Questions tagged [stochastic-volatility]

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When to use a Local Vol model vs Stochastic Vol Model?

I'm new to volatility modeling, I'm struggling to understand when to use a Local Vol model and when to use Stochastic Vol Model, Also now we use a hybrid model combining the two models ? Can someone ...
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Eulero discretization [closed]

Write the Euler discretization of the 1-dimensional stochastic equation $dXt = b (t, X_t) \space dt + \sigma (t, X_t) \space dW_t$ For this part I would say all right because it is a purely ...
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3 votes
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278 views

ATM Implied Volatility and Expected Variance

This answer claims that $$\sigma^2_{ATM}\approx E^Q\left(\frac{1}{T}\int_0^T\sigma^2_t dt\right)$$ ie implied ATM vol = risk-neutral expectation of integrated variance. Is there some proof available? ...
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Help needed in replicating FX Implied Vol Surface

I am relatively new to this area and am doing some self studying on SLV model. I am however getting stuck on trying to replicate this implied vol surface (which I will use to calculate the local vol) ...
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Single barrier options in stochastic volatility models

In this note/sketch, I derive among others a closed-form formula for an up and in put (UIP) in stochastic volatility models of the form $$ dS(t) = \sigma(t) S(t) \left[ \rho dW(t) + \sqrt{1-\rho^2} dZ ...
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1 vote
1 answer
94 views

Calibration and pricing with the Stochastic Local Volatility model

I'm reading the stochastic local volatility model literature, e.g., the Heston Stochastic Local Volatility model (https://ir.cwi.nl/pub/22747/22747D.pdf); but I'm a bit unsure about its calibration ...
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Deriving vol of vol from volatility futures price

From Colin Bennet's trading volatility (pg 117), he says: "A forward on a volatility future is short vol of vol. This means it is possible to back out the implied vol of vol from the price of ...
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Best Way To Compute the Volatility Risk Premium

I'm trying to come up with a measure for the volatility risk premium (VRP) for a strategy I want to implement, but I'm not entirely sure how to proceed. My situation is as follows. The underlying is ...
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SABR LMM for RFR

Is there a research showing a way to use SABR LMM with new RFRs such as SOFR, i.e. pricing exotic path-dependent RFR derivatives with volatility smile and skew? I'm aware that Looking Forward to ...
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  • 301
2 votes
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95 views

Pricing a put-option in the Heston Model

Assume the Heston Model with dynamics under the martingale measure $Q$ given by \begin{align} dS_t &= (r-q)S_t dt + \sqrt{v_t}S_tdW_{1,t}^Q\\ dv_t &= \kappa(\theta-v_t)dt + \sigma\sqrt{v_t}dW_{...
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  • 197
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GARCH option pricing

I have been trying to implement GARCH(1,1) model for pricing call options. Suppose I have calibrated Garch(1,1) model for modelling the conditional volatility using the historical data of an equity ...
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Python/Matlab code to price options under Heston-Hull-White (or Heston-CIR) using sparse grid/finite difference methods

I am looking for Python (or Matlab) code to price options under the Heston-Hull-White (or the Heston-CIR) using sparse grid/finite difference approach. I can find code just for Heston, or just for ...
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89 views

Calculating model-free implied volatility [closed]

I am trying to come up with model-free implied volatility as in Britten-Jones, M. and Neuberger, A. (2000) Option Prices, Implied Price Processes, and Stochastic Volatility, Journal of Finance, 55, ...
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2 answers
201 views

Local Vol vs Stoch Vol Option Pricing

This is an interview question: Imagine you have a double knock-out barrier option: the current spot is 100, the lower barrier is 80, and upper barrier is 120. The barrier is continuous, meaning that ...
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1 answer
115 views

Calculating Expectation of Stochastic Volatility

I have a question while reading THE NELSON–SIEGEL MODEL OF THE TERM STRUCTURE OF OPTION IMPLIED VOLATILITY AND VOLATILITY COMPONENTS by Guo, Han, and Zhao. I don't understand why the above equations ...
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Comparison of Option-Pricing Models (volatility models) vs Product-Mapping

I scoured this forum, looking for some indicative (updated as of year 2021) comparison of volatility/option-pricing models. There were some, but they seem dispersed and lacking in general details... ...
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187 views

How do you hedge volatility risk?

