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 ...
- 6,924
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 ...
- 15.4k
14
votes
Accepted
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 ...
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 ...
- 561
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 ...
- 5,959
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 ...
- 1,446
8
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 ...
- 172
7
votes
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 ...
- 346
7
votes
Accepted
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}...
- 21k
7
votes
Accepted
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{...
- 14.5k
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 $\tau$ ...
- 15.4k
7
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 ...
- 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 ...
- 15.4k
6
votes
Accepted
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 ...
- 226
6
votes
Accepted
How much would one pay for the max of two stocks?
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,...
- 21k
6
votes
Accepted
Prove $E_{\mathbb Q}[X_t | \mathscr F_u] = X_u$ given $Y_t$ is a martingale
Bayes' rule for conditional expectation (or here) gives us
$$E_{\mathbb Q}[X_t | \mathscr F_u] E[L_T| \mathscr F_u] = E[X_tL_T| \mathscr F_u]$$
Use martingale property and iterated expectation:
$$E_{\...
- 921
6
votes
Accepted
Quantile normal and lognormal
Quantiles are preserved under monotonic transformations, hence the quantile for $Y$ is simply the exponential of the quantile of $X$, no need for corrections whatsoever (see here for instance).
Put ...
- 14.5k
6
votes
Accepted
Produce the random variable for an asset from a uniformly distributed random varible
The question requires you to provide a method which uses uniform random variables and transforms them to generate realizations of the described asset values.
To give a bit more general answer: this ...
- 76
6
votes
Accepted
How to test signifcance of a sharpe ratio
The answer above is not correct.
Let's go by parts:
Denote the mean of returns $\mu$. Denote the standard deviation of returns: $\sigma$.
Therefore the sharpe ratio is:
$$ SR = \frac{\mu-r_f}{\sigma} $...
- 8,062
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 ...
- 1,707
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 ...
- 2,209
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 ...
- 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).
- 1,507
5
votes
KMV-Merton Probabilties of Default vs Moody's EDF
I understand that Moody's uses an empirical distribution while KMV
uses a normal distribution in order to calculate these probabilities
KMV doesn't use a normal distribution to map distance to ...
- 1,549
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....
- 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 ...
- 7,999
5
votes
Geometric Brownian Motion - Price Probabilities
knowing that the log of the prices in a GBM follows the following normal distribution:
$$\operatorname{ln}(S_t) \sim N\left(\operatorname{ln}S_0 + T*\left( \mu - \frac{\sigma^2}{2} \right), \sigma^2 ...
- 171
5
votes
Accepted
Option and probability of finishing in the money?
You got to be careful with $\mathbb{P}$ and $\mathbb{Q}$. Indeed, $N(d_2)$ is the probability of the event $\{S_T\geq K\}$ in the risk-neutral world. Note that $r$ (or $r-q$) is the drift in the risk-...
- 15.4k
5
votes
Accepted
Proving $\mathbb{P}(S_t<0|S_0=s_0)=0$ for Geometric BM
I think the easiest way to derive the solution to the GBM is via Ito's Lemma.
The GBM: $dS_t = \mu S_t dt + \sigma S_t dW_t$ is a short hand for:
$$ S_t = S_0 + \int_{h=0}^{h=t}\left(\mu S_h\right)dh ...
- 5,998
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) $...
- 15.4k
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