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17 votes
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Problems with local volatility models (vs stochastic volatility models)

1. What does it mean by the vol surface is the current view of vol? The local volatility model is calibrated to vanillas prices (and equivalently their implied volatilities), which reflect the market'...
byouness's user avatar
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16 votes
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Local vol, stochastic vol, implied vol

Along with Gatheral's book, I'd recommend reading Lorenzo Bergomi's "Stochastic Volatility Modelling". The first 2 chapters are available for download on his website. That being said, let me try to ...
Quantuple's user avatar
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15 votes
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SSR definition in Bergomi in relation to sticky strike and sticky delta

Some Notations It's easy to get lost so let's introduce some notations and let $$ \sigma : (t, S, K, \tau) \to \sigma(K,\tau; S, t) $$ denote the implied volatility smile prevailing at time $t$ ...
Quantuple's user avatar
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12 votes

Is the local volatility linear if smile is linear?

To simplify the problem, let us consider normal local volatilities $ \sigma \left ( S_t, t \right) $ and implied volatilities $ \sigma_i \left ( K, T \right) $ such that the model is: $$ dS_t = \...
Antoine Savine's user avatar
12 votes

Mixed local-stochastic volatility model in Quantlib

Stochastic-Local Vol (SLV) is an attempt to mix the strengths and weaknesses of both Stochastic Vol and Local Vol models. Below, I'll quickly summarise each model and their strengths and weaknesses, ...
StackG's user avatar
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11 votes
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For pricing, what types of Exotic Options are suitable using Local Volatility Model or a Stochastic Volatility Model?

Whenever you use any model to price anything, all you need to do is make sure you model the underlying dynamics that the product you're pricing actually depends on. Any product will be dependent on ...
will's user avatar
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10 votes
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What are the benefits of using Dupire model

Some points below as food for thought: Suppose you possess an implied volatility surface over a continuous strike cross time to expiry domain (how to get there from the discrete market specification ...
Quantuple's user avatar
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10 votes
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Different volatility surface ( Local vol, Stochastic vol etc.)

I'll answer both of your questions in one go: Your ideas are correct. If the Black-Scholes model was true, the implied volatility surface would be flat but it is not in real life. Thus, the geometric ...
Kevin's user avatar
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10 votes

Book/ Articles recommendation for Volatility models

I have also currently started to learn about the subject. This is some of the material I have encountered: Many people recommend the book "The Volatility Surface: A Practitioner's Guide" by ...
Jesper Tidblom's user avatar
9 votes
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Forward skew generated by Local Vol model

We can demonstrate this via a pricing experiment using QuantLib-Python. I've defined several utility functions in the code block at the bottom of the answer that you will need to replicate the work. ...
StackG's user avatar
  • 3,016
8 votes

Problems with local volatility models (vs stochastic volatility models)

The following paper is helpful for understanding the point you raise: Hagan et al.: Managing Smile Risk, January 2002, Wilmott 1:84-108 The main point is given in the paper: [...] the dynamics ...
vonjd's user avatar
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7 votes

In Dupire's paper, why is $(S_t, t)$ in the $(K, T)$ space?

This is merely a question of notation, you should simply read $$ \sigma(K,T) = \sigma(S_t=K, t=T) $$ For an easy to follow derivation see this excellent note from Fabrice Rouah Some intuition behind ...
Quantuple's user avatar
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7 votes
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Mixture models of Stochastic Volatility and Local Volatility

Stochastic local volatility model means $dS_t/S_t=...dt+\sigma_t L(S_t,t)dW_t$ with $\sigma_t$ the stochastic part (modeled for instance as in the Heston model, or any other dynamics deemed ...
Antoine Conze's user avatar
7 votes
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Interpolation of FX Vol Surface from non-uniform strike vs tenor grid

I tried something along these lines in Quantlib python a few weeks ago. Slightly more simple compared to your approach I think: start with a standard delta quote convention for FX vols (10D puts, 25D ...
user35980's user avatar
  • 1,386
7 votes

