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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Simulating volatility process in the Heston model using the relation between the CIR Process and Ornstein–Uhlenbeck processes

I am trying to simulate the volatility process in the Heston model using the relation between the CIR Process and Ornstein–Uhlenbeck processes. In fact, giving $\mathbf{X}$ a $n$-dimensional vector ...
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1answer
44 views

Difference between modelValue from HestonModelHelper and NPV() from VanillaOption

I am trying to calibrate an Heston model and price vanilla option using Quantlib 1.15 and Python 2.7. I use the following code ...
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51 views

Difference between characteristic function and Fourier transform

I'm struggling to understand the difference between this two functions. I have this condition: $P_j:=\mathbb{Q}(S_T>K):=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{+\infty}Re[\frac{e^{iuK}f_j(u,x,v)}{iu}]\...
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49 views

Sticky Delta Property - Heston Model

Given the model in the picture, how can you verify the sticky delta property without any computational methods? I was told that it is possible to deduce it just looking at the model but frankly i have ...
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40 views

What is the model behind Heston-Nandi functions in the fOptions R package?

I am dealing with Heston model in R and for this purpose I am using the package fOptions from RMetrics. The calibration formula requires the specification of some parameters (omega, lamda, alpha, ...
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33 views

What adjustments need to be made to Heston model to price futures options? [duplicate]

My understanding for the Black Scholes model is that a few adjustments need to be made so that the BS model can be used to price futures. Hence the Black-76 model. What adjustments, if any, do we ...
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112 views

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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37 views

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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119 views

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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59 views

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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240 views

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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101 views

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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1answer
157 views

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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1answer
115 views

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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56 views

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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1answer
122 views

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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1answer
151 views

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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1answer
227 views

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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147 views

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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97 views

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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1answer
190 views

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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84 views

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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1answer
148 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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258 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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51 views

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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123 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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144 views

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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81 views

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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1answer
260 views

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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1answer
278 views

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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178 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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1answer
169 views

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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1answer
356 views

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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1answer
284 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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60 views

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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1answer
242 views

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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307 views

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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522 views

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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2answers
318 views

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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119 views

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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1answer
120 views

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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1answer
56 views

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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1answer
394 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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1answer
191 views

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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1answer
108 views

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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2answers
111 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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1answer
145 views

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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403 views

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 ...