Questions tagged [density]

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missing part of code for calculation of bondarenko positive convolution method for risk neutral density

I am adjusting code provided to Wm. Burknecht (see this post) to calculate risk neutral density using Bondarenko positive convolution method. The SLQSP minimization problem has the form minimize sum ...
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1answer
116 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 ...
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2answers
82 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: ...
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28 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 ...
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47 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 ...
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39 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 $T$ ...
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47 views

Second derivative w.r.t. strike as risk-neutral density?

According to Breeden & Litzenberger(1978), the second derivative of the call-value-function, with respect to the strike price, can be seen as the risk-neutral density of the underlying asset at ...
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1answer
47 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-...
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2answers
129 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?
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1answer
149 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 ...
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2answers
350 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 ...
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0answers
370 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 ...
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1answer
145 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 $c$)...
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184 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 ...
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1answer
312 views

Generating surface of Kernel Density Estimates over time

I have a 1-minutely OHLC dataset indexed by time as follows: ...
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0answers
37 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: ...
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1answer
945 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:...
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1answer
7k 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 ...
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1answer
378 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 ...
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1answer
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 ...
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2answers
160 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 ...
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1answer
980 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} = \...
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1answer
226 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! ...
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674 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 :...
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4answers
18k 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 ...
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1answer
57 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 ?
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1answer
2k 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 ...