Interpolation of FX Vol Surface from non-uniform strike vs tenor grid

In the end I found that fitting a SABR smile to each tenor (borrowing a result from this answer) was sufficient to build a local vol surface that was smooth and well-behaved enough to build a variance ...
StackG's user avatar
  • 3,016
7 votes
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Introductory material for getting started with local and stochastic volatility modelling

If you are looking for a short introduction into various concepts used in volatility modeling without too much mathematical derivations (although written by a mathematician), I would recommend 'Smile ...
BEQuant's user avatar
  • 428
6 votes

Local Volatility with Monte Carlo Simulation

Let the risk-neutral dynamics under your LV model be given by $$ \frac{d S_t }{S_t } = \mu_t dt + \sigma(t,S_t) dW_t $$ Let's drop the drift contribution (not relevant here) and apply Itô's lemma to ...
Quantuple's user avatar
  • 14.6k
6 votes

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

First, what is the SLV? It combines LV (not really a model, just uses vanilla surface to get a grid) with SV (in a nutshell, BSM with a separate stochastic process for vol, hence multiple dynamic ...
AKdemy's user avatar
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6 votes

Introductory material for getting started with local and stochastic volatility modelling

You may find A Short Note on Volatility Models an interesting summary providing bird's-eye overview of general ideas in volatility modeling. I would highly recommend SABR and SABR LIBOR Market Models ...
Hasek's user avatar
  • 814
5 votes

Proof of arbitrage-free implied volatility surface in relation to local volatility surfaces

This is not quite true, in either direction. If you have an arbitrage free implied vol surface, you might not have a well-defined local vol surface. An example comes from a discrete model. Consider ...
q.t.f.'s user avatar
  • 1,885
5 votes

Local vol, stochastic vol, implied vol

Gatheral's book is one of the best reference around so it's worth bearing with it, especially as he covers the relationship between implied, local, and stochastic volatility: local volatility ...
Antoine Conze's user avatar
5 votes

Local Volatility vs. Stochastic Volatility

A similar question posted after this one has a very good answer. Especially the section that explains that the calibrated leverage surface is typically observed to flatten with maturity (a shortcoming ...
AKdemy's user avatar
  • 8,739
5 votes

Problems with local volatility models (vs stochastic volatility models)

The following source contains detailed answers to your questions in a research paper from ETH Zürich. van der Weijst, Roel (2017). "Numerical Solutions for the Stochastic Local Volatility Model" http:/...
Alistar's user avatar
  • 51
5 votes
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Break even Levels Local volatility

The LV model is a particular kind of model where the implied volatility of a European vanilla of given strike and maturity emerges a deterministic function of time, spot level and the local volatility ...
Quantuple's user avatar
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5 votes

Correct Monte Carlo simulation of local volatility models

[I think] the problem is with the SDE, rather than the numerical scheme At a glance, and as I commented, I think the issue you are coming up against stems more from the underlying SDE rather than the ...
oliversm's user avatar
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5 votes
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Calibration Heston Local Stochastic Volatility (LSV) Model

Under Heston LSV (HLSV) dynamics, Gatheral's equality is: $$ \sigma_{LV}^{HLSV}(S_t,t) = \sqrt{E^{HSLV}\left[V_tL(S_t,t)^2 | S_t \right]} = L(S_t,t)\sqrt{E^{HSLV}\left[V_t | S_t \right]}, $$ as $L(S_t,...
ir7's user avatar
  • 5,043
5 votes

Transformation of local volatility model

Consider a function $f(X_t)$. Ito's lemma gives: $$df(X_t)=\text{time terms}+f'(X_t)\sigma(X_t)dW_t$$ Now any $f$ satisfying: $$f'(X_t)\sigma(X_t)=\text{constant}$$ gives a constant volatility for $f(...
fes's user avatar
  • 1,727
5 votes

Does Gatheral formula for local volatility translate to a constraint on implied volatility

This is actually explained in the article "Arbitrage-free SVI volatility surfaces" by Gatheral : It means that the surface is free of butterfly arbitrage.
servabat's user avatar
  • 151
4 votes

Local volatility SVI parametrization

For short maturity SPX option chain, the analytic form of the V-shape volatility smile has been fully worked out in my latest paper on SSRN. You can take a look.
Steve Lihn's user avatar

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