Questions tagged [local-volatility]
The local-volatility tag has no usage guidance.
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Dupire Formula question
I want to calculate the local volatility from Dupire's formula:
$\sigma _{VL}^{2} (K,T,S_{0}) = \frac{\frac{\partial C}{\partial T}}{\frac{1}{2} K^{2} \frac{\partial^2 C}{\partial K^2}}$
So I use ...
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0answers
54 views
Local Volatility in Python [closed]
I have studied Local Volatility based on Dupire Formula. I would like to know If anyone can share how to implement this in python with real world example with background of Local Volatility that would ...
2
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1answer
187 views
LSV model calibration with only few quotes per maturity
At this link I have asked what is the market standard when pricing options in different asset classes. Based on the answers, the standard for FX and equities seems to be the local-stochastic ...
1
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0answers
82 views
Mixed local-stochastic volatility model in Quantlib
At a conference the speaker mentioned that it is a standard approach today to use a mix of local and stochastic volatility model in equity, FX and interest rates.
Can you please suggest the most ...
2
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0answers
77 views
How do you numerically solve the Dupire Local Volatility PDE in log moneyness-time space?
I am trying to implement a numerical solution to price vanilla calls. I am using the Dupire equation in log moneyness-time (k = ln(F/T)) space as per below PDE
I have tried solving it using a fully ...
8
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1answer
201 views
Vanilla Option Prices from Local Vol Surface (using neither MC nor PDE)
There are numerous papers that describe the derivation of the Local-Vol equation using available market prices of options. For example:
Dupire's formula (see e.g. OpenGamma (2013)) gives us LV in ...
2
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2answers
211 views
Probability Density Function sign problem when using Call Price
Given the call price $C_t = e^{-\int_t^Tr(s)ds}\int (s-K)^{+}\phi_{S_T}(T,s)ds $
we know that $$\frac{dC}{dK}=-e^{-\int_t^Tr(s)ds}\int_K^{\infty} \phi_{S_T}(T,s)ds$$
Now when I use dirac delta ...
1
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1answer
118 views
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\...
5
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2answers
215 views
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 ...
4
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2answers
475 views
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 ...
0
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1answer
260 views
how to price barrier option under local vol model using QuantLib
I use QuantLib in Python. Now I have implied volatility surface data. How can I get the local vol surface than using finite difference method to price a barrier option in QuantLib?
0
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1answer
114 views
Drift of Local Volatility Model - Dupire
i understand that the local volatility function can be computed from the implied volatility surface.(i.e there is no calibration to option prices, we just need the full implied volatility surface only)...
6
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2answers
267 views
Vega of exotic options
I'am wondering if there is a standard definition to the Vega of an exotic product when the underlying model is not Black-Scholes.
Let me give some examples :
What is the Vega if the price is ...
3
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0answers
138 views
Finite Difference with SVI Vol Model
I am attempting to implement a local vol pricing model in finite difference for equity index options.
I have followed Gatheral's Lectures and fitted an SVI Model bringing me to the following local ...
4
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1answer
133 views
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%?
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0answers
82 views
Dynamically adjusting the size of a Constant Range Bar (on an intraday fx chart)
Constant Range Bars (CRB) is a type of candlestick charting method that does not draw a new candle every unit of time (like every 1/5/15/30 minutes), but every time the range (high-low) of the ...
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0answers
34 views
Trying to understand Strike Adjusted Spread, can someone explain using a simple example?
I should start by saying that I am not a quant, I am someone interested in options but I perhaps lack the mathematics background to always follow along. I recently stumbled upon a terrific article ...
3
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0answers
204 views
Calibration of Cox-Ingersoll-Ross process that hits zero (Feller condition violation)
I'm considering a Cox-Ingersoll-Ross (CIR) process
$$
dx_{t} = \alpha\left(\theta - x_{t}\right)dt + \sigma \sqrt{x_{t}}\,dW_{t}\,,\qquad \alpha,\beta,\sigma > 0
$$
which by assumption has $2\...
0
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1answer
257 views
Forward Skew in the Local Volatility Model
How does the local volatility model cause a forward skew? How is this different to the skew observed for future tenors in the vol surface?#
Also how do LV models underestimate vol of vol?
5
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3answers
2k views
Problems with local volatility models (vs stochastic volatility models)
Why is pricing with local volatility models are problem with exotics, mainly due to "the volatility surface is the market's current view of volatility and this will change in the future meaning the ...
5
votes
2answers
990 views
Autocallable pricing under stochastic vs. local volatility
I am interest in the reason why an Autocallable (structured product) is cheaper under local volatility compared to stochastic volatility.
I thought this was due to the following:
when thinking in ...
5
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0answers
202 views
Dupire's calibration
I'm trying to implement a method for calibrating the local volatility model (Dupire's one). I'm working on the paper from Andreasen and Huge : Volatility interpolation (SSRN). Is this considered to be ...
2
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0answers
72 views
Local variance derivation by Gatheral
I've bought Gatheral's book on Local Volatility and I have troubles with understanding a part where he shows that local variance is a conditional expectation of instantaneous variance.
Why in the ...
3
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1answer
150 views
Proof for ATM delta with Local col
I am looking at a time-homogeneous local volatility model where
ATM implied volatility equals ATM local volatility: $\sigma_{imp}(S_0)=\sigma_{local}(S_0)$
ATM IV Skew = half of LV slope
In general $\...
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0answers
119 views
How to compute SABR's probability density function
I am trying to compute the probability density function of the forward rate implied by the SABR formula approximation in order to see how the density implied by the approximation has negative ...
