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Transformation of local volatility model

Assume we have an SDE $$dX_t=\mu(X_t)dt + \sigma(X_t)dW_t$$ where $\sigma>0$ and $W_t$ is a Wiener process. Is there a transformation $y(X_t)$ that will make the dynamics of the transformed process ...
Qwerty's user avatar
  • 43
2 votes
2 answers
1k views

Local Vol vs Stoch Vol Option Pricing

This is an interview question: Imagine you have a double knock-out barrier option: the current spot is 100, the lower barrier is 80, and upper barrier is 120. The barrier is continuous, meaning that ...
bahahaha's user avatar
2 votes
0 answers
180 views

Implied Volatility is the harmonic average of Local Volatility

I am trying to demonstrate the famous result that states that when $T \rightarrow 0$, the Implied Volatility is the harmonic average of Local Volatility. I am st the final stage, and I have the ...
Joanna's user avatar
  • 863
6 votes
0 answers
263 views

Closed formula for computing Implied Volatility from Local Volatility function

The main result of this paper (Asymptotics and Calibration in Local Volatility Models, Berestycki, Busca, and Florent. Quantitative Finance, 2002) is equation (16) on page 63, that states that: In the ...
Joanna's user avatar
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1 vote
0 answers
108 views

local volatility not reasonable

We are going to generate synthetic option prices using a Heston model, i.e., $$ \begin{gather*} dS_t = \sqrt{v_t} S_t dZ_t,\\ dv_t = \lambda (\mu - v_t) d_t + \eta \sqrt{v_t} dW_t, \end{gather*} $$ ...
user61159's user avatar
1 vote
1 answer
529 views

Monte Carlo: How to interpolate Dupire's Local Volatility

I am trying to price barrier options which can have daily or monthly observations. I first calibrated by Black vols into smooth SVI vols (with linear interpolation along time in variance) to obtain ...
boom's user avatar
  • 11
2 votes
1 answer
289 views

Dupire's Formula by a Replicating Portfolio

I understand that the BS equation can be explained by a replicating portfolio, e.g., short an option and long $\Delta$ shares of the underlier [Bergomi's Stochastic Volatility Modeling]. I also ...
Michael's user avatar
  • 301
1 vote
1 answer
484 views

Calibrate Local Volatility model to price quanto options

I have a Local Volatility model. I compute the LV surface $\sigma_{S}^{local}$ on vanilla option of $S$. Assume the vol of foreign exchange is constant and know, and the correlation equity/FX is known....
Joanna's user avatar
  • 863
1 vote
1 answer
954 views

Why Local Volatility model underestimate price of double no touch options

By reading this great answer, on points 2 and 3, it is stated that the Local Volatility model is not adapted to price barrier double-no-touch options. But I don't understand exactly why. Could you ...
Joanna's user avatar
  • 863
1 vote
2 answers
808 views

Dupire pricing equation derivation vs Black Scholes PDE

I know the Dupire pricing equation is derived in similar way to Black Scholes PDE, but it is not exactly the same equation. Dupire equation reads: $\boxed{\frac{\partial C}{\partial T} = \frac{\sigma^...
Joanna's user avatar
  • 863
0 votes
0 answers
737 views

Forward volatility smile: Local Volatility vs Stochastic volatility

I was reading this great answer: What are the advantages/disadvantages of these approaches to deal with volatility surface? And I have the following question: How to show that the forward volatility ...
Joanna's user avatar
  • 863
3 votes
1 answer
904 views

Correlation Spot Vol - when is it important?

I know that a local volatility model does not allow to control the correlation between Spot and Vol. I know also that the correlation Spot Vol is important for products like autocalls. Why is ...
Joanna's user avatar
  • 863
4 votes
1 answer
579 views

How is Emanuel Derman's implied tree model implied volatility skew derived?

I am reading Emanuel Derman's paper Patterns of Volatility Change. The section, Implied Volatility In The Sticky Implied Tree Model has the linear skew approximation near the old underlying $S_0$ $$\...
Hans's user avatar
  • 2,806
3 votes
0 answers
275 views

Impact of Discrete and linear dividends on Local Volatility model

I am trying to understand the assumptions and weaknesses of a Dupire Local Volatility model. If dividends are assumed linear, is it a problem for model calibration? If yes, why? Why would large values ...
Joanna's user avatar
  • 863
1 vote
1 answer
206 views

How to extract volatility smile implied by a mixture model?

