Questions tagged [stochastic-volatility]
The stochastic-volatility tag has no usage guidance.
110
questions with no upvoted or accepted answers
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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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22
votes
0
answers
2k
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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12
votes
0
answers
476
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 ...
- 2,746
10
votes
0
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434
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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 ...
- 101
9
votes
0
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770
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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 ...
- 111
7
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0
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135
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Implied vol bounded if and only if instantaneous vol bounded
I'd like to show that in diffusion models IV is bounded iff instantaneous vol is bounded if there is to be no arbitrage. So, assume a model under the pricing measure of the form
$$
dS_u = \sigma_u S_u ...
user34971
6
votes
0
answers
551
views
Libor Market Model with SABR Calibration
What is the industry practice in calibrating SABR Libor Market Model? Do you first calibrate the SABR model using market data and then implement the libor market model with the calibrated parameters?
...
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5
votes
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271
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Hedging : effect of not matching the term structure of skew
Let us assume that we construct a pure stochastic volatility model calibrated to the implied volatility surface, but that the model does not replicate accurately the observed term structure of the ...
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4
votes
0
answers
341
views
Why calibrate volatility Models to volatility surfaces rather than underlying's historical price data?
I'm trying to grasp the rationale for calibrating stochastic volatility models (i.e. Heston model) to empirical IV data from market prices. Doesn't this assume that the options are fairly priced and ...
- 41
4
votes
0
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316
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Characteristic function of the Bates model
I have a misunderstanding concerning the derivation of the SVJ model :
Firsty,I understand how to reach the final differential equation from :
\begin{gather}
dS_t = (r - q - \lambda t (e^{m-\frac{\nu}{...
- 446
4
votes
0
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279
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Angular bracket notation (physics)
In a few papers I have seen the following notation:
$$
\langle X_t \rangle
$$
Also, in Bergomi's book, at page 8, we have the following equality:
$$
\biggr\langle \int_0^T e^{-rt}s^2 \frac{d^2P_{\hat{\...
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4
votes
0
answers
162
views
Why is the Schöbel-Zhu model affine?
In the Schöbel-Zhu model, the stochastic volatility process is $dv_t=\kappa(\theta-v_t)dt+\sigma dW_t$.
The characteristic function of the stock process can be found by arguing that the model is ...
- 41
4
votes
0
answers
271
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....
- 1,627
4
votes
0
answers
227
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, ...
- 41
3
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0
answers
32
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Separability of Stochastic Volatility Model
After having read the article of Trolle & Schwartz regarding their general stochastic volatility term structure model (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=966364), it is not clear ...
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3
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0
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327
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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 ...
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3
votes
0
answers
85
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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 $\mathcal{F}_t$.
...
- 1,009
3
votes
0
answers
154
views
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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3
votes
1
answer
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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3
votes
0
answers
151
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Single barrier options in stochastic volatility models
In this note/sketch, I derive among others a closed-form formula for an up and in put (UIP) in stochastic volatility models of the form
$$
dS(t) = \sigma(t) S(t) \left[ \rho dW(t) + \sqrt{1-\rho^2} dZ ...
user34971
3
votes
0
answers
292
views
SABR LMM for RFR
Is there a research showing a way to use SABR LMM with new RFRs such as SOFR, i.e. pricing exotic path-dependent RFR derivatives with volatility smile and skew?
I'm aware that
Looking Forward to ...
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3
votes
0
answers
111
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Comparison of Option-Pricing Models (volatility models) vs Product-Mapping
I scoured this forum, looking for some indicative (updated as of year 2021) comparison of volatility/option-pricing models. There were some, but they seem dispersed and lacking in general details...
...
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3
votes
0
answers
86
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What is the relationship between the estimated GARCH(1,1) conditional volatility and the true conditional volatility
Suppose that the data has been generated by a GARCH(1,1) model, i.e.
\begin{align}
y_t &= h_t \epsilon_t, \; \epsilon_t \sim N(0,1) \\
h_t &= \alpha_0 + \alpha_1 \epsilon_{t-1}^2 + \...
- 2,436
3
votes
0
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116
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Fractional Brownian Motion's Covariance Proof
Let's have the non independent Brownian motion such :
$B_{H}(r)=\frac{1}{A(H)} \int_{R}\left[\left\{(r-s)_{+}\right\}^{H-1 / 2}-\left\{(-s)_{+}\right\}^{H-1 / 2}\right] \mathrm{d} B(s), \quad r \in R$
...
