Questions tagged [stochastic-volatility]

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35
votes
0answers
957 views

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 ...
29
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0answers
1k views

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)dt+ \...
16
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0answers
644 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 ...
12
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1answer
464 views

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{...
9
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0answers
282 views

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 ...
7
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0answers
258 views

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 ...
5
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0answers
138 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 ...
4
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0answers
57 views

Stochastic Long-Run Mean Instantaneous Variance in Heston Model (and extensions)?

I'm working on my dissertation in Financial Economics, focusing on the topic of Stochastic Volatility Jump Diffusion models; and I'm playing around with some ideas for model extensions. In particular, ...
4
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1answer
212 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 ...
3
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0answers
68 views

Simulating volatility process in the Heston model using the relation between the CIR Process and Ornstein–Uhlenbeck processes

I am trying to simulate the volatility process in the Heston model using the relation between the CIR Process and Ornstein–Uhlenbeck processes. In fact, giving $\mathbf{X}$ a $n$-dimensional vector ...
3
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0answers
274 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\...
3
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1answer
213 views

Mixing Black Scholes with SABR

I am new to the whole concept of stochastic volatility so I am experimenting with option pricing. I think the concept is really difficult to understand / grasp. I was wondering if the following ...
3
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0answers
440 views

Relationship between SABR and Heston

What is the relationship between SABR parameters $\sigma, \alpha, \beta, \rho$ and heston parameters $\nu, \kappa, \theta, \xi, \rho$? How do they influence the smile; skewness, kurtosis, etc? And ...
3
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0answers
96 views

delta hedging with stochastic volatility

In my thesis I want to work with delta hedging with stochastic volatility using Black-Scholes model. How will you suggest I implement numerical solutions using data from the real world? Beside Monte ...
3
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0answers
211 views

When to use SV or a GARCH model

So i have been searching for this answer for a question if there is a rule or something that would say when to use GARCH type model or use an stochastic volatility model to predict the volatility of ...
3
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0answers
128 views

Approximate asian geometric option with Heston

I am trying to implement Theorem 1 from this Journal in RStudio. The journal says the it is possible to find a approximate price of a geometric asian option in a Heston setup this way: $$X_{1cGAO}=e^{...
2
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0answers
52 views

Finding Jump Probability For Time Series Data

I'm relatively new here, so if it seems like I'm asking a bad question, go easy on me. So I was looking at the Merton Jump Diffusion Stochastic Model on Turing Finance's article. Instead of creating ...
2
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0answers
113 views

Using SVI model for IV surface

I am using well-known paper of J. Gatheral & A. Jacquier Arbitrage-free SVI volatility surface to explore SVI model. on the page 6 in the bottom is statet that The SVI-Jump-Wings (SVI-JW) ...
2
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0answers
45 views

The Free Boundary SABR: Natural Extension to Negative Rates

In the paper by Antonov, Konikov and Spector An alternative approximation for the SABR model is presented. I'm interested to implement the formula for the ATM swaptions implied volatilities in the ...
2
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0answers
38 views

Forward looking estimation of market price of risk of stochastic volatility

I would like to estimate the market price of stochastic volatility by forward looking methods, such as option values. The stochastic volatility model I have in mind is the Heston model or some other ...
2
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0answers
100 views

Suggestions to build a copula to price Quanto options

I am willing to price a quanto option through the use of copulas. I will follow the following procedure: 1) Obtain the marginal distributions of the underlying asset and the exchange rate from ...
2
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0answers
60 views

Heston Model Maximum Return Distribution

What is the joint probability distribution of the maximum of the return between time $0$ and $t$ and the return at $t$, for the Heston model, when the return drift is $0$ and the correlation between ...
2
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0answers
47 views

what kind of test for volatility and where find the data

I am working on a model for stochastic volatility. In short, the model try to capture that the volatility goes up suddenly after a shock (war, policy, financial events, etc) and then goes down slowly, ...
1
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0answers
49 views

The error term of Hagan's approximation of Black's vol in SABR

Hagans approximation of Black's implied vol in SABR is very! difficult to understand fully. But I want to ask in here if anyone can tell me more about the error term. Consider the paper: http://web....
1
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0answers
36 views

Who came up with 3/2 SV model

Sorry, not a very quantitative question, but does anybody know who was the first person to write down and publish the 3/2 stochastic volatility model? I need this for a reference/bibliography.
1
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0answers
64 views

Taylor expansion of stochastic variables with dynamics of the form $dX_t=b(\sigma_t,X_t)dW_t$

https://www.math.nyu.edu/~cai/Courses/Derivatives/compfin_lecture_5.pdf In the above document stochastic taylor expansions are nicely explained. Let us now consider a typical SDE model in finance ...
1
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0answers
40 views

Why can't we create a “magic” basket of options to sell for no-arbitrage pricing in SVJ model?

I am learning how to price SVJ options and am reading some stuff on no-arbitrage pricing for SVJ model using the typical approach you would use (like in BSM option pricing) of creating a risk free ...
1
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0answers
61 views

What are good TEXTBOOK on stochastic volatility and interest rate theory?

