Questions tagged [stochastic-volatility]

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What are some effective and easily implementable volatility smile/skew smoothing models?

Inspired by another post on Bakshi et al. (1997), the paper talks about the feasibility of option pricing models, particularly the SVSI-J variant. I would like to ask the Quant community if there are ...
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Independence of log-returns under Heston model

I first precise that I am new to QuantSE. Heston is a widely used moodel but I have some doubts on it and I couldn't find a proper answer on the internet. How can we prove that Heston in not and ...
NancyBoy's user avatar
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Areas of research in calibration of stochastic volatility models

I am working on a thesis in deep calibration of the Heston model, and I wanted to include a section on the historical work, before the use of neural networks in this area. Thus, I was wondering what ...
sxminho's user avatar
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GSABR model vs SABR model

I've read about the SABR model for pricing options, however I am told there is a variant called GSABR. Does anyone know how this model differs from the original SABR model?. Any papers would be really ...
David's user avatar
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Implied forward volatility definition

What is the rigourous definition of the 'implied forward volatility' and how is it calculated? I couldn't find a rigorous definition as would be the case for 'implied volatility'. Also, could anyone ...
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QuantLib error: `RuntimeError: negative local vol^2 at... the black vol surface is not smooth enough` for calibrating the SLV model

I am trying to generate the SLV process using QuantLib on real SPX data. The issue that I am having is that calendar arbitrage is being violated. I put my data in a list in my code, and am using $r\...
Xerium's user avatar
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Are there standardized measures to characterize the volatility skew?

Might be too simple a question, but I saw in Gatheral & Jacquier (2014) that commonly used features to match volatility skews are (and then I subsequently ChatGPTed some commonly used industry ...
KaiSqDist's user avatar
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Value of the logcontract $Q^T(t,S)$ with payoff $Q(T,S)=-2lnS_T$

Why is the value of the log-contract (Neuberger ,1990) with payoff $Q(T,S) = -2\ln S$ given by $$ Q^T(t,S)=-2e^{-r(T-t)}\left(\ln S + (r-q)(T-t)-\frac{\hat\sigma^2}{2}(T-t)\right) $$ ? It is reported ...
Mr Frog's user avatar
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In the Stochastic-Local Volatility (SLV/LSV) calibration procedure, which surface is used when calibrating the Leverage function

Before we match the leverage function $L(S_t,t)$ to the implied volatility surface generated from the market, we are supposed to calibrate the pure Heston parameters, $(\theta, \kappa, v_0, \rho, \xi)$...
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Gatheral clock time explanation

At the beginning of his well known book, Gatheral writes the following [...] Moreover, unlike alternative models that can fit the smile (such as local volatility models, for example), SV models ...
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Vol-Vol Breakeven (MC Estimation)

I am currently reading the paper Computation of Break-Even for LV and LSV Models. This paper defines the vol-vol breakeven \begin{align*}\tag{1} B_t(T,K,T',K') &\ :=\ d\langle \ln \sigma^{T,K}...
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Simulate Spot Process with Forward Variance (Bergomi)

I am reading Bergomi's book (Stochastic Volatility Modeling), and in section 8.7 The two-factor model (page 326), the following dynamics are given: \begin{align} dS_t &= \sqrt{\xi_t^t}\,S_t\,...
Phil-ZXX's user avatar
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Heston model characteristic function

The characteristic function of $x=ln(S_T)$ in the framework of Heston model is guessed to be: $$f_j(\phi,x,v)=e^{C_j(\tau,\phi)+D_j(\tau,\phi)+i\phi x}$$ The call price is guessed to have the form: $$...
lukada's user avatar
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smile dynamics IV appendix 4

I am having difficulty in recovering some result in smile dynamics of Bergomi https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1520443, the paper gives $(1-3\alpha x +(6\alpha^2 - \frac{5}{2}\beta)...
opsle's user avatar
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Finite difference method for the Heston model using the ADI scheme

I am trying to implement the ADI FDM scheme for the heston and I am following The Heston Model and Its Extensions in Matlab and C#. They have the scheme: $$U'(t) = \textbf{L}U(t),$$ $$\textbf{L} = A_0 ...
Xerium's user avatar
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Feller Condition in the Heston Model

I understand that for MC simulations we require the Feller Condition otherwise the simulation becomes unstable when approximating the events when $v_t<0$, but for semi-analytical solution and using ...
Xerium's user avatar
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Separability of Stochastic Volatility Model

After having read the article of Trolle & Schwartz regarding their general stochastic volatility term structure model (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=966364), it is not clear ...
stokboi's user avatar
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A book that has exercises which closely resembles the content of Lorenzo's Stochastic Volatility Modeling book?

