12

Assume the price follows a lognormal process. We can convert it into a problem of finding the probability of a standard Brownian motion particle starting from $0$ and hitting $x$ before time $t$, or its first passage time $\tau_x$ being less than $t$. This can be derived through the reflection principle. The paths crossing $x$ are exactly paired up by the ...


10

In the derivatives context, "arbitrage free" means almost surely for the probability measure under consideration. This is in opposition with statistical arbitrage used at high frequencies for example. More precisely the assumption is that there is no $T\geq 0$ and self-financed portfolio $V$ such that $V_0 = 0$, $P(V_T < 0) = 0$ and $P(V_T > 0) > ...


9

You can't determine this with just the correlation; you need to know the joint probability.


9

Why not using the so simple Monte-Carlo estimator $ \hat{p}_N =\frac{ \sum_{i=1}^N 1_{|A_{i+1}-A_i|>0 \cap |B_{i+1}-B_i|>0}} {\sum_{i=1}^N 1_{|A_{i+1}-A_i|>0 }}$ where $1_{|A_{i+1}-A_i|>0}$ is $1$ if stock $A$ has moved at time $i+1$


9

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) = \Phi(...


9

If you want to address interesting problems that are interesting for financial mathematics, I do not believe you have the good list. Pricing. For instance, most of explicit formulas for pricing that are not available yet will never be. In this direction, you should have a look at simulation techniques. See for instance Nonlinear Option Pricing. Interesting ...


9

The way that I understand your question is that you are looking to fit the market prices of European plain vanilla options of a single maturity and then back out the corresponding implied probability density function. There are multiple ways that you could approach your problem. 1) Modelling the Market Prices The market prices of European plain vanilla ...


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 ...


8

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 instrument,...


8

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: Cox-...


8

$\mathbb{P}$ is the true probability measure. Measure $\mathbb{Q}$ is a measure of convenience that allows risk neutral pricing. Stochastic discount factor $M$ takes you between the two. If you care about prices you can either: (1) work under $\mathbb{Q}$ or (2) work under $\mathbb{P}$ with a stochastic discount factor $M$. There's an isomorphic ...


8

In order to estimate a probability of an event with small probability, you might want to try to estimate the probability for a changed random variable that allocates a larger probability mass for the event happening. So, in your case, you might want to change the original $N(0, 1)$ to $N(100, 1)$ because for the second r.v. the probability of it being higher ...


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

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 ...


7

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 ...


7

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 a\}...


7

In this case, the t-statistic is used to determine if the returns are statistically different from zero (the theoretical mean). A small t-statistic would imply that the null hypothesis (no significant excess return) cannot be rejected. Newey-West standard errors are used to correct for the correlations of error terms over time. I have written a Matlab ...


7

you should have a look at implied probability densities. They do exactly what you are asking - extracting the pricing density from option prices. This is done by differentiating the option price with respect to the call. Here are two links. The first one explains the procedure the second one deals with where such densities can be applied Implied state ...


7

I can clarify 100% that $(dw)^2$= $dt$ and recommend you to accept it as a fact. Like any other differential, this differential is defined in terms of its integral: $$ \int_{t_{0}}^{t_{1}}(dW)^{2}\equiv\lim_{n\rightarrow\infty}\sum_{k=0}^{n-1}[W(t_{k+1})-W(t_{k})]^{2} $$ Where $t_{k}=t_{0}+k(t_{1}-t_{0})/n$. Since $$ W(t_{k+1})-W(t_{k})=\sqrt{t_{k+1}-t_{k}}\...


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

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

Not sure about all of the complicated math and programming above, but I can tell you that, if you want to calculate for 1 Standard Deviation from the current stock price X days away, the following calculation will give you a +/- value from the current stock price. 1 StdDev Move = (Stock Price X Implied Volatility X the Square Root of 'how many days') all ...


6

One approach is to take the entire option chain, and calculate the prices for adjacent butterflies along the chain. The risk / reward of each of the butterflies represent the empirical probability that the market is pricing for the underlying to move between the strikes of the butterfly. To make sure it is a proper probability distribution, you will want ...


6

Consider two jointly normal random variables $X_1 \sim N(u_1, \sigma_1^2)$ and $X_2 \sim N(u_2, \sigma_2^2)$. Note that, \begin{align*} \max(X_1, X_2) = X_2 + \max(X_1-X_2, \ 0). \end{align*} Moreover, $X_1-X_2$ is a normal random variable with mean $\mu=\mu_1-\mu_2$ and variance \begin{align*} \sigma^2 = \sigma_1^2+\sigma_2^2 - 2 \rho\sigma_1\sigma_2, \end{...


6

Let $\tau = T-t$. Then \begin{align*} S_T = S_t e^{(\mu - \frac{1}{2}\sigma^2) \tau + \sigma \sqrt{\tau}\, Z}, \end{align*} where $Z$ is a standard normal random variable, independent of $\mathcal{F}_t$. Moreover, \begin{align*} E\left(S_T 1_{\{S_T >K\}}\mid \mathcal{F}_t \right) &= E\left(S_t e^{(\mu - \frac{1}{2}\sigma^2) \tau + \sigma \sqrt{\tau}\, ...


6

IMHO the problem isn't stated correctly indeed, in the sense that the Radon-Nikodym derivative provided as the "solution" is not the unique way to define a measure $\mathbb{Q}$ equivalent to $\mathbb{P}$ and under which $X_t$ is a martingale. Just take $$\frac {d\mathbb{Q}}{d\mathbb{P}} =\mathcal{E}\left(-\int_0^t \cos(s) dW_s + a\right)$$ for any $a \in \...


6

Quantiles are preserved under monotonic transformations, hence the quantile for $Y$ is simply the exponential of the quantile of $X$, no need for corrections whatsoever (see here for instance). Put otherwise, let $q$ denote the quantile $\alpha$ of $X$ i.e. $$\Bbb{P}(X \leq q) = \alpha$$ then \begin{align} \Bbb{P}( X \leq q ) &= \Bbb{P}( \underbrace{\...


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.


Only top voted, non community-wiki answers of a minimum length are eligible