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Questions tagged [probability]

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1
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0answers
753 views

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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1answer
189 views

$\mathbb{P}$ 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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0answers
66 views

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 $...
0
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1answer
808 views

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 ...
2
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1answer
432 views

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 ...
2
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1answer
201 views

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 ...
5
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2answers
141 views

Probability of Closing Stock Price Over a Defined Period

$$ p(S,t;S',t') = \frac{1}{\sigma S'\sqrt{2\pi (t'-t)}} \exp\left(-\frac{(\log(S/S') + (\mu-1/2\sigma^2)(t'-t))^2}{2\sigma^2(t'-t)}\right) $$ I found this equation when I was reading "Paul Wilmots on ...
2
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0answers
92 views

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 ...
2
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3answers
293 views

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: ...
2
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3answers
287 views

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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0answers
272 views

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 ...
2
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1answer
477 views

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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0answers
147 views

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 ...
7
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1answer
826 views

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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1answer
143 views

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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1answer
50 views

$P(S_T > S_u \mid S_v = s_*)$

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 $$ P(S_T > S_u \mid S_v = s_*) $$ ...
1
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0answers
53 views

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 ...
1
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1answer
173 views

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 ...
2
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0answers
487 views

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 ...
0
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1answer
54 views

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 ...
1
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2answers
194 views

Symmetry of option-implied probability density

I was wondering whether the option implied probability density of the log returns: $x = \ln\left(\frac{S}{S_0}\right)$ with S the value of a certain stock, is always symmetric ? I was asking myself ...
3
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2answers
210 views

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 $W_T$...
5
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1answer
677 views

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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0answers
110 views

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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2answers
440 views

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 ...
8
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1answer
641 views

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 \...
12
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2answers
986 views

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) = \...
1
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0answers
136 views

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 ...
5
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2answers
373 views

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 ...
12
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1answer
909 views

Quantum Mechanics and Economics… What

I was reading this paper: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2002698&download=yes The author has the model presented here: http://modelingcommons.org/browse/one_model/3443#...
1
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1answer
445 views

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 ...
3
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1answer
395 views

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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1answer
80 views

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 ...
1
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1answer
229 views

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 ...
3
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2answers
835 views

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\...
3
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1answer
174 views

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 ...
0
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1answer
71 views

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 ...
1
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1answer
162 views

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 ...
4
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1answer
65 views

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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0answers
92 views

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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0answers
16 views

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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0answers
115 views

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 ...
8
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1answer
233 views

pdf 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 ...
1
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1answer
151 views

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 $...
3
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1answer
2k views

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 ...
2
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1answer
505 views

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: ...
0
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1answer
54 views

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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3answers
2k views

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 ...
0
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1answer
295 views

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 ...
0
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0answers
286 views

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 ...