28 votes
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Why dynamics of local volatility is wrong?

A general model (with continuous paths) can be written $$ \frac{dS_t}{S_t} = r_t dt + \sigma_t dW_t^S $$ where the short rate $r_t$ and spot volatility $\sigma_t$ are stochastic processes. In the ...
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  • 3,826
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 ...
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  • 13.9k
16 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'...
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  • 2,110
13 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 = \...
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12 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 $t$ ...
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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 ...
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9 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 ...
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  • 2,366
9 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, ...
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  • 2,836
9 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 ...
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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 ...
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  • 26.7k
8 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 ...
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  • 13.7k
8 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. ...
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  • 2,836
7 votes

A Difference between Local Vol and Stochastic Vol Models

The local vol model has exactly enough freedom to match the individual densities $X_t.$ There is no additional freedom in the local vol model to match even a joint density for a pair of times $(X_t,...
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  • 1,855
7 votes

Local volatility surface corresponding to the implied volatility surface

You should not expect the local vol to be equal to the implied vol except in the trivial case where both are constant (Black-Scholes model). I haven't read the Derman articles but it is quite clear ...
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  • 3,826
7 votes

Why dynamics of local volatility is wrong?

Here "dynamics" means the assumed future behaviour of the spot process, namely that it follows the SDE $$ dS/S = r dt + \sigma_{loc}(S,t) dW_t .$$ There are various ways to see that these dynamics ...
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  • 1,855
6 votes
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Local volatility SVI parametrization

Gatheral and Jacquier discuss this issue in section 4 of the paper. Instead of using the raw parameterization of the SVI, they use the natural parameterization of the total implied variance: $$ w(k) = ...
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  • 2,310
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 ...
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  • 13.9k
6 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(...
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  • 1,222
6 votes
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Transformation of local volatility model

Yes it is called the Lamperti transform. This document, in particular Theorem 2, page 7, describes what the Lamperti transform is.
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5 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 ...
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5 votes
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Local volatility pricer

You can view the price of an option as the cost to dynamically replicate it. The more volatility, the more costs you will have trading the underlying to keep your delta equal to 0 (I'm assuming you ...
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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 ...
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  • 1,855
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 ...
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5 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 ...
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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:/...
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  • 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 ...
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  • 13.9k
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 ...
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  • 1,359
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,...
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  • 4,963
4 votes

Time-independent local volatility

Yes, there is a unique time homogeneous local vol model. This is proven in http://www.sciencedirect.com/science/article/pii/S0304414912002487. There is a slight generalization required that if the ...
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  • 1,855
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.
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