Questions tagged [stochastic-volatility]

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38 views

“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,...
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1answer
333 views
+100

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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0answers
86 views

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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0answers
36 views

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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0answers
97 views

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{\...
3
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1answer
125 views

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 ...
6
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0answers
142 views

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 ...
4
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1answer
165 views

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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0answers
35 views

Can the Heston model be used to price ANY option?

I've been reading through Heston's work and different Monte Carlo extensions of it and it seems very interestingly flexible. I've mainly used an application of it for pricing Memory Autocalls. Am I ...
2
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1answer
115 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 ...
0
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0answers
39 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 ...
6
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0answers
67 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 ...
3
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1answer
119 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 ...
2
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3answers
161 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 $v_t$...
2
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2answers
176 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 ...
4
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0answers
70 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. ...
3
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1answer
76 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 ...
4
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1answer
85 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.
0
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1answer
91 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 ...
1
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0answers
40 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
31 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 ...
3
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0answers
126 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 ...
1
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1answer
72 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, ...
2
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2answers
97 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
74 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 ...
4
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0answers
109 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? ...
0
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0answers
48 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, $\sqrt{V_{t-1}}$ ...
1
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1answer
77 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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0answers
52 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 ...
4
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1answer
152 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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1answer
111 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 ...
0
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0answers
27 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 ...
1
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0answers
50 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 ...
3
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3answers
295 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 ...
4
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1answer
208 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 ...
2
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1answer
104 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 ...
0
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2answers
462 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 ...
0
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0answers
29 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 - ...
0
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0answers
72 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 ...
1
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1answer
186 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
80 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 ...
1
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1answer
118 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, ...
0
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1answer
121 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}^{...
0
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2answers
80 views

Are there volatility models dependent on returns?

When I look at the relationship between volatility and price, I see a clear negative correlation as shown in this figure (SPY and VIX prices today looking back 1 year). The common volatility models (...
2
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0answers
128 views

Stochastic Volatility Models Real World Calibration

I am trying to find some research pertaining to the historical (or real world) calibration of stochastic volatility models. For example, in applications such as counterparty credit risk (IMM) or ...
11
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2answers
400 views

Solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$

Let $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$ be a stochastic differential equation where $a$, $b$, and $c$ are positive constants, so I tried to solve it but I got stuck in ...
3
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0answers
98 views

Robust bounds or approximations on implied volatility skew when $\lvert \rho \rvert \rightarrow 1$

Are there any robust / non-parametric results for pure stochastic volatility models, in terms of bounds or preferably accurate approximation, for the implied volatility skew $\partial IV(k) / \partial ...

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