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18 votes

$\mathbb{P}$ 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 ...
Matthew Gunn's user avatar
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16 votes
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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 ...
Kevin's user avatar
  • 16k
13 votes
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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 ...
ragoragino's user avatar
10 votes
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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 ...
LocalVolatility's user avatar
9 votes

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 ...
Cettt's user avatar
  • 1,456
7 votes
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Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

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{...
Quantuple's user avatar
  • 14.7k
7 votes
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Understanding the solution of this integral

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}...
Gordon's user avatar
  • 21.1k
7 votes
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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 $\tau$ ...
Kevin's user avatar
  • 16k
7 votes

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 ...
kaddy's user avatar
  • 152
7 votes
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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 ...
xnor's user avatar
  • 186
7 votes

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 ...
Kevin's user avatar
  • 16k
6 votes
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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 ...
Quantuple's user avatar
  • 14.7k
6 votes
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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-...
Kevin's user avatar
  • 16k
6 votes
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 ...
Oskar's user avatar
  • 76
6 votes
Accepted

How to test signifcance of a sharpe ratio

The answer above is not correct. Let's go by parts: Denote the mean of returns $\mu$. Denote the standard deviation of returns: $\sigma$. Therefore the sharpe ratio is: $$ SR = \frac{\mu-r_f}{\sigma} $...
phdstudent's user avatar
  • 8,381
6 votes
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What is the distribution of the risk-free asset?

The standard way to think about this is that at time $t$ the riskless asset gives you known return of $r_{f,t}$ over a short time period. However, this rate may itself be time-varying and stochastic ...
fes's user avatar
  • 1,727
6 votes
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Requesting for price?

As @noob2 noted, nobody is going to quote you a price unless you're a customer. And when I say "customer", I mean "customer of the desk", not just of the bank. Would require an ...
user42108's user avatar
  • 2,262
5 votes

Quantum Mechanics and Economics... What

I have had a read through the paper that you quoted and have the following comments which you might find helpful: (I am formally trained in QM, so hopefully there shouldn't be any errors in the ...
oliversm's user avatar
  • 1,389
5 votes

Lévy alpha-stable distribution and modelling of stock prices.

I asked this question 6 years ago, and in the meantime I came across this little volume: Lévy Processes in Finance: Pricing Financial Derivatives by Wim Schoutens (2003).
Raskolnikov's user avatar
  • 1,527
5 votes

How can we have negative probabilities in finance? Can we have negative payments in bonds? If not, how else can we have negative probabilities?

The answer is NO, with very few exceptions There might be bonds with negative coupon(s), and the Bloomberg search even finds some, but there are plenty of reasons why negative coupons are impractical....
vanguard2k's user avatar
  • 2,915
5 votes
Accepted

Equivalent martingale measure exists if and only if $a < S_0^1(1+r)< b$

Assume that: $$ S_0^1(1+r)\leq a,b $$ Arbitrage for a portfolio $V_t$ is defined as: $$V_0\leq0, \quad P(V_1\geq0)=1, \quad P(V_1>0)>0$$ Consider borrowing at rate $r$ to buy the risky asset ...
Daneel Olivaw's user avatar
5 votes

Geometric Brownian Motion - Price Probabilities

knowing that the log of the prices in a GBM follows the following normal distribution: $$\operatorname{ln}(S_t) \sim N\left(\operatorname{ln}S_0 + T*\left( \mu - \frac{\sigma^2}{2} \right), \sigma^2 ...
ricmarchao's user avatar
5 votes
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 ...
Jan Stuller's user avatar
  • 6,118
5 votes
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Real world probabilities from option implied risk neutral density?

Great question! Unfortunately, it's not easy. We can use option prices to get the $\mathbb{Q}$-distribution. However, the probability measure $\mathbb{Q}$ merges the stochastic discount factor (SDF) $...
Kevin's user avatar
  • 16k
5 votes
Accepted

Conditional probability of Brownian motion (with drift and scaling) hitting barrier

For part 1 of your question, the short answer is no, calculating conditional density is a looong way of doing it. Possible but not the easiest. Here is the sketch for a shorter version. We note that $(...
piterbarg's user avatar
  • 940
5 votes

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 ...
KaiSqDist's user avatar
  • 1,312
4 votes

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 ...
mbison's user avatar
  • 1,578
4 votes
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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....
nbbo2's user avatar
  • 11.4k
4 votes
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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 ...
Quantuple's user avatar
  • 14.7k

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