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13 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 ...
Quantuple's user avatar
  • 14.7k
10 votes
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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, \,...
Gordon's user avatar
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9 votes
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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 ...
Daneel Olivaw's user avatar
9 votes
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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. ...
StackG's user avatar
  • 3,046
9 votes
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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, ...
Kevin's user avatar
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9 votes
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Recovery Rates in CDS valuation

Apparently, you refer to this passage from Prof John C. Hull (11th edition, 2021): It's confusing because Hull is referring to the market conventions before the "Big Bang". This is no ...
Dimitri Vulis's user avatar
8 votes
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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 ...
Daneel Olivaw's user avatar
8 votes

Pricing and hedging caps and floors on illiquid emerging markets

It could be worse. You're not asked to price rate exotics like accreters that might need more inputs besides implied vol cube :) and you're only asked to make markets. I.e., if I understand the ...
Dimitri Vulis's user avatar
7 votes
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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 ...
Quantuple's user avatar
  • 14.7k
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 ...
Antoine Savine's user avatar
7 votes
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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 ...
SI7's user avatar
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7 votes
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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 ...
d_797's user avatar
  • 394
7 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, ...
demully's user avatar
  • 5,081
6 votes
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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 $(v_0,\theta,\kappa)$, ...
Quantuple's user avatar
  • 14.7k
5 votes
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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 ...
Alex C's user avatar
  • 9,382
5 votes
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What is the definition of "co-terminal swaptions"? why they are important in the calibration process?

This question has partially already been answered here. Let's do a simple example to illustrate the idea though. Take a 5y Bermudan callable S/A USD bond. How would you reconstruct the multi-call ...
oronimbus's user avatar
  • 1,906
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) ...
Antoine Savine's user avatar
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 ...
David Addison's user avatar
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
learningIR's user avatar
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 ...
Gordon's user avatar
  • 21.2k
4 votes
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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 ...
Bernd's user avatar
  • 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 ...
Antoine Conze's user avatar
4 votes
Accepted

How frequently is local volatility calibrated to implied vol surface, in practice?

I worked on a single name Equity Derivatives trading desk. Implied volatility is remarked at least once per day, but that depends also on market movements, volatility movements, volumes, etc. For this ...
Vitomir's user avatar
  • 821
4 votes
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How to calibrate models with unbounded parameter space

Usually multidimensional objective function of calibration error of stochastic volatility models (Heston , bergomi etc) have many local minima, thus you would get similar calibration error for very ...
alexprice's user avatar
  • 861
4 votes
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How to choose the martingale measure in incomplete markets

The incompleteness property says that there are infinitely many martingale measures producing an interval of arbitrage-free prices. In reality one has to charge a reasonable price for partial hedging (...
ir7's user avatar
  • 5,043
4 votes
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Hull-White Monte Carlo simulation - mean reversion function

Given a initial discount bond $P^M(0, T)$ curve, the expression for $\theta(t)$ in the Hull White Short Rate model is a know result given by: $$ \theta(t) = \frac{1}{\kappa} \cdot f'(0, t) + f(0, t) + ...
rvignolo's user avatar
  • 741
4 votes
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Statistical metric to measure how well does the volatility surface fit the market

I suspect you want to use a weighted norm: https://math.stackexchange.com/questions/394237/understanding-weighted-inner-product-and-weighted-norms Generally, your volatility surface (or volatility ...
Attack68's user avatar
  • 11k
4 votes

Sabr Calibration not fitting the market volatility

Well, it looks pretty good to me. The SABR model has only 4 parameters and there is only so much you can do with them. If you have a lot of volatilities, especially if they have a quite irregular ...
Jesper Tidblom's user avatar
4 votes

The Holy Grail of Volatility Modelling: The SPX & VIX - Why?

Recall that the VIX can be expressed as a weighted portfolio of European call and put options on the S&P500. Thus, there is a relationship between the VIX and the S&P implied volatility, so ...
Daneel Olivaw's user avatar

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