Questions tagged [stochastic-volatility]
The stochastic-volatility tag has no usage guidance.
51
questions with no upvoted or accepted answers
35
votes
0answers
992 views
How to show that this weak scheme is a cubature scheme?
Weak schemes, such as Ninomiya-Victoir or Ninomiya-Ninomiya, are typically used for discretization of stochastic volatility models such as the Heston Model.
Can anyone familiar with Cubature on ...
29
votes
0answers
2k views
Law of an integrated CIR Process as sum of Independent Random Variables
It is known (see for example Joshi-Chan "Fast and Accureate Long Stepping Simulation of the Heston SV Model" available at SSRN) that for a CIR process defined as :
$$dY_t= \kappa(\theta -Y_t)dt+ \...
17
votes
0answers
746 views
Local Stochastic Volatility - Break even levels
In Chapter 12 of his excellent book Stochastic Volatility Modeling, Lorenzo Bergomi discusses the topic of local-stochastic volatility models (LSV).
As most of you are probably aware of, the idea is ...
12
votes
1answer
506 views
Transformation of Volatility - BS
I have recently seen a paper about the Boeing approach that replaces the "normal" Stdev in the BS formula with the Stdev
\begin{equation}
\sigma'=\sqrt{\frac{ln(1+\frac{\sigma}{\mu})^{2}}{t}}
\end{...
9
votes
0answers
305 views
Jim Gatheral's ansatz
In the Ansatz section of Jim Gatheral's book Volatility Surface (page 32), he assumes $$\mathbb E[x_s|x_T]=x_T\frac{\hat w_s}{\hat w_T}$$
where $\hat w_t:=\int_0^t \hat v_s ds$ is the expected total ...
8
votes
0answers
284 views
Transition densities in the Heston model
Knowing the Characteristic function $\Phi_{T,t} = \mathbb{E} [ e^{i u S_T} | S_t, V_t]$ (or equivalently, the Laplace transform) of an affine process, it's possible to know the distribution of the ...
5
votes
0answers
157 views
SABR Question: Why does the market take the beta parameter as a constant?
SABR Question
Why does the market take the $\beta$ parameter as a "constant"?
I see most brokers quoting SABR parameters nowadays.
I've seen many banks use $\beta$=0.5 as a rule.
I've seen quants ...
5
votes
1answer
372 views
Calibration of Cox-Ingersoll-Ross process that hits zero (Feller condition violation)
I'm considering a Cox-Ingersoll-Ross (CIR) process
$$
dx_{t} = \alpha\left(\theta - x_{t}\right)dt + \sigma \sqrt{x_{t}}\,dW_{t}\,,\qquad \alpha,\beta,\sigma > 0
$$
which by assumption has $2\...
4
votes
0answers
177 views
The error term of Hagan's approximation of Black's vol in SABR
Hagans approximation of Black's implied vol in SABR is very! difficult to understand fully. But I want to ask in here if anyone can tell me more about the error term.
Consider the paper:
http://web....
4
votes
0answers
79 views
Stochastic Long-Run Mean Instantaneous Variance in Heston Model (and extensions)?
I'm working on my dissertation in Financial Economics, focusing on the topic of Stochastic Volatility Jump Diffusion models; and I'm playing around with some ideas for model extensions. In particular, ...
3
votes
0answers
92 views
Robust bounds or approximations on implied volatility skew when $\lvert \rho \rvert \rightarrow 1$
Are there any robust / non-parametric results for pure stochastic volatility models, in terms of bounds or preferably accurate approximation, for the implied volatility skew $\partial IV(k) / \partial ...
3
votes
0answers
53 views
What models are used for pricing cliquet options (esp. for Asian Equity underliers)? How good is Bergomi model?
What are the most common models, actually used by trading desks for Asian underliers, for pricing cliquet options?
I would like to know both - (1) the production model used for daily P&L, and ...
3
votes
0answers
80 views
Simulating volatility process in the Heston model using the relation between the CIR Process and Ornstein–Uhlenbeck processes
I am trying to simulate the volatility process in the Heston model using the relation between the CIR Process and Ornstein–Uhlenbeck processes. In fact, giving $\mathbf{X}$ a $n$-dimensional vector ...
3
votes
0answers
105 views
delta hedging with stochastic volatility
In my thesis I want to work with delta hedging with stochastic volatility using Black-Scholes model. How will you suggest I implement numerical solutions using data from the real world? Beside Monte ...
