Questions tagged [stochastic-volatility]
The stochastic-volatility tag has no usage guidance.
367
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How to show that this weak scheme is a cubature scheme?
Weak schemes, such as Ninomiya-Victoir or Ninomiya-Ninomiya, are typically used for discretization of stochastic volatility models such as the Heston Model.
Can anyone familiar with Cubature on ...
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1
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Law of an integrated CIR Process as sum of Independent Random Variables
It is known (see for example Joshi-Chan "Fast and Accureate Long Stepping Simulation of the Heston SV Model" available at SSRN) that for a CIR process defined as :
$$dY_t= \kappa(\theta -Y_t)...
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Local Volatility vs. Stochastic Volatility
Are there any empirical observations or practices when to prefer Local Volatility Model for pricing over Stochastic Model or vice versa?
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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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Bergomi: Skew arbitrage
In his paper "Smile Dynamics IV" (https://www.fields.utoronto.ca/programs/scientific/09-10/finance/derivatives/bergomi.pdf) as well as in his book "Stochastic Volatility Modeling" (...
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Can VIX be interpreted as a proxy for instantaneous volatility?
Bakshi et al., (2006) Estimation of continuous-time models with an application to equity volatility dynamics (Table 2) estimate the following Cox-Ingersoll-Ross model for market variance, $\sigma^2_t$:...
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4
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How to prove that markets are incomplete under the Stochastic Volatility model?
Has anyone ever formally proved that Markets are incomplete under the stochastic volatility model?
I know that if there are more random sources than traded assets, then the market is incomplete but ...
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For which instruments performs SABR/LMM better than LMM?
For which class of instruments the SABR/LIBOR Market Model does perform better than the classical LIBOR Market Model?
The LIBOR Market Model
The LIBOR Market Model — also known as Brace, Gatarek, ...
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1
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How do different models impact option Greeks?
If I trade an option using delta, vega, Prob OTM, etc. these are derived from a model. How do leading models impact valuations in terms of the Greeks?
I suppose to form a baseline it would have to be ...
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14
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How to, from various hypotheses on the P&L, get known models (BS, Heston etc ...)
Usually models in quantitative finance are taught by giving, let's say, stochastic differential equations, initial conditions, and then pricing, under the model, various derivatives written on the ...
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Why is the SABR volatility model not good at pricing a constant maturity swap (CMS)?
I have heard that the SABR volatility model was not good at pricing a constant maturity swap (CMS). How is that?
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2
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Realized variance in SVJJ (Heston with jumps) model
I am working with the stochastic volatility model with jumps in both the price and volatility dynamics, ie. the risk neutral dynamics are of the form:
$$\mathrm{d}V_t = \kappa(\theta - V_t)\mathrm{d}t ...
- 131
13
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1
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Transformation of Volatility - BS
I have recently seen a paper about the Boeing approach that replaces the "normal" Stdev in the BS formula with the Stdev
\begin{equation}
\sigma'=\sqrt{\frac{ln(1+\frac{\sigma}{\mu})^{2}}{t}}
\end{...
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Jim Gatheral's ansatz
In the Ansatz section of Jim Gatheral's book Volatility Surface (page 32), he assumes $$\mathbb E[x_s|x_T]=x_T\frac{\hat w_s}{\hat w_T}$$
where $\hat w_t:=\int_0^t \hat v_s ds$ is the expected total ...
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Solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$
Let $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$ be a stochastic differential equation where $a$, $b$, and $c$ are positive constants, so I tried to solve it but I got stuck in ...
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Can the Heston model be shown to reduce to the original Black Scholes model if appropriate parameters are chosen?
Summary
For Heston model parameters that render the variance process constant, the solution should revert to plain Black-Scholes. Closed from solutions to the Heston model don't seem to do this, even ...
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Why is there a stong intraday-correlation between spot and vol?
Fig.1 shows an intraday scatterplot of the DAX future against its volatility index VDAX on 6-Jan-2016.
The data suggest a strong negative correlation between the two.
There are various models ...
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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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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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3
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clarification to use collocation methods to get arbitrage free sabr
I'm reading the following two papers (first, second) which suggest a so called "stochastic collocation method" to obtain an arbitrage free volatility surface very close to an initial smile stemming ...
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Transition densities in the Heston model
Knowing the Characteristic function $\Phi_{T,t} = \mathbb{E} [ e^{i u S_T} | S_t, V_t]$ (or equivalently, the Laplace transform) of an affine process, it's possible to know the distribution of the ...
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Why is it useless to model stochastic volatility when pricing Vanilla style derivatives?
With respect to the answer by user AFK
in Ideas about Stochastic volatility models.
I am specifically interested in interest rate options (IR Caps/Floors and Swaptions).
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For pricing, what types of Exotic Options are suitable using Local Volatility Model or a Stochastic Volatility Model?
I understand that stochastic volatility models should be used when the exotic option payoff is volatility dependent (such as variance swaps and volatility swaps).
Stochastic volailtiy models should ...
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3
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450
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Why does the price of a derivative not depend on the derivative with which you hedge volatility risk?
