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 ...
- 7,024
16
votes
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 ...
- 16.3k
13
votes
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 ...
- 571
10
votes
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 ...
- 6,094
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 ...
- 1,476
8
votes
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 ...
- 196
7
votes
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-...
- 16.3k
7
votes
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 $\tau$ ...
- 16.3k
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 ...
- 151
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 ...
- 16.3k
6
votes
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 ...
- 14.8k
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 ...
- 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:
$$ SR = \frac{\mu-r_f}{\sigma} $...
- 8,621
6
votes
Accepted
What is the distribution of the risk-free asset?
The standard way to think about this is that at time $t$ the riskless asset gives you known return of $r_{f,t}$ over a short time period. However, this rate may itself be time-varying and stochastic ...
- 1,747
6
votes
Accepted
Requesting for price?
As @noob2 noted, nobody is going to quote you a price unless you're a customer. And when I say "customer", I mean "customer of the desk", not just of the bank. Would require an ...
- 2,292
5
votes
Lévy alpha-stable distribution and modelling of stock prices.
I asked this question 6 years ago, and in the meantime I came across this little volume:
Lévy Processes in Finance: Pricing Financial Derivatives by Wim Schoutens (2003).
- 1,537
5
votes
Accepted
Equivalent martingale measure exists if and only if $a < S_0^1(1+r)< b$
Assume that:
$$ 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$$
Consider borrowing at rate $r$ to buy the risky asset ...
- 8,219
5
votes
Geometric Brownian Motion - Price Probabilities
knowing that the log of the prices in a GBM follows the following normal distribution:
$$\operatorname{ln}(S_t) \sim N\left(\operatorname{ln}S_0 + T*\left( \mu - \frac{\sigma^2}{2} \right), \sigma^2 ...
- 181
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 ...
- 6,480
5
votes
Accepted
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) $...
- 16.3k
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 $(...
- 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 ...
- 2,231
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 ...
- 1,578
4
votes
Accepted
Probability of exercise in the Black-Scholes Model
As a random variable, the terminal asset price has a semi-infinite support, bounded at zero. Intuitively, this means that when increasing volatility while keeping all other parameters unchanged (same ...
- 14.8k
4
votes
Accepted
Subadditivity of Expected Shortfall
The expected shortfall is defined by
\begin{align*}
ES_{\alpha} = \frac{1}{1-\alpha}\int_{\alpha}^1 VaR_{p}(L) dp,
\end{align*}
where $L$ is the loss function. For the case with 500 scenarios, the $\...
- 21.3k
4
votes
Compare two distributions for forecasting returns
Answer
If you assume your returns are independent (yes your models might loosen this assumption) then the two models, $Q_1$ and $Q_2$ assign probability distributions to the returns on any given day, ...
- 12.1k
4
votes
Importance of filtrations that are NOT natural filtrations
The natural filtration, as you said, refers to the filtration of a particular process. The filtration generated by two different processes is not necessarily the same as the natural filtration of a ...
- 280
4
votes
stock specific volatility
The stock specific volatility (also known as idiosyncratic volatility) is the volatility that remains after controlling for beta. I suppose you have $$R_i = R_f + \beta_i \cdot \big(R_m-R_f\big) + \...
- 16.3k
4
votes
Probability that the price of stock following a brownian motion goes under a certain value
Let $(S_t)$ be the price process of your stock such that $S_t = S_0+ \mu t + \sigma B_t$ where $(B_t)$ is a standard Brownian motion. Then, since $B_t\sim N(0,t)$, we get $S_t\sim N(S_0+\mu t, \sigma^...
- 16.3k
4
votes
Probability that the price of stock following a brownian motion goes under a certain value
I think there is a typo in the previous answer- assuming arithmetic brownian is meant- here is my working:
$P\left[S_t \le 8\right]=P\left[S_0+\mu t+\sigma B_t \le 8\right]$
$=P\left[S_0+\mu t+\...
- 6,242
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