3
votes
0answers
218 views
When to use SV or a GARCH model
So i have been searching for this answer for a question if there is a rule or something that would say when to use GARCH type model or use an stochastic volatility model to predict the volatility of ...
3
votes
0answers
136 views
Approximate asian geometric option with Heston
I am trying to implement Theorem 1 from this Journal in RStudio.
The journal says the it is possible to find a approximate price of a geometric asian option in a Heston setup this way:
$$X_{1cGAO}=e^{...
2
votes
0answers
55 views
Hagan et. al original argument for SABR
In the original SABR paper (Hagan et al 2002 ), the introduction of the famous model is motivated by the observation that local volatility models spot dynamics work the wrong way. As the spot ...
2
votes
0answers
89 views
Stochastic Volatility Models Real World Calibration
I am trying to find some research pertaining to the historical (or real world) calibration of stochastic volatility models.
For example, in applications such as counterparty credit risk (IMM) or ...
2
votes
0answers
39 views
Volatility of a perpetuity $E\Big[\Big(\int_0^\infty e^{-ks+mz_s}ds\Big)^\eta\vert\mathcal{F}_t\Big]$
Let $z$ be a brownian motion, let $\mathcal{F}$ be the filtration it generates. For $k>0$ and $m\in\mathbb{R}$, I define the process $Y$ as
$$Y_t=E\Big[\Big(\int_0^\infty e^{-ks+mz_s}ds\Big)^\eta\...
2
votes
0answers
75 views
Bates Model Jump Percentage Parameters
I am trying to calculate the jump parameters for the Bates volatility jumps, specifically, the mean of the jump percentages, $\mu_j$. For the value of $J$, I am using jumps $|\frac{s_{i}-s_{i-1}}{s_{i-...
2
votes
0answers
69 views
Finding Jump Probability For Time Series Data
I'm relatively new here, so if it seems like I'm asking a bad question, go easy on me.
So I was looking at the Merton Jump Diffusion Stochastic Model on Turing Finance's article. Instead of creating ...
2
votes
0answers
72 views
Taylor expansion of stochastic variables with dynamics of the form $dX_t=b(\sigma_t,X_t)dW_t$
https://www.math.nyu.edu/~cai/Courses/Derivatives/compfin_lecture_5.pdf
In the above document stochastic taylor expansions are nicely explained.
Let us now consider a typical SDE model in finance ...
2
votes
0answers
70 views
The Free Boundary SABR: Natural Extension to Negative Rates
In the paper by Antonov, Konikov and Spector
An alternative approximation for the SABR model is presented.
I'm interested to implement the formula for the ATM swaptions implied volatilities in the ...
2
votes
0answers
42 views
Forward looking estimation of market price of risk of stochastic volatility
I would like to estimate the market price of stochastic volatility by forward looking methods, such as option values. The stochastic volatility model I have in mind is the Heston model or some other ...
2
votes
1answer
135 views
Question about derivation of SABR volatility formula in original paper 'Managing Smile Risk' by Hagan et al
I have a question regarding the starting point of the derivation of SABR volatilities formulas in the appendix of the famous paper 'Managing Smile Risk' by Hagan et al.
To derive SABR volatility ...
2
votes
0answers
112 views
Suggestions to build a copula to price Quanto options
I am willing to price a quanto option through the use of copulas. I will follow the following procedure:
1) Obtain the marginal distributions of the underlying asset and the exchange rate from ...
2
votes
0answers
65 views
Heston Model Maximum Return Distribution
What is the joint probability distribution of the maximum of the return between time $0$ and $t$ and the return at $t$, for the Heston model, when the return drift is $0$ and the correlation between ...
2
votes
0answers
48 views
what kind of test for volatility and where find the data
I am working on a model for stochastic volatility. In short, the model try to capture that the volatility goes up suddenly after a shock (war, policy, financial events, etc) and then goes down slowly, ...
1
vote
1answer
31 views
How to project 1 Year ATM Implied volatility for SPX 500 1Year from now? Final goal is to calculate 1 Year Call prices on SPX 500 1 year from now?
I have the historical data for 1Year ATM Implied Volatility on SPX 500. I want to simulate the 1 year call option prices 1 year from now. What methods and approaches do I need to use? (Heston,GARCH, ...
1
vote
0answers
79 views
How to correctly simulate volatility shocks?
I am working on the comparison of different volatility timing/target strategies on portfolios starting from different conditions (data, asset classes, calculation of realized volatility, different ...
1
vote
1answer
125 views
Local volatility and Stochastic Volatility
Please help me understand similarity and differences between local volatility and Stochastic Volatility both intuitively and mathematically.
