Questions tagged [stochastic-volatility]

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115 views

Methods of quantifying shifts in return distributions [closed]

I am studying and running some experiments on minute-resolution asset returns and visualizing shifts in the return distribution across a moving window. The returns have fatter tails than if one used a ...
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1answer
95 views

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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0answers
34 views

Price volatility short-term (10 seconds) forecast

Dataset: list of all realized trades (BTCUSDT) from a certain cryptoexchange with timestamps (15 days worth of data) Problem: predict the "price volatility" (standard deviation of realized ...
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0answers
51 views

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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1answer
162 views

How many options would be required to dynamically replicate the VIX nowadays?

The VIX is a portfolio of OTM options on the SPX with non-zero quotes. From CBOE white-paper: Only SPX options quoted with non-zero bid prices are used in the VIX Index calculation. [...] As ...
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1answer
89 views

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 ...
3
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1answer
117 views

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 ...
1
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1answer
54 views

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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0answers
51 views

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 $v_t$...
3
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1answer
541 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 ...
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1answer
569 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{...
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1answer
523 views

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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1answer
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)...
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1answer
446 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\...
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2answers
153 views

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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2answers
298 views

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 ...
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0answers
68 views

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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0answers
53 views

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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1answer
69 views

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 ...
1
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1answer
61 views

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, ...
4
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1answer
75 views

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.
4
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1answer
195 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 ...
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1answer
90 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, ...
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1answer
213 views

Local volatility and Stochastic Volatility

Please help me understand similarity and differences between local volatility and Stochastic Volatility both intuitively and mathematically.
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1answer
79 views

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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0answers
37 views

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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0answers
114 views

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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0answers
25 views

Dupire formula and stochastic rates

I was reading a proof of Dupire's formula from Ch. 2 of Lorenzo Bergomi's Stochastic Volatility modeling and a question came up: what if the repo rate and the risk-free rate are stochastic? Do we have ...
2
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2answers
75 views

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....
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2answers
544 views

Realized variance in SVJJ (Heston with jumps) model

I am working with the stochastic volatility model with jumps in both the price and volatility dynamics, ie. the risk neutral dynamics are of the form: $$\mathrm{d}V_t = \kappa(\theta - V_t)\mathrm{d}t ...
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2answers
67 views

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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0answers
9 views

Is the rate of reversion of spot variance smaller or greater than the rate of reversion of long-term mean of spot variance?

Is the rate of reversion of spot variance smaller or greater than the rate of reversion of long-term mean of spot variance? In other words, is kappaM>kappa or kappa>kappaM of a two-factor affine ...
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0answers
77 views

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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1answer
69 views

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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47 views

How to code Heston’s Square-Root Volatility Model?

I’m currently trying to code the Heston square-root volatility model with the aim to sample from its posterior with MCMC. However, I couldn't add the lagged volatility term, that is, $\sqrt{V_{t-1}}$ ...
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0answers
49 views

Implied vol expansion for $\lambda$-SABR

Is anyone aware of a good implied volatility expansion formula for $\lambda$-SABR (SABR with mean reversion)? I am not sure if there is a formula as simple (or just slightly more complex) as the ...
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2answers
6k views

Strike / delta relationship for FX options

I am trying to find out how to go from delta to strike. If we look at the Bloomberg I am looking at 1M ATM volatility. I have included the Bloomberg data as a picture where we have following ...
4
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1answer
142 views

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 ...
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0answers
26 views

Correlation in GARCH model

I don't think I have ever come across the concept of stochastic correlation so I imagine it's not very widespread, but I had the idea to implement a Monte Carlo VaR model for a portfolio of stocks by ...
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0answers
48 views

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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3answers
274 views

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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1answer
102 views

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 ...
4
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1answer
189 views

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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0answers
28 views

Deriving coupling equation(s) for Heston Stochastic Volatility Model

In Bergomi Smile Dynamics (2003) Section 2.1 we are given the following coupled equations for the mean and for the variance of the hedger's portfolio: $ \begin{align*} \frac{dm}{dt} + \mathcal{L}m - ...
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0answers
58 views

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 ...
0
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1answer
100 views

Numerical simulation of Bates model (Monte Carlo)

I'm trying to build Bates model in Python! $$dS_{t} = \mu S_{t} dt + \sqrt{V_{t}}S_{t}dW_{t}^{1} + J_{t}dQ_{t}$$ $$dV_{t} = \kappa(\theta - V{t})dt + \eta \sqrt{V_{t}}dW_{t}^{2}$$ $$dW_{t}^{1}dW_{t}^{...
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1answer
183 views

Going from $\mathcal{P}$ to $\mathcal{Q}$

Under $\mathcal{P}$, we have the Heston Model given by: $$ d S_{t}=\mu S_{t} d t+\sqrt{\nu_{t}} S_{t} d W_{t}^{S},\\ d \nu_{t}=\kappa\left(\theta-\nu_{t}\right) d t+\xi \sqrt{\nu_{t}} d W_{t}^{\nu}. $...
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0answers
74 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 ...
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2answers
1k views

SVCJ (SVJJ) Duffie et. al Model implementation in Matlab

I'm attempting to implement aforementioned SVCJ model by Duffie et al in MATLAB. so far without success. It's supposed to price vanilla (european) calls . parameters provided, the expected price is: ~...
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2answers
962 views

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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