# Tag Info

### $\mathbb{P}$ vs $\mathbb{Q}$ Probabilities - Transitioning Between Measures

$\mathbb{P}$ 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 ...
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### What is the connection between the risk neutral implied density and the real world density?

I'll outline how you can estimate the (implied) real-world density function from (observed) option prices. Having found this real-world density, you can then compute all sorts of probabilities and ...
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### open problems in mathematical finance

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 ...
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### Probability in different measures

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 ...
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### Modeling Call Price w.r.t. Strike w Models that Capture Vol Smile

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 ...

### Throwing a dice and risk neutral probability

the information you provided is not sufficient to deduce risk neutral probabilities. You have to provide something like a price process from which risk neutral probabilities can be computed. Here are ...

### Probability Puzzle from a Quant Interview

Answering my own question because I figured out a semi-elegant way to approach this analytically and get the closed-form solution. TL;DR The probability is $\frac{5}{28}$ Method The first claim is ...

### Pricing when arbitrage is possible through Negative Probabilities or something else

You cannot use negative probabilities in this context. When there is no unique probability measure, there can be no unique price. You only know that it is in [0, 0.6] range, if you want to tighten ...
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Accepted

### Probability Density Function of a Wiener Process Minimum

Firstly, $m_T=\min\limits_{t\in[0,T]} B_t = -\max\limits_{t\in[0,T]} -B_t \overset{Law}{=} -\max\limits_{t\in[0,T]} B_t = -M_T$. So, you can either consider the running maximum or minimum. Let $\tau$ ...
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### Probability Puzzle from a Quant Interview

Here's an attempt at a more intuitive argument, at the risk of losing some rigor, based on the solution I wrote out in the section after the dividing bar. Assume that in the first 70 draws, we saw 5 ...

### Probability of success given expected return and volatility

It's probably a simple textbook example to illustrate Taleb's point. Suppose $\text{d}S_t=\mu S_t \text{d}t+\sigma S_t \text{d}W_t$ with $\mu=0.15$ and $\sigma=0.1$. By Itô's lemma, the log return ...
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### Pricing when arbitrage is possible through Negative Probabilities or something else

I believe there is not a unique price if you can't short. Say, instead of buying the option you spent 0.5 on a half a unit of the asset $S^2_1$ This asset pays out $[0.4, 0.6, 0.8]$ which first order ...
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,...