Questions tagged [stochastic-volatility]
The stochastic-volatility tag has no usage guidance.
370
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Fractional Brownian Motion's Covariance Proof
Let's have the non independent Brownian motion such :
$B_{H}(r)=\frac{1}{A(H)} \int_{R}\left[\left\{(r-s)_{+}\right\}^{H-1 / 2}-\left\{(-s)_{+}\right\}^{H-1 / 2}\right] \mathrm{d} B(s), \quad r \in R$
...
- 446
2
votes
1
answer
169
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Variance swaps and the Log-Moment formula
I was looking at the paper of Raval and Jaquier The Log Moment Formula For Implied Volatility
available here : https://arxiv.org/pdf/2101.08145.pdf
On the page 4 they wrote(with $<logS>_T$ and $&...
- 446
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vote
2
answers
550
views
What is wrong in my Heston model's code
I am trying to code a heston model pricer.However,it seems correct at the beginning but when inserting extreme data I retrieve myself with negative probabilities or negative prices.
There is the code :...
- 446
4
votes
1
answer
224
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forward variances under rough bergomi
I have seen in several papers on rough volatility using the following expression for the forward variances
$$ d\xi_t(u) = \xi_t(u) \eta \sqrt{2H} (u-t)^{H-1/2}dW_t $$
Can anyone explain to me how this ...
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0
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187
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Is the market price of risk deterministic or stochastic in the Heston model?
I am recently digging into the Heston model and I have noticed that every author refers to the market price of risk simply as $\lambda$, or sometimes it is more clearly specified to be bi-dimensional ...
- 221
5
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1
answer
1k
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Calibration Heston Local Stochastic Volatility (LSV) Model
The Heston Local Stochastic Volatility (LSV) model has the following dynamics:
$$dS_{t}=r S_{t} d t+L\left(S_{t}, t\right) \sqrt{V_{t}} S_{t} d W_{t},$$
$$d V_{t}=\kappa\left(\theta-V_{t}\right) d t+\...
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1
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506
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Negative Density in Local Stochastic Volatility (LSV) Model Calibration
I'm trying to calibrate Local stochastic volatility model using finite difference method, and I'm mainly following this referece: Tian (2015).
I met a problem when calibrating leverage function - the ...
- 121
0
votes
0
answers
57
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generating synthetic asset prices
I would like to use geometric brownian motion (gbm) in order to generate artificial asset prices. I know that gbm has constant volatility, therefore I somehow converted it to stochastic in a very ...
1
vote
0
answers
139
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Local vol vs stochastic vol in the context of American digital options
I have two models of some spot. One is under local vol and the other is under stoch vol. Both are calibrated to the prevailing vanilla prices. I then consider the option that pays $1$ if the spot ...
- 595
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1
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102
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Heston model with underlying BS dynamics always gives 1/2 of the right value, what am I doing wrong?
Just as an exercise I'm trying to follow this paper: https://arxiv.org/ftp/arxiv/papers/1502/1502.02963.pdf
In the section 2.2 it calculates the value of a Call using the characteristic function of ...
- 135
2
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1
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300
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Is pricing options using the volatility surface implied by the Heston model equivalent to pricing using the Heston model directly for all options?
Given Heston model parameters calibrated from vanilla put/call options it is possible to imply a volatility surface by pricing calls or puts for different strikes and maturities and solving the ...
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1
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649
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Stochastic Volatility vs Vanna-Volga
I'm working on the calibration of the Heston Stochastic Volatility Model for some FX option data for my bachelor thesis and I was asked "Why should people use Heston instead of other simple ...
0
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424
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Heston model vs. GARCH
Heston model is a stochastic volatility extension of the Black-Scholes model. On the other hand, there is also closed-form expression for option pricing that uses GARCH stochastic volatility model. ...
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2
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312
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Black Scholes implied vol of SVJ model
Under the SVJ model https://en.wikipedia.org/wiki/Stochastic_volatility_jump, what is the formula of the Black Scholes (log-normal) implied vol for an option with strike $K$ and time to maturity $T$ (...
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Numerical/integration methods within dynamic SABR
I have a question regarding volatility estimates in the dynamic SABR model. It is well known that the original Hagan et al. (2002) approximation formula for the SABR model does not work good for ...
