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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Bayesian trade probability with factors
I have a strategy Y which is influenced by some factors X1, ..., Xn (for example asset volatility, distribution of macroeconomic factors). At moment t0 I have historical distribution(prior) of X1, ...,...
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Expectation of the product of two Brownian motions [closed]
Could you please let me know the steps to follow to get to the solution?
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Hedging Value-Financial Mathematics
EXERCISE
We consider a free from arbitrage financial market $(Ω,F,P,S_0,S_1)$ with $α<S_0^{1}\cdot(1+r)<β$,where $$0<α:=min_{ω \in Ω} S_1^{1}(ω), β:=max_{ω \in Ω}S_1^{1}, α<β$$
Let C be a ...
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Equivalent martingale measure exists if and only if $a < S_0^1(1+r)< b$
Exercise :
We consider a market of one period $(\Omega, \mathcal{F}, \mathbb P, S^0, S^1)$, where the sample space $\Omega$ has a finite number of elements and the $\sigma-$algebra $\mathcal{F} = 2^\...
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What is the Probability Distribution of Max-Drawdown?
How to obtain the probability distribution of Maximum Drawdown, starting from the probability distribution of Daily Returns? Here the details:
Suppose I have a time serie of N=1000 daily returns.
...
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The duality of the free energy and relative entropy used to deduce deduce the stochastic game between an agent and the market
I am reading the article Pricing via utility maximization and entropy by Richard Rouge and Nicole El Karoui. They talk about the relative entropy of a probability measure $Q$ with respect to the ...
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Conditional Probability - Geometric Brownian Motion
Background
I am trying to find a way to price a variant of a gap option by using closed-end expressions. What makes this option a bit tricky is that it can be exercised at four predetermined dates (t=...
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How skew in vertical put spreads change the payoff?
An spx four strikes wide Put Spread from at the money has a payoff ratio of 1 to 2 meaning if the Premium on the spread is \$10 your reward is \$20; yet the corresponding Call Spread with the same ...
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From Butterfly Price to Probability of $S_T$ Falling within a Range
If a butterfly in the limit represents a probability (by the Breeden-Litzenberger result), what can be said about the relative likelihood of a random variable $S_0$ from the price of a vanilla-option ...
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Probability of Implied Volatility Move [closed]
I want to see the probability of Implied Volatility of an underlying moving up or down from its current position. Would it just be 50% probability of going up and 50% of it going down? Because I've ...
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Detecting butterfly spread arbitrage for American options through European option prices
It's easy to demonstrate that if European option prices are concave with strike, then an arbitrage exists. For example, the risk-neutral probability density is the second derivative of European put ...
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How to determine the default probability of a county in a bond that is not in its native currency?
Disclaimer: This post is cross posted in here also.
Consider the following case:
Country P uses the currency Euro and gives p percent interest on a one year bond issued in Euro.
Country Q uses the ...
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Prove that $F(s,x_0)=0$, $F(t,x)=1$ and $\frac{\partial F}{\partial t}+\frac{1}{2}\frac{\partial^2 F}{\partial x^2}=0$
Using the Dynkin's formula, prove that $F(s,x_0)=0$, $F(t,x)=1$ and $\frac{\partial F}{\partial t}+\frac{1}{2}\frac{\partial^2 F}{\partial x^2}=0$
where $F(s,t)=2\int_{x-x_0}^{\infty}\frac{1}{\sqrt{2\...
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How are Risk indices linked to Physical Trading returns?
Ref to my previous question here:
Physical trading spot transaction analysis-Quantified
I have been able to narrow down my aim to defining a physical trading strategy P&L. My question is, how ...
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Girsanov's Theorem for Multiple Risky Assets
Girsanov's theorem provides the measure transformation from probability measure P to Q such that-
$dW_t^Q=dW_t^P+\lambda dt\implies \xi_tW_t^Q$ is a martingale under the P measure where $\xi_t=e^{-\...
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credit risk - marginal default probability
I have been working on an assignment trying to calculate marginal/conditional probability of default. Using a logistic regression framework, I was able to compute the 12-month unconditional PD for ...
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Uniqueness of data metric [closed]
Is there a metric that calculates "uniqueness of data"? For example if i have two sets of 200 observations,
DataSet 1 has 70 unique values but 4 values take up the next 130 observations.
DataSet 2 ...
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Distribution in Heston
$$dV_t=-k(V_t-1)dt+ \epsilon\sqrt{V_t}dW_t$$
$W_t$ is wiener process and the rest is just some parameters.
For $T_{i+1}>T_{i}$ how do I find the expectation and variance of $V_{T_{i+1}}$ ...
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How to compute SABR's probability density function
I am trying to compute the probability density function of the forward rate implied by the SABR formula approximation in order to see how the density implied by the approximation has negative ...
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Can you determine USD swap rate movement probability from OTM swaption premiums?
E.g., the USD 1y x 4y swap rate is currently 2.84%.
ATM receiver swaption , European exercise is currently at ATM premium of 1.15% while swaption premium at strike 1.5% is 0.15% or about 90% lower ...
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conditional probability of default
I would like to ask the following question.
I would appreciate if someone could help me out.
On what argument is based that states that conditional default rates ( loans of corporate borrowers) ...
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Stochastic Calculus: How to test for dependency of random variables
If I let $g(x)$ be a deterministic function of a real variable $x$ and define $X(t)$ as:
$$X_T=\int_{0}^{T}f(u)dW_u$$ with $W_t$ being a wiener process.
For $s<t$, Will $X_s$ and $X_s-X_t$ then be ...
