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Questions tagged [local-volatility]

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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 ...
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63 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 ...
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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 ...
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1answer
193 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 ...
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2answers
210 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 ...
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1answer
112 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\...
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2answers
174 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 ...
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2answers
390 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 ...
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1answer
217 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?
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1answer
107 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)...
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2answers
252 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 ...
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122 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 ...
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1answer
126 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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78 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 ...
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189 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\...
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1answer
235 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?
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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 ...
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2answers
890 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 ...
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0answers
198 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 ...
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0answers
69 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 ...
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1answer
144 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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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 ...
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1answer
320 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\...
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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 (...
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1answer
252 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 ...
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1answer
434 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 ...
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2answers
703 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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510 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 ...
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1answer
116 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 ...
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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"...
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0answers
127 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) : \...
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244 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). ...
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1answer
206 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[\...
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1answer
128 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 ...
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2answers
356 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 ...
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1answer
317 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 ...
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1answer
467 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."
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1answer
647 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 ...
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1answer
873 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
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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?
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464 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 $K=S$...
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1answer
266 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 ...
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0answers
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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...)? ...
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0answers
98 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 ...
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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 ...
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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" ...
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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 ...
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2answers
717 views

Pricing variance swaps using Monte Carlo

For pricing variance swaps there is the well-known formula as sum of OTM options weighted by the inverse of the squared strike (see e.g. here). Would it also be valid to derive the local-volatility ...
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Implied volatility

I have a question about calculating the implied vol. Assuming the implied vol that a option will expire in 1 day is $\sigma_1$, and the implied vol that the option will expire in 2 days is $\sigma_2$. ...