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31
votes
5answers
2k views

Lévy alpha-stable distribution and modelling of stock prices.

Since Mandelbrot, Fama and others have performed seminal work on the topic, it has been suspected that stock price fluctuations can be more appropriately modeled using Lévy alpha-stable distrbutions ...
24
votes
2answers
4k views

How useful is Markov chain Monte Carlo for quantitative finance?

Naively, it seems that Bayesian modeling, structural models particularly, would be quite useful in finance because of their ability to incorporate market idiosyncrasies and produce accurate ...
22
votes
5answers
3k 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 ...
17
votes
2answers
663 views

How are distributions for tail risk measures estimated in practice?

Let's say you want to calculate a VaR for a portfolio of 1000 stocks. You're really only interested in the left tail, so do you use the whole set of returns to estimate mean, variance, skew, and shape ...
14
votes
5answers
1k views

How to estimate the probability of drawdown / ruin?

A fairly naive approach to estimate the probability of drawdown / ruin is to calculate the probabilities of all the permutations of your sample returns, keeping track of those that hit your drawdown / ...
14
votes
8answers
4k views

Probability of touching

For a vanilla option, I know that the probability of the option expiring in the money is simply the delta of the option... but how would I calculate the probability, without doing monte carlo, of the ...
13
votes
2answers
659 views

How do you distinguish “significant” moves from noise?

How do you distinguish between losses that are within the normal range for day-to-day shifts and situations with a real potential for loss? The specific application I have in mind is pattern ...
11
votes
2answers
3k views

How does left tail risk differ from right tail risk?

How does left tail risk differ from right tail risk? In what context would an analyst use these metrics?
9
votes
1answer
408 views

Fixed income modeling

I am currently working on my research paper and trying to explain a two-dimensional variable: volume and instrument of corporate debt financing. Independent variables that I believe must be included ...
8
votes
4answers
1k views

How do I estimate the joint probability of stock B moving, if stock A moves?

I have two stocks, A and B, that are correlated in some way. If I know (hypothetically) that stock A has a 60% chance of rising tomorrow, and I know the joint probability between stocks A and B, how ...
8
votes
1answer
2k views

How to estimate probability of default from bond prices?

How do you use bond prices/yields to infer probabilities of default? I would think of it as follows: Create a relationship between default free (e.g., Germany) and defaultable (e.g., Greece) bond ...
8
votes
2answers
1k views

What are some examples of Compound Poisson processes in insurance?

I'm writing the Bachelor thesis but I need some information. I need to find some practical examples and applications of the Compound Poisson Process in insurance. Does anyone have any good examples?
7
votes
3answers
1k views

How to estimate real-world probabilities

In the world of finance, Risk-neutral pricing allow us to estimate the fair value of derivatives using the risk free rate as the expected return of the underlyings. However, the behavior of ...
7
votes
1answer
335 views

Simulating the joint dynamics of a stock and an option

I want to know the joint dynamics of a stock and it's option for a finite number of moments between now and $T$ the expiration date of the option for a number of possible paths. Let $r_{\mathrm{s}}$ ...
7
votes
1answer
611 views

If stock A has a 60% chance of rising, and stocks A and B have 80% correlation, what is the chance of stock B rising?

As in the subject, I'm interested in a math puzzle of sorts: If stock A has a 60% chance of rising, and stocks A and B have an 80% correlation, what is the chance of stock B rising? Would it be ...
7
votes
1answer
379 views

Do people use unbounded interest rate models, and what alternatives exist?

A simple interest rate model in discrete time is the autoregressive model, $$ I_{n+1} = \alpha I_n+w_n $$ where $\alpha\in [0,1)$ and $w_n\geq 0$ are i.i.d. random variables. When working with ruin ...
6
votes
1answer
366 views

What distribution should I apply to estimate the likelihood of extreme returns?

Say I have a limited sample, a month of daily returns, and I want to estimate the 99.5th percentile of the distribution of absolute daily returns. Because the estimate will require extrapolation, I ...
6
votes
2answers
165 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 ...
5
votes
2answers
876 views

Strategies for Liar's Poker

This question is only tangentially related to quantitative finance. Scott Patterson's book The Quants describes how a quant at Kidder Peabody figured out a strategy to playing Liar's Poker in the late ...
5
votes
3answers
352 views

Calculate the expectation of a shift CDF

Suppose $X$ is a normal random variable with mean 0, and variance $\sigma^2$. $F(x)$ is the CDF(cumulative distribution function) of a standard normal random variable(mean 0 and variable 1), how to ...
5
votes
5answers
901 views

How to fit probability density function from sample moments?

