Questions tagged [probability]

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1answer
44 views

Can a stochastic process be neither adapted to filtration nor previsible?

The idea behind the question arises from my intuition about the concepts of 'adapted to filtration' and 'previsbility'. If a process is adapted, it essentially means that the evolution of the ...
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1answer
33 views

stock specific volatility

I was unsure about the precise definition of "stock specific volatility". Used in this question "A stock has beta of 2.0 and stock specific daily volatility of 0.02. Suppose that yesterday's closing ...
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1answer
37 views

Show that the variance of the market portfolio is the weighted average of the ovariances between each constituent and the market portfolio itself

Let us assume that the market portfolio consists of n assets. Given that the return of the market portfolio can be written as $r_m = \sum_{j=1}^{n} w_jr_j$, we have that $\sigma^2_m = E(\sum_{j=1}^{n} ...
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0answers
36 views

Credit spread model

Let $c(t,T):=-\frac{1}{T-t}[\mathrm{ln}(P_1(t,T))-\mathrm{ln}(P_0(t,T))]$, with: $c$ measure of how a company is prone to fail; $P_0(t,T):=e^{-r(T-t)}$ price of no-defaultable bond. $P_1(t,T):=\...
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0answers
62 views

How to solve these SDE Problems

Quuestion1. I make a solution $r(t)$ used by Ito's lemma $r(t)=e^{-a t}r(0)+\int _{0}^{t}e^{a (s-t)}\theta (s)ds+\sigma e^{-a t}\int _{0}^{t}e^{a u}\,dB^{1}(u)$ Is this right? and I try to make ...
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1answer
76 views

Geometric Brownian Motion - Price Probabilities

I am modeling a stock price that follows Geometric Brownian Motion and have the following: $E(X)$ = .16 (16%) $\sigma$ = .24 (24%) $X_0$ = 95 $T$ = 1 (12 months) I am trying to find the ...
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1answer
68 views

Does E[max(x, y)] equal to E[x|x>y]*P(x>y) + E[x|x<y]* P(x<y) when x and y are not independent?

Suppose x and y are discrete random variable, I can write them in summation. And it seems like they are equal. Any ideas?
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0answers
37 views

How to work with vine copula in R?

I have returns of 4 stocks: stock1, stock2, stock3, stock4. And I use R and library(VineCopula) to do: ...
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1answer
30 views

A quick and dirty loss distribution and Credit VaR

I need to create a loss distribution for a credit portfolio as the first steps to estimate the portfolio Credit VaR. I have historical monthly account snapshots (payment history) of all accounts ...
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0answers
36 views

Problem in copula fitting

I have returns of 2 stocks: stock1 and stock2. And I want to fit pair copula. I use this libraries library(VineCopula) library(copula) then I select an ...
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0answers
46 views

Convolution of Dependent Random Variables with Copulas

Lets say I have 2 different observations which are fitted to a parametric distribution. And lets say that they are dependent and can be modeled by one of the copulas. I want to calculate “a value” ...
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0answers
42 views

Unconditional Expectation vs. Conditional Expectation at time $0$

In most mathematical finance books I have read (all of them actually), the expectation, with respect to the sigma algebra at time $0$, $\mathcal F_0$, is considered the same as the unconditional ...
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1answer
404 views

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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1answer
101 views

Introduction of a stochastic discount factor in martingale pricing

The example below is taken from Björk (2009). Let Radon-Nikodym derivative be $$L=\frac{dP}{dQ} \;\; \text{on} \; \mathcal F$$ or written analogously $$P(A) = \int_AL(\omega)dQ(\omega) \;\; \text{for ...
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31 views

Need help figuring out probability that price will be touched in a specific time period

I have a formulas for figuring out probability the price will be struck within T days. Now what I need help with is figuring out the probability price will be stuck with in a given (T) minutes, or (T) ...
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2answers
122 views

Importance of filtrations that are NOT natural filtrations

I know the natural filtration intuitively represents the history of the process as the process evolves over time, and hence can be used to talk about conditional probabilities and conditional ...
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1answer
70 views

Reference material (EV/ betting game questions) for Quant Hedge Funds Interviews [closed]

I need material to practice EV games questions.But I lack practice in betting questions where a set-up of a game is given and one has to respond to the best strategy or best bet to take. A good book ...
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0answers
35 views

Is every filtration a natural filtration of some stochastic process?

We have a notion of natural filtrations, which intuitively represents the history of the process as the process evolves over time. We also have a notion of filtrations in general, which are ...
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1answer
224 views

Probability and statistics in Quantitative Finance

Certain types of traders attempt to repeatedly buy and sell the same asset for a profit over a short time period, such as high-frequency “market makers”. For example, if you can repeatedly sell a ...
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2answers
94 views

Compare two distributions for forecasting returns

Let's imagine that we have two separate models, both used to forecast the return for the next period. Both models are estimated everyday, and both models outputs a probability distribution. How can we ...
3
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3answers
160 views

How to prove that $X_s=\int^s_0 f(u)dW_u$ is independant from $X_t-X_s$

I am asked to prove that $X_s$ and $X_t-X_s$ are independant for $s<t$ then $$X_t=\int^t_0f(u)dW_u$$ for a deterministic function $f$ and brownian motion $W_t$. For the proof I am giving a hint to ...
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1answer
66 views

Finding the limit $\lim_{n \to \infty} P_0^n$ for a European Cash-or-Nothing put option with $P=K^2\cdot \mathbf{1}_{\{S_T < K\}}$

Exercise : Let $K>0$. A European Cash-or-Nothing put option $P$ has the following pay-out profile : $$P=K^2\cdot \mathbf{1}_{\{S_T < K\}}$$ Let $P_0^n$ be the no-arbitrage value at time $...
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1answer
59 views

In search of nice (approx) function forms of the volatility of cumulative simple returns

Let's consider a period $t\in[0,T]$, and let the simple return over year $t$ ($1\le t\le T$) be $r_t$. Assume $r_t$ are iid normal. The cumualative simple return over the whole period $[0,T]$ is $$R_T=...
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1answer
144 views

Probability ITM formula for options

Given a stock of price price and annual volatility annual_volatility, and given an option with strike price ...
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1answer
74 views

How to calculate number of round trips given volatility?

Suppose we know stock price volatility is normally distributed with mean = 0 and annual volatility say 20%. Let's assume markets never close and we can trade at 1 second intervals. Let's assume stock ...
3
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1answer
176 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 ...
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1answer
76 views

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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1answer
135 views

Expectation of the product of two Brownian motions [closed]

Could you please let me know the steps to follow to get to the solution?
3
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1answer
83 views

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 ...
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1answer
212 views

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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0answers
940 views

Fitting Student t-distributions to log-returns

It seems that some tail-risk centric groups are bent on using Paretian and t-distributions to account for tail risk when fitting log-returns. It has been observed, however, that with and without ...
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1answer
467 views

$\mathbb{P}$ 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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1answer
295 views

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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2answers
128 views

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 $c$. ...
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1answer
108 views

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 ...
3
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1answer
182 views

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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3answers
286 views

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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1answer
42 views

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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2answers
368 views

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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2answers
269 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 ...
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0answers
124 views

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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0answers
200 views

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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0answers
66 views

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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0answers
46 views

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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0answers
39 views

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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1answer
103 views

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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1answer
91 views

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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0answers
63 views

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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2answers
2k views

KMV-Merton Probabilties of Default vs Moody's EDF

Moody's used to publish probability of default estimates from their Moody's EDF model, but they have temporarily discontinued it. I understand that the Moody's EDF model is closely based on the Merton ...
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0answers
137 views

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 ...