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Questions tagged [heston]

A type of stochastic volatility model developed by associate finance professor Steven Heston in 1993 for analyzing bond and currency options. The Heston model is a closed-form solution for pricing options that seeks to overcome the shortcomings in the Black-Scholes option pricing model related to return skewness and strike-price bias.

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Pricing in the Heston Model

The dynamics of the Heston Model is \begin{align*} \frac{dS}{S} & = \lambda \sqrt{\nu} d W^S \\[0.5em] d \nu & = k (1- \nu )dt + \epsilon \sqrt{\nu} dW^\sigma \end{align*} where $\lambda$...
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Negative theta in Log-linear stochastic volatility model

I was asked to simulate the following geometric Brownian motion to get paths for the SPX stock price. the process follows a Log-Linear stochastic volatility. $dS_t = \mu S_tdt+e^VS_tdW_1 $ where ...
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Feller Condition (Cox-Ingersoll-Ross) source

For the Cox-Ingersoll-Ross model $$\text{d}r_t = a(b-r_t)\text{d}t+\sigma\sqrt{r_t}\text{d}W_t$$ the condition (referred to as "Feller condition") $$2ab\geq\sigma^2$$ ensures that the solution is ...
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Quanto basket payoff

I have a payoff that is the worst of the returns two indices: S&P500 (SPX) and Euro Stoxx 50 (SX5E). $\pi = \min \left\{\left(\frac{\text{SPX}_\tau-\text{SPX}_0}{\text{SPX}_0}\right),\left(\frac{\...
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Using QuantLib Python to value FX options using stochastic volatility

I would like to use QuantLib (and in particular the python wrapper) to value FX option using the Heston model. Thanks to http://gouthamanbalaraman.com and all of the articles therein : in particular ...
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Zero-rebate barrier option pricing under the Heston model

I'm trying to derive an approximation for the zero-rebate barrier option under the Heston model: $$dS_t=\mu S_tdt+\sqrt{v_t}S_tdW^S_t$$ $$dv_t=\kappa(\bar{v}-v_t)dt+\eta\sqrt{v_t}dW^v_t,\quad d\langle ...
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How to determine the risk-neutral measure in a Heston model?

To clarify, I'm quite familiar with the risk-neutral pricing framework, and I know one can efficiently Monte-Carlo a Heston model via the non-central $\chi^2$ distribution approach. But so far we're ...
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Good references on Heston Model?

I am looking for good bibliographic references on Heston Model and Stochastic volatility models in general. Does anyone know any good introductory/intermediate references on this topic?
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ATM strike Heston model

I'm thinking about the heston model. price of the asset $S^1=(S_t^1)_{t \leq T}$ fullfills the differential equation $dS_t^1=S_t^1(\mu dt + \sqrt{V_t} dB_t^1)$ the stochastic volatility is given by ...
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Heston model computations

In the Heston model the dynamics of a single-asset $S$ are given by: $dS_t = rS_tdt+S_t \sqrt{V_t}dW^S$ where $W^s$ is a brownian-motion $W^S$ and the square root variance process $V$ is given by ...
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Options on realized volatility / variance

If I'd like to price options on variance/volatility in the Heston model. Is MC simulation and/or finite difference the only way to do it? Or is there an analytical expression for the probability ...
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Intuition for the Effect of Vol of Vol in Heston Model on Volatility Surface

I was hoping someone could describe the economic/mathematical intuition behind the effect that the vol of vol parameter has on the volatility surface, in particular the slope to maturity. Take for ...
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Forward implied vol vs Instantaneous vol

In the Discrete Stochastic Implied Volatility Model which is from the standard Heston Model, the model shows the evolution of forward implied volatilities with time. I thought forward implied ...
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Equivalent martingale measure in time changed Levy models

I am investigating time changed Levy models. As far as I have seen, these models are usually directly described under the risk neutral measure $\mathbb{Q}$. However, I'm interested in first modelling ...
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Hybrid Heston-Hull White Model

