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.

Filter by
Sorted by
Tagged with
0
votes
0answers
189 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
votes
1answer
1k 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:...
-1
votes
1answer
194 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 ...
2
votes
1answer
547 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 ...
1
vote
1answer
51 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 $$ P(S_T > S_u \mid S_v = s_*) $$ ...
2
votes
0answers
56 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 ...
2
votes
1answer
195 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 ...
3
votes
0answers
674 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
votes
1answer
56 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 ...
2
votes
3answers
542 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
votes
2answers
4k views

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 ...
3
votes
2answers
220 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 $W_T$...
2
votes
1answer
217 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 ...
1
vote
0answers
190 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 }{\...
1
vote
1answer
495 views

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 $P(B_t>0,B_s<0)$?
6
votes
2answers
298 views

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 ...
1
vote
1answer
213 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-...
9
votes
1answer
755 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 \in ...
7
votes
2answers
475 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 ...
1
vote
0answers
159 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
votes
2answers
381 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 ...
1
vote
1answer
717 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
votes
1answer
536 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
votes
1answer
82 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
vote
1answer
302 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 ...
4
votes
2answers
1k 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
votes
1answer
191 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 ...
1
vote
1answer
191 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 ...
1
vote
0answers
98 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] $...
3
votes
1answer
178 views

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 ...
1
vote
0answers
17 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 ...
4
votes
0answers
119 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 ...
3
votes
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
votes
1answer
652 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
votes
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 ...
4
votes
1answer
72 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 ...
0
votes
1answer
75 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
vote
1answer
373 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 ...
1
vote
0answers
404 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 ...
4
votes
2answers
131 views

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 ...
3
votes
3answers
156 views

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 ...
0
votes
0answers
152 views

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) ...
3
votes
1answer
208 views

Prove $E_{\mathbb Q}[X_t | \mathscr F_u] = X_u$ given $Y_t$ is a martingale

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 $\mathbb ...
1
vote
1answer
202 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 $...
5
votes
2answers
666 views

Pricing when arbitrage is possible through Negative Probabilities or something else

Assume that we have a general one-period market model consisting of $d+1$ assets and $N$ states. Using a replicating portfolio $\phi$, determine $\Pi(0;X)$, the price of a European call option, with ...
2
votes
1answer
70 views

Asymmetric Random Walk / Prove that $T:= \inf\{n: X_n = b\}$ is a $\{\mathscr F_n\}_{n \in \mathbb N}$-stopping time

Given random variables $Y_1, Y_2, ... \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \...
3
votes
2answers
155 views

Asymmetric Random Walk / Prove that $E[T:= \inf\{n: X_n = b\}] < \infty$

Given random variables $Y_1, Y_2, ... \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \...
5
votes
1answer
410 views

Radon-Nikodym: Changing Distribution vs Changing Random Variable

Let $X \sim \mathcal{N}(\mu,\sigma^2)$ under the probability measure $P$ on the measurable space $(\Omega, \mathcal{F})$. We may define a Radon-Nikodym derivative $Z$, also defined on $(\Omega, \...
1
vote
1answer
266 views

Negative risk neutral probabilities economic argument

We know of plenty ways to extract risk neutral distirbutions from option prices (for example Breeden Litzberger) but there is no real analysis on how to interpret negative state prices (Haug 2007 for ...
10
votes
1answer
4k views

open problems in mathematical finance

What are open problems in mathematical finance that use fundamental concepts of mathematics (functional analysis, geometry and topology, algebra and number theory etc.) and not data-driven. I have ...