Quantitative Finance Stack Exchange
Quantitative Finance Stack Exchange

Questions tagged [local-volatility]

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Local volatility surface corresponding to the implied volatility surface

In Derman/Kani/Zou paper about local vol they rebuilt a local vol surface from an implied vol surface. Each implied volatility depicted in the surface of the "implied Vol" is the Black-Scholes implied volatility. Bascially the volatility you have to…
Catchitup
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Is the local volatility linear if smile is linear?

Assume $dS = S_t\sigma(S_t,t)dW$. Given a implied volatility smile which is linear in, say, $(K - S_0)$, (we know its intercept and slope), we wish to calibrate $\sigma(S_t, t)$ to it. Will it too be linear? If so, with what intercept and slope? If…
Saidu
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Break even Levels Local volatility

I came across a presentation where it is stated that using a local volatility model the PnL of an option is and What does he mean by spot/vol correl = -100%?
JiLight
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Local volatility implied by implied vol surface

In his book volatility and correlation, Rebonato tries to explain intuitively the shape of local volatility surface (depending on stock level and time) from the implied volatility surface in the OTM put side. See below. However his explanation…
ababoua
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What are the benefits of using Dupire model

I'm trying to understand what is the point of the local volatility model in practice. Rather than asking a question I will explain what is what for me hoping someone will spot where I'm wrong: The point of using another model than Black-Scholes is…
loyd.f
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Methods to compute Local Volatility surface and price

I am trying to wrap my head around how exactly Dupire's formula is implemented in practice. We need $\sigma(S,T)$ for every possible $S$ and $T$. If we had that, then we can just run a monte carlo scheme. So, do we run monte carlo, and then during…
Calibrator
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Hagan's 2002 SABR paper "Managing Smile Risk" on Dupire local vol model

I'm reading Hagan's 2002 paper Managing Smile Risk originally published on the WILMOTT magazine, and got something confusing on his comment on Dupire's local volatility model. The set up: Consider a European call option on an asset $A$ with exercise…
athos
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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 understand how to derive the Dupire's formula using…
Michael
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Dupire equation at T = 0

Is Dupire's equation defined when the T input (the expiration) is equal to 0 (the current time)? If it's not, how would you determine an appropriate local volatility to use at that time when modelling stock price movements?
kev
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local volatility formula

I try to compute the local volatility in python in both formula, i.e. in terms of call price surface and total variance surface. However, I get 2 different values. What did I do wrong? import numpy as np from scipy.stats import…
StupidMan
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how to understand Black-Scholes volatility is average of local volatilities

Black-Scholes volatility is average of local volatilities. It is from: https://bookdown.org/maxime_debellefroid/MyBook/all-about-volatility.html First what's the meaning of the average of all the paths between spot and the maturity and strike of…
user6703592
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Deriving Dupire's Volatility Formula : Why $\lim_{s \rightarrow \infty} (s-K) \frac{d}{ds} \big[ \sigma^2(T,s)s^2\phi(T,s)\big] = 0 $

In deriving Dupire's formula for the local volatility, using European call option, this is used in the integration by part : $$\lim_{s \rightarrow \infty} (s-K) \frac{d}{ds} \Big[ \sigma^2(T,s) s^2\phi(T,s)\Big] = 0 $$ Why is it the case? Notation…
user30614
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Conditional expectation and Dirac delta function

In the proof of Dupire equation we end up with an identity involving the Dirac delta function. How to prove that $$\dfrac{E[\sigma_T^2\delta(S_T-K)]}{E[\delta(S_T-K)]}=E[\sigma_T^2|S_T = K].$$ where $\delta(x)$ is the Dirac delta function. $S_T$ is…
A.Oreo
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Reference request: local volatility and time-dependent volatility

Is there a good practitioners guide to time-dependent and local volatility models?
Jaood
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Dupire's formula explanation

In Stochastic Volatility Modeling (Lorenzo Bergomi) the Dupire's formula is: $\sigma (t,S)^2$ $=$ $2$${dC\over dT}$ $+$ $qC$ $+(r-q)K$${dC \over dK}$ $x$ ${1 \over K^2 {d^2C \over dK^2}}$ with $K=S$ and $T=t$ Then he says this equation expresses…
Axel Haddar
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