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18

I recently read "Modeling financial data with stable distributions" (Nolan 2005) which gives a survey of this area and might be of interest (I believe it was contained in "Handbook of Heavy Tailed Distributions in Finance"). Another more recent reference is "Alpha-Stable Paradigm in Financial Markets" (2008). I'm not aware of anything covering "risk of ...

17

Google for this paper "Financial applications of random matrix theory: Old laces and new piece" from Marc Potters, Jean-Philippe Bouchaud, and Laurent Laloux. You can also check Prof. Gatheral presentation about Random Matrix Theory http://www.math.nyu.edu/fellows_fin_math/gatheral/RandomMatrixCovariance2008.pdf In R, the package "tawny" has an ...

16

There are several application of Lévy alpha-stable distributions to finance, especially in insurance and reinsurance. I believe that Embrechts-Kluppelberg-Mikosh's "Modelling Extremal Events for Insurance and Finance" is still an excellent reference. However, in the modeling of stock prices, this line of research is essentially inactive. The reason is that ...

11

Check out page 55 in "Quantitative Equity Investing: Techniques and Strategies," Fabozzi et al. Section is titled "Random Matrix Theory" - very intro. The context pertains to the estimation of a large covariance matrix. Also, see work at Capital Fund Management, filed under: Random Matrix and Finance : correlations and portfolio optimisation

10

I highly recommend the Maximum Entropy Bootstrap for time series, implemented by the meboot package in R. In my work, I've stopped using both the block bootstrap and residuals bootstrap in favor of meboot, and I am pleased with the results. Hrishikesh Vinod, the researcher behind meboot, described it in his talk at UseR/2010 last year. The algorithm is ...

10

I am still a beginner to this topic, and have been working through Cont and Tankov's textbook Financial Modelling With Jump Processes (2003), which is a fairly elementary treatment of the subject. I think a revised second edition is to come out later this year. One interesting area of applications that has become more prominent with a recent wave of papers ...

9

There is a simple solution if there is no drift, as the probability $p(x,t)$ obeys a simple diffusion equation: $\mathrm{d}(p)/\mathrm{d}t = \frac{1}{2} \sigma^2 \frac{\mathrm{d}(\mathrm{d}(p))}{\mathrm{d}x^2}$, here $x$ is the price difference $\text{price}(t) - \text{price}(t=0)$. Of course there is a simple solution to the diffusion equation (using ...

9

As far as I know MCMC and also (PMCMC) can be usefull for (bayesian) estimation of parameters of some Hidden process like in the Heston Model case based on observations of the Stock (filtering). But the problem here is that those estimates are not matching those based on calibration of vanilla options of the Risk Neutral measure. So as an econometric tool it ...

8

MCMC can be used for Bayesian inference of other models with hidden variables. Gibbs sampling, for example, is used in Hidden Markov Models. Here is a paper that discuss the differences between MCMC and the more classical approach using the EM algorithm. The question is: Are HMMs a useful model in finance? Some academics argue that they have predictive ...

8

I just ran across an interesting presentation comparing the effectiveness of Normal, Cauchy, and Student's-t distributions in modeling the S&P. It concludes that the normal distribution underestimates extreme movements, the Cauchy overestimates them (although a comment on the presentation points out that Mandelbrot used different parameters than the ...

8

Perhaps you may want to consider article by D. Levine - Modeling Tail Behavior with Extreme Value Theory who gives practicale example on how EVT can be used to calculate probabilities on returns in tails with use of the Pickands-Balkema-de Haan Theorem and generalized Pareto distribution. It also contains some criterias and points on other methods that can ...

8

One approach is Conditional Value at Risk (CVaR) a.k.a. Expected Shortfall (ES). It does, as you suggest, take into account the whole set of returns. However, instead of traditional VaR which asks "what is the worst 1% or 5% loss I can expect" in a given time frame, conditional VaR asks "assuming I sustain losses of at least 95% or 99% (and perhaps am ...

8

In this scenario, the "joint dynamics" are trivially computed since the option value is a known deterministic function of stock price. For example, the mean of the option value for time $\tau$ is $$\mu_O = \int_0^\infty BSM( S_\tau ) p(S_\tau) dS_\tau$$ which is best computed using quadrature as available in standard numerical libraries like scipy. The ...

7

So you want to calculate $\mathbb{P}[B_1 > B_0 + \varepsilon \;|\; A_1 > A_0 + \varepsilon]$? If you truly have the joint distribution of $A_1$ and $B_1$ and the current prices $A_0$ and $B_0$, this just becomes a simple exercise in integration, by the definition of probability density. Are you asking how to find a conditional probability in general, ...

