Questions tagged [stochastic-volatility]
The stochastic-volatility tag has no usage guidance.
390
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Volatility smile vs Vol of Vol smile
Is the vol of vol smile due to the correlation spot/vol similarly to vol smile ?
It seems that if we see the correl as the impact of vol on spot we get the vol smile. And if we see the same correl as ...
- 181
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Sabr extension with stochastic params
I recall reading an article that discussed an extension of the SABR model, where the model's parameters were suggested to be stochastic, leading to improved volatility surface modeling. Unfortunately, ...
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71
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Difference between Stochastic Volatility (SV) model and Stochastic-Local Volatility (SLV) Model
I cannot get what's the difference between SV and SLV models.
The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component.
Based on ...
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81
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Log-normal mean reversion SDE
I study the Tataru-Fisher 2003 LSV model (implemented in the Bbg terminal for FX exotics pricing), the volatility has the following dynamics : $$dV_t = \kappa (1 - V_t) dt + \xi V_t dB_t $$
In the ...
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Local volatility from stochastic volatility: implications for hedging
This is something I've been wondering about:
Given a stochastic volatility model with (stochastic) spot variance $\sigma^2_t$, according to Gyöngy's theorem there exists a local volatility $\sigma^2(K,...
- 2,153
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83
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Discretisation of Heston SV with Jumps (SVJ - Bates)
I want to simulate a price path of SVJ model (Bates) in Excel to see how it works in real time but I need help on how to discretise and construct the jump part with a Poisson process into Heston model ...
- 23
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53
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Predict future Implied Volatility Surface with LSV models
From my understanding, Local Stochastic Volatility (LSV) models (such as the Heston-LSV for instance) are ones of the most used diffusion models used for exotic pricing. One of their advantages (by ...
- 81
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Equivalence time-changed models and stochastic volatility models
I suspect that time changed models can be written as a stochastic volatility model (and vice versa) if the random time is independent of the Brownian motion. Specifically, suppose under the risk-...
- 2,153
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Issues with a time-dependent market price of risk
I have a time-dependent market price of risk of an asset as:
$$ \lambda(t) = \frac{\mu(t)-r(t)}{(T-t)\sigma}
$$
where $t$ is the current time and $T$ is a constant maturity time of an asset.
Here, $\...
- 161
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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 ...
- 1,670
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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 ...
- 121
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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 ...
- 57
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88
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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 ...
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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\...
- 99
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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 ...
- 1,670
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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 ...
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153
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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 ...
- 853
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88
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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}...
- 1,052
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164
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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\,...
- 1,052
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310
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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: $$...
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222
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smile dynamics IV appendix 4
I am having difficulty in recovering some result in smile dynamics of Bergomi https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1520443,
the paper gives $(1-3\alpha x +(6\alpha^2 - \frac{5}{2}\beta)...
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90
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Finite difference method for the Heston model using the ADI scheme
I am trying to implement the ADI FDM scheme for the heston and I am following The Heston Model and Its Extensions in Matlab and C#. They have the scheme:
$$U'(t) = \textbf{L}U(t),$$
$$\textbf{L} = A_0 ...
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113
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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 ...
- 99
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68
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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 ...
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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 ...
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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) ...
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284
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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 $\bar{v}$. ...
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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 ...
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57
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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 ...
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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 ...
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120
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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 $...
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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 ...
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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 ...
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130
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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, ...
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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 ...
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189
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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 ...
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366
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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 ...
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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 ...
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458
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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 ...
- 2,153
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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 ...
- 2,153
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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
$$
dS_t = \mu_t S_tdt + \sqrt{V_t}S_tdW_t, \\
dV_t = \alpha(S_t,V_t,t)dt + \eta \beta(S_t,V_t,t)\...
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263
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Time-shifted power law in path dependent volatility
I can't understand a function which is part of a volatility model.
This is all explained in an open access paper titled "Volatility is (mostly) path-dependent" by Guyon and Lekeufack. My ...
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Construction of stochastic volatility model from a given local volatility model
The Gyongy's theorem:
Let $X_t$ be a stochastic process satisfying
$$dX_t = \mu_t dt+\sigma_tdW_t$$
where $\mu_t, \sigma_t$ are bounded stochastic
process adapted to the filtration $\mathcal{F}_t$.
...
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114
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Drift of stochastic variance as slope of the short end of the forward variance curve
I was re-reading Chapter 6 of Stochastic Volatility Modeling by Lorenzo Bergomi. On page 203, he considers a forward variance of the following form:
$$
d\xi_t^T=\lambda_t^T dZ_t^T,
$$
where $Z_t^T$ ...
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166
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Stochastic volatility estimation in R
Can anyone help me with the stochvol package in R? I estimated the volatilities using this package but I am not being able to understand how to download the ...
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188
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Smile Dynamics - forward variance
I was reading Smile Dynamics II by Lorenzo Bergomi. It is clear to me that on page 2
$$
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}
$$
is the fair strike of a forward-starting ...
- 747
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173
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Characteristic function of Gamma-OU process
Consider the Gamma-Ornstein-Uhlenbeck process defined in the way Barndorff-Nielsen does, but consider a different long running mean $b$ which may be bigger than zero:
$$dX(t) = \eta(b - X(t))dt + dZ(t)...
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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 ...
