Questions tagged [stochastic-volatility]
The stochastic-volatility tag has no usage guidance.
359
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how to reflect spot and implied vol relationship in vol curve
There is much evidence about the correlation between spot price and option implied vol in the empirical. This is very important in risk management(i.e. delta hedge). I want to know how to add this ...
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53
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Volatility Surface Construction: Ask IV, Bid IV and Mid IV
I am presently engaged in a project wherein my objective is to construct a volatility surface utilizing either the SVI parameterization or the SABR model, leveraging real market data. Initially, I ...
- 75
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1
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147
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Vanna Volga Price of an Up and In Put
In the Vanna-Volga approach to pricing first generation exotics, such as single barriers, as I understand it the pricing is as follows:
Let $K,S_t < B$. I'll choose the ATM IV $I_{ATM}$ as the ...
- 1,594
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312
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Rough Volatility and Change of Measure
When deriving the rough Bergomi model, Bayer et al in "Pricing Under Rough Volatility" (2015) perform a change of measure to ensure the price process is a martingale as shown in the ...
- 31
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1
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Typical values Heston parameters for FX options
I am not as familiar with FX options as I am with equity index options.
For the purposes of numerical testing/experiments I'd appreciate if somebody could tell me what are typical parameter values for ...
- 1,594
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88
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Is homogeneity preserved under change of measure?
In a paper, Joshi proves that the call (or put) price function is homogeneous of degree 1 if the density of the terminal stock price is a function of $S_T/S_t$. In the paper I think Joshi is silently ...
- 1,594
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59
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How to replicate a claim in a stochastic volatility model?
Given a Markovian stochastic volatility model with an asset $S$ and a variance process $V$ given by
$$
dS_t = \mu_t S_tdt + \sqrt{V_t}S_tdW_t, \\
dV_t = \alpha(S_t,V_t,t)dt + \eta \beta(S_t,V_t,t)\...
- 195
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120
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Time-shifted power law in path dependent volatility
I can't understand a function which is part of a volatility model.
This is all explained in an open access paper titled "Volatility is (mostly) path-dependent" by Guyon and Lekeufack. My ...
- 452
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Construction of stochastic volatility model from a given local volatility model
The Gyongy's theorem:
Let $X_t$ be a stochastic process satisfying
$$dX_t = \mu_t dt+\sigma_tdW_t$$
where $\mu_t, \sigma_t$ are bounded stochastic
process adapted to the filtration $\mathcal{F}_t$.
...
- 994
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Drift of stochastic variance as slope of the short end of the forward variance curve
I was re-reading Chapter 6 of Stochastic Volatility Modeling by Lorenzo Bergomi. On page 203, he considers a forward variance of the following form:
$$
d\xi_t^T=\lambda_t^T dZ_t^T,
$$
where $Z_t^T$ ...
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94
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Stochastic volatility estimation in R
Can anyone help me with the stochvol package in R? I estimated the volatilities using this package but I am not being able to understand how to download the ...
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106
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Smile Dynamics - forward variance
I was reading Smile Dynamics II by Lorenzo Bergomi. It is clear to me that on page 2
$$
V_t^{T_1,T_2}=\frac{(T_2-t)V^{T_2}_{t}-(T_1-t)V^{T_1}_{t}}{T_2-T_1}
$$
is the fair strike of a forward-starting ...
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110
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Characteristic function of Gamma-OU process
Consider the Gamma-Ornstein-Uhlenbeck process defined in the way Barndorff-Nielsen does, but consider a different long running mean $b$ which may be bigger than zero:
$$dX(t) = \eta(b - X(t))dt + dZ(t)...
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143
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Barrier options in LSV (local stochastic volatility) / Austing's Smile pricing explained
In his book (chapters 9.5 to 9.7), Peter Austing argues that barrier options are insensitive to the details of the stochastic volatility model used in a LSV model, except for the level of vol of vol. ...
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82
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Calibration of LSV models to vanna/volga break-even
In this paper, Labordère, the author computes a probabilistic representation of the the vanna/vomma(volga) break-even levels. He mentions that they can be used to calibrate LSV models to historical ...
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Non-stationarity and repricing as a source of idiosyncratic and systematic "risk"?