Suppose I model an asset $S_1(t)$ under a stochastic volatility model. To price an option on $S_1$, I must assume the existence of an asset $S_2$ that is used to hedge against changes in the ...
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7 votes
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118 views

Implied vol bounded if and only if instantaneous vol bounded

I'd like to show that in diffusion models IV is bounded iff instantaneous vol is bounded if there is to be no arbitrage. So, assume a model under the pricing measure of the form $$ dS_u = \sigma_u S_u ...
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122 views

Are Stochastic Differential Equation diffusion terms always invariant under a change of measure?

I'm struggling with learning change of numeraire, and stochastic differential equations. I'm reading the beginning of Brigo and Mercurio's Interest Rate Models- Theory and Practice, and I'm on the ...
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Sabr model asset classes

For the SABR stochastic volatility model, can it be used for any asset classes that exhibit a volatility smile (not skew) or is it just restricted to interest rate derivatives? Thanks
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1 vote
1 answer
218 views

Rogers Satchell Volatility

I am trying to implement Roger Satchell volatility in Go, but my results do not match reality... I have been at this all day, but cannot find my error. The 30 day Rogers Satchell vol is at 8.75%, but ...
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1 vote
1 answer
129 views

Stochastic (volatility) models with the elements of fundamental analysis - are there such models and why not?

I read about stochastic volatility models (e.g. https://en.wikipedia.org/wiki/Stochastic_volatility) and those models are quite simple, but the most important feature is that parameters are quite ...
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Forward volatility smile: Local Volatility vs Stochastic volatility

I was reading this great answer: What are the advantages/disadvantages of these approaches to deal with volatility surface? And I have the following question: How to show that the forward volatility ...
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Correlation Spot Vol - when is it important?

I know that a local volatility model does not allow to control the correlation between Spot and Vol. I know also that the correlation Spot Vol is important for products like autocalls. Why is ...
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Are there any public implementations of realized kernels? (preferably in Python)

looking to implement a realized kernel model to forecast realized variance of around ~140 equities and indices in Python given order book data. I have read "Realised Kernels in Practice: Trades ...
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Reduced volatility in local stochastic volatility model

in Local Stochastic Volatility models I always read or hear "first the stochastic volatility model is calibrated to reduced vols and then the local volatility model corrects it" also I head ...
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2 votes
1 answer
183 views

Question about pricing forward start option with Heston Monte Carlo

I'm trying to price a forward start option with payoff $\Big(\dfrac{S_{T_2}}{S_{T_1}}-1\Big)^+$ with Heston Monte Carlo. Heston Model: $$ dS_t = rS_tdt + \sqrt{v_t}S_tdW_t^1$$ $$ dv_t = \kappa(m-v_t) +...
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2 votes
1 answer
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HJM drift condition problem: Show that the HJM drift condition implies $b(t) \equiv b, \rho^{2}(t) \equiv a$

I need your help with understanding and solving the HJM framework. I am hoping I can get some help as I feel so lost with HJM and learning online because of the pandemic is adding more stress. Anyway ...
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3 votes
0 answers
71 views

What is the relationship between the estimated GARCH(1,1) conditional volatility and the true conditional volatility

Suppose that the data has been generated by a GARCH(1,1) model, i.e. \begin{align} y_t &= h_t \epsilon_t, \; \epsilon_t \sim N(0,1) \\ h_t &= \alpha_0 + \alpha_1 \epsilon_{t-1}^2 + \...
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126 views

Why calibrate volatility Models to volatility surfaces rather than underlying's historical price data?

I'm trying to grasp the rationale for calibrating stochastic volatility models (i.e. Heston model) to empirical IV data from market prices. Doesn't this assume that the options are fairly priced and ...
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7 votes
2 answers
367 views

Heston: Variance of Integrated Variance

Consider the standard Heston model\begin{align*} dX&=\left(r-\frac{1}{2}v\right)dt+\sqrt{v}dB,\\ dv&=\kappa(\theta-v)dt+\xi\sqrt{v}dW, \\ dBdW&=\rho dt. \end{align*} Computing $\mathbb{E}\...
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0 answers
41 views

