Questions tagged [probability]

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2
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1answer
816 views

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 ...
1
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1answer
628 views

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 ...
0
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1answer
59 views

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 ...
2
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1answer
279 views

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 {...
2
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0answers
67 views

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 = ...
1
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0answers
60 views

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} = ...
3
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0answers
34 views

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 ...
1
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0answers
151 views

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, ...
11
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1answer
499 views

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 ...
3
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1answer
412 views

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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0answers
52 views

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: ...
3
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1answer
166 views

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 ...
0
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1answer
128 views

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 ...
2
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1answer
107 views

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 ...
0
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1answer
4k views

Probability of a return from historical average and standard deviation

I have a question from a sample exam paper that I'm having some trouble figuring out. The question is: Bavarian Sausage stock has an average historical return of 16.3% and a standard deviation of 5....
13
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6answers
32k views

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 ...
1
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0answers
834 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 ...
1
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1answer
191 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
67 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
899 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
482 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
203 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
149 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
94 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
356 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
305 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 &...
8
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0answers
284 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
492 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
165 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
875 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
157 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 ...
0
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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 $$ P(S_T > S_u \mid S_v = s_*) $$ ...
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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
177 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
520 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
197 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
213 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
698 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
122 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 }{\...
7
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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 ...
9
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1answer
659 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
1k 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) = \...
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0answers
150 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
377 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
956 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
481 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
434 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 ...
-1
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1answer
81 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
250 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 ...