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Local volatility from stochastic volatility: implications for hedging

This is something I've been wondering about: Given a stochastic volatility model with (stochastic) spot variance $\sigma^2_t$, according to Gyöngy's theorem there exists a local volatility $\sigma^2(K,...
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resolve the DUPIRE Forward PDE

I want to implement the "Dupire forward PDE". I will use this PDE to calibrate a parametric local volatility model. Could you recommend an article or link that explains concretely how I can ...
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Implied Vol under CEV model

Consider the following steps: Suppose the underlying equity follows a CEV model $dS_t = rS_t dt + \sigma S^{0.5} dW_t$. Use the above CEV model to simulate Monte Carlo paths and price a large set (...
Michael's user avatar
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LSV leverage function calibration

Introduction I try to understand how to calibrate an Heston-LSV model, and I have trouble with how to use Gyongy theorem. Here is the model (1): \begin{align} dS_t &= rS_t dt + \sigma_{\text{LSV}}(...
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Predict future Implied Volatility Surface with LSV models

From my understanding, Local Stochastic Volatility (LSV) models (such as the Heston-LSV for instance) are ones of the most used diffusion models used for exotic pricing. One of their advantages (by ...
Noomkwah's user avatar
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Changing parameterization in Dupire's Formula

In chapter 2 of Bergomi's book on stochastic volatility, we have dupire's formula given as $$\sigma(S, t)^2 = \left|\frac{\frac{\partial C}{\partial T} + (r-q)K\frac{\partial C}{\partial K} + qC}{\...
I_cosine_this's user avatar
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1 answer
51 views

Negative Dupire Variance

I want to compute Dupire Local volatility using the identity that links Dupire local variance to BS implied total variance. I calibrated an SVI on options data to get the implied total variance ...
M2000's user avatar
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How does delta adjustment relate to skew stickiness ratio (SSR)?

The correct delta hedging of a derivative $V$ in a model where volatility $\sigma$ is a function of the underlier $S$ requires a stock holding of an amount $$ \frac{dV}{dS}=\frac{\partial V}{\partial ...
Mr Frog's user avatar
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GSABR model vs SABR model

I've read about the SABR model for pricing options, however I am told there is a variant called GSABR. Does anyone know how this model differs from the original SABR model?. Any papers would be really ...
David's user avatar
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Implied forward volatility definition

What is the rigourous definition of the 'implied forward volatility' and how is it calculated? I couldn't find a rigorous definition as would be the case for 'implied volatility'. Also, could anyone ...
Petra Di Mario's user avatar
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QuantLib error: `RuntimeError: negative local vol^2 at... the black vol surface is not smooth enough` for calibrating the SLV model

I am trying to generate the SLV process using QuantLib on real SPX data. The issue that I am having is that calendar arbitrage is being violated. I put my data in a list in my code, and am using $r\...
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In the Stochastic-Local Volatility (SLV/LSV) calibration procedure, which surface is used when calibrating the Leverage function

Before we match the leverage function $L(S_t,t)$ to the implied volatility surface generated from the market, we are supposed to calibrate the pure Heston parameters, $(\theta, \kappa, v_0, \rho, \xi)$...
Xerium's user avatar
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Vol-Vol Breakeven (MC Estimation)

I am currently reading the paper Computation of Break-Even for LV and LSV Models. This paper defines the vol-vol breakeven \begin{align*}\tag{1} B_t(T,K,T',K') &\ :=\ d\langle \ln \sigma^{T,K}...
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Arbitrage-Free implied/local volatility surface with Cubic Spline Interpolation

I am trying to create a local volatility surface using a cubic spline interpolated implied volatility surface. In other words, I have a function $\sigma(T,K)$, that is arbitrage-free $\forall T,K$ (...
THATS MY QUANT MY QUANTITATIVE's user avatar
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Does Gatheral formula for local volatility translate to a constraint on implied volatility

In Gatheral's "The Volatility Surface : A Practitioner's Guide", Equation (1.10) page 13, the following relation linking squared local volatility and squared implied volatility is expressed :...
servabat's user avatar
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Creating the local volatility surface from the IV surface

I have been using the dupire equation: $$ \sigma_{LV} (K,T) = \frac{\sigma_{i m p}^2+2 \sigma_{i m p} T\left(\frac{\partial \sigma_{i m p}}{\partial T}+(r-q) K \frac{\partial \sigma_{i m p}}{\partial ...
Xerium's user avatar
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smile dynamics IV appendix 4

I am having difficulty in recovering some result in smile dynamics of Bergomi https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1520443, the paper gives $(1-3\alpha x +(6\alpha^2 - \frac{5}{2}\beta)...
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When getting the local vol surface from the implied vol surface, do we interpolate the strikes?

