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9
votes
1answer
353 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) = ...
7
votes
1answer
249 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 ...
0
votes
1answer
18 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 ...
9
votes
0answers
236 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: ...
7
votes
0answers
109 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 ...
6
votes
0answers
97 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 ...
6
votes
0answers
88 views

Transition densities in the Heson 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 ...
5
votes
0answers
281 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 ...
3
votes
0answers
118 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 ...
3
votes
0answers
139 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 ...
3
votes
0answers
104 views

Particular Conditional Expectation of Geometric Brownian Motion

If we have the density function $$f_{Y}(y,t)=\frac{1}{y \sqrt {2\pi\sigma^2t}}exp(-\frac{(ln \ y - \mu t)^2}{2\sigma^2t})$$ Then the mean of $Y(t)=e^{X(t)}$ conditional on $Y(0)=y_0$ is found to be ...
3
votes
0answers
136 views

default probability

Suppose the hazard rate is $\lambda$ the default probability density function follow exponential $f(t) = \lambda e^{-\lambda t}$ and cumulative probability function is $F(t) = 1 - e^{-\lambda t}$ ...
3
votes
0answers
372 views

Monty Hall Model

Given a fixed time period,say 3 days, the stock/market can go up,down or stay sideways. A hedge fund can long, short or use rangebound(options strategy) to bet for that 3 days closing level. Hedge ...
2
votes
0answers
299 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 ...
2
votes
0answers
100 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
votes
0answers
181 views

Probability Density of Returns of Bonus Certificates

Could anyone please help me with the following? I need to generate a histogram (resp. probability density) of returns of a bonus-certificate. A bonus-certificate can be replicated by an underlying ...
1
vote
0answers
15 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 ...
1
vote
0answers
17 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 ...
1
vote
0answers
152 views

Distribution of Brownian Bridge

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 ...
1
vote
0answers
247 views

Quadratic utility function

May you can help me undertanding the following conclusion: Suppose we have an agent who has preferences over contingent claims, represented by a concave function $U$. This simply means that ...
1
vote
0answers
218 views

Modeling asset performance to Bitcoin revenue

I'm attempting to model asset performance to Bitcoin revenue, which is a driving force in the Bitcoin community. Question Is there any model, or research being done that tracks "hashes per second" ...
0
votes
0answers
18 views

Altman Z-Score to Probability of default

I have computed the Altman Z Score for approximatly 2500 companies. I was wondering if mathematically I am allowed to use a logistic function ? Such as: ...
0
votes
0answers
66 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 ...
0
votes
0answers
22 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) ...
0
votes
0answers
16 views

Selling two uncorrelated OTM options lowers the over all probability of profit?

I am trying to simulate shorting two uncorrelated put options, I wrote a python program and used monte carlo method to simulate the PnL on expiration: gist It seems the probability of profit is ...
0
votes
0answers
46 views

How to fit a copula to empirical data?

There are numerous types of Copulas one can choose to fit empirical data. My question is wow to select the 'best' copula to fit the data. More specifically, let's assume empirical data ...
0
votes
0answers
75 views

How do you calculate the asset drift rate in the Merton model? (used in N(-d2))

Can I replace it with the internal rentability rate? I only have the financial statements and some market data, but I can't find the expected returns anywhere. The goal is to calculate the probability ...
0
votes
0answers
18 views

Value of sequential mutually-exclusive options (A variant on the Secretary Problem)

How much should you offer a potential hire in a signing bonus? Imagine you are interviewing a list of candidates for a particular job. Each candidate has a "lifetime value", and probability of ...
0
votes
0answers
111 views

How to calculate probability of option expiring in the money?

Given the following values ...
0
votes
0answers
60 views

Inferring the maximum drawdown depth for a different sample size

Let's say there's a trading system that has a 10 % chance of getting a maximum drawdown >= 50 % over a sample of ...