1.Assuming a one period economy with two assets in which cash flows are assigned certain probabilities, using the CAPM, we can derive the P0 given the E(CF) at t1. Within this distribution, we have ...
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226
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Heston Riccati equation
Let
$$
\begin{align*}
dY_{t} &= \left(r - \frac{1}{2} V_{t}\right) dt + \sqrt{V_{t}}dW_{t}\\
dV_{t} &= \kappa(\theta - V_{t}) dt + \rho \sigma \sqrt{V_{t}}dW_{t} + \sigma\sqrt{1-\rho^{2}}\sqrt{...
- 107
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213
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Vega hedge of a barrier option
I was re-reading Lorenzo Bergomi's paper Smile Dynamics I. On the first page, he makes the point that it is necessary for a model to match the vanilla smile observed in markets in order to incorporate ...
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119
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Can you use a forward rate curve to infer the SABR model parameters?
I am currently doing a thesis on a class of SDE parameter inference methods and using the SABR model as an example for inference. I want to extend the application to market data. My question is does ...
2
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1
answer
136
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Pricing Quantos with Local-Stochastic Volatility model
I would like to price equity quanto options with the Heston Local-Stochastic Volatility model (LSV) but I am having hard time understanding how to apply quanto adjustment in such complex setup.
When ...
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Trading options - risk adjusted return
I have often wondered what kind of risk restrictions do traders of options in Hedge funds have but have not managed to find any information on this matter. I presume there must be some kind of measure ...
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1
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322
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How to hedge a dual digital option
Let us assume we have two FX rates: $ 1 EUR = S_t^{(1)} USD$ and $ 1 GBP=S_t^{(2)} USD $. Let $K_1>0, K_2>0$ be strictly positive values and a payoff at some time $ T>0 $ (called maturity) ...
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119
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Barrier on realized volatility
I am trying to understand the risk exposures of vanilla options that also have a European barrier on realized volatility. For example, the option could knock out if the realized volatility over the ...
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3
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950
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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 ...
- 161
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68
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Affine Jump Diffusion
I'm currently looking into affine jump-diffusions. I would like to get to know the literature better and I know the paper by Duffie, Pan, and Singleton (2000) is a very celebrated paper. Although I ...
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65
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Heston Process: Accept-Reject Sampling to Alleviate the Problem of Negative Variances
I've read even in recent papers, and on page 21 of the book "The Volatility Surface" by Jim Gatheral (2006), all the debate over whether to reflect or truncate negative variances whilst ...
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87
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Hurdle Barrier option
Let us denote the upper and lower barrier levels by $B$ and $b$, respectively. The payoff of the knock-out hurdle double knock-in barrier call option can be defined as follows:
$$\left(S_T-K\right)^+\...
- 1,012
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Stochastic volatility with jumps [closed]
I'm reading the Duffie, Pan, and Singleton (2000) paper now and I've stumbled upon something that seems to me as an inconsistency. Whenever I look up the SVJJ model, I see that its log-transform is ...
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147
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How calculate expectation and variation of stochastic integral Based on Heston model?
I was calculated Heston volatility model. But I think it is wrong.
$dS_t = \mu dt + \sqrt V_t dW_t^s$
$dV_t = k(\theta - V_t)dt + \sigma \sqrt V_t dW_t^v$.
$dW^s_t dW^v_t = \rho dt$
take integral to ...
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34
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Surface SVI and value-at-risk computation [duplicate]
Currently interested in some Value-at-risk calculation methods, I understood in a video by Claude Martini (https://youtu.be/_OZvk-G92EQ), that it is now common to see SSVI-based VaR calculation models ...
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SABR model - beta
In the SABR model, the parameter beta largely controls the back-bond behaviour of the model. How do people estimate beta?
One approach is to regress atm vol vs forward, i.e.
$$\ln(\textrm{atm vol}) = \...
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Why is the market price of risk a non-entity according to Bergomi?
I am reading Bergomi's book Stochastic Volatility Modelling. In the chapter 6 dedicated to the Heston model, page 202, he describes the traditional approach to first generation stochastic volatility ...
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Strange use of dynamical programming principe
I am in a finance seminar and yesterday evening we had a lecture from a quant in a big bank about shortcomings of Heston model.
