Questions tagged [stochastic-volatility]
The stochastic-volatility tag has no usage guidance.
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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, ...
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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 ...
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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 ...
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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 ...
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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: $S_t$. ...
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How to determine the risk-neutral measure in a Heston model?
To clarify, I'm quite familiar with the risk-neutral pricing framework, and I know one can efficiently Monte-Carlo a Heston model via the non-central $\chi^2$ distribution approach. But so far we're ...
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Drift term in rough volatility models
I'm studying rough volatility papers and was wondering, why the drift term is always missing.
See for example the paper Pricing under rough volatility by Bayer, Friz, Gatheral.
On page 2, the ...
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When a stochastic volatility model is calibrated?
In an Investment Bank, how often a stochastic volatility model is calibrated ? Is it calibrated daily ? Is it calibrated whenever a pricing is required ?
Thanks.
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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 ...
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Intuition for the Effect of Vol of Vol in Heston Model on Volatility Surface
I was hoping someone could describe the economic/mathematical intuition behind the effect that the vol of vol parameter has on the volatility surface, in particular the slope to maturity. Take for ...
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Why Can I not estimate a CVAR from Heston Model
I fit the parameters of Heston model, using option data for SPX. Now I have the process S and P 500 is expected to follow. I make 100,000 simulations of this process and then calculate the expected ...
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Approximate Hagan formula for SABR model with negative beta
While looking into fixing the $\beta$ parameter (based the following regression: $\text{ln } \sigma^{ATM}_t = \text{ln } \alpha - (1-\beta)\text{ln }F_t$, as explained in West (2004), page 6) before ...
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Why do we require a continuous volatility calibration while pricing Options [closed]
On pricing Options the volatility surface is represented by a mathematical model (with parameters).
What does it mean to calibrate the volatility surface
How often has the volatility surface to be ...
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How to show that SABR is log-normal for $\beta=1$ and normal for $\beta=0$?
For $\beta = 1$ SABR is log-normally distributed and for for $\beta = 0$ SABR is normally distributed. This is a very common property mentioned in almost every paper about SABR. But I can't find the ...
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SABR ATM volatility
The ATM implied volatility is important in SABR when calibrating the model. Let's consider the ATM vol (for a european call option):
$$\sigma =
\frac{\alpha}{f^{1-\beta}} \left[ 1+ \left(\frac{(1-\...
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Rough Volatility Prediction - Gatheral, Jaisson, Rosenbaum Paper
I just read through the paper "Volatility Is Rough" by Gatheral, Jaisson and Rosenbaum. There is a website (link: http://tpq.io/p/rough_volatility_with_python.html) that details the simulations they ...
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How to deduce the formula of the wealth process of a stochastic volatility model?
I am reading the paper Solution of the HJB Equations Involved
in Utility-Based Pricing from Daniel Hernandez and Shuenn Jyi Sheu.
The authors consider the utility function $U: \mathbb{R} \to \...
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The positivity of the market price of risk
Does the market price of risk, be it of stochastic volatility, interest rate or equity return, have to be positive? What is the rationale if it does?
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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 ...
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Vega of exotic options
I'am wondering if there is a standard definition to the Vega of an exotic product when the underlying model is not Black-Scholes.
Let me give some examples :
What is the Vega if the price is ...
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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 ...
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Discretizing a Continuous Time Stochastic Volatility Model
How does the discrete time stochastic volatility model arise from the continuous time one?
Also, forgive me for cross-posting.
I have the following continuous time SDE for a stochastic volatility ...
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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 ...
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Pricing an option with sparse data, high underlying volatility and returns
I'm currently pricing American and European options on an underlying with sparse data (interpolated), high annual volatility and returns over the last year around 300%. The product isn't similar to ...
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140
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stochastic vol modelling not enough for smile
It seems in practice models that include Stochastic Volatility alone do not have enough power to produce actual observed implied vol surfaces. Is there recent empirical literature documenting this?
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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
$$
which by assumption has $2\...
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Literature recommendation subordinator models
I'm looking for relevant papers covering subordinator models for stock price modelling. I have alreay read the paper 'A Subordinated Stochastic Process Model with Finite Variance for Speculative ...
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3
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Problems with local volatility models (vs stochastic volatility models)
Why is pricing with local volatility models are problem with exotics, mainly due to "the volatility surface is the market's current view of volatility and this will change in the future meaning the ...
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Autocallable pricing under stochastic vs. local volatility
I am interest in the reason why an Autocallable (structured product) is cheaper under local volatility compared to stochastic volatility.
I thought this was due to the following:
when thinking in ...
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Mixing Black Scholes with SABR
I am new to the whole concept of stochastic volatility so I am experimenting with option pricing. I think the concept is really difficult to understand / grasp.