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1
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770
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Boundary conditions Heston's stochastic volatility model
I'm trying to derive the following boundary conditions for heston's stochastic volatility model.
This is p. 289 of Shreve's Stochastic calculus for finance
\begin{align}
c(T, s, v) &=(s-K)^{+} \...
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0
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"Pricing European Options in a Stochastic-Volatility-Jump Diffusion Model" ,does anyone have this article?
I can't find the article "Pricing European Options in a Stochastic-Volatility-Jump Diffusion Model" of
Thomas Knudsen and Laurent Nguyen-Ngoc, Journal of Financial and Quantitative Analysis,...
18
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1
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Bergomi: Skew arbitrage
In his paper "Smile Dynamics IV" (https://www.fields.utoronto.ca/programs/scientific/09-10/finance/derivatives/bergomi.pdf) as well as in his book "Stochastic Volatility Modeling" (...
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Are rough stochastic volatility models used on the street for equity derivatives ? (2020)
I'm building out some stochastic vol models for pricing exotic equity derivatives. What's the state of the art on the street?
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159
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Characteristic function for heston model with jumps in price and variance
I need the characteristic function of the Heston model with jumps in price and variance, or in other words, the characteristic function of the Bates model (1996) adding jumps in the variance dynamics.
...
4
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279
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Angular bracket notation (physics)
In a few papers I have seen the following notation:
$$
\langle X_t \rangle
$$
Also, in Bergomi's book, at page 8, we have the following equality:
$$
\biggr\langle \int_0^T e^{-rt}s^2 \frac{d^2P_{\hat{\...
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993
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Vega in the Heston model
I'm trying to calculate the hedging quantities of the Heston model. I undestand that the replicating portfolio consist of one option, $V = V(S,v,t)$, $\Delta$ stocks and $\phi$ units of the option to ...
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271
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Hedging : effect of not matching the term structure of skew
Let us assume that we construct a pure stochastic volatility model calibrated to the implied volatility surface, but that the model does not replicate accurately the observed term structure of the ...
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4
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1
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1k
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Deriving the solution for European call option in the Heston Model
I'm deriving the solution for European call option in the Heston Model. I follow the original paper by
Heston and Fabrice Douglas Rouah's derivations in his book The Heston Model and Its Extensions in ...
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3
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1
answer
372
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Calibrate Stochastic Volatility Model
For stochastic volatility models, and any vol model I know, it seems the standard approach is to calibrate the model from option prices. As other user said, this seems a chicken egg problem - how do I ...
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770
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Autocallable option Delta
There have been numerous exotic trading desk blow ups lately, related to various reasons. However, in particular, one bank had some issues where they were pricing autocallable notes with Local ...
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3
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1
answer
347
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LIBOR market model with stochastic volatility
I have read that there are 3 types of pricing models: local volatility, stochastic volatility and stochastic-local volatility models (LSV).
I am now looking at interest rates exotics pricing models ...
- 41
2
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3
answers
560
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Simulating the Rough Heston
I found this paper here https://arxiv.org/abs/1810.04868, "The Lifted Heston", but since I'm not an expert in stochastic volterra processes , nor in fractional ricatti equations, the math is ...
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Can you approximate stochastic volatility processes using GARCH processes?
Let me specific. Suppose that you have the following process:
\begin{align}
z_t &= \sigma_t \epsilon_t \\
\sigma_t &= \sigma \exp \left( \frac{v_t}{2} \right)
\end{align}
where $v_t$...
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2
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Strike Arbitrage
In Stochastic Volatility Modelling, Chapter 2, the author derived the Dupire equation
$$\mathbb{E}[\sigma_T^2|S_T = K] = 2\frac{\frac{dC}{dT} + qC +(r-q)K\frac{dC}{dK}}{K^2 \frac{d^2C}{dK^2}}.$$
The ...
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162
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Why is the Schöbel-Zhu model affine?
In the Schöbel-Zhu model, the stochastic volatility process is $dv_t=\kappa(\theta-v_t)dt+\sigma dW_t$.
The characteristic function of the stock process can be found by arguing that the model is ...
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0
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124
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Modelling volatility for higher frequency data
I'm doing some academic work on volatility forecasting. I've got 1-minute bar data. It is not clear to me what model is best suited for forecasting volatility when higher frequency data is available.
...