He was deriving the Heston PDE. (I know how to derive the Heston PDE ...
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277
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Implied volatility to local volatility via Dupire
I am doing some self study on stochastic local volatility modelling and am having a hard time replicating some results from the paper "FX Option Pricing with Stochastic-Local Volatility Model&...
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1
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105
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Impact of stochastic rates on varswaps and volswaps
Let us consider that we are looking at issuing some varswaps or volswaps on some FX rate. By longer term I mean something longer than 3 months. Different from this time two years ago, now the interest ...
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1
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167
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Initial forward variance curve calibration
Let $V_t^{T_1, T_2}$ be the forward variance swap rate for the period $[T_1, T_2]$, seen from $t$ (see for instance Lorenzo Bergomi's Smile Dynamics II) and let $\xi_t^T = V_t^{T,T} = \frac{\partial}{\...
- 1,223
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66
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Mixing formula for SVJ models
I am trying to understand the mixing formula (Hull and White formula) for stochastic volatility models with jumps in the asset price. One article which discusses this is Lewis, The mixing approach to ...
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122
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Useful methods to avoid degenerate calibration? (Heston model in my case)
I have implemented a Levenberg-Marquardt(LM) based method to calibrate the Heston model against market data by minimizing a weighted $L^2$-norm of differences of market vs model prices. Pretty ...
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log-normal random walk VS mean reverting random walk?
I'm reading Wilmott's book. He talks about several model as : log normal random walk and mean reverting random walk. I don't find answer to these questions :
In this chart : let's assume that Part 1 ...
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Reference request: Approximate mapping of a multi-factor stochastic volatility model to single-factor stochastic volatility model
I am looking for approaches to transform a more complicated stochastic volatility model such as the one shown in Section 2.2 of Smile Dynamics II to a single-factor model such as the one shown in ...
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467
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Mid-curve swaption pricing - how to get the spread vol?
I believe I understand the following (from the accepted answer to the Quantitative Finance question called "volatility of a mid curve option"):
A swaption in which the underlying swap ...
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189
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Implied volatility skew decay over expiry
I seem to remember the implied volatility skew of European options decreases as the expiry increases. It is true for the Heston model under some approximation. What are the good references that prove ...
- 2,706
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107
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Ito's lemma in stochastic volatility models [closed]
I couldn't help but notice that in all stochastic volatility models articles I consulted, whenever Ito lema is applied with a process of the sort
$$\frac{d S_t}{S_t} = \sigma_t d W_t $$
With $(\...
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317
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Model based PnL explain for FX Options
In FX options the vol surface for a given maturity is usually described by three or five points, I.e. Atm, 25 delta risk reversal and butterfly and 10 delta risk reversal and butterfly. Then models ...
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270
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When calculating VIX, how to deal with the problem of asymmetry of put and call data?
I'm trying to calculate the VIX index according to the methodology of CBOE. I am looking at commodity options. I found that at some time, like at this minute, there are 13 call options out of the ...
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166
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Stochastic vs. local volatility model choices for greeks
As a follow-up of another question (which is I feel slightly separate, hence a new question). Assume we want to fit a volatility surface with the goal of calculating good greeks, not prices. We can ...
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350
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Delta hedging when volatility is stochastic
From my understanding in a BSM world you can make a bet on volatility using options and delta hedging with the underlying.
If you think realized volatility of the underlying will be higher than the ...
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220
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How to price american barrier with Local-Stochastic Volatility
I have attended a conference where one speaker mentioned that the market standard to price FX and Equity derivatives is now the Local-Stochastic volatility model.
I understand this class of model is a ...
- 113
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221
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Independence vs correlation in stochastic vol models
I am struggling a bit with some basic stuff lately:
Consider a SV model
\begin{align}
dS_t &= \sigma_t S_t dW_t \\
d\sigma_t &= b(\sigma_t,t) dZ_t
\end{align}
with $dW_t dZ_t = 0$.
I know that ...
user34971
4
votes
2
answers
622
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Introductory material for getting started with local and stochastic volatility modelling
Are you able to provide some suggestions for resources to get started with non-flat volatility modelling? The models I am interested in are the likes of CEV, Heston, SABR etc.
I have tried looking ...
- 152