I was wondering if the following ...
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Calibrating Heston paremeters based on market data for Implied Vol for Call options
Several questions have been asked in here regarding calibration in Heston yet I have not found what I have been looking for, so I will ask:
I am looking at a Heston model:
$$dS_t=\lambda \sqrt{v_t}...
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439
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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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SSR definition in Bergomi in relation to sticky strike and sticky delta
In Bergomi [Stochastic Vol Modelling] (Sec. 2.5.2), in the section on surface dynamics, the following definition of the "Skew Stickiness Ratio" (SSR) is made:
$$ SSR = \dfrac{1}{\mathcal{S}_T}\frac{d\...
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Volatility swap hedge
What are the hedging methods for volatility swap (rather than variance swap)? What are the possibilities of setting up a static, semi-static or dynamic hedging?
I am aware of but have not yet read ...
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Interpretation and intuition behind the Put-Call symmetry under the Heston Model
I am currently working on a report regarding the put-call symmetry relations under the Heston model. I did all the math and managed to prove the relations using PDE approach. However, I wish to have a ...
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Mixture models of Stochastic Volatility and Local Volatility
As far as I can see on this website the stochastic volatilty models seem to be preferred to local volatility models, mainly due to the fact that stochastic volatility is 2D diffusive process whilst ...
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Local Stochastic Volatility - Break even levels
In Chapter 12 of his excellent book Stochastic Volatility Modeling, Lorenzo Bergomi discusses the topic of local-stochastic volatility models (LSV).
As most of you are probably aware of, the idea is ...
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Motivation of the singular perturbation solution formulation for local volatility model
I am puzzled by the motivation of the particular choice of the (singular) perturbation method used in Equivalent Black Volatilities. Equation (A.6a) sets
$$\epsilon:= A(K)\ll 1.$$
What is the ...
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What is the name of this VaR calculation strategy?
Here's a question on a passage from this paper I'm reading. Here's the quote:
Given the vector of portfolio weights $w$, and the estimate of the
conditional variance, $\Sigma_{t,k}$, the ...
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How to derive the change in portfolio value as given by Gatheral in The Volatility Surface?
I’m trying to follow Gatheral’s Volatility Surface Ch. 1, i.e. the text (pg. 5 and 6) linked to in this question, with further text discussed in this question. I can’t figure out how to arrive at the ...
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Jim Gatheral's ansatz
In the Ansatz section of Jim Gatheral's book Volatility Surface (page 32), he assumes $$\mathbb E[x_s|x_T]=x_T\frac{\hat w_s}{\hat w_T}$$
where $\hat w_t:=\int_0^t \hat v_s ds$ is the expected total ...
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Two Wiener process under same martingale measure Q
Let $W_1,$ $W_2$ be to Wiener processes under the martingale measure $Q$. What can be said about $dW_1*dW_2$? I know that $$(dW_i)^2=dt$$ but what about the case with two different wiener processes?
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Terminal Variance in the Heston Model
I am trying to understand the basics of financial models.
Random Walk as a model for asset prices.
We use gaussian random numbers to generate a Gaussian Random walk. The variance of the terminal value ...
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Suggestions to build a copula to price Quanto options
I am willing to price a quanto option through the use of copulas. I will follow the following procedure:
1) Obtain the marginal distributions of the underlying asset and the exchange rate from ...
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Local vol, stochastic vol, implied vol
I've been studying volatility modelling over past the few days; in particular, the connections between local vol, stochastic vol, implied vol. I've been reading Gatheral's book "The volatility surface"...
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994
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Hedging error in a stochastic volatility model
I would like to find how much error I make when I hedge a call option using Black Scholes model in a market which is actually governed by a stochastic volatility process such as
$$dS_t = rS_tdt + \...
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SABR: how often is tuning parameters needed?
This questions is regarding the behaviour of banks and other financial institutions who deal with FX products and use SABR model volatilities to price options.
How often do they change/tune ...
user25295
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Ljung_Box Statistic of R and R^2 values in Return analysis
I have found a result that I find truly puzzling. Here is an extract from a GARCH-Analysis I have performed:
Test______________Statistic_______p-Value
Ljung-Box Test_____R Q(10)_____0.4047773
...
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Gatheral's change of variables for stochastic volatility PDE
This is taken from Gatheral's book "The Volatility Surface", where he tries to go from equation 2.3 to equation 2.4.
We have the following PDE,
$$
\frac{\partial V}{\partial t}+\frac{1}{2}vS^2\frac{...
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How can I compare 30 day implied volatility forecasts with GARCH forecasts?
I'm trying to understand whether there is a good way to compare forecasts for volatility from different sources i.e., implied volatility and GARCH. I'll outline a few statements that I believe and if ...
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