If I have calculated the sample mean, variance, skew and kurtosis of a set of data, how would I go about fitting a probability distribution to match these moments (i.e. choosing a probability ...
5
votes
2answers
620 views

How do you synthesize a probability density function (pdf) from equally weighted price data?

What I'm working with: I have a collection of prices that has very few to no repeating values (depending on the look back period) ie each price value is unique, some prices are clustered and some can ...
5
votes
0answers
66 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 ...
4
votes
5answers
5k 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 ...
4
votes
1answer
180 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 ...
4
votes
1answer
137 views

pricing of heat rate-linked derivative

It's a simplified model. Suppose $U_t$ is a random variables subject to Lognormal($x_1$, $z_1^2$)distribution. $V_t$ is a random variables subject to Lognormal($x_2$, $z_2^2$)distribution. Suppose ...
4
votes
1answer
477 views

Coin Toss System

Coin Toss Runs Calculator The expected number of runs for two consecutive heads or tails is 3. Is there an edge if we place a progressive constant size bet(limited to 3 times)for consecutive ...
4
votes
2answers
660 views

Heuristics for calculating theoretical probabilities of being ITM at time T for listed options

I'm looking for a heuristic way to calculate the probabilities of being in the money at expiry for non-defined risk options combinations (listed options). I use delta as a proxy for this probability ...
3
votes
2answers
168 views

Random Brownian Simulation Startling Results

I was playing around in Excel the other day, simulating possible equity curve/P&L paths for a simple game I designed. The game is really trying to find an optimal risk managment strategy. I start ...
3
votes
2answers
575 views

on “recovering probability distributions from option prices” - how to subtract influence of stochastic volatility?

This is based on a 1995 paper by Rubinstein/Jackwerth by the above title where the authors produces a distribution of stock prices inferred from option prices. But their approach only produces a joint ...
3
votes
1answer
124 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 ...
3
votes
1answer
3k 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
votes
1answer
76 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 ...
3
votes
2answers
764 views

Risk neutral probability in binomial lattice option coming greater than 1…what's wrong?

I am substituting reasonable values in the below fomula (like r=0.12, T=20, nColumn=16, sigma=0.004)...why is probability coming out to be greater than 1? Any help? Thanks! ...
3
votes
0answers
49 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
149 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
260 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
2answers
300 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 ...
2
votes
3answers
92 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 ...
2
votes
1answer
72 views

Convolution copula?

Using copula formulation for the following probability: $$\mathbb{P}(X\leq x,y_{1}\leq Y\leq y_{2})=\mathbb{P}(X\leq x,Y\leq y_{2})-\mathbb{P}(X\leq x,Y\leq y_{1})$$ ...
2
votes
1answer
90 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). ...
2
votes
1answer
221 views

What are $d_1$ and $d_2$ for Laplace?

What are the formulae for d1 & d2 using a Laplace distribution?
2
votes
1answer
115 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 ...
2
votes
1answer
415 views

Monte Carlo Options Probability Calculation

I have a fairly simple problem for an application I am writing currently. How do you calculate the options probability of being in the money or touching a certain strike price. I know there are at ...
2
votes
0answers
100 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 $F(t) = 1 - e^{-\lambda t}$ ...
2
votes
1answer
69 views

Creditworthiness indicator for copula one-factor model

In this paper in equation 15 on page 261 dealing with one factor copula model, one is using creditworthiness indicator as one of a variables. It is defined as \begin{equation} Y_c = \sqrt{\rho_c} Z ...
2
votes
0answers
141 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 ...
2
votes
1answer
160 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) = ...
1
vote
3answers
1k views

Probability - Generating fair outcome using unfair coin

I have been thinking a lot about the following puzzle. But, could not arrive at a solution. Can someone explain me how can you get a fair (equal probability) outcome using only an unfair coin (where ...
1
vote
4answers
386 views

Proof that the number of trades done (successfully) matters for whether or not a strategy was lucky

Me and a friend is trying to settle and argument in relation to the following quote by Nassim Nicholas Taleb: I don’t want to spend too much time on Buffett. George Soros has 2 million times more ...