I'm trying to derive the valuation equation under a general stochastic volatility model.
What one can read in the literature is the following reasoning:
One considers a replicating self-financing ...
- 131
9
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0
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759
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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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4
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Stock Price Behavior and GARCH
In my (limited) understanding, the behavior of a stock price can be modeled using Geometric Brownian Motion (GBM). According to the Hull book I'm currently reading, the discrete-time version of this ...
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2
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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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Confusion with volatility smiles implied by different models
I am reading a book "The concepts and practice of mathematical finance" by Mark Joshi. In Chapter 18 he discusses the shapes and dynamics of smiles under different models. I do not understand what is ...
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Does it make sense to use upward and downward volatility in option pricing?
Historically stocks have a higher likelihood to increase in price than to fall in price. As such would it make sense to split a stocks volatility measurement into upward and downward components?
For ...
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SABR Model Closed Form Solution
I've been researching the SABR model and one of the main benefits it seems is that you can obtain a closed for solution of the implied BS volatility in certain cases.
In all the papers I've read, I ...
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Hedging error in a stochastic volatility model
I would like to find how much error I make when I hedge a call option using Black Scholes model in a market which is actually governed by a stochastic volatility process such as
$$dS_t = rS_tdt + \...
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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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How can I compare 30 day implied volatility forecasts with GARCH forecasts?
I'm trying to understand whether there is a good way to compare forecasts for volatility from different sources i.e., implied volatility and GARCH. I'll outline a few statements that I believe and if ...
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Estimate rolling stochastic volatility forecast using stochvol in R
I want to use the R package stochvol to fit a SV model to a DAX training set and use the output to estimate a rolling one-step-ahead forecast:
...
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8
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1
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Are there "live" uses of the Generalized Method of Moments or are they all academic?
I see the Generalized Method of Moments suggested in numerous academic papers as a way to calibrate stochastic volatility models. However, any decent trading shop is going to calibrate to observable ...
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1
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Why do we fit volatility surfaces implied from a option pricing model to the empirical data?
As far as I understand volatility surface. It is made of 2 components, the volatility skew/smile and the volatility term structure. Together they form something like
Implied volatility is ...
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1
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What tradeoff is there to using an accurate estimate with a large confidence interval?
I am working on calibrating a Heston model from simulated historical stock data.
After obtaining an accurate estimate of the model parameters I found very large 95% confidence intervals for these ...
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8
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1
answer
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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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What is an acceptable error on implied volatility?
Given an implied volatility surface (on equity indexes) and a calibrated model, what is the range of error on implied volatility a trader would accept ?
This obviously depends on the model used to ...
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3
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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 ...
- 2,462
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Is variance swap long volatility of volatility?
In JPM's note on variance swaps, on page 29, they say "... a long variance swap is also long volatility of volatility".
In Bennett's book Trading Volatility, on page 115, he says "... a ...
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Modeling Call Price w.r.t. Strike w Models that Capture Vol Smile
I am trying to model $C(K)$, the price of the call $C$ as a function of strike $K$. Because this is tied to Prob ITM - and in fact the probability density function of that particular expiration (https:...
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How to use a stochastic volatility model to price a quanto option
I want to price a quanto option using a Stochastic Volatility model (like Heston model, 1993).
Normally, what we do is:
Calibrate the stochastic volatility model,
draw a binomial tree consistent ...
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7
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1
answer
997
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Interpretation and intuition behind the Put-Call symmetry under the Heston Model
I am currently working on a report regarding the put-call symmetry relations under the Heston model. I did all the math and managed to prove the relations using PDE approach. However, I wish to have a ...
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SABR Calibration: Normal vs Log-Normal Market Data
This question is about getting some clarification as to how to understand market quotes for normal & log-normal vols together with certain model assumptions.
So let us define
$C_{BS}(F_0,K,T,\...
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1
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SABR calibration: simple explanation and implementation
I would like to learn more about the SABR model and ho it is used in modeling smiles in equity, FX and rates markets.
How would you explain the process and its implementation in simple steps? Any web ...
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Calculating 6-minute, 20-minute, 45-minute, and 3-hour volatility
I am looking to measure the volatility from the open of the market until a trade takes place and use that volatility in post-trade regressions to help explain transaction costs. A simple regression ...
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2
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on "recovering probability distributions from option prices" - how to subtract influence of stochastic volatility?
This is based on a 1995 paper by Rubinstein/Jackwerth by the above title where the authors produces a distribution of stock prices inferred from option prices. But their approach only produces a joint ...
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Market price of volatility risk
Reading Gatheral's The volatility surface, page 7.
The model they are talking about is
$$\begin{align}dS_t&=\mu_tS_tdt+\sqrt{\nu_t}S_tdZ_1\\d\nu_t&=\alpha(S_t,\nu_t,t)dt+\eta\beta(S_t,\nu_t,...
7
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1
answer
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Vega in the Heston model
I'm trying to calculate the hedging quantities of the Heston model. I undestand that the replicating portfolio consist of one option, $V = V(S,v,t)$, $\Delta$ stocks and $\phi$ units of the option to ...
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