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'...
- 2,230
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 ...
- 14.7k
16
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 $t$ ...
- 14.7k
14
votes
Accepted
Bergomi: Skew arbitrage
Great question. Let me try to provide some insights and thoughts regarding the points and questions you raised. It may not be a full answer but hopefully it will help connecting the contents in the ...
- 863
13
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, ...
- 3,056
13
votes
Accepted
Solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$
Let
\begin{align*}
Y_t = e^{(a+\frac{c^2}{2})t-cW_t}.
\end{align*}
Then
\begin{align*}
dY_t = Y_t\left[\big(a+c^2\big)dt -c dW_t \right].
\end{align*}
Moreover,
\begin{align*}
d(X_tY_t) &= Y_t ...
- 21.2k
11
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 ...
- 2,591
11
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 ...
- 16.2k
10
votes
Accepted
Modeling Call Price w.r.t. Strike w Models that Capture Vol Smile
The way that I understand your question is that you are looking to fit the market prices of European plain vanilla options of a single maturity and then back out the corresponding implied probability ...
- 6,064
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 ...
- 731
10
votes
Accepted
Deriving the solution for European call option in the Heston Model
Itô's Lemma
The standard version of Itô's Lemma applies to a single Itô process $\text{d}X_t=\mu(t,X_t)\mathrm{d}t+\sigma(t,X_t)\mathrm dW_t$. Then,
$$\mathrm{d}f(t,X_t) = \left(f_t+\mu(t,X_t)f_x + \...
- 16.2k
8
votes
Confusion with volatility smiles implied by different models
In the context of option pricing, "implied volatility" always refers to the equivalent diffusion coefficient in the geometric Brownian motion (GBM) dynamics that is necessary to match an observed ...
- 6,064
8
votes
Problems with local volatility models (vs stochastic volatility models)
The following paper is helpful for understanding the point you raise:
Hagan et al.: Managing Smile Risk, January 2002, Wilmott 1:84-108
The main point is given in the paper:
[...] the dynamics ...
- 27.5k
8
votes
Accepted
Stochastic Volatility and Sticky Delta
Intuitively, in a (log)-space homogenous diffusion model
$$ S_t \propto S_0, \forall t \geq 0 $$
such that implied volatilities will only depend on the moneyness level and not on the absolute spot ...
- 14.7k
7
votes
Accepted
SABR Calibration: Normal vs Log-Normal Market Data
I think you did something wrong in translating the input to numerics. As pointed out by dm63 normal vols are quoted in basis points.
Using equation A.67a) from the Hagan paper you linked we see (...
- 1,738
7
votes
SABR beta range
The SABR process is a strict martingale for all values of beta < 1 (in particular, negative betas are fine). If beta = 1, the process is a strict martingale if and only if rho < 0. Under all ...
- 126
7
votes
Accepted
Strike / delta relationship for FX options
In FX world, the ATM strike is the delta-neutral strike, that is, the absolute delta values of a call and the corresponding put are the same. Moreover, the delta can be premium adjusted or not ...
- 21.2k
7
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 ...
- 5,692
7
votes
Accepted
Interpretation and intuition behind the Put-Call symmetry under the Heston Model
This is a consequence of transforming a Put on $S_T$ with strike $K$ into a Call on $(K S_0)/S_T$ with strike $S_0$ under the stock measure. The new set of parameters $r_p$, $q_p$, $\kappa_p$, ... etc ...
- 5,692
7
votes
What's the point of stochastic volatiliy models if you can use local volatility?
Well, what you find is that the introduction of stochastic vol changes the delta of your options. So what does this mean? If the new delta reduces the variance of your hedged portfolio versus the ...
- 17.5k
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
volatility input for black scholes formula
The best authority I have seen on this stuff is Natenberg: Option Volatility and Pricing. I can't do much better than check my copy. He says: "Note that there are a variety of ways to calculate ...
- 1,156
6
votes
Accepted
The positivity of the market price of risk
No, it can be negative. The price of risk is what you agree to receive on average in exchange for positive returns when the risk measure is high, and determined by the covariance of the risk measure ...
- 932
6
votes
Accepted
Intuition for the Effect of Vol of Vol in Heston Model on Volatility Surface
Maybe it would help you to think of it the following way.
The strike $\sigma^2(T)$ of a fresh-start variance swap of maturity $T$ in the Heston model only depends on parameters $(v_0,\theta,\kappa)$, ...
- 14.7k
6
votes
How many options would be required to dynamically replicate the VIX nowadays?
Gonzalez-Perez (2015) Model-free volatility indexes in the financial literature: A review makes some remarks on this topic in section 2.2.
Andersen, Bondarenko & Gonzalez-Perez (2013) identify a ...
6
votes
Is Local Stochastic Vol needed in order to price barrier options?
Local volatility models capture skew today but not dynamics tomorrow. Stochastic vol captures dynamics tomorrow but not necessarily skew today (how well does your calibrated vol surface match ...
- 755
6
votes
Accepted
Boundary conditions Heston's stochastic volatility model
You can't really derive or prove boundary conditions. You impose them and try to economically motivate them.
Let's consider a European-style call option and go through the boundary conditions step by ...
- 16.2k
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