I'm currently going through Lorenzo Bergomi's book Stochastic Volatility Modeling. The one issue I have is that it does not contain exercises to test your knowledge and learn. Is there a textbook ...
THAT'S MY QUANT MY QUANTITATIV's user avatar
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Intuition behind the benefits of Stochastic Local Volatility (SLV) models [duplicate]

There have been various posts on this topic, but they don't really discuss the intuition behind the benefits of the stochastic local volatility (SLV) models over normal stochastic volatility (SV) ...
THAT'S MY QUANT MY QUANTITATIV's user avatar
1 vote
0 answers
130 views

Calibration of $\rho$ in the heston model

When calibrating the Heston model, the gradient of the price of the call/cost function wrt $\rho$ (correlation between $S$ and $V$), is a lot less than the other parameters like $v_0$ and $\bar{v}$. ...
THAT'S MY QUANT MY QUANTITATIV's user avatar
1 vote
0 answers
92 views

Efficient Method of Moments(EMM) for Stochastic volatility model

We are attempting to calibrate the parameters of the Heston model via EMM on historical stock price returns. However, we are first trying a simple stochastic volatility model using EMM. We have come ...
AJ van Niekerk's user avatar
1 vote
1 answer
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Change of expansion point for singular perturbation solution in Equivalent Black Volatilities

In the paper Equivalent Black Volatilities, an peturbative solution is derived for the equivalent Black volatility of a vanilla call option under the dynamics $dF_t = a(t) A(F_t) dW_t$ by Taylor ...
Zach Effman's user avatar
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1 answer
161 views

Calibrating the Heston with the Levenberg-Marquardt algorithm

I am trying to implement the Levenberg-Marquardt algorithm similarly to Cui et al. Full and fast calibration of the Heston stochastic volatility model, 2017 here (although using a different method to ...
THAT'S MY QUANT MY QUANTITATIV's user avatar
1 vote
1 answer
98 views

Incomplete market

How to prove that market with one risky asset $S_t$ and interest rate $r = 0$ is incomplete: $$dS_t = S_t (\mu dt + \sigma_t dW_t^{1}), \quad S_0 = 1,$$ $$\sigma_t = 1 + |W_t^{2}|,$$ $W_t^{1}$ and $...
Strike's user avatar
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COS method for Wishart Heston Model

NOTE: This code is a piece of code I am using for a master's thesis, so I do not expect someone to do the work for me, but I gladly accept suggestions of any kind. However, I am trying to get the ...
SimoPape's user avatar
2 votes
1 answer
138 views

Characteristic Function for Wishart Heston Model

I don't know if this is the right place (at most they will close the post). Anyway, I am trying to implement the characteristic function of the Heston Wishart Stochastic Volatility model illustrated ...
SimoPape's user avatar
2 votes
1 answer
118 views

Volatility Mismatch in SABR Calibration

Problem Statement Hi, I am trying to calibrate SABR on a new asset, which is not 'forward swap rate'. While using the vanillaSABR calibration, I find the parameter 'sigma' (one of model parameters, ...
anmo's user avatar
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how to reflect spot and implied vol relationship in vol curve

There is much evidence about the correlation between spot price and option implied vol in the empirical. This is very important in risk management(i.e. delta hedge). I want to know how to add this ...
aicer's user avatar
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2 votes
0 answers
132 views

Volatility Surface Construction: Ask IV, Bid IV and Mid IV

I am presently engaged in a project wherein my objective is to construct a volatility surface utilizing either the SVI parameterization or the SABR model, leveraging real market data. Initially, I ...
Starlord22's user avatar
3 votes
1 answer
261 views

Vanna Volga Price of an Up and In Put

In the Vanna-Volga approach to pricing first generation exotics, such as single barriers, as I understand it the pricing is as follows: Let $K,S_t < B$. I'll choose the ATM IV $I_{ATM}$ as the ...
Frido's user avatar
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Rough Volatility and Change of Measure

When deriving the rough Bergomi model, Bayer et al in "Pricing Under Rough Volatility" (2015) perform a change of measure to ensure the price process is a martingale as shown in the ...
NavStoke's user avatar
2 votes
1 answer
274 views

Typical values Heston parameters for FX options

I am not as familiar with FX options as I am with equity index options. For the purposes of numerical testing/experiments I'd appreciate if somebody could tell me what are typical parameter values for ...
Frido's user avatar
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2 votes
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Is homogeneity preserved under change of measure?

In a paper, Joshi proves that the call (or put) price function is homogeneous of degree 1 if the density of the terminal stock price is a function of $S_T/S_t$. In the paper I think Joshi is silently ...
Frido's user avatar
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How to replicate a claim in a stochastic volatility model?

