# Questions tagged [density]

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### 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?
133 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 ...
240 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 ...
272 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 ...
104 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 $c$)...
164 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 ...
282 views

### Generating surface of Kernel Density Estimates over time

I have a 1-minutely OHLC dataset indexed by time as follows: ...
36 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: ...
864 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:...
6k 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 ...
355 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 ...
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 ...