10
votes
Accepted
Breeden-Litzenberger formula for risk-neutral densities
I assume that for approximating the second derivative of the call price $C (K,T) $ at the bounds of the strike domain (see first 2 "if" cases of the last for loop of your code) you tried to set up ...
- 13.9k
9
votes
Accepted
How to get set the theta function in the Hull-White model to replicate the current yield curve
Concerning your first question, this depends on what curve, currency, etc. you are interested in. The general method for constructing yield curves is called bootstrapping which allows you to derive ...
- 7,091
8
votes
Accepted
Proof behind solution for theta in Hull-White with time-dependent volatility and mean reversion?
We assume that the process $\{r(t), \, t \ge 0\}$ satisfies an SDE of the form
\begin{align*}
dr(t) = \big( \theta(t) - a(t) r(t) \big)dt + \sigma(t) dW_t, \quad t > 0,
\end{align*}
where $\{W_t, \,...
- 20.4k
8
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.
...
- 2,836
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 ...
- 1,809
7
votes
Calibrating Hull-White model
I find your approach to calibration (training an ANN to learn the inverse function f-1 from a training set of 'market_prices = f(model_parameters)' interesting, novel (at least this is the first time ...
- 608
7
votes
Accepted
Calibration by monte carlo, should I fix my seed?
It is not cheating. It allows you to make your results (e.g. prices, calibrated parameters) 'reproducible' which is good. However, fixing the seed can hide convergence issues. When the variance of ...
- 13.9k
7
votes
Accepted
Implied vol and model calibration for an american option on a dividend paying stock - is there a market standard pricing model?
Theoretically, this is a more difficult problem than it looks like at first glance. Unfortunately, existing literature taking into account a proper dividend consideration is rare (at least from a ...
- 721
7
votes
Accepted
What does it mean to "calibrate vols"
You are an investment bank. You trade a multitude of vanilla and exotic options. You want to make sure the option prices you quote as a client are arbitrage-free with respect to liquid option prices ...
- 7,091
7
votes
Accepted
Calibration when characteristic function is not known
You are likely thinking of affine stochastic volatility (SV) or Levy models which have characteristic functions that can be obtained via semi-analytic expression. For these types of models Fourier ...
- 394
7
votes
Accepted
Risk Neutral Valuation, Drifts and Calibration
There are two parts to your question which I try to answer separately. The first one is about what calibration actually is whereas the second question deals with risk-neutral pricing.
As an example, ...
- 13.8k
6
votes
Accepted
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) = ...
- 2,310
6
votes
Accepted
Commonly used vol surface calibration model in the industry
The most used equity volatility models in the industry are the Black-Scholes model (including its time dependent version) and the local volatility model. It always come along with stochastic rates, ...
- 535
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 $(v_0,\theta,\kappa)$, ...
- 13.9k
6
votes
Theoretical and practical drawbacks of using Deep Learning for calibration and pricing
The essence of the problem is the "bias-variance" problem in machine learning. Which you can wiki (or find dozens of papers on; it's a famous issue).
You can, with ever greater complexity, ...
- 4,896
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 ...
- 10.5k
5
votes
How to calibrate a volatility surface using SVI
I will just answer your first question as I do not know the details of SSVI.
Total variance is more intrinsic than volatility.
The BS formula can be rewritten in terms of 3 parameters: the log-...
- 3,826
4
votes
Accepted
Calibration of nested pricing models consistently on two different classes of derivatives
It seems that implicitly you have a multi-objective optimization in mind, hence of course it may happen that you are not able to achieve all the objectives simultaneously. Let's say that output of a ...
- 1,474
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.
- 191
4
votes
How does one estimate theta in the Ho-Lee model from a yield curve?
Given the Ho-Lee interest rate model of the form
\begin{align*}
dr_t = \theta_t dt + \sigma dW_t,
\end{align*}
the price at time $t>0$ of a zero-coupon bond, with maturity $T$ and unit face, has ...
- 20.4k
4
votes
Accepted
Implied Vol vs. Calibrated Vol
The main difference is that one approach assumes that a certain dynamical structure properly describes the underlying instrument, while the other approach is really only a re-writing of the price in ...
- 1,809
4
votes
Libor Market Model Calibration
Market practitioners do the following: Correlation is calibrated most often by looking at historical correlations between liquid par swap rate pairs. One could look at implied correlations within ...
- 13.7k
4
votes
Accepted
SVI model and Greeks calculation
The SVI is simply a function (empirically fit to the data) which given a maturity and a strike price K, computes a BS implied volatility $\sigma$. Once you have that implied volatility you can plug it ...
- 9,077
4
votes
ATM i.r. Caps - Black vol calibration
Because these are ATM Swaps, strike rates should be equal to the Swap rates which can be computed off the forward curves
- 41
4
votes
Is a common approach to calibration reasonable?
I am intrigued by this question because it gets at the heart of so many grey areas of the financial system in which it becomes almost impossible to know how many assets derive their values from some ...
- 2,825
4
votes
Why Hull White 2 Factor model can't capture vol skew?
Local and/or stochastic vol extensions of HW (incl. multi-factor) were produced around the mid 1990s, more or less independently in a number of research papers, the most notable being Cheyette (1992) ...
- 608
4
votes
Accepted
Instruments for calibrating Hull White Model
I assume you are asking for the popular Hull/White one-factor model.
You could eiter calibrate them to Cap/Floor Volas or to swaption volas. Don't try to fit a model to both at the same time. You ...
- 342
4
votes
Accepted
why calibrate volatility and fix the mean reversion
Fixing the mean reversion, and parameterizing the volatility as a step function or as a piecewise linear function, the volatility can be bootstrapped exactly to a set of vanilla options sorted by ...
- 5,532
Only top scored, non community-wiki answers of a minimum length are eligible