17
votes
Accepted
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'...
- 2,200
16
votes
Accepted
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 ...
- 14.5k
15
votes
Accepted
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 $t$ ...
- 14.5k
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 = \...
- 618
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, ...
- 2,996
10
votes
Accepted
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 ...
- 2,531
10
votes
Accepted
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 ...
- 14.5k
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 ...
- 661
9
votes
Accepted
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 ...
- 15.4k
9
votes
Accepted
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.
...
- 2,996
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 ...
- 27.3k
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) $$
For an easy to follow derivation see this excellent note from Fabrice Rouah
Some intuition behind ...
- 14.5k
7
votes
Accepted
Interpolation of FX Vol Surface from non-uniform strike vs tenor grid
I tried something along these lines in Quantlib python a few weeks ago. Slightly more simple compared to your approach I think:
start with a standard delta quote convention for FX vols (10D puts, 25D ...
- 1,231
7
votes
Interpolation of FX Vol Surface from non-uniform strike vs tenor grid
In the end I found that fitting a SABR smile to each tenor (borrowing a result from this answer) was sufficient to build a local vol surface that was smooth and well-behaved enough to build a variance ...
- 2,996
7
votes
Accepted
Introductory material for getting started with local and stochastic volatility modelling
If you are looking for a short introduction into various concepts used in volatility modeling without too much mathematical derivations (although written by a mathematician), I would recommend 'Smile ...
- 428
6
votes
Accepted
Mixture models of Stochastic Volatility and Local Volatility
Stochastic local volatility model means $dS_t/S_t=...dt+\sigma_t L(S_t,t)dW_t$ with $\sigma_t$ the stochastic part (modeled for instance as in the Heston model, or any other dynamics deemed ...
- 5,632
6
votes
Local Volatility with Monte Carlo Simulation
Let the risk-neutral dynamics under your LV model be given by
$$ \frac{d S_t }{S_t } = \mu_t dt + \sigma(t,S_t) dW_t $$
Let's drop the drift contribution (not relevant here) and apply Itô's lemma to ...
- 14.5k
6
votes
Introductory material for getting started with local and stochastic volatility modelling
You may find A Short Note on Volatility Models an interesting summary providing bird's-eye overview of general ideas in volatility modeling.
I would highly recommend SABR and SABR LIBOR Market Models ...
- 764
5
votes
Local vol, stochastic vol, implied vol
Gatheral's book is one of the best reference around so it's worth bearing with it, especially as he covers the relationship between implied, local, and stochastic volatility:
local volatility ...
- 5,632
5
votes
Proof of arbitrage-free implied volatility surface in relation to local volatility surfaces
This is not quite true, in either direction.
If you have an arbitrage free implied vol surface, you might not have a well-defined local vol surface. An example comes from a discrete model. Consider ...
- 1,875
5
votes
Problems with local volatility models (vs stochastic volatility models)
The following source contains detailed answers to your questions in a research paper from ETH Zürich.
van der Weijst, Roel (2017). "Numerical Solutions for the Stochastic Local Volatility Model"
http:/...
- 51
5
votes
Accepted
Break even Levels Local volatility
The LV model is a particular kind of model where the implied volatility of a European vanilla of given strike and maturity emerges a deterministic function of time, spot level and the local volatility ...
- 14.5k
5
votes
Correct Monte Carlo simulation of local volatility models
[I think] the problem is with the SDE, rather than the numerical scheme
At a glance, and as I commented, I think the issue you are coming up against stems more from the underlying SDE rather than the ...
- 1,389
5
votes
Accepted
Calibration Heston Local Stochastic Volatility (LSV) Model
Under Heston LSV (HLSV) dynamics, Gatheral's equality is:
$$ \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]}, $$
as $L(S_t,...
- 5,008
5
votes
Transformation of local volatility model
Consider a function $f(X_t)$. Ito's lemma gives:
$$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(...
- 1,707
5
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 ...
- 8,193
4
votes
Downward sloping smile in normal model
The implied Black-Scholes skew will be downward sloping in the limit on both the left and the right. (I believe @Gordon's derivation claiming upward slope may have a sign error somewhere).
Left Side
...
- 14.7k
4
votes
Accepted
Downward sloping smile in normal model
Since $S_T = S_0 + \sigma W_T$,
\begin{align*}
C &:= E\left((S_T-K)^+ \right)\\
&= E\left((S_0+\sigma W_T-K)^+ \right)\\
&=\int_{\frac{K-S_0}{\sigma \sqrt{T}}}^{\infty}(S_0+\sigma\sqrt{T} ...
- 21k
4
votes
Downward sloping smile in normal model
Although it's a bit different story, there are VERY accurate approximation formulas for the implied volatility under normal model (so-called basis point volatility). Using them, you can obtain the ...
- 1,143
Only top scored, non community-wiki answers of a minimum length are eligible