If one had to extract the implied volatility smile from a local volatility model, one can simply use the relationship: $\sigma^2_{imp}(t, x)T = \int_t^T \sigma^2_{loc}(s, x)ds$ with $\sigma_{loc}$ the ...
user56787's user avatar
  • 125
1 vote
0 answers
114 views

Question on model recalibration upon a spot shift scenario analysis

I am given a plot of the fair value of a complex derivative against a scenario spot shift for a range odd possible shifts (-40% to 40%). Let us say the pricing model is a local vol model. I am unable ...
Arshdeep's user avatar
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2 votes
1 answer
502 views

Finding Option Probability Density Using Local Volatility from Dupire Model

This question is different than pricing using dupire local volatility model and Is Dupire's local volatility model path independent to recover historical option price? I also asked this on Math ...
curious123456789's user avatar
1 vote
0 answers
229 views

Price difference digital option : constant vol vs local vol

I got the following interview question: Consider a digital option, it will be priced by using two approaches: 1)constant volatility; 2)local volatility. At the strike, both volatilities are equal. (...
Joanna's user avatar
  • 863
0 votes
0 answers
249 views

Implied and Local Volatility relation in Monte Carlo

I am implementing a Monte Carlo engine with the local volatility model based on Dupire. Obviusly, I obtain the local volatility surface from the implied volatility surface and that surfaces has ...
Nico Blanco's user avatar
5 votes
1 answer
1k views

Calibration Heston Local Stochastic Volatility (LSV) Model

The Heston Local Stochastic Volatility (LSV) model has the following dynamics: $$dS_{t}=r S_{t} d t+L\left(S_{t}, t\right) \sqrt{V_{t}} S_{t} d W_{t},$$ $$d V_{t}=\kappa\left(\theta-V_{t}\right) d t+\...
ffbzona's user avatar
  • 358
7 votes
1 answer
559 views

Negative Density in Local Stochastic Volatility (LSV) Model Calibration

I'm trying to calibrate Local stochastic volatility model using finite difference method, and I'm mainly following this referece: Tian (2015). I met a problem when calibrating leverage function - the ...
Dovie Chu's user avatar
  • 121
1 vote
0 answers
354 views

How to use log moneyness in a local volatility context

I am implementing a monte carlo to price various options using a local volatility model. The implied volatility surface from which the local volatility is derived is a function of logmoneyness and ...
sigma1988's user avatar
0 votes
0 answers
751 views

Local volatility (dupire equation) for monte Carlo, discretisation - need help understanding spatial and TEMPORAL dimensions

As proven in Gatheral notes (or in this discussion) The equations of the local volatility as a function of the vanilla calls can be written as $$ \sigma^2(T,K) = \frac{\frac{\partial C}{\partial T} +...
user25844's user avatar
  • 365
1 vote
0 answers
48 views

When is the effect of skew most potent for an early exercise option?

Let us say I have a Bermudan option which I can terminate at 3 possible dates. When can I expect the discrepancy between a local vol and a stochastic vol model to be highest (assuming both are ...
Arshdeep's user avatar
  • 2,451
1 vote
0 answers
151 views

Local vol vs stochastic vol in the context of American digital options

I have two models of some spot. One is under local vol and the other is under stoch vol. Both are calibrated to the prevailing vanilla prices. I then consider the option that pays $1$ if the spot ...
Calculon's user avatar
  • 595
2 votes
1 answer
2k views

calibration of a local volatility model

Generally speaking, when calibrating a local volatility model a la Dupire to European vanilla calls, should I use the numerically (PDE or Monte Carlo) solved price for the vanilla call in the cost ...
sigma1988's user avatar
2 votes
1 answer
220 views

Local Volatility Monte Carlo option price - different starting index level

I constructed the local volatility surface of S&P 500 from implied vols and was able to price the options accurately using Monte-Carlo. Let's say I priced a 80% of S0 put option with S0 = 4000. ...
Sarat Muppana's user avatar
6 votes
2 answers
1k views

Correct Monte Carlo simulation of local volatility models

I am using Monte Carlo simulation to evolve the following SDE over a grid of timepoints $0,t_1,...,t_N$. \begin{equation} dS(t)=\sigma(t, S(t))dw(t) \end{equation} Here $\sigma(t_i,S(t_i)), i=1,...,N$ ...
hs16022021's user avatar
1 vote
0 answers
149 views

calibration of local volatility to

I'm looking to understand the practical details of calibrating local volatility to option prices for a range of different expiries using the Dupire local volatility equation. Would appreciate some ...
hs16022021's user avatar
0 votes
1 answer
145 views

Price a contingent claim with payoff $(S_T-K)1_{\{S_T>K\}}1_{\{L\leq X_T\leq U\}}$

I'd like to price the following contingent claim using a copula model. $$V_T = (S_T-K)1_{\{S_T>K\}}1_{\{L\leq X_T\leq U\}}$$ where $S$ and $X$ are two stock price processes which follow a non-flat ...
John Doe's user avatar
  • 387
0 votes
1 answer
353 views

Is there a Dupire's Formula for put options?

Generally, Dupire's formula is taking derivatives on the call option prices. Here it only uses information of the call options. If now we have the data including both call and put options, is there a ...
Lefair's user avatar
  • 101
3 votes
2 answers
1k views

Question About SVI and SSVI Tradeoff between Fitness and No-Arbitrage

I’m currently working on a project to build a local volatility model out of implied volatility data and am struggling in the selection of an appropriate method to interpolate the volatility surface. I ...
Dovie Chu's user avatar
  • 121
4 votes
0 answers
387 views

SABR vs Dupire: when to use what?

