# Questions tagged [stochastic-volatility]

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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,\... 2answers 785 views ### 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 ... 2answers 531 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 ... 1answer 523 views ### 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 + \... 1answer 391 views ### Filtering out AR(1) effects before using stochastic volatility model I wonder if I first filter out AR(1) (autoregressive model with lag 1) effects from univariate time series and then fit stochastic volatility model does above procedure introduce any bias at first or ... 1answer 615 views ### 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 ... 1answer 801 views ### 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 ... 3answers 7k 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 ... 2answers 3k 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"... 4answers 10k views ### relation between asset's and equity volatilities - merton model In terms of Merton credit risk model need to find the initial value of counterparty's assets and the volatility of the assets. Both value are not directly observable thus we have to approximate them ... 3answers 570 views ### Stochastic volatility model with exponential OU volatility I have a friend in the industry who said they are interested in the model I gave in the title. Whether they use it, idk. dS_t= S_t(rdt+ \sigma_t dW_t) And \sigma_t is the exponential of an OU ... 1answer 2k views ### 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 ... 2answers 902 views ### SKEW and VIX relations? My question is about the CBOE published index VIX and SKEW. To start with, I consider working on the variance dynamics. I calibrate the market data (such as VIX and VIX futures) into the Heston model.... 1answer 1k views ### 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,... 1answer 535 views ### How to determine the risk-neutral measure in a Heston model? To clarify, I'm quite familiar with the risk-neutral pricing framework, and I know one can efficiently Monte-Carlo a Heston model via the non-central\chi^2$distribution approach. But so far we're ... 0answers 161 views ### SABR Question: Why does the market take the beta parameter as a constant? SABR Question Why does the market take the$\beta$parameter as a "constant"? I see most brokers quoting SABR parameters nowadays. I've seen many banks use$\beta$=0.5 as a rule. I've seen quants ... 1answer 416 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\...
In his paper Gatheral presents the following parametrization of the implied total variance $w(k,T) = \sigma_{BS}(k,T)^2T$ $$w(k) = a + b\{\rho (k-m) + \sqrt{(k-m)^2 + \sigma^2} \}.$$ Assuming that ...