I wanted to learn stochastic volatility modelling and interest rate modelling. On this site, a answer recommended me the books "Stochastic Volatilty Modelling" by Lorenzo Bergmo and "Interest Rate ...
1
vote
1answer
273 views

Understanding the ZABR model (an extension of SABR)

http://janroman.dhis.org/finance/SABR/ZABR%20Andreasen.pdf In this acticle the SABR model is first presented in another form ( see equation 7 in the article ) and then extended to the so called ZABR ...
1
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0answers
113 views

Implied volatility as break-even delta hedge volatility

There have been some posts on this topic, but not what I am looking for, so a new post on an old topic.. I think some/most of us here are familiar with the following formula expressing implied ...
1
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0answers
102 views

Fitting a forecasting S&P500 roll volatilities

I have a time series of S&P500 prices, for which I have calculated log-returns and roll-volatility. My goal is to forecast daily realized volatility and test a straddle strategy based on it (I ...
1
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0answers
173 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 ...
1
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0answers
142 views

Is SABR being used in practice for Equity options

Just to be clear: By "in practice" I mean what the banks and other financial companies do. Do financial companies use SABR for pricing equity options? Consider a stock with price $t$ being: $S_t$. ...
1
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0answers
67 views

Does the Asian Option (average Option) depend on the forward implied vol

I can easily understand that the forward starting Option and Barrier Option depend on the forward implied vol smile at resetting date, so we always choose the stochastic vol model for underlying to ...
1
vote
1answer
132 views

Rough Volatility Prediction - Gatheral, Jaisson, Rosenbaum Paper

I just read through the paper "Volatility Is Rough" by Gatheral, Jaisson and Rosenbaum. There is a website (link: http://tpq.io/p/rough_volatility_with_python.html) that details the simulations they ...
1
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0answers
67 views

Effect of Volatility Regime on Volatility Smile

For short-term FX options, I find empirically that the degree of curvature of the smile (OTM/ATM in %) is higher in low volatility environments. Similar results are found by Pena et al. ("Why do we ...
1
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0answers
87 views

Question about derivation of SABR volatility formula in original paper 'Managing Smile Risk' by Hagan et al

I have a question regarding the starting point of the derivation of SABR volatilities formulas in the appendix of the famous paper 'Managing Smile Risk' by Hagan et al. To derive SABR volatility ...
1
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0answers
37 views

Literature recommendation subordinator models

I'm looking for relevant papers covering subordinator models for stock price modelling. I have alreay read the paper 'A Subordinated Stochastic Process Model with Finite Variance for Speculative ...
1
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0answers
83 views

Calibrating Heston paremeters based on market data for Implied Vol for Call options

Several questions have been asked in here regarding calibration in Heston yet I have not found what I have been looking for, so I will ask: I am looking at a Heston model: $$dS_t=\lambda \sqrt{v_t}...
1
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0answers
141 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 ...
1
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0answers
65 views

How to derive the change in portfolio value as given by Gatheral in The Volatility Surface?

I’m trying to follow Gatheral’s Volatility Surface Ch. 1, i.e. the text (pg. 5 and 6) linked to in this question, with further text discussed in this question. I can’t figure out how to arrive at the ...
1
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0answers
418 views

Stochastic Vol simulation - Quant job interview question

this is a question from a quant interview (FO quant for IR Exotics for a big 4). First it might be useful when preparing your interviews, second, any brainstorming will be appreciated. Note that no ...
1
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0answers
164 views

School project about Black Scholes with stochastic volatility

In a university project I am looking at Black Scholes model with a stochastic volatility. I’m still not quite sure about my focus (I am in the beginning 'Idea phase'). I want to explain the theory ...
1
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0answers
289 views

Heston model - Andersen scheme implementation

I would like to implement Andersen scheme for Heston simulation. On the following snipped is my code for generating asset path: ...
1
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0answers
178 views

Reference request about stochastic volatility model

I'm fiddling with estimation of stochastic volatility models and have build up a somewhat flexible framework using indirect inference. I would like to try and throw a lot of different continuous ...
0
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0answers
17 views

Bates Model Jump Percentage Parameters

I am trying to calculate the jump parameters for the Bates volatility jumps, specifically, the mean of the jump percentages, $\mu_j$. For the value of $J$, I am using jumps $|\frac{s_{i}-s_{i-1}}{s_{i-...
0
votes
1answer
79 views

Compute implied volatility surface of a put option from a call option

Suppose the function double bsCall(double S0, const double &K, double T, double r, double sigma) computes analytically the Black-Scholes price of a call option ...
0
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0answers
40 views

Gatheral's SVI implementation in Java/Scala

I am trying to fit equity option implied vols using SVI model in Java, and I am using apache math commons library. Some of the option expiries fit very well, but others are completely off, and I am ...
0
votes
0answers
35 views

Vega computation in a stochastic volatility model

What are the possible strategies to compute analytically the Vega (not numerically) in a stochastic volatility model? The goal is to vega-hedge in a generic stochastic volatility model if possible, ...
0
votes
0answers
273 views

Are there any papers measure the accuracy of various option pricing models against real market price?

There are many stochastic volatility option models not only require significant more computation/simulation comparing to the standard BSM model but also introdue large source of possible problems at ...