6
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1answer
355 views
SSR definition in Bergomi in relation to sticky strike and sticky delta
In Bergomi [Stochastic Vol Modelling] (Sec. 2.5.2), in the section on surface dynamics, the following definition of the "Skew Stickiness Ratio" (SSR) is made:
$$ SSR = \dfrac{1}{\mathcal{S}_T}\frac{d\...
2
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0answers
221 views
Implied Vol skew VS Local Vol skew (as presented by Derman 1995)
I am reading Derman's article/notes regarding local volatilty:
http://www.emanuelderman.com/writing/entry/the-local-volatility-surface.
I am examining the graph on page 13.
The Implied volatility (...
1
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1answer
263 views
Mixture models of Stochastic Volatility and Local Volatility
As far as I can see on this website the stochastic volatilty models seem to be preferred to local volatility models, mainly due to the fact that stochastic volatility is 2D diffusive process whilst ...
3
votes
1answer
506 views
Local Volatility implementation
The Dupire equation is well-known and mentioned in thousands of articles. Although I could not find a lot of documentation about a consistent and proper way of implementing the formula (The difficulty ...
1
vote
2answers
755 views
Marking implied vol surface daily with sticky strike and sticky delta
Suppose that implied vol surfaces are calibrated once per month due to data restrictions (i.e. option data is only available at month end). How can a trading desk remark their vol surfaces on a daily ...
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0answers
534 views
Local Stochastic Volatility - Break even levels
In Chapter 12 of his excellent book Stochastic Volatility Modeling, Lorenzo Bergomi discusses the topic of local-stochastic volatility models (LSV).
As most of you are probably aware of, the idea is ...
6
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1answer
121 views
Motivation of the singular perturbation solution formulation for local volatility model
I am puzzled by the motivation of the particular choice of the (singular) perturbation method used in Equivalent Black Volatilities. Equation (A.6a) sets
$$\epsilon:= A(K)\ll 1.$$
What is the ...
5
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2answers
1k views
Local vol, stochastic vol, implied vol
I've been studying volatility modelling over past the few days; in particular, the connections between local vol, stochastic vol, implied vol. I've been reading Gatheral's book "The volatility surface"...
3
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0answers
132 views
Problem of negative local volatility:
Consider the displaced log-normal process: $$dS(t) = \lambda(t)(a(t)+b(t)S(t))dW(t), S(0) = S_0>0, $$ where $W(t)$ is a one-dimensional Brownian motion.
We suppose that $(\forall t \ge 0) : \...
3
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0answers
260 views
Ill-posed problem: Local volatility calibration. Regularization vs Smoothing
I have asked my question on Mathematics site of Stack Exchange but maybe I will get the answer rather here.
I am working on inverse problem - calibration of local volatility (financial application). ...
1
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1answer
217 views
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[\...
0
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1answer
130 views
Why/When local volatility is preferred over implied distribution sampling?
Let's say we have an option whose payoff is path dependent (let's say it's asian option with observations every month). Then why these are usually priced with local vol instead of sampling from ...
6
votes
2answers
374 views
Difference between Local Vol and Copula
Let's assume we have ATM European call on a basket of two stocks and price it with:
1) Multivariate Local Vol with constant correlation
2) Gaussian copula
Assuming we use the same correlation ...
0
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1answer
333 views
Approximations for Quanto Options pricing
On page 4 of this paper, the auhor provides two good approximations for quanto options pricing: $V^d_{black}$ and $V^d_{blackATM}$. These approximations consist of using the ATM and/or stike ...
5
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1answer
503 views
Proof of arbitrage-free implied volatility surface in relation to local volatility surfaces
I'm looking for proof of the following statement:
"The existence of an arbitrage-free implied volatility surface is
equivalent to the existence of a well-defined local volatility
surface."
3
votes
1answer
706 views
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 ...
7
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1answer
914 views
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 ...
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0answers
41 views
Reference request: local volatility and time-dependent volatility
Is there a good practitioners guide to time-dependent and local volatility models?
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0answers
510 views
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 $K=S$...
1
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1answer
275 views
Confusion about CEV model
Under the CEV model the stock price has the following dynamics:
$dS_t=\mu S_tdt+\sigma S_t^\gamma dW_t$, where $\sigma\geq0, $ $\gamma\geq0$.
According to Wikipedia, if $\gamma <1$ the ...
2
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0answers
105 views
from local volatility function to implied volatility for a given strike
Assuming we know the local volatility function σ(S,t) for all S and t, how can we recover the corresponding implied volatility for a given strike K (and the other necessary parameters, S, r, T, t...)? ...
2
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0answers
99 views
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 ...
3
votes
2answers
1k views
pricing using dupire local volatility model
I am reading about Dupire local volatility model and have a rough idea of the derivation. But I can't reconcile the local volatility surface to pricing using geometric brownian motion process. If I'm ...
5
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4answers
2k views
Downward sloping smile in normal model
We consider an stock price $S$ following a normal model: $dS_t = \sigma dW_t$
We can write this as $\frac{dS_t}{S_t}=\frac{\sigma}{S_t}dW_t$
Hence we can see that $S$ follows a "log-normal" ...
2
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0answers
445 views
Code for quasi-Gaussian model (Cheyette model)
I'm looking into the quasi-Gaussian model with linear local volatility as explained by Andersen and Piterbarg (Interest Rate Modeling, Volume 2). I'm trying to calibrate this model and implement it. I ...
