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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Explanation on the application of CLT in bionomial tree model
We have a stock price binomial tree model of $n$ steps, with step length $\Delta t=T/n$, stock price volatility $\sigma$ s.t. $u_n=e^{\sigma\Delta t}$ and $d_n=1/u_n$, and the risk neutral probability ...
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Forward price - T-forward martingale
I have a problem figuring out some of the calculations in the book: Fixed Income modelling
In the chapter on forwards the author makes an argument that the forward is a martingale under the T-forward ...
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Cox-Ross-Rubinstein - getting volatility
i have exam coming on financial engineering, and need help asap with this thing.
Basically there's a European put option ex dividend. We know that the stock price is $S_t = 85$, the exercise price is $...
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Why is a martingale a risk-neutral measure
We have the risk-free valuation formula $$ \pi^X_i = B_T^{-1}B_iE_{P^*}[X|F_i]$$
Where $P^*$ is an equivalent martingale measure.
Why is this martingale measure considered risk-neutral? All I know is ...
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Why do we have zero drift when switching to a martingale measure?
I am told that this is a consequence of the Girsanov theorem, yet I do not see how it it is.
Consider some standard model with $dS_i = \mu S_i dt + \sigma S_i dW^P$. Let $Q$ be an equivalent ...
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Portfolio diversification on default risk
A portfolio of 13 different companies have loans. Company $i$ default on their loan with probability $p_i$ and survive with prob $q_i=1-p_i$. Let $Y_i=1$ denote default. Question: How could I get to a ...
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Probability default calculation
I want to calculate default of probability of internal ratings for a particular bank. I have only the following data:
Liquidity Ratio
short-term assets / short-term liabilities = 2.6
Profitability ...
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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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Estimating Recovery Rates
What are some methods for estimating recovery rates for an entity? For example, say I am trying to find the recovery rate that would be used to price a single name CDS on JPMorgan.
The true ...
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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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Implied Probability Density with Puts
The second derivative of the call price at K gives the probability of that strike (implied probability density).
In practice, what adjustments or acknowledgements (if any) need to be made to produce ...
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How to estimate the probability of a scenario in general
For my finance lecture we are currently on the topic of operation risk.
Scenarios play a vital role in the estimation of low frequency (or probability), high impact (or severity) events. How could ...
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$P(S_T > S_u \mid S_v = s_*)$
Let $u < v < T$ and assume $S_t$ follows a lognormal $((\mu - \sigma^2/2)t, \sigma^2 t)$ process. I'm interested in computing the conditional probability
$$
P(S_T > S_u \mid S_v = s_*)
$$
...
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A priori selection of acceptable backtesting errors (type I and II)
Is it possible to a priori select an acceptable values of type I and II errors in backtesting (f.e. in case of the unconditional coverage test)? Type I error is directly connected to the significance ...
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How do I find this Expectation?
I have an expectation given as:
$\mathbb{E}\left(S_{T}\mathbb{1}_{S_{T}\geq K} \right)$
where $K$ is just an arbitrary number (i.e. the strike price, but that's unimportant) and $S$ can be modelled ...
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interview question : replication strategy of a betting game
Here is a question I found in a book I am not able to finish. Your help will be much appreciated! I also included where I have been so far.
Q: Team A plays team B in a series of 7 games, whoever wins ...
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Probability Distribution that fits my parameters?
I'm trying to create a PDF that has the max values at its tails, and a P(x) of 0 at its mean.
Essentially it would be something like two normal distributions lined up side to side.
Is there any ...
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How to calculate Empirical Cumulative Probability in R
I have a dataset of S&P500 returns. How can I calculate the value of $F(X ⩽ x)$. My code is as below:
...
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Box-Muller Method Proof
Here we want to show that the Box-Muller method generates a pair of independent standard Gaussian random variables. But I don't understand why we use the determinant? For me when you have two ...
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How much to invest to reach a target?
Your current wealth is $W$. Each day you can invest some of it; there's a probability $p$ that you will win as much as you invested, $1-p$ that you will lose it. You want to reach a target wealth $W_T$...
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Girsanov theorem and default rates in bond credit rating
Default rates are kind of probabilities, right?
Is it possible to use the Girsanov theorem in that context? For example if we have a table of real world probabilities, could we use the Girsanov ...
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methodology confirmation for computing implied risk-neutral CDF from option prices
In this question, the risk-neutral probability distribution $q(S_T=s)$ for the underlying at time $t = T$ is given by the Breeden-Litzenberger identity as:
$$ \frac{1}{P(0,T)} \frac{ \partial^2 C }{\...
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Given Brownian motion $B_t,B_s$ and $t>s$, how to calculate $P(B_t>0,B_s<0)$?
As stated, this is an interview question.
Given Brownian motion $B_t,B_s$ and $t>s$, how to calculate $P(B_t>0,B_s<0)$?
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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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$\mathbb{P}$ and $\mathbb{Q}$ probability measure/distribution interpretations
I'm trying to understand probability distributions implied from market prices and was reading through this reference explaining the interpretation of $N(d_1)$ and $N(d_2)$ in the log-normal vol Black-...