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4
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1
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152
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How to calibrate models with unbounded parameter space
I am calibrating the Heston model with sequential quadratic programming algorithm. It turns out that the volatility surfaces I am calibrating to can be fit very well with extreme values of mean ...
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2
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1k
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how to calculate vega in stochastic vol?
since vega is defined as option value changes regarding the implied vol parallel shift, how is vega defined or calculated in stochastic vol models since implied vol is not an input there? thank you.
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1
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346
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Book/ Articles recommendation for Volatility models
I am looking for references on volatility models. I want to gain more insights on these models but have a little background as of now. Thus, looking for references that can pick the topic from basics ...
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0
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164
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Intuition behind local volatility curve shapes in interest rate environments
I have some questions regarding the intuition behind shapes for the local volatility (LV) curve as seen in quite popular models. Let's say we have the following generalized stochastic-local volatility ...
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0
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498
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Rigorous proof of Dupire formula (e.g. using Gyöngy's theorem)
Where can I find a rigorous proof of the Dupire formula (for example, using using Gyöngy's theorem)? I imagine this would be covered by a paper or by a standard financial math text, but I could not ...
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1
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236
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Serial correlation, quadratic variation and variance of returns
On p. 3 of Lorenzo Bergomi's book on Stochastic Volatility Modeling, there is the following assertion:
Indeed, to a good approximation, the variance of returns scales linearly with their time scale, ...
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2
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2
answers
642
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What is the difference between parametric and non-parametric models?
I'm reading about volatility modelling and I came across the concept of parametric and non-parametric models. For example, GARCH is a parametric model and Realized Volatility is a non-parametric model....
- 452
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votes
2
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495
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Stochastic Volatility Models - are they complete markets?
I'm reading about stochastic volatility models - the ones which resulted after Wiggins proposed in 1986/7 that $\sigma$ in Black-Scholes should be a stochastic process rather than a constant.
In ...
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551
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Libor Market Model with SABR Calibration
What is the industry practice in calibrating SABR Libor Market Model? Do you first calibrate the SABR model using market data and then implement the libor market model with the calibrated parameters?
...
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1
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160
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Do all stochastic volatility models capture volatility smile?
I started reading SABR model recently.
In Wiki page, it states that the SABR model can capture volatility smile in derivative market.
However, I do not see how it does so.
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1
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What is vega, really?
Assume for now we are working in a stohastic volatility (SV) setting,
$$
dS_r = \sqrt{v_r} S_r dW
$$
and
$$
dv_r = a(v_r,r)dr + b(v_r,r) dZ
$$
with
$$
dWdZ = \rho dr
$$
Let $C(S_t,v_t,t)$ denote the ...
user34971
2
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1
answer
442
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How do you handle implied volatility performing a VaR Monte-Carlo simulation using a stochastic volatility process calibrated on the underlying
Say you have a portfolio consisting of options each having a market implied volatility. If you now use some stochastic volatility model like GARCH to calibrate the real world volatility of the ...
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0
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107
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Dupire Vomma and Stochastic volatility
Suppose that you are short an option on asset $X_t$ following a pure diffusion. Suppose you are hedging your position using (Dupire) Local volatility model. Suppose that the option is concave with ...
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3
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559
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Are there any books/articles on how to use options to be long volatility (implied or realized)? [duplicate]
Given the market turmoil of late I have become fixated with this idea of using options to be long volatility (realised and implied). However, I dont know where to start, what to read, who to follow ...
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685
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What stochastic volatility models are industry standard for option pricing and how do they work?
I've started reading up on stochastic volatility models and it seems very difficult to discern which ones are used in practice and which have been mostly left alone in theory. What are the popular ...
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1
answer
280
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Rigorous proof that volatility target strategies actually tend to the target
I'm working on a paper about volatility timing and target strategies, practical implementation included.
While writing down the mathematical description of the model I wanted to include a rigorous ...
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2
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5k
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Heston volatility surface in Python QuantLib
Does anyone have experience with the Python QuantLib function HestonBlackVolSurface?
I'm trying to produce a 3D plot of the volatility surface as done in the ...
- 1,012
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373
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Discretizing Bates SVJ Model to simulate paths
I am trying to simulate a path for Bates Stochastic-Volatility-Jump model. It has the following dynamics:
I've managed to implement the Heston model by following Gatheral's books the Volatility ...
- 1