Questions tagged [density]
The density tag has no usage guidance.
32
questions
3
votes
1
answer
114
views
implied-information in american option
I have recently been researching European options versus American options implied information. For European options, an overview article is Christoffersen(2012). But for American options, I only found ...
- 31
0
votes
0
answers
95
views
Move from risk-neutral probability to historical probability
I am working on a density forecasting project using options. Using the Breeden-Litzenberger formula it is possible to find the implied density at maturity under the risk neutral probability of an ...
5
votes
0
answers
225
views
Probability density from COS method too sensitive to truncation range
I have a long-standing confusion around the truncation range of the COS method proposed by Fang and Oosterlee because I find that the results are highly volatile given the different truncation ranges.
...
- 51
2
votes
2
answers
334
views
Option implied risk neutral distribution vs BKM risk neutral moments
I am doing some research on the option implied risk neutral distribution and methods calculate it, and so far have come across two ways to do so.
The first way is through the Breeden-Litzenberger ...
- 93
11
votes
1
answer
308
views
Trading strategy for a misspecified density
I am trying to implement a strategy that exploits potential misspecifications in density predictions (e.g.: long states with too-low probability; short states with too-high probability).
In particular,...
- 1,451
1
vote
1
answer
317
views
Best way to trade probability density
From the option chain of a security, we can calculate the implied probability density at the maturity
$T$ (assume the options are European. Now suppose we have our own view/prediction on the ...
- 281
2
votes
1
answer
212
views
COS Method and existence of density
Hey in the COS method we use characteristic function of $\ln{S_T}$ to price european options (by recovering density from characteristic function). But how do we know that density exists? For example I ...
- 71
0
votes
1
answer
218
views
Which financial time series have a PDF and/or CDF?
Consider the following types of financial time series for a single publicly-listed stock:
Price data
Log returns
Cumulative returns
Each is computed from the item listed before it: log returns are ...
- 2,980
1
vote
2
answers
125
views
Calculate moments given density values
Suppose I have given a finite number of grid values belonging to a probability density function. Moreover, I have the associated values of the density support. For instance:
...
- 165
2
votes
0
answers
34
views
Does equity premium puzzle affect option-implied RWDs using Arrow-Debreu equilibrium?
I am researching and learning about option-implied RNDs (risk neutral densities) and transformation to RWDs (risk world densities) using expected utility theory to compute risk aversion values.
This ...
- 61
4
votes
0
answers
330
views
R: How do i finish the tails in the risk neutral density, obtained from option prices
Im currently working on constructing the risk neutral probability distribution of a stock, based on the option prices. In doing so, i calculate the implied volatilities from the option prices, and ...
- 63
2
votes
0
answers
215
views
Stability of Finite Difference method for Breeden-Litzenberger
I am trying to derive a risk-neutral density from European call option prices using a second order finite difference scheme. Let $C(K,T)$ be the price of a European call with strike $K$ and expiry $T$ ...
- 431
1
vote
1
answer
61
views
Which models have non-smooth densities?
By smooth, I mean a density $f$ that lies in the space $C^\infty$, infinitely differentiable.
Are there, in the literature, some known models where the underlying density of the state process is non-...
- 11
1
vote
2
answers
201
views
Are densities used in finance square integrable?
Let $f$ be the density of the stock asset under some model (Heston, SABR, Black Scholes, Variance-Gamma, etc).
Is $f$ square-integrable in these models?
- 11
2
votes
1
answer
295
views
How to quantify the Variance Risk Premium (VRP) with probability density functions?
The VRP is usually displayed by charts like this one:
It's easy to see that, for most of the time, options are priced by using volatility which will reveal itself larger than the realized one. So VRP ...
- 2,111
5
votes
2
answers
685
views
Estimation of Risk-Neutral Densities Using Positive Convolution Approximation - Python
I'm trying to estimate the risk-neutral density through positive convolution approximation (introduced by Bondarenko 2002: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=375781). I'm currently ...
