Questions tagged [stochastic-volatility]

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32 views

Problems with Monte Carlo simulation of the Heston model in R

I am trying to perform European option pricing in the Heston framework by Monte Carlo simulation but I get almost 70% of NaN outcomes for the stock price by the ...
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2answers
119 views

What is wrong in my Heston model's code

I am trying to code a heston model pricer.However,it seems correct at the beginning but when inserting extreme data I retrieve myself with negative probabilities or negative prices. There is the code :...
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1answer
111 views

forward variances under rough bergomi

I have seen in several papers on rough volatility using the following expression for the forward variances $$ d\xi_t(u) = \xi_t(u) \eta \sqrt{2H} (u-t)^{H-1/2}dW_t $$ Can anyone explain to me how this ...
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19 views

Is the market price of risk deterministic or stochastic in the Heston model?

I am recently digging into the Heston model and I have noticed that every author refers to the market price of risk simply as $\lambda$, or sometimes it is more clearly specified to be bi-dimensional ...
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1answer
62 views

Calibration Heston Local Stochastic Volatility (LSV) Model

The Heston Local Stochastic Volatility (LSV) model has the following dynamics: $$dS_{t}=r S_{t} d t+L\left(S_{t}, t\right) \sqrt{V_{t}} S_{t} d W_{t},$$ $$d V_{t}=\kappa\left(\theta-V_{t}\right) d t+\...
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1answer
137 views

Negative Density in Local Stochastic Volatility (LSV) Model Calibration

I'm trying to calibrate Local stochastic volatility model using finite difference method, and I'm mainly following this referece: Tian (2015). I met a problem when calibrating leverage function - the ...
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33 views

generating synthetic asset prices

I would like to use geometric brownian motion (gbm) in order to generate artificial asset prices. I know that gbm has constant volatility, therefore I somehow converted it to stochastic in a very ...
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23 views

Local vol vs stochastic vol in the context of American digital options

I have two models of some spot. One is under local vol and the other is under stoch vol. Both are calibrated to the prevailing vanilla prices. I then consider the option that pays $1$ if the spot ...
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1answer
53 views

Heston model with underlying BS dynamics always gives 1/2 of the right value, what am I doing wrong?

Just as an exercise I'm trying to follow this paper: https://arxiv.org/ftp/arxiv/papers/1502/1502.02963.pdf In the section 2.2 it calculates the value of a Call using the characteristic function of ...
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1answer
105 views

Is pricing options using the volatility surface implied by the Heston model equivalent to pricing using the Heston model directly for all options?

Given Heston model parameters calibrated from vanilla put/call options it is possible to imply a volatility surface by pricing calls or puts for different strikes and maturities and solving the ...
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1answer
118 views

Stochastic Volatility vs Vanna-Volga

I'm working on the calibration of the Heston Stochastic Volatility Model for some FX option data for my bachelor thesis and I was asked "Why should people use Heston instead of other simple ...
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82 views

Heston model vs. GARCH

Heston model is a stochastic volatility extension of the Black-Scholes model. On the other hand, there is also closed-form expression for option pricing that uses GARCH stochastic volatility model. ...
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2answers
150 views

Black Scholes implied vol of SVJ model

Under the SVJ model https://en.wikipedia.org/wiki/Stochastic_volatility_jump, what is the formula of the Black Scholes (log-normal) implied vol for an option with strike $K$ and time to maturity $T$ (...
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34 views

Numerical/integration methods within dynamic SABR

I have a question regarding volatility estimates in the dynamic SABR model. It is well known that the original Hagan et al. (2002) approximation formula for the SABR model does not work good for ...
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1answer
164 views

Boundary conditions Heston's stochastic volatility model

I'm trying to derive the following boundary conditions for heston's stochastic volatility model. This is p. 289 of Shreve's Stochastic calculus for finance \begin{align} c(T, s, v) &=(s-K)^{+} \...
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56 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
589 views

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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113 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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55 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. ...
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110 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{\...
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1answer
278 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 ...
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157 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
299 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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46 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 ...
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1answer
149 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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47 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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132 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
142 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 ...
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3answers
238 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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56 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 $v_t$...
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2answers
194 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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78 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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58 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
94 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 ...
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1answer
102 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.
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1answer
110 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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48 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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40 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 ...
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157 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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1answer
100 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, ...
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2answers
161 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
148 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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10 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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226 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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55 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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1answer
102 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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60 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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1answer
177 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
180 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 ...
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60 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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