20 votes

How do different models impact option Greeks?

This is an interesting and not so easy question. Here's my 2 cents: First, you should distinguish between mathematical models for the dynamics of an underlying asset (Black-Scholes, Merton, Heston ...
  • 14.1k
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'...
  • 2,120
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 ...
  • 14.1k
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 ...
  • 20.5k
12 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 $t$ ...
  • 14.1k
12 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 ...
  • 721
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 ...
  • 2,446
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 ...
10 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, ...
  • 2,856
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 + \...
  • 14.1k
9 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 ...
  • 14.1k
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 ...
8 votes
Accepted

2 Ito processes - $d(X_{t} + X^{'}_{t})^2 = (Y_t + Y^{'}_{t})^2 dt$ why it is true?

$X_t$ being a stochastic process, one cannot use ordinary calculus to express the differential of a (sufficiently well-behaved) function $f$ of $t$ and $X_t$. Instead one should turn to Itô's lemma, ...
  • 14.1k
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 ...
  • 27k
8 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 ...
  • 14.1k
7 votes
Accepted

SABR calibration: simple explanation and implementation

There are lots of papers online and here are a few I would suggest math.umn riskworx G. Dimitroff, J. de Kock Nowak, Sibetz I you have matlab there is an step step example to calibrate SABR ...
  • 891
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,...
  • 1,865
7 votes
Accepted

Stochastic volatility

[Short answer] No closed-form formula in general. You need to resort to numerical methods. Monte Carlo is preferred by most practitioners but you could also use Finite Difference schemes (and ...
  • 14.1k
7 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 ...
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

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 ...
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 ...
  • 14.3k
7 votes
Accepted

ATM Implied Volatility and Expected Variance

I am not so sure about the ATM approximation from the other answer (i.e. I don't think it's a great approximation). I think it comes from the following for $T \ll 1$: \begin{align} E \left[ \frac{1}{T}...
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

Implied volatility and pricing of vanilla options

CRR is just a numerical approximation to Black--Scholes. Its main use is in getting American option price. There is no real difference other than slight inaccuracy when using it for Europeans. So no ...
  • 6,763
6 votes
Accepted

Covariance matrix and Cholesky decomposition

I am not sure if I understood your question correctly but I will try to answer it anyway. If you have a standard normal random vector $z \sim N(\mathbb{0},I_n)$ (where $z,0 \in \mathbb{R}^{n\times1}$ ...
  • 2,894
6 votes

Calculating 6-minute, 20-minute, 45-minute, and 3-hour volatility

Quick summary: Your model should still be well specified, as long as: 1) You do the analysis on a heavily traded asset, e.g. IBM on NYSE, and 2) You use heteroskedasticity-consistent standard errors ...
6 votes
Accepted

Why is there a stong intraday-correlation between spot and vol?

This effect is coming from the supply and demand in the options markets. Many portfolio managers want (or need) to buy out of the money put options, and many are willing to sell out of the money call ...
  • 14.3k
6 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 (...
  • 1,638

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