15
votes
Accepted
Difference between GARCH and Heston Volatility model
Heston gives an expression for the characteristic function, from which option prices can be computed. Therefore it can be calibrated (statically) on a set of vanilla option prices with different ...
- 4,267
15
votes
Pricing VIX Futures
Heston - Change of measure
Consider the following Heston dynamics written under the real world measure $\Bbb{P}$
\begin{gather}
\frac{dS_t}{S_t} = \mu_t dt + \sqrt{v_t} dW_S^{\Bbb{P}}(t),\ S(0) = S_0 \...
- 14.3k
11
votes
Accepted
Delta of an option under Heston model
Bad news: Your calculation is not quite correct
As you say, the initial price of a European call option is
$$C(S_0;K,T)= S_0e^{-qT}\Pi_1-Ke^{-rT}\Pi_2. \tag{$\star$}$$
However, the exercise ...
- 14.9k
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 ...
- 5,821
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 + \...
- 14.9k
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 ...
- 631
7
votes
Option Pricing Model Calibration In Practice
The typical approach is: you only use option data from the last day. Furthermore, you only include those points that are liquid enough. One approach to this is to weigh the modelling error of an ...
- 1,869
7
votes
Heston Model Integration Oscillations
There has been a huge amount of work on this. Generally a Fourier transform approach is used.
First, be careful to use the form of the characteristic function that does not wind about zero in order ...
- 6,823
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 ...
- 5,592
7
votes
Accepted
What is the Radon-Nikodym derivative in the Heston model?
Let
\begin{align*}
\mathrm{d}S_t&=\mu S_t\mathrm{d}t+\sqrt{v_t}S_t\mathrm{d}B_{S,t}, \\
\mathrm{d}v_t&=\kappa(\bar{v}-v_t)\mathrm{d}t+\xi\sqrt{v_t}\mathrm{d}B_{v,t},
\end{align*}
where $\...
- 14.9k
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
CIR Process from Ornstein–Uhlenbeck process
I don't think that the statement you reference is correct for general $n \in \mathbb{R}$ but only for $n \in \mathbb{N}$.
The intuition behind this is that each Ornstein-Uhlenbeck (OU) process is ...
- 5,821
6
votes
Accepted
Jim Gatheral's assertion on ATM implied volatility vs. square root variance
Below are my 2 cents only, but this was too long for a comment.
As he shows in the next lines (see also Variance Swaps chapter of Bergomi's book)
$$ \sigma_{VS}^2(T) = \int_{-\infty}^{+\infty} \tilde{\...
- 14.3k
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 $(v_0,\theta,\kappa)$, ...
- 14.3k
6
votes
Accepted
Most accurate Fourier transform method for extreme OTM options
I'll give it a start and stick with Fourier methods. The approaches from Carr and Madan (1999) and Fang and Oosterlee (2009) are indeed known to be inaccurate for highly OTM options. I'd suggest to ...
- 14.9k
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 ...
- 14.9k
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 ...
- 636
5
votes
Option Pricing Model Calibration In Practice
I know two papers explaining how to calibrate this kind of models, and one of them explain the impact of the quality of the fit on a pricing model:
Aït-Sahalia, Y. (2002, January). Maximum likelihood ...
- 11k
5
votes
Accepted
parameters in Heston model and their impact on volatility smile
Intuition: You can think of the vol smile as a reflection of the risk neutral distribution (compared to the Black Scholes Gaussian density). A fat tailed distribution creates the smile: fat tail -> ...
- 4,267
5
votes
Accepted
About the Feller Condition in Heston Calibration
You should not use the Feller condition as a constraint. In many cases its violation will be required for a good fit to the market data.
- 1,875
5
votes
Accepted
Analytical Solution for Heston Model
This equation is unrelated to the Heston model. It is simply the value of a European call under the a constant coefficient geometric Brownian motion, i.e. the Black and Scholes (1973) model. Here $\nu$...
- 5,821
5
votes
Terminal Variance in the Heston Model
From the equations of the model it is clear that $v_t$ is the instantaneous variance of the log-returns, not the terminal annualised variance of the log-asset price.
Put differently, you are you ...
- 14.3k
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,...
- 4,948
5
votes
Accepted
Heston: Variance of Integrated Variance
Studying zero-coupon bond prices in the CIR (1985) short rate model, $\text{d}r_t=\kappa(\theta-r_t)\text{d}t+\xi\sqrt{r_t}\text{d}W_t$, Hirsa (2013, Section 1.2.6.2) states that the characteristic ...
- 14.9k
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