Industry standards for vol control index options

Consider an index of the type: $I(t)/I(t-1) = 1+ a(t) (S(t)/S(t-1)-1)+(1-a(t))r(t-(t-1))$ It is arbitrarily initialized. $r$ is the risk free rate. a(t) is determined piecewise as: $a(t)=s_{target}/s_{...
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0 answers
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Derivation of Bergomi model

In Stochastic Volatility Modeling, L. Bergomi introduces in Chapter 7 the pricing equation (7.4) : $$ \frac{dP}{dt}+(r-q)S\frac{dP}{dS}+\frac{\xi^t}{2}S^2\frac{d^2P}{dS^2}+\frac{1}{2}\int_t^Tdu\int_t^...
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  • 510
2 votes
1 answer
95 views

Can you shift a standard libor market model with regard to only at-the-money options?

Suppose I have an LMM defined using the spot measure as in Brigo and Mercurio: $dF_k(t) = \sigma_k(t)F_k(t)\sum^k_{j=\beta(t)}\frac{\tau_j\rho_{j,k}\sigma_j(t)F_j{t}}{1+\tau_jF_k(t)}dt + \sigma_k(t)...
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  • 103
1 vote
1 answer
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Fourier transform of a European put

In book The concepts and practice of mathematical finance, in the context of illustrating the stochastic volatility model, the Fourier transform $\hat{P}(\xi, V, T)$ of a European put $P(x, V, T)$ is ...
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  • 346
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1 answer
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In what cases characteristic function of (log-)price process is known?

Hey I know that we can use characteristic function of log-price process to price different options. But when we know the characteristic function? I know that we can take Levy processes and constant ...
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How to Discretize this SDE found in finance? (cross-posted)

Continuous-Time In continuous-time form, the "Heston model" is written as $$ dS_t = \mu S_t dt + \sqrt{\nu_t} S_t dW_t^S \\ d\nu_t = \kappa (\theta - \nu_t) dt + \xi \sqrt{v_t} dW_t^{\nu} $$ ...
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4 votes
0 answers
172 views

Characteristic function of the Bates model

I have a misunderstanding concerning the derivation of the SVJ model : Firsty,I understand how to reach the final differential equation from : \begin{gather} dS_t = (r - q - \lambda t (e^{m-\frac{\nu}{...
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  • 446
2 votes
0 answers
102 views

Is $C(K,S_t)$ a (local) martingale if PCS is broken?

When put-call symmetry holds $$ P(S_t,K) = C(K,S_t) = \frac{K}{S_t} C \left( S_t, \frac{S_t^2}{K} \right) $$ where $P$ is the market price of a put option and $C$ is the market price of a call option. ...
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3 votes
1 answer
218 views

Bergomi Volatility Model

I was studying on the Bergomi volatility model(using forward variance represented as $\xi_{t}^{T}$).However I don't understand how the author passes from the sde to the first step by only integrating ...
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  • 446
3 votes
1 answer
128 views

Non-constant Volatility of the Volatility in Stochastic Volatility Models

In pricing financial derivatives, we often first assume that the volatility of the stock price is constant. $$\mathrm{d}S(t) = \alpha S(t) \mathrm{d}t + \sigma S(t) \mathrm{d}W(t)\text{.}$$ The ...
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277 views

Heston Nandi Garch Implementation Problem for Python

I have a coded my own Garch class in order to implement the Heston-Nandi Garch model. ...
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3 votes
0 answers
101 views

Fractional Brownian Motion's Covariance Proof

Let's have the non independent Brownian motion such : $B_{H}(r)=\frac{1}{A(H)} \int_{R}\left[\left\{(r-s)_{+}\right\}^{H-1 / 2}-\left\{(-s)_{+}\right\}^{H-1 / 2}\right] \mathrm{d} B(s), \quad r \in R$ ...
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  • 446
2 votes
1 answer
106 views

Variance swaps and the Log-Moment formula

I was looking at the paper of Raval and Jaquier The Log Moment Formula For Implied Volatility available here : https://arxiv.org/pdf/2101.08145.pdf On the page 4 they wrote(with $<logS>_T$ and $&...
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  • 446
1 vote
2 answers
239 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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  • 446
4 votes
1 answer
150 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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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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  • 221
4 votes
1 answer
401 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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  • 348
7 votes
1 answer
251 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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0 votes
0 answers
43 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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