Using the dupire method: $$\sigma(T, K)=\sqrt{\frac{\sigma_{\mathrm{imp}}^2+2 \sigma_{\mathrm{imp}} T\left(\frac{\partial \sigma_{\mathrm{imp}}}{\partial T}+(r-q) K \frac{\partial \sigma_{\mathrm{imp}}...
Xerium's user avatar
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Intuition behind the benefits of Stochastic Local Volatility (SLV) models [duplicate]

There have been various posts on this topic, but they don't really discuss the intuition behind the benefits of the stochastic local volatility (SLV) models over normal stochastic volatility (SV) ...
THATS MY QUANT MY QUANTITATIVE's user avatar
2 votes
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Hagan's 2002 SABR paper "Managing Smile Risk" on Dupire local vol model

I'm reading Hagan's 2002 paper Managing Smile Risk originally published on the WILMOTT magazine, and got something confusing on his comment on Dupire's local volatility model. The set up: Consider a ...
athos's user avatar
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1 vote
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Change of expansion point for singular perturbation solution in Equivalent Black Volatilities

In the paper Equivalent Black Volatilities, an peturbative solution is derived for the equivalent Black volatility of a vanilla call option under the dynamics $dF_t = a(t) A(F_t) dW_t$ by Taylor ...
Zach Effman's user avatar
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What would be the practitioner way of hedging jump risks?

I have developed a keen interest in volatility strategies and have implemented various approaches based on practitioner delta. This delta is meticulously calibrated using a no-arbitrage implied ...
Frank's user avatar
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244 views

Dupire Formula with Discrete Cash Dividend

For stocks when there is cash dividend, the Dupire formula should still hold according to Bergomi. In the book "Stochastic Volatility Modeling", he says: In the presence of cash-amount ...
happydog's user avatar
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Local Volatility Derivation

In Jim Gatheral's book (The Volatility Surface), in the first chapter, it says below while deriving Dupire Equation for Local Volatility. I do not understand how the drift term got removed. Can ...
Pranav's user avatar
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Neural Network to learn Heston Model parameters

I am trying to solve this question: Write down pseudocode to learn a local stochastic volatility for finitely many given option prices: assume a Heston stochastic variance and parametrize local ...
Niko's user avatar
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0 answers
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Valuation via decomposition or via simulation of the underlying?

My question might be very straight forward but I have seen both approaches being followed in practice so I am curious to see if there are arguments in favor or against each one. I am explaining my ...
Kostas's user avatar
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3 votes
0 answers
127 views

Construction of stochastic volatility model from a given local volatility model

The Gyongy's theorem: Let $X_t$ be a stochastic process satisfying $$dX_t = \mu_t dt+\sigma_tdW_t$$ where $\mu_t, \sigma_t$ are bounded stochastic process adapted to the filtration $\mathcal{F}_t$. ...
NN2's user avatar
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From model vega matrix to market vega matrix

I'm reading Antonie Savine's fascinating book Modern Computational Finance AAD and Parallel Simulations. However, I got a bit confused while reading and couldn't make sense of how it works in his work....
James's user avatar
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Questions on limitations of local volatility model

I am currently studying local volatility for equity models and I am trying to understand some limitations of the model: 1. under local volatility, the forward smile gets flatter and higher. Lorenzo ...
StochasticMan's user avatar
2 votes
0 answers
103 views

Good resources about Volatility Calibration with code Snippet

As I just landed in the quantitative finance world, I would like to dig deeper into Volatility Surfaces construction. I have a good theoritical background ( I'm familiar with volatility models ) but I'...
StochasticMan's user avatar
4 votes
1 answer
333 views

Vega hedge of a barrier option

I was re-reading Lorenzo Bergomi's paper Smile Dynamics I. On the first page, he makes the point that it is necessary for a model to match the vanilla smile observed in markets in order to incorporate ...
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2 votes
1 answer
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Pricing Quantos with Local-Stochastic Volatility model

I would like to price equity quanto options with the Heston Local-Stochastic Volatility model (LSV) but I am having hard time understanding how to apply quanto adjustment in such complex setup. When ...
justLeito's user avatar
2 votes
0 answers
205 views

Barrier on realized volatility

I am trying to understand the risk exposures of vanilla options that also have a European barrier on realized volatility. For example, the option could knock out if the realized volatility over the ...
fwd_T's user avatar
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2 votes
2 answers
213 views

Use `LocalVolTermStructureHandle` in Python QuantLib

I would like to simulate a local volatility underlying $$ dS_t = S_t\sigma(t, S_t)dW_t $$ and have looked at QuantLib's LocalVolTermStructureHandle to do so. So far:...
rwicl's user avatar
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Is the time derivative of asset returns expressible as an SDE?