I am wondering if anyone could recommend a few good papers on hybrid heston-hull white models, in particular with respect to the approximation of model European options for calibration. Literature on ...
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Pricing a Path-Dependent Option with Heston

I want to price a path-dependent option (let's say for example an arithmetic average Asian option) under a Heston model. In a Black-Scholes setup, I use forward volatilities to do so. I want to apply ...
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Deriving the set where the moment generating function is well-defined from the structure of the assets

So lets assume I have the following structure \begin{align} dS_1 = & \ rS_1 dt + S_1 \left( \sqrt{V_1} dW_{11} + \sigma _{1m} \sqrt{V_m} dW_{12} \right) \nonumber \\ dS_2 = & \ rS_2 dt + S_2 \...
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Heston Wrapper for QuantLib

As a follow up to my question here, I have written a (hopefully) easy to use Python 3 Wrapper around the excellent QuantLib library. My wrapper abstracts away all the QuantLib machinery under the hood ...
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139 views

How to do QE scheme for n correlated assets?

I'm trying to simulate correlated assets under Heston model. I coded the QE scheme for a single asset but i dont understand the next step: How should i set the correlation matrix given my n-asset ...
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169 views

Robust and free Heston pricing software

For a PhD research paper, I need to calibrate Heston. So far, I use Dale Roberts R-Code: https://github.com/daleroberts/heston/blob/master/heston.r for computing prices from which I invert IV ...
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Positive Heston model theta

Under the Heston model, is there a concrete example where the theta $\frac{\partial C}{\partial t}>0$ where $C$ is the price of a European call?
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106 views

Monte Carlo simulation error estimation

How does one estimate the error of a Monte Carlo simulation, for example, of the price of a European call under the Heston model with a given step size and number of paths?
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Multi-Asset Heston Model

How does one best define the correlation structure in a Multivariate Heston Setup? (i.e. Correlations between the Wiener processes of the Stocks / their Variance Processes)
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Calibrating Heston paremeters based on market data for Implied Vol for Call options

Several questions have been asked in here regarding calibration in Heston yet I have not found what I have been looking for, so I will ask: I am looking at a Heston model: $$dS_t=\lambda \sqrt{v_t}...
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Jim Gatheral's assertion on ATM implied volatility vs. square root variance

In Jim Gatheral's book The Volatility Surface Section Dependence on Skew and Curvature on page 138, he asserts that We know that the implied volatility of an at-the-money forward option in the ...
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Interpretation and intuition behind the Put-Call symmetry under the Heston Model

I am currently working on a report regarding the put-call symmetry relations under the Heston model. I did all the math and managed to prove the relations using PDE approach. However, I wish to have a ...
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169 views

Quasi Monte Carlo method and Heston model

I want to run a quasi monte carlo simulation for Heston model in matlab. Obviously there exists a lot of literature regarding the theoretical aspects of the topic, for example by Baldeaux and Roberts, ...
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How to derive the change in portfolio value as given by Gatheral in The Volatility Surface?

I’m trying to follow Gatheral’s Volatility Surface Ch. 1, i.e. the text (pg. 5 and 6) linked to in this question, with further text discussed in this question. I can’t figure out how to arrive at the ...
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what's the difference between market implied volatility and implied volatility?

what's the difference between market implied volatility and implied volatility, how it could be calculated? also what's the quoted implied volatility? thanks.
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Jim Gatheral's ansatz

In the Ansatz section of Jim Gatheral's book Volatility Surface (page 32), he assumes $$\mathbb E[x_s|x_T]=x_T\frac{\hat w_s}{\hat w_T}$$ where $\hat w_t:=\int_0^t \hat v_s ds$ is the expected total ...
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Pricing VIX Futures

In a 2006 paper Zhang and Zhu propose a model for VIX and VIX Futures based on Heston. I am struggling in understanding how they get equation 6 and 8 (where they define the parameters). Can anyone ...
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256 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 ...
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Determine GARCH(1,1) from a mean reverting time series recursion

Let $(v_t)$ be a discrete time series of variance obeying a mean-reverting variance process $v_t$, which is actually the discrete version of the Heston model in finance. \begin{align} x_t &= \sqrt{...
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Heston model reparametrisation