1
vote
0answers
41 views
Who came up with 3/2 SV model
Sorry, not a very quantitative question, but does anybody know who was the first person to write down and publish the 3/2 stochastic volatility model? I need this for a reference/bibliography.
1
vote
0answers
45 views
Why can't we create a “magic” basket of options to sell for no-arbitrage pricing in SVJ model?
I am learning how to price SVJ options and am reading some stuff on no-arbitrage pricing for SVJ model using the typical approach you would use (like in BSM option pricing) of creating a risk free ...
1
vote
0answers
65 views
What are good TEXTBOOK on stochastic volatility and interest rate theory?
I wanted to learn stochastic volatility modelling and interest rate modelling.
On this site, a answer recommended me the books "Stochastic Volatilty Modelling" by Lorenzo Bergmo and "Interest Rate ...
1
vote
0answers
313 views
Implied volatility as break-even delta hedge volatility
There have been some posts on this topic, but not what I am looking for, so a new post on an old topic..
I think some/most of us here are familiar with the following formula expressing implied ...
1
vote
0answers
112 views
Fitting a forecasting S&P500 roll volatilities
I have a time series of S&P500 prices, for which I have calculated log-returns and roll-volatility. My goal is to forecast daily realized volatility and test a straddle strategy based on it (I ...
1
vote
0answers
290 views
Mixed local-stochastic volatility model in Quantlib
At a conference the speaker mentioned that it is a standard approach today to use a mix of local and stochastic volatility model in equity, FX and interest rates.
Can you please suggest the most ...
1
vote
0answers
233 views
Is SABR being used in practice for Equity options
Just to be clear: By "in practice" I mean what the banks and other financial companies do.
Do financial companies use SABR for pricing equity options?
Consider a stock with price $t$ being: $S_t$. ...
1
vote
0answers
93 views
Does the Asian Option (average Option) depend on the forward implied vol
I can easily understand that the forward starting Option and Barrier Option depend on the forward implied vol smile at resetting date, so we always choose the stochastic vol model for underlying to ...
1
vote
0answers
71 views
Effect of Volatility Regime on Volatility Smile
For short-term FX options, I find empirically that the degree of curvature of the smile (OTM/ATM in %) is higher in low volatility environments. Similar results are found by Pena et al. ("Why do we ...
1
vote
0answers
37 views
Literature recommendation subordinator models
I'm looking for relevant papers covering subordinator models for stock price modelling. I have alreay read the paper 'A Subordinated Stochastic Process Model with Finite Variance for Speculative ...
1
vote
0answers
90 views
Calibrating Heston paremeters based on market data for Implied Vol for Call options
Several questions have been asked in here regarding calibration in Heston yet I have not found what I have been looking for, so I will ask:
I am looking at a Heston model:
$$dS_t=\lambda \sqrt{v_t}...
1
vote
0answers
167 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 ...
1
vote
0answers
75 views
How to derive the change in portfolio value as given by Gatheral in The Volatility Surface?
I’m trying to follow Gatheral’s Volatility Surface Ch. 1, i.e. the text (pg. 5 and 6) linked to in this question, with further text discussed in this question. I can’t figure out how to arrive at the ...
1
vote
0answers
464 views
Stochastic Vol simulation - Quant job interview question
this is a question from a quant interview (FO quant for IR Exotics for a big 4). First it might be useful when preparing your interviews, second, any brainstorming will be appreciated. Note that no ...
1
vote
0answers
173 views
School project about Black Scholes with stochastic volatility
In a university project I am looking at Black Scholes model with a stochastic volatility. I’m still not quite sure about my focus (I am in the beginning 'Idea phase'). I want to explain the theory ...
1
vote
0answers
309 views
Heston model - Andersen scheme implementation
I would like to implement Andersen scheme for Heston simulation. On the following snipped is my code for generating asset path:
...
1
vote
0answers
182 views
Reference request about stochastic volatility model
I'm fiddling with estimation of stochastic volatility models and have build up a somewhat flexible framework using indirect inference.
I would like to try and throw a lot of different continuous ...
0
votes
0answers
57 views
Gatheral's SVI implementation in Java/Scala
I am trying to fit equity option implied vols using SVI model in Java, and I am using apache math commons library. Some of the option expiries fit very well, but others are completely off, and I am ...
0
votes
0answers
44 views
Vega computation in a stochastic volatility model
What are the possible strategies to compute analytically the Vega (not numerically) in a stochastic volatility model? The goal is to vega-hedge in a generic stochastic volatility model if possible, ...