I was wondering for which products one would use the (AF)SABR model and for which ones Dupire's Local Volatility model. If I understand correctly, Dupire is by construction Arbitrage-Free but produces ...
victorinux's user avatar
5 votes
0 answers
281 views

Hedging : effect of not matching the term structure of skew

Let us assume that we construct a pure stochastic volatility model calibrated to the implied volatility surface, but that the model does not replicate accurately the observed term structure of the ...
fwd_T's user avatar
  • 747
0 votes
1 answer
303 views

Basket option: volatility surface

I would like to calculate a basket European option with Black Scholes local volatility model. I want to simplified the basket option into a single underlying European option. Should we get the local ...
StupidMan's user avatar
  • 170
4 votes
0 answers
130 views

$\frac{\partial C_{BS}}{\partial T}$ in local volatility derivation in terms of implied volatility

In Gatheral's book, in the derivation of local volatility in terms of implied volatility, we use the regular Dupire formula $$ \frac{\partial C}{\partial T} = \frac{1}{2} \sigma^{2}K^{2}\frac{\partial^...
Ruse's user avatar
  • 109
3 votes
1 answer
414 views

Calibrate Stochastic Volatility Model

For stochastic volatility models, and any vol model I know, it seems the standard approach is to calibrate the model from option prices. As other user said, this seems a chicken egg problem - how do I ...
TomDecimus's user avatar
9 votes
0 answers
909 views

Autocallable option Delta

There have been numerous exotic trading desk blow ups lately, related to various reasons. However, in particular, one bank had some issues where they were pricing autocallable notes with Local ...
ellie_cat's user avatar
  • 111
0 votes
1 answer
358 views

Implied Volatility vs Actual Volatility Calculation

To build a term structure I need different volatilities; as I don't get them at every strike, I use interpolation technique to calculate the rest and plot. This is how I calculate the implied vols. ...
syed tabrez's user avatar
4 votes
2 answers
2k views

Forward skew generated by Local Vol model

I'm digging into the properties of the Local Vol model and I become confused with statements made by authors in papers/textbooks (without explanations) like, "The forward skew in local vol model ...
noob-mathematician's user avatar
3 votes
1 answer
385 views

LIBOR market model with stochastic volatility

I have read that there are 3 types of pricing models: local volatility, stochastic volatility and stochastic-local volatility models (LSV). I am now looking at interest rates exotics pricing models ...
Diuoo's user avatar
  • 41
5 votes
2 answers
3k views

Interpolation of FX Vol Surface from non-uniform strike vs tenor grid

TL;DR I'm trying to fit a vol surface to market FX options quotes in order to build a local vol model to price with. Unlike listed options that typically have a nice rectangular grid of strikes and ...
StackG's user avatar
  • 3,036
6 votes
1 answer
822 views

Local Volatility Model Error

I am implementing my local volatility pricer using the finite difference method in MATLAB. I parametrise the implied volatility surface using the SSVI parametrisation (Gatheral & Jacquier), which ...
ffbzona's user avatar
  • 358
3 votes
0 answers
143 views

Infinitesimal Generators and Expectation of First Hitting Time as Solution of Differential Equation

I've been learning about Linear Diffusions and how their infinitesimal generators can be used to relate expectations and deterministic differential equations. Let $X$ be an one-dimensional diffusion ...
Victor Felipe's user avatar
4 votes
1 answer
905 views

Euler Discretization to use with Monte Carlo simulation and Local Volatility Model

Like in the title, I am working on running Monte Carlo simulations to price options with the Local Volatility model as a project. I just want to make sure that I am understanding the process, ...
Bach Pham's user avatar
0 votes
1 answer
428 views

Book/ Articles recommendation for Volatility models

I am looking for references on volatility models. I want to gain more insights on these models but have a little background as of now. Thus, looking for references that can pick the topic from basics ...
User's user avatar
  • 93
3 votes
1 answer
939 views

Three questions regarding local volatility implementation (based on the Andreasen, Huge article "Volatility interpolation")

I am new to the area of local volatility interpolation and I am trying to make a decent implementation for calculating the local volatility surface from option prices using the basic methodology from ...
Jesper Tidblom's user avatar
1 vote
0 answers
179 views

Intuition behind local volatility curve shapes in interest rate environments

I have some questions regarding the intuition behind shapes for the local volatility (LV) curve as seen in quite popular models. Let's say we have the following generalized stochastic-local volatility ...
Charlie Shuffler's user avatar
3 votes
0 answers
540 views

Rigorous proof of Dupire formula (e.g. using Gyöngy's theorem)

Where can I find a rigorous proof of the Dupire formula (for example, using using Gyöngy's theorem)? I imagine this would be covered by a paper or by a standard financial math text, but I could not ...
fwd_T's user avatar
  • 747
2 votes
2 answers
806 views

What is the difference between parametric and non-parametric models?

I'm reading about volatility modelling and I came across the concept of parametric and non-parametric models. For example, GARCH is a parametric model and Realized Volatility is a non-parametric model....
s5s's user avatar
  • 442