Questions tagged [local-volatility]

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When to use a Local Vol model vs Stochastic Vol Model?

I'm new to volatility modeling, I'm struggling to understand when to use a Local Vol model and when to use Stochastic Vol Model, Also now we use a hybrid model combining the two models ? Can someone ...
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Local volatility implied spot vol correlation

I have a question about local volatility models. In a lot of articles it is stated that the implied spot vol correlation of this model is -1 and we usually compare this with stochastic volatility ...
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1 vote
1 answer
94 views

Calibration and pricing with the Stochastic Local Volatility model

I'm reading the stochastic local volatility model literature, e.g., the Heston Stochastic Local Volatility model (https://ir.cwi.nl/pub/22747/22747D.pdf); but I'm a bit unsure about its calibration ...
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4 votes
2 answers
360 views

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 ...
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how to evaluate numerically the equation of Dupire with the method of Implicit Euler?

I shoud evaluate numerically the equation of Dupire with the method of Implicit Euler and then studying the convergence of the implied volatilities predicted by the model toward those of market to the ...
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2 votes
2 answers
201 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 ...
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3 votes
0 answers
74 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 ...
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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 ...
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local volatility not reasonable

We are going to generate synthetic option prices using a Heston model, i.e., $$ \begin{gather*} dS_t = \sqrt{v_t} S_t dZ_t,\\ dv_t = \lambda (\mu - v_t) d_t + \eta \sqrt{v_t} dW_t, \end{gather*} $$ ...
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1 vote
1 answer
182 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 ...
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1 answer
153 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 ...
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1 answer
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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....
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1 answer
152 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 ...
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1 answer
131 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^...
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93 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 ...
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2 votes
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72 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 ...
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4 votes
1 answer
181 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 $S_0$ $$\...
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Local volatilty models for fixed maturity curves with sparse equity data

I'm implementing an options analytics platform - with sparse market data going out a few months. Mostly equities and fx options. Building a fixed maturity curve like the one on quikstrike by bantix is ...
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2 votes
0 answers
83 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 ...
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1 vote
1 answer
116 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 ...
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1 vote
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50 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 ...
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2 votes
1 answer
162 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 ...
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1 vote
0 answers
87 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. (...
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120 views

Vega Surface with Local Volatility Model

I am trying to obtain the Vega of some equities with the Dupire local volatility model. For this I have already validated the pricing model (I am using Monte Carlo) and now I am able to obtain the ...
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0 answers
50 views

How to match simulated local vol european prices with closed formula

Say an implied volatility is given by $\sigma_{imp}(T, log(F_T/K))$ and we note the Dupire local volatilty $\sigma_{loc}(T, log(F_T/K))$ with $F_t$ the forward rate and $K$ the strike. The price of a ...
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0 votes
0 answers
124 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 ...
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4 votes
1 answer
401 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+\...
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7 votes
1 answer
251 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 ...
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1 vote
0 answers
161 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 ...
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106 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} +...
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34 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 ...
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1 vote
0 answers
60 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 ...
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2 votes
1 answer
512 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 ...
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1 vote
1 answer
127 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. ...
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6 votes
2 answers
551 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 $\sigma(t_i,S(t_i)), i=1,...,N$ ...
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1 vote
0 answers
72 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 ...
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0 votes
1 answer
114 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 ...
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0 votes
1 answer
167 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 ...
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3 votes
2 answers
578 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 ...
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4 votes
0 answers
182 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 ...
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6 votes
0 answers
210 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 ...
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0 votes
1 answer
123 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 ...
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4 votes
0 answers
109 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^...
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  • 99
3 votes
1 answer
231 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 ...
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7 votes
0 answers
412 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 ...
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0 votes
1 answer
257 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. ...
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4 votes
2 answers
818 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 ...
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3 votes
1 answer
224 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 ...
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3 votes
2 answers
1k 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 ...
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  • 2,836
6 votes
1 answer
453 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 ...
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