# Questions tagged [density]

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### 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 ...
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 ...
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. ...
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 ...
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 vote
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 ...
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 ...
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 ...
1 vote
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: ...
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 ...
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 ...
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 $T$ ...
1 vote
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-...
1 vote
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?
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 ...
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 ...
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 ...
1 vote
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 vote
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 ...
362 views

### Generating surface of Kernel Density Estimates over time

I have a 1-minutely OHLC dataset indexed by time as follows: ...
1 vote
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: ...
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:...
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 ...
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 ...
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 ...