It is well-known that calibrating Heston to the vanilla market is not as easy as it seems: some parameters are "interdependent" and the objective function exhibit plateaus in the parameter space (at ...
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Heston Model: Quadratic exponential scheme

I am having trouble understanding the QE scheme of Andersen. Leif Andersen: Efficient Simulation of the Heston Stochastic Volatility Model, 2006 Is it possible for the variance process to become ...
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Heston Model Calibration

I've calibrated the Heston Model using options data and I was wondering if the parameters I've obtained are stable enough. Also, is Feller condition imposed, when calibrating the Heston Model, in the ...
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How can I improve the numerical integration accuracy in Heston model?

I am trying to perform the numerical integration in the Heston using Gaussian quadrature but I obtain an error of 4e-3 while some of the deep out-of-the-money near expiry Call prices are smaller than ...
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Heston (1997) paper

Does anyone know where I can find this paper ? A Simple New Formula for Options With Stochastic Volatility - (Heston,1997) Thanks
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Gatheral's change of variables for stochastic volatility PDE

This is taken from Gatheral's book "The Volatility Surface", where he tries to go from equation 2.3 to equation 2.4. We have the following PDE, $$ \frac{\partial V}{\partial t}+\frac{1}{2}vS^2\frac{...
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Heston with Forward Dynamics

I'm just curious if it's possible to use forward dynamics and work out the pricing formulas. Does anyone know if there's a reference (paper/url) I can look at?
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371 views

The Heston Solution For European Option - Jim Gatheral

I have this equation (Eq. (2.4) "The Volatility Surface - A Practitioner's Guide" by Jim Gatheral (Ed. 2006)): $$-\frac{\partial C(v, x, \tau)}{\partial \tau}+\frac{1}{2}v \frac{\partial^2 C(v,x,\tau)}...
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Estimate the mean reversion level of the variance process under the real world measure

This paper gives on equation 22 an estimator for the mean reversion level of the variance process under the real world measure. The context is the Heston model, where the variance is stochastic and ...
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Heston ITM and OTM options pricing

In the Carr and Madan (1999) methodology exploiting the fast Fourier transform, the quasi-analytical price of a call is given by: $$C(t,T,K)=e^{-r(T-t)}\frac{e^{-\alpha \log (K)}}{\pi}Re\left[\int_0^\...
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106 views

Shifted heston call price

If we take the heston model but change it slightly by introducing a new parameter $\alpha$ such that is there a way to price the call option within this model as, maybe, a function of the call price ...
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What parameters give a smile (not smirk) in Heston?

I am trying to create a smile in Heston model, however, as of yet, I have only been able to get smirks (i.e., big negative slope ITM that flattens out ATM, and then a very small positive slope OTM). ...
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Relationship between SABR and Heston

What is the relationship between SABR parameters $\sigma, \alpha, \beta, \rho$ and heston parameters $\nu, \kappa, \theta, \xi, \rho$? How do they influence the smile; skewness, kurtosis, etc? And ...
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Calibration of Monte Carlo value?

I wish to calibrate the Heston model parameters to a given smile. Trouble is, I have Heston implemented as a Monte Carlo simulation, and not some deterministic pricing function. So, how do we ...
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Heston & Nandi GARCH model, parameters estimation from option data

I wonder if anybody has code for the HN-GARCH model where the parameters is NOT estimated with maximum likelihood and instead estimated by looking at the option data where an loss function is chosen ...
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Simulation of the Vega in Heston model (for Asian Option)

I'm new here and I hope you guys can help me. I want to calculate/simulate the Vega for my Asian option in the Heston model. The only source I found is the paper of Broadie/Kaya (2004) but they just ...
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how to understand the zero vol condition in Heston stochastic vol model

I can't understand one of the boundary conditions in Heston's model: $$c(t,s,0) = (s-e^{-r(T-t)}K)^+$$ Why the current vol is zero can deduce such result. here $c(t,s,v)$ $s$ is current price and $v$ ...