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Is a wiener proces measurable? (exercise from Bjork)
I will claim $$E[W(T) \vert F_t] = 0$$ for $t<T$. Anyway, in an exercise in Bjork the results requires that $$E[W(t) \vert F_t] = 0$$ But why? Isn't $W(t)$ measurable at time $t$ and hence not ...
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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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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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When predicting Forex price using HMM what, typically, are the states and what are the observations?
I understand their abstract definition but having trouble applying the HMM method to Forex prices. What should the observations be? Then what should the states be (like "hot", "cold", etc.)?
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Spot Interest Rate at time $t$
I know that the general model for the dynamics of the spot interest rate is
$$dr(t)=\mu(r,t)dt+\sigma(r,t)dB(t)$$
My question is, if $P(t,T)$ is the bond value at time $t$, how would I derive $dP$?
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Deriving $dR(t)$ For Reverse Exchange Rate
Say $Q(t)$ is the exchange rate at time $t$.
It's the price in domestic currency of one unit of foreign currency and converts foreign currency into domestic currency.
The model for the dynamics of ...
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R Calculate future price range and plot the result
First I want to say that I've read this post (How to calculate future distribution of price using volatility?) but it doesn't help much.
Here is what I'm trying to do (values are not real)
Let's ...
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Computing the PDF of the sum of N moves of an empirical PDF for USDJPY 1-minute moves
Per-minute tick data for USDJPY is available here. Suppose we download this file to usdjpy.txt and then save it into a Numpy array in Python 3 as follows:
...
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Am I calculating my Kelly Criterion correctly?
I'm taking a look at my trading history over a particular time period and have 500 trades on with an win rate of 82%. My average win is $W$. My average loss is $L$. So am I correct in assuming the ...
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Girsanov Transform and Likelihood Process Domestic to Foreign
Working two exercises relating to $Q^d$ and $Q^f$. I'm comfortable working with transforms and likelihood processes on a risky asset between $Q$ and $Q^s$, and also on an exchange rate $X$ between $Q$ ...
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What's the relationship between $VaR_{\alpha}(X)$ and $VaR_{1-\alpha}(X)$ if the probability distribution function is not symmetric?
If the probability distribution function $f(x)$ is not symmetric, is there any relationship between $VaR_{\alpha}(X)$ and $VaR_{1-\alpha}(X)$?
Here, $VaR$ is defined as
$$
VaR_{\alpha}(X) := \inf\...
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Probability in different measures
I'm having some troubles understanding a problem.
The problem:
"Show how a measure change can be used to estimate the probability for $Y > 100$ when $Y \sim \mathcal{N}(0, 1)$.
The book I'm using ...
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Can a Kelly Criterion Percent be very high?
This is my personal record trading options (selling spreads) over a certain time period:
Win Rate: 83.94%
Average Win: $299
Average Loss: $1,181.40
The formula for the Kelly Criterion is:
$$
f=\frac{...
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Subadditivity of Expected Shortfall
I am able to see why Expected Shortfall will be subadditive for normal distribution or a uniform distribution. I am trying to prove the result for any generic distribution. I came across many proofs ...
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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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Quantile with periodic investing
Short Version
Can I get a quantile of such an expression?
\begin{equation}
\sum_{k=1}^{n} A_k\exp(\mathcal{N}(t_k\mu-\sigma\sqrt{t_k}/2,\sigma)))
\end{equation}
I know I can do it for one part of ...
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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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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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Laplace Exponent of a Jump-Diffusion Process
I'm currently reading a paper (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2543702) which uses the following process to describe the dynamics of a firm's asset value:
\begin{equation}
V_t = ...
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Solving for roots of a stochastic pay-off function
I have a pay-off function for a derivative which is defined by the Heaviside difference between $G$ and $B$ shifted by $-F$. To find the value of $V_{t=0}$, I need to find $\tau$ when $\frac{dV}{dt} = ...
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Binary probit model: relevant which outcome is 1?
I'm currently working on predicting bear and bull phases with a dynamic probit model in the form of $y_t=\beta_1X_t+\gamma_1y_{t-1}+\epsilon_t$. So far I've written all my code in matlab and it works ...
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Is it possible to calculate implied probability of >=X% return based on implied volatilities from options
My question is: Is it possible to imply either the upside or downside (one sided) probability from looking at implied volatilities of stock options?
Let's take an example: say you had Stock A at $50, ...
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How is the "probabilities sum to $1$" rule enforced in betting exchanges?
Suppose that I am interested in a market on a betting exchange for the outright winner of some event, with three competitors, $A, B$ and $C$ with corresponding probabilities of winning $a, b$ and $c$. ...
Alex Seaton
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Is this the right formula to use implied volatility to gauge probability of a stock being within a certain range? [duplicate]
I read online somewhere, and I can't find it now, that to find the probability of a stock hitting a certain price within a certain time frame, we can use Implied Volatility:
...
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First passage probability formula
I recently read an article and they provide a formula for the first-passage probability as
$$Z = {1 \over \sigma }\left[ {\log S/{S_t} + (r - {1 \over 2}{\sigma ^2})t} \right]$$
${{S_t}}$ value of ...
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Expectation of N(d2)?
I am trying to find out the Pricing Equation for certain type of Options under Risk-Neutral pricing. This is the equation I am getting, but I am not sure if this can be solved or not. Any help is ...
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First passage probability in american option pricing
In an article i recently read (The American Put Option and Its Critical Stock Price by David S. Bunch and Herb Johnson link) the authors presented this formula as something very general and as common ...
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Probability of default
I have to calculate probability of default (PD) rates for our clients (I am working in a Bank) based on clients' financials. Could you, please, advise me how to do that?
I think we have two Options:
...
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