2
votes
0
answers
844
views
Detecting butterfly spread arbitrage for American options through European option prices
It's easy to demonstrate that if European option prices are concave with strike, then an arbitrage exists. For example, the risk-neutral probability density is the second derivative of European put ...
- 121
1
vote
1
answer
437
views
Probability density function of the sum of two independent Levy-distributed random variables?
I posted the following questions in math stack exchange
https://math.stackexchange.com/posts/2762047/edit
Here's the text:
Prove that the sum of two independent Levy-distributed (having parameter $c$)...
- 1,012
1
vote
0
answers
441
views
How to compute SABR's probability density function
I am trying to compute the probability density function of the forward rate implied by the SABR formula approximation in order to see how the density implied by the approximation has negative ...
- 463
2
votes
1
answer
362
views
Generating surface of Kernel Density Estimates over time
I have a 1-minutely OHLC dataset indexed by time as follows:
...
- 229
1
vote
0
answers
42
views
Computing the PDF of the sum of N moves of an empirical PDF for USDJPY 1-minute moves
Per-minute tick data for USDJPY is available here. Suppose we download this file to usdjpy.txt and then save it into a Numpy array in Python 3 as follows:
...
- 187
7
votes
1
answer
2k
views
Modeling Call Price w.r.t. Strike w Models that Capture Vol Smile
I am trying to model $C(K)$, the price of the call $C$ as a function of strike $K$. Because this is tied to Prob ITM - and in fact the probability density function of that particular expiration (https:...
- 735
7
votes
1
answer
15k
views
Breeden-Litzenberger formula for risk-neutral densities
Based on this topic:
How to derive the implied probability distribution from B-S volatilities?
I am trying to implement the Breeden-Litzenberger formula to compute the market implied risk-neutral ...
- 431
3
votes
1
answer
463
views
Dupire's formula proof
I just have a question for the beginning of a proof:
Suppose
$\frac{dS_{t}}{S_{t}}=(r_{t}-q_{t})dt+\sigma(t,S_{t})dW_{t}$
with $r,q,S$ stochastic.
In the book I read, it is written:
We define the ...
- 617
3
votes
1
answer
2k
views
Density plot of the skew-t distribution
I am using the sgt package in R to recreate the plot from Hansen's paper ( available here http://www.ssc.wisc.edu/~bhansen/papers/ier_94.pdf on page 8) using random ...
- 491
2
votes
2
answers
212
views
Problem with obtaining densities
For my research I need to obtain a series of densities, however, I am encountering some problems.
The first problem is perhaps very simple, but the answer eludes me. Let's say I have an observation ...
- 491
8
votes
1
answer
1k
views
Density forecast of a GARCH model
I am currently working on developing a series of density forecasts and I am encountering some problems. I am working on weekly S&P 500 returns and the returns process is described as
$r_{t} = \...
- 491
5
votes
1
answer
317
views
Density of Geometric BM via Fokker-Planck
Attempting to derive density of a GBM (which we know is log-normal) the long way, using the Fokker Planck-equation. Can't figure out where I went wrong - would appreciate a few sets of extra eyes!
...
- 2,778
12
votes
1
answer
981
views
"Extract" the density of the underlying, given the implied volatility "surface"
Suppose given implied volatility quotations $\widehat{\sigma}(T_i,K_j)$ of call options on an underlying $S$ for various expiries $T_i$'s and strikes $K_j$'s. I am interested in the following problem :...
- 1,223
13
votes
5
answers
23k
views
Copulas simply explained
I try to understand the basic idea of copulas, however I am still struggling and hope that someone can help me.
I understood that in general a copula is a function which links several marginal ...
- 539
0
votes
1
answer
112
views
y-axis unity of density probability function
What is the unity/interpretation of the y-axis of a density distribution function?
The X-axis is the values of the random variable, the area is the probabilty what about the y-axis ?
- 113
3
votes
1
answer
3k
views
Implied probability density (Question 2 - Applications and Interpretation)
Using the second derivative of the Call-Option-Price one can try to recover the pricing density.
Formally: Assuming a constant interst rate $r$ and also not making any assumptions on the model ...
- 3,377