Given a Markovian stochastic volatility model with an asset $S$ and a variance process $V$ given by $$ dS_t = \mu_t S_tdt + \sqrt{V_t}S_tdW_t, \\ dV_t = \alpha(S_t,V_t,t)dt + \eta \beta(S_t,V_t,t)\...
julian2000P's user avatar
2 votes
1 answer
194 views

Time-shifted power law in path dependent volatility

I can't understand a function which is part of a volatility model. This is all explained in an open access paper titled "Volatility is (mostly) path-dependent" by Guyon and Lekeufack. My ...
s5s's user avatar
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0 answers
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Construction of stochastic volatility model from a given local volatility model

The Gyongy's theorem: Let $X_t$ be a stochastic process satisfying $$dX_t = \mu_t dt+\sigma_tdW_t$$ where $\mu_t, \sigma_t$ are bounded stochastic process adapted to the filtration $\mathcal{F}_t$. ...
NN2's user avatar
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3 votes
1 answer
107 views

Drift of stochastic variance as slope of the short end of the forward variance curve

I was re-reading Chapter 6 of Stochastic Volatility Modeling by Lorenzo Bergomi. On page 203, he considers a forward variance of the following form: $$ d\xi_t^T=\lambda_t^T dZ_t^T, $$ where $Z_t^T$ ...
fwd_T's user avatar
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1 vote
1 answer
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Stochastic volatility estimation in R

Can anyone help me with the stochvol package in R? I estimated the volatilities using this package but I am not being able to understand how to download the ...
nusratecon's user avatar
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1 answer
140 views

Smile Dynamics - forward variance

I was reading Smile Dynamics II by Lorenzo Bergomi. It is clear to me that on page 2 $$ V_t^{T_1,T_2}=\frac{(T_2-t)V^{T_2}_{t}-(T_1-t)V^{T_1}_{t}}{T_2-T_1} $$ is the fair strike of a forward-starting ...
fwd_T's user avatar
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4 votes
1 answer
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Characteristic function of Gamma-OU process

Consider the Gamma-Ornstein-Uhlenbeck process defined in the way Barndorff-Nielsen does, but consider a different long running mean $b$ which may be bigger than zero: $$dX(t) = \eta(b - X(t))dt + dZ(t)...
Tom's user avatar
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0 answers
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Barrier options in LSV (local stochastic volatility) / Austing's Smile pricing explained

In his book (chapters 9.5 to 9.7), Peter Austing argues that barrier options are insensitive to the details of the stochastic volatility model used in a LSV model, except for the level of vol of vol. ...
greg's user avatar
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0 answers
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Calibration of LSV models to vanna/volga break-even

In this paper, Labordère, the author computes a probabilistic representation of the the vanna/vomma(volga) break-even levels. He mentions that they can be used to calibrate LSV models to historical ...
Greyearl's user avatar
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1 answer
53 views

Non-stationarity and repricing as a source of idiosyncratic and systematic "risk"?

1.Assuming a one period economy with two assets in which cash flows are assigned certain probabilities, using the CAPM, we can derive the P0 given the E(CF) at t1. Within this distribution, we have ...
Leonid Konoplev's user avatar
4 votes
2 answers
341 views

Heston Riccati equation

Let $$ \begin{align*} dY_{t} &= \left(r - \frac{1}{2} V_{t}\right) dt + \sqrt{V_{t}}dW_{t}\\ dV_{t} &= \kappa(\theta - V_{t}) dt + \rho \sigma \sqrt{V_{t}}dW_{t} + \sigma\sqrt{1-\rho^{2}}\sqrt{...
Marc Allan's user avatar
4 votes
1 answer
291 views

Vega hedge of a barrier option

I was re-reading Lorenzo Bergomi's paper Smile Dynamics I. On the first page, he makes the point that it is necessary for a model to match the vanilla smile observed in markets in order to incorporate ...
fwd_T's user avatar
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1 vote
1 answer
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Can you use a forward rate curve to infer the SABR model parameters?

I am currently doing a thesis on a class of SDE parameter inference methods and using the SABR model as an example for inference. I want to extend the application to market data. My question is does ...
FledglingQuant's user avatar
2 votes
1 answer
224 views

Pricing Quantos with Local-Stochastic Volatility model

I would like to price equity quanto options with the Heston Local-Stochastic Volatility model (LSV) but I am having hard time understanding how to apply quanto adjustment in such complex setup. When ...
justLeito's user avatar
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0 answers
77 views

Trading options - risk adjusted return

I have often wondered what kind of risk restrictions do traders of options in Hedge funds have but have not managed to find any information on this matter. I presume there must be some kind of measure ...
fwd_T's user avatar
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1 vote
1 answer
623 views

How to hedge a dual digital option

Let us assume we have two FX rates: $ 1 EUR = S_t^{(1)} USD$ and $ 1 GBP=S_t^{(2)} USD $. Let $K_1>0, K_2>0$ be strictly positive values and a payoff at some time $ T>0 $ (called maturity) ...
fwd_T's user avatar
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2 votes
0 answers
158 views

Barrier on realized volatility

I am trying to understand the risk exposures of vanilla options that also have a European barrier on realized volatility. For example, the option could knock out if the realized volatility over the ...
fwd_T's user avatar
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