7

A block bootstrap makes sense to me. (If the term doesn't make sense to you, I explain it at the end.) In order to pick the block size, I would essentially do a grid search: pick the largest feasible block size pick a smallest reasonable block size pick how many block sizes you feel like testing I'd run the selected bootstraps and see if there was a ...

7

Measuring expected shortfall (also known as conditional value-at-risk) answers the simpler question of "what is my average expected loss at the i-th quantile?" given the empirical distribution of returns. A variation is value-at-risk which measures the loss at the i-th quantile. Arguably you could leave at this this and you have your answer. You probably ...

6

Here's a partial answer: This partly depends on the return characteristics. One way to look at this is to analyze the skewness and kurtosis of the returns. Most strategies have a negative skewness, which roughly means that they have mostly consistent small positive returns, with the occasional large negative return. Alternatively, some strategies have ...

6

There are certainly (short-rate) models which assume bounded interest rates. I suppose I should clarify - the design of the model prohibits negative interest rates. Further, some models asymptotically reach some target, or mean rate which is considered mean reversion, the most famous perhaps the Vasicek. Short rate models where rates cannot go negative: ...

6

The most basic strategy is beta-based quantiles. That is to say, you first control for losses on your individual stock versus overall market performance. (Your trading strategy may or may not wish to hedge away the market factor using, say, SPX futures). Then you choose a quantile, call it the 5th percentile, beyond which you consider a move to be ...

6

What you refer to as the 99.5th percentile is known as the "Value-at-Risk." You are correct that you will need to make a distributional assumption, and there is a popular and well-researched approach to this problem, though I'm not certain it could be called "standard." I would recommend you use the "truncated Levy flight" distribution. James Xiong at ...

6

You can point out to your friend that, statistically speaking, having more observations reduces uncertainty in estimators. Mathematically, $SE_\bar{x}\ = \frac{s}{\sqrt{n}}$, showing that the standard error of a statistical estimator decreases with increased observations. This argument is concise and consistent with the Taleb quote. From wikipedia on ...

6

Its a simple expected value question: Probability of throwing a 6 is 1/6 Probability of not throwing a 6 is 5/6 thus expected pay off per roll: 10 dollars * 1/6 + (-1 dollar) * 5/6 = 5/6 dollars Edit: And several of your above assumptions are plain wrong: "Probability tells me that every 6 throws I get one 6 and 5 different numbers." -> That is NOT what ...

6

This leads to the same result as Alexeys answer. However, my reasoning is different. $$E[F_X(X+a)]=\int_{-\infty}^{\infty} F_X(x+a) f_X(x)dx=\int_{-\infty}^{\infty} \int_{-\infty}^{x+a}f_X(y)dy f_X(x)dx=\\ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} 1_{(-\infty,x+a]}(y) f_X(y) f_X(x)dydx= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} 1_{_{\{y-x\le ... 5 ...do you know of any standard libraries that can calculate the joint probability of stocks A and B, given a time series of historical data? Using R and the LSPM package with the code posted here might be what you are looking for. 5 In practice, I would begin with the recovery assumption. In the case of Greece, dealers are probably already quoting recovery swaps, allowing you to set this parameter directly. In general, you have to be willing to make assumptions based on history or on conversations with bankruptcy experts. Once I have the recovery assumption, I can take any ... 5 N(d2) is near to the probability the option will expire in the money; I have a video showing how d2 is similar to distance to default in the Merton here on youtube. N(d1) is the delta. The technical issue is that N(d2) is a risk-neutral probability; the input in d2 is the riskfree rate, although the theory is more involved. But, if you replace the ... 5 Short answer: there are multiple optimal mixed (i.e. non-deterministic) strategies. Long answer: There is an equivalent game, called bluff or liar's dice, which is played using dice. Each player has a number of dice, and can see only their own rolls. The game consists of claims on the whole pool of the dice of all players, for example "there are at least 4 ... 5 When I run this simulation I see the same results, and it makes sense. For the straight 50%/50%, I found that my win ration was about 38% and my loss ratio 61%. The reason it wasn't 50/50 was that if I had consecutive up flips my value could keep going up, but if I had consecutive down flips I would 0 out and the sequence would have to end as I had lost ... 4 You could try measuring autocorrelation at varying lags, as described here, and then choose your optimal block size according to the results of this test, i.e. if there is significant autocorrelation up to and including lag 5, your block size should be no larger than 5. 4 If you use a risk-neutral pricing model and consider the probability there, then you get the probability with respect to a risk neutral measure, in addition that probability depends on the chosen numeraire. For example, in Black-Scholes model taking the risk-neutral measure with respect to the bank account B gives$$P(S(T)<K) = Q^{B}(S(T)<K) = ...

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