Questions tagged [probability]

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290 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 ...
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2answers
129 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 ...
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3answers
144 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 ...
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123 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) ...
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2answers
145 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 \...
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1answer
169 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 ...
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1answer
65 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 \...
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1answer
365 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, \...
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1answer
215 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 ...
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1answer
3k 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 ...
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2answers
690 views

Proving Derivative Property of Moment-Generating Function

In Shreve II, exercise 1.8, he walks the reader through proving the derivative of a moment-generating function $\phi$ is equal to the expectation $\mathrm{E}[Xe^{tX}]$; i.e., $$ \phi^\prime(t) = \...
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1answer
497 views

Empirical copula

I am trying to find the empirical copula linking two random variables $X$ and $Y$. I have some data available but it's limited with respect to the variable $Y$ and I am not convinced it's enough data ...
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0answers
21 views

Multiple similar values simulation

Perhaps some of you came across the following task that I am trying to automate for @RISK, VOSE or other simulation software? I have a question as we are trying to use the software to estimate the ...
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1answer
229 views

Probability of Stock breaching barrier

If a stock has a process: $dS(t) = sigma*dB(t)$, where $B(t)$ is a standard Brownian motion, and current stock price is $S(0)$. There is a barrier $H>S(0)$. What is the probability that the stock ...
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2answers
278 views

Future spot price versus current forward price

Which are the two conditions necessary to claim that the future spot price will have as many chances to be above or below the current forward price?
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160 views

Is there a countably infinite Sigma-Algebra? Why?

Assume $\,\mathcal{F}$ be a nonempty collection of subsets of $\Omega$. $\,\mathcal{F}$ is called a $\sigma$-Algebra whenever if $A\in\mathcal{F}$ then $A^c\in\mathcal{F}$, and if $A_1,A_2,...\in\...
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1answer
796 views

Beta between stock and option

In Black Scholes model I would like to compute $$ \beta_K = \frac{\mathrm{cov}(C_{K,T},S_T)}{\mathrm{cov}(S_T,S_T)} = \frac{\mathrm{cov}((S_T - K)^+,S_T)}{\mathrm{cov}(S_T,S_T)} $$ with respect to say ...
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2answers
440 views

Do futures follow physical or risk-neutral distributions

I've spent a while looking for an answer to this question and while I feel it is a simple question I have not found an answer. I know prices of option contracts follow an implied, risk-neutral ...
4
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2answers
132 views

Joint probability distribution only measures product sets?

According to these notes (top of p 133), "We say that random variables $X_1, X_2, \ldots X_n : \Omega \to \mathbb{R}$ are jointly continuous if there is a joint probability density function $p(x_1, ...
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1answer
133 views

Proving $\mathbb{E}(g(X)) = \int_{\mathbb{R}} g(x) f(x) dx$

Let $X$ be a random variable on a probability space $(\Omega, \mathcal{F}, P)$ and let $g$ be a Borel-measurable function on $\mathbb{R}$. In Shreve II (p 28) he proves, using the standard machine, ...
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2answers
2k views

Confidence Intervals of Stock Following a Geometric Brownian Motion

In preparation for my Options, Future's and Risk Management examination next week, I have been presented with a series of questions and their answers. Unfortunately, my lecturer, one of the less ...
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1answer
319 views

Lipschitz condition in mathematical finance

I am interested in a rigorous explanation on why the Lipschitz condition plays a major part in stochastic calculus, most significantly in mathematical finance. To be specific, suppose we want to ...
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2answers
9k views

t-statistics for the mean return, using Newey-West standard errors

I have seen that in several papers, where the aim was to evaluate the performance of a certain investment strategy, they use t-statistics to test for significance in the results. However, this seems a ...
3
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2answers
114 views

Sums of random variables and independence

I'm having troubles with this proof: Let $\{Z_i\}_{i\in\mathbb{Z}}$ be i.i.d. random variables with zero mean and unit standard deviation. For $(a_0, a_1, ..., a_r)$ a sequence of $r$ real numbers ...
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3answers
261 views

Direct use of implied volatility

I am not sure to understand exactly the direct use of implied volatility. Let's take an example: if an instrument has a daily volatility of $\sigma$, there is a 68% probability that its value will be ...
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1answer
134 views

probability that the stock price is below the strike price

How can I prove that under the risk-neutral probability: $\mathbb{P}[S_{t}<K]=-\frac{\partial{C}}{\partial{K}}(K,T)$ where $S_{t}$ is the stock price, K is the strike price, C is the call ...
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1answer
114 views

To lump or not to lump

Suppose I have a very simple asset whose price takes only three possible values: $X_t\in \{-1,0,1\}$. I also got some discrete time series $X = (X_t)_{t\geq 0}$ and I would like to come up with a ...
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2answers
161 views

arbitrage opportunity in a two period model

I have a little problem evaluating an european call. I Suppose the following: in $$t=0 : S_0 = 10$$ $$t = 1 : S_1 = \{10,11\}~with ~p=0.5$$ riskless rate : $(1+r)=\beta=1.049$ Strike ...
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1answer
191 views

If the risk neutral probability measure and the real probability measure should coincide

Sorry if this may be a stupid question. I have not had that much mathematical finance, I've only learned about discrete time models. But lets for the argument say that you have a stochastic process ...
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1answer
162 views