Consider the following SDE for $(S_t)_{t\geq 0}$ under $\mathbb{Q}$, \begin{equation} \mathop{dS_t}=S_t\left(r\mathop{dt}+\sigma(t,S_t)\mathop{dW_t}\right), \end{equation} which (in Langevin form) may ...
UNOwen's user avatar
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0 answers
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Quantlib Monte Carlo using regular Volatility Surface, not Local Volatility surface

I am trying to run a Quantlib Python Monte Carlo simulation using either the ql.BlackScholesMertonProcess or the ql.GeneralizedBlackScholesProcess. I have a vol surface that I have generated using ql....
vman's user avatar
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Implied volatility to local volatility via Dupire

I am doing some self study on stochastic local volatility modelling and am having a hard time replicating some results from the paper "FX Option Pricing with Stochastic-Local Volatility Model&...
APMATH24's user avatar
3 votes
1 answer
556 views

Quantlib vol surface issue 'the black vol surface is not smooth enough'

I create a vol surface from the market and do smoothing(interpolation and extrapolation), and explicitly correct for any total variance decreasing on a given strike as we increase maturity. I create a ...
vman's user avatar
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1 vote
1 answer
137 views

Impact of stochastic rates on varswaps and volswaps

Let us consider that we are looking at issuing some varswaps or volswaps on some FX rate. By longer term I mean something longer than 3 months. Different from this time two years ago, now the interest ...
fwd_T's user avatar
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1 vote
0 answers
231 views

how to understand Black-Scholes volatility is average of local volatilities

Black-Scholes volatility is average of local volatilities. It is from: https://bookdown.org/maxime_debellefroid/MyBook/all-about-volatility.html First what's the meaning of ...
user6703592's user avatar
1 vote
0 answers
173 views

Implied volatility from local volatility versus market implied volatility

Does the local volatility flattens the (existing not forward) skew faster than what we observed in the implied volatility surface? The process is: Get market implied volatilities Fit a IV model (i.e. ...
zeke's user avatar
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4 votes
2 answers
842 views

Is Local Volatility a function of the Strike or the Underlying price?

Long story cut short: I am asking why the Local Volatility function can be thought of as a function of the underlying, when in fact it appears to be a function of the strike. Additionally, I wonder ...
Jan Stuller's user avatar
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How do I proceed with pricing after calculating local vol volatilities?

What are the next steps for pricing under the local volatility formula? It feels like I'm missing the trick (as I only see the numerical methods used to obtain prices after calculating local vol). I.e....
userPrimeNumber's user avatar
5 votes
0 answers
311 views

Very close local volatility and implied volatility using Dupire's equation

I used Dupire's equation to calculate the local volatility as in https://www.frouah.com/finance%20notes/Dupire%20Local%20Volatility.pdf and Numerical example of how to calculate local vol surface from ...
nickzhy's user avatar
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1 vote
2 answers
2k views

How to calculate the local volatility from implied volatility in practice

The local volatility can be derived from the implied volatility. But in practice how we deal with the first-order and second-order derivatives? I have seen this formula $$ \sigma_{\mathrm{Dup}}(T, K)^{...
nickzhy's user avatar
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0 answers
258 views

Stochastic vs. local volatility model choices for greeks

As a follow-up of another question (which is I feel slightly separate, hence a new question). Assume we want to fit a volatility surface with the goal of calculating good greeks, not prices. We can ...
freistil90's user avatar
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298 views

How to price american barrier with Local-Stochastic Volatility

I have attended a conference where one speaker mentioned that the market standard to price FX and Equity derivatives is now the Local-Stochastic volatility model. I understand this class of model is a ...
Goo Gle's user avatar
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0 answers
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how to interpolate and extrapolate the local volatility surface?

Local volatility can be computed in terms of call prices using Dupire's formula. Assume we have a rectangle call price surface, let's say $I = [30,60]\times[1 day, 1year]$. For interpolation, should ...
StupidMan's user avatar
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1 answer
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Adequate model to payoff

Consider a payoff that pays a certain amount N of a vanilla Call (underlying: S, Maturity= T, strike:K). Every semester date Ts before T, if S>K(Ts), then N is increased by 1. This product seems ...
user25844's user avatar
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5 votes
1 answer
240 views

Independence vs correlation in stochastic vol models

I am struggling a bit with some basic stuff lately: Consider a SV model \begin{align} dS_t &= \sigma_t S_t dW_t \\ d\sigma_t &= b(\sigma_t,t) dZ_t \end{align} with $dW_t dZ_t = 0$. I know that ...
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