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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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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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logistic regression multivariable fractional ploynomials stata vs. R
I a going through Hosmer, Lemenshow and Sturdivant's (HLS) Applied Logistic Regression (2013) and trying to interpret the difference between what STATA is doing and what R is doing. Concerning the fit ...
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Brexit implied probability
It is possible to bet on the Brexit e.g. on this page:
https://sports.ladbrokes.com/en-gb/betting/politics/british/eu-referendum/uk-european-referendum/220800266/
The quotes are 8/15 for remain, and ...
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Paper on the use of probability theory in finance?
I have taken probability theory course in college and want to see how it is used practically in finance.
What papers should I read? I want it to be not too difficult (undergraduate probability theory ...
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CIR model - nth moment generation $E^*[r_T^n]$
I am analyzing the nth moment generation process for $r_t$ with dynamics defined by CIR model
$r_t$ has following dynamics
$$dr_t=a(b-r_t)dt+\sigma \sqrt{r_t} dW_t^* \quad \quad (1)$$ for some ...
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Determining confidence level of directional signals
With regards to technical analysis, are there ways of determining the confidence level of a directional signal? Taking a relative strength index (RSI) as an example, can the extent to which an asset ...
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Option delta - Conditional probability definition?
Can someone help me interpret this definition of delta?
Delta is a conditional probability of terminal value (St) being greater than the Strike (X) given that St > X for a call option.
Is the ...
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Understanding the solution of this integral
The following integral represents an expected value of a geometric brownian motion for $S_T>K$ (i.e. part of the Black-Scholes call option price):
$$\int_{z^*} (S_te^{\mu\tau-\frac{1}{2}\sigma^2\...
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What are the answers to these questions on card deck and option pricing?
here are 3 questions I have some trouble dealing with. Your help will be greatly appreciated!
1 - We have a deck card: 26 red, 26 black. we play a game: you draw a card from the deck without putting ...
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BSM Model - Actual probability
Actual probability of exercise of put option under BSM model is:
PD = N(-d2(u))
(using expected return of stock, u)
Risk-neutral equivalent is
...
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On the construction of a Brownian motion from a Gaussian process
Let $X$ a Gaussian process defined by
$$
X_t=\int_{0}^{t}\left(\frac{1}{\sigma}\left(r_s-\frac{\sigma^2}{2}\right)-\rho\sigma_P(s,T)\right)\mathrm{d}s+\sqrt{1-\rho^2}Z_2(t)+\rho Z_1(t);\;\;t\in[0,T]
$...
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Modeling Interest-only Mortgages
Can we infer a range of future all-in costs for I/O ARMs with current index forward curves?
Essentially, just taking a worksheet like this and adding some type of ramping capability after the fixed ...
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Affect of choosing different combinations of variables for multivariate regression [closed]
If I have variables x1,x2,x3,and x4 that have correlation coefficients −0.9,−0.5,0.5, and 0.9 to another variable y, what is the effect of choosing different combinations of them in a multivariate ...
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How do I calculate the present value of a credit default swap?
I am paid 20 million every time a bond drops to a new low over a 120 month period. I need to know how to find the present value of such an arrangement if there is a continuously compound interest of 5 ...
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Density plot of the skew-t distribution
I am using the sgt package in R to recreate the plot from Hansen's paper ( available here http://www.ssc.wisc.edu/~bhansen/papers/ier_94.pdf on page 8) using random ...
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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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How were the probabilities of recession over the next four quarters calculated in this table?
http://www.bloomberg.com/news/articles/2016-02-08/goldman-sachs-says-defy-mr-market-as-recession-risk-still-low
The probability of a slump in the U.S. is just 18 percent and 23
percent over the ...
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Analytical Bond Price under Rendlemen-Bartter?
Assuming the short rate $r_t$ follows the risk-neutral (so $W_t$ is a $Q$-Brownian motion) process
$$
dr_t = ar_t dt + \sigma r_t dW_t,
$$
does anyone know of an analytical bond price formula? We ...
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Scaling of probability mass function
Given a histogram and the probability mass function values for each observation, when plotting the histogram and the curve (this is bell curve since the data is assumed to be normal) on the same ...
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Creating the histogram for the distribution of the portfolio returns
Given log returns for some stocks $A$ and $B$, which are the constituents of our hypothetical portfolio in equal weights, how does one actually come up with a distribution of the log returns of the ...
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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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Calculating probability of Yuan's slump from options market
http://www.bloomberg.com/news/articles/2016-01-06/if-options-traders-are-right-the-yuan-s-slump-is-far-from-over
Contract prices indicate a 79 percent probability that the currency
will weaken ...
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How much would one pay for the max of two stocks?
I'm trying to figure out if stochastic calculus is the right approach for this problem... but I only vaguely understand it and I am trying to gauge if I need to spend the time learning measure theory ...
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Compute stock price probability distribution from option data (IB method & negative probabilities issue)
I'm using a procedure as described in the interactive brokers article here (https://www.interactivebrokers.com/en/index.php?f=5910&ns=T) to compute a probability distribution from option (call) ...
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