What can you say about the probability of a 1(or ...
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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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