# 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 ...
• 6,964
Accepted

### 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 ...
• 16k
Accepted

### 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 ...
• 561
Accepted

### 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 ...
• 6,044

### 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 ...
• 1,456
Accepted

• 21.1k
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 $\tau$ ...
• 16k

### 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 ...
• 152
Accepted

### 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 ...
• 186

### 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 ...
• 16k
Accepted

### Quantile normal and lognormal

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 ...
• 14.7k
Accepted

### Option and probability of finishing in the money?

You got to be careful with $\mathbb{P}$ and $\mathbb{Q}$. Indeed, $N(d_2)$ is the probability of the event $\{S_T\geq K\}$ in the risk-neutral world. Note that $r$ (or $r-q$) is the drift in the risk-...
• 16k
Accepted

### Produce the random variable for an asset from a uniformly distributed random varible

The question requires you to provide a method which uses uniform random variables and transforms them to generate realizations of the described asset values. To give a bit more general answer: this ...
• 76
Accepted

• 181
Accepted

### Proving $\mathbb{P}(S_t<0|S_0=s_0)=0$ for Geometric BM

I think the easiest way to derive the solution to the GBM is via Ito's Lemma. The GBM: $dS_t = \mu S_t dt + \sigma S_t dW_t$ is a short hand for:  S_t = S_0 + \int_{h=0}^{h=t}\left(\mu S_h\right)dh ...
• 6,118
Accepted

• 940

### Probability Puzzle from a Quant Interview

Here to share my answer. I did not make use of Bayes law or any other probability theorems. I just worked with numbers, which is my usual approach towards these quizzes (there are tons of super smart ...
• 1,312

### Can the concept of negative probabilities be used to price a call option?

Without looking at probabilities or quasiprobabilities, I think your market will always allow for arbitrage opportunities. Suppose you have a security that pays [1;0;0], this arrow security can be ...
• 1,578
Accepted

### Brexit implied probability

The general formula for conversion of "a to b" odds to a probability is $p=\frac{b}{a+b}$ http://www.calculatorsoup.com/calculators/games/odds.php So 8/15 remain implies remain with probability 0....
• 11.4k
Accepted

### Option delta - Conditional probability definition?

IMHO the 'definition' you mention is not a mathematical definition per se, but rather an approximation used by some practitioners. Mathematically, it is $N(d_2)$ in the BS formula which figures the ...
• 14.7k

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