Stochastic Differential

Let $W_t$ be a Wiener process. It is clear to me that $dW_t$ is of size $\sqrt{dt}$. This can be seen because $$ \mathrm{Var}(W_{t+\Delta} - W_{t})=\Delta. $$ But am I allowed to actually write $(...
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843 views

Law of a geometric brownian motion first hitting time (formula dont match Monte Carlo Simulation)

I posted this question before on MSE I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all ...
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0answers
251 views

negative transition probabilities in the heston model

I've been trying to implement a bivariate tree for pricing american options with the heston model in R using the paper of Beliaeva and Nawalkha (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=...
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1answer
54 views

y-axis unity of density probability function

What is the unity/interpretation of the y-axis of a density distribution function? The X-axis is the values of the random variable, the area is the probabilty what about the y-axis ?
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175 views

Beta distribution - Holding period

Let's say I have a risk factor that is defined between [0,1], such as recovery rates. Assuming I have daily data, I can estimate the "daily VaR", i.e. the tails over 1 day period, since the data is ...
2
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1answer
108 views

Distribution of minimum of hazard functions

Suppose I have two random variables, $X_1$ and $X_2$, that are independent (but not identically distributed) and assume both have hazard functions $\lambda_1(s)$ and $\lambda_2(s)$, for $s > 0$. ...
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1answer
101 views

Properties of a Symmetric Copula

I am working with the following copula, and have a few questions about it: $C(x,y) = xy + \theta (1-x)(1-y)xy$ Here $\theta \in [-1,1]$ and $x,y \in [0,1]$ First, I am trying to show this copula is ...
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2answers
117 views

Joint distribution from expectations

Given two random variables $X$ and $Y$ and let $K$ be a constant value. Assume the expectation $\mathbb{E}[X(Y-K)^{+}]$ is given for all possible values of $K\geq 0$. Is there a way to derive the ...
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1answer
2k views

Arbitragefree Pricing: Q vs. P

I read that the Fundamental Theorem of Asset Pricing states, that a market is arbitrage-free if and only if there exists an equivalent martingale measure Q, under which the discounted asset price ...
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1answer
39 views

How to define the median for bivariate function?

I know if we define a function f(x) and its cdf is F(x). The inverse function of cdf is inverseF. I can define its median as follows: median = inverseF(0.5). But if I want to get the median for a ...
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3answers
1k views

Difference betweem martingale property and adapted filteration

What is the difference between a random process that is adapted to a filteration and one that had the martingale property. It seems the two notions are quite similar and would be helpful to construct ...
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214 views

2-state HMM / ARMA process?

I have issues with this problem: Let $\{X_t, t\in \Bbb N\}$ be a 2-state stationary Markov chain, with transition $M$ (and $M(1,2)\neq 0 \neq M(2,1)$), let $\{W_t, t\in \Bbb N\}$ be a strong Gaussian ...
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1answer
350 views

Convolution copula?

Using copula formulation for the following probability: $$\mathbb{P}(X\leq x,y_{1}\leq Y\leq y_{2})=\mathbb{P}(X\leq x,Y\leq y_{2})-\mathbb{P}(X\leq x,Y\leq y_{1})$$ $$=C(F_{X}(x),F_{Y}(y_{2}))-C(F_{...
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1answer
8k views

Baye's rule for conditional expectations (Proof review)

The Baye's rule for conditional expectations states $$ E^Q[X|\mathcal{F}]E^P[f|\mathcal{F}]=E^P[Xf|\mathcal{F}] $$ With $f=dQ/dP$ - thus being the Radon-Nikodyn derivative and $X$ being ...
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1answer
147 views

Where does this copula come from?

In a paper I encountered the following notation $$P(Z\leq z,u\leq Y\leq v)=C(F_{Z}(z),F_{Y}(v)-F_{Y}(u))$$ However I don't see why this holds in relation to uniform random variables. Usually $$P(Z\...
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5answers
5k views

Random matrix theory (RMT) in finance

The new kid on the block in finance seems to be random matrix theory. Although RMT as a theory is not so new (about 50 years) and was first used in quantum mechanics it being used in finance is a ...
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1answer
233 views

Effects of random-generator-choice on derivative's price

There is a plethora of pseudo-random-generators out there. Some of them are definetly better and some of them severily underperform. My standard tool is Mersenne Twister - when I need to generate ...
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1answer
814 views

Definition of orthogonality and independence for a stochastic processes

Somehow I can't find the explicit definition of when two processes are supposed to be orthogonal or independent anywhere. I think orthogonality and independence should mean the same thing in this ...
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2answers
7k views

How do I calculate probability distribution of stock prices given option prices?

I'd like to calculate a probability distribution for prices given the option prices for that stock? Any ideas how to do this? My desire is to do this daily and then see how the price PD changes over ...
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2answers
594 views

Normally Distributed Returns Become Leptokurtic Due to Compounding

I was running a bunch of simple simulations in excel the other day in excel. Using the NORM.INV(RAND(),0,1) to simulate daily stock returns I noticed that the more compounded the returns, ie, the more ...
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1answer
178 views

Summary statistic for the average probability of default?

I have the following scenario: Let $X_i$ denote the event where some institution $i$ 'defaults' (don't worry about the exact definition of a default here, it is not relevant to the question at hand). ...