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3
votes
0
answers
498
views
Rigorous proof of Dupire formula (e.g. using Gyöngy's theorem)
Where can I find a rigorous proof of the Dupire formula (for example, using using Gyöngy's theorem)? I imagine this would be covered by a paper or by a standard financial math text, but I could not ...
- 747
3
votes
0
answers
119
views
Robust bounds or approximations on implied volatility skew when $\lvert \rho \rvert \rightarrow 1$
Are there any robust / non-parametric results for pure stochastic volatility models, in terms of bounds or preferably accurate approximation, for the implied volatility skew $\partial IV(k) / \partial ...
user34971
3
votes
0
answers
212
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What models are used for pricing cliquet options (esp. for Asian Equity underliers)? How good is Bergomi model?
What are the most common models, actually used by trading desks for Asian underliers, for pricing cliquet options?
I would like to know both - (1) the production model used for daily P&L, and ...
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3
votes
0
answers
217
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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 ...
- 131
3
votes
0
answers
182
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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 ...
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3
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0
answers
240
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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0
answers
179
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^{...
- 31
2
votes
0
answers
85
views
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 ...
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2
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0
answers
99
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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 ...
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2
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0
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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)\...
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2
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0
answers
93
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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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2
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0
answers
130
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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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2
votes
0
answers
208
views
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 ...
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2
votes
0
answers
348
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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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2
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Pricing a put-option in the Heston Model
Assume the Heston Model with dynamics under the martingale measure $Q$ given by
\begin{align}
dS_t &= (r-q)S_t dt + \sqrt{v_t}S_tdW_{1,t}^Q\\
dv_t &= \kappa(\theta-v_t)dt + \sigma\sqrt{v_t}dW_{...
- 548
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0
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208
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How do you hedge volatility risk?
Suppose I model an asset $S_1(t)$ under a stochastic volatility model. To price an option on $S_1$, I must assume the existence of an asset $S_2$ that is used to hedge against changes in the ...
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2
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0
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173
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Are Stochastic Differential Equation diffusion terms always invariant under a change of measure?
I'm struggling with learning change of numeraire, and stochastic differential equations. I'm reading the beginning of Brigo and Mercurio's Interest Rate Models- Theory and Practice, and I'm on the ...
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2
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0
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378
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Are there any public implementations of realized kernels? (preferably in Python)
looking to implement a realized kernel model to forecast realized variance of around ~140 equities and indices in Python given order book data.
I have read "Realised Kernels in Practice: Trades ...
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2
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0
answers
106
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Is $C(K,S_t)$ a (local) martingale if PCS is broken?
When put-call symmetry holds
$$
P(S_t,K) = C(K,S_t) = \frac{K}{S_t} C \left( S_t, \frac{S_t^2}{K} \right)
$$
where $P$ is the market price of a put option and $C$ is the market price of a call option. ...
user34971
2
votes
0
answers
171
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Are rough stochastic volatility models used on the street for equity derivatives ? (2020)
I'm building out some stochastic vol models for pricing exotic equity derivatives. What's the state of the art on the street?
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2
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0
answers
159
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Characteristic function for heston model with jumps in price and variance
I need the characteristic function of the Heston model with jumps in price and variance, or in other words, the characteristic function of the Bates model (1996) adding jumps in the variance dynamics.
...
2
votes
0
answers
265
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Stochastic Volatility Models Real World Calibration
I am trying to find some research pertaining to the historical (or real world) calibration of stochastic volatility models.
For example, in applications such as counterparty credit risk (IMM) or ...
- 81
2
votes
0
answers
52
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Volatility of a perpetuity $E\Big[\Big(\int_0^\infty e^{-ks+mz_s}ds\Big)^\eta\vert\mathcal{F}_t\Big]$
Let $z$ be a brownian motion, let $\mathcal{F}$ be the filtration it generates. For $k>0$ and $m\in\mathbb{R}$, I define the process $Y$ as
$$Y_t=E\Big[\Big(\int_0^\infty e^{-ks+mz_s}ds\Big)^\eta\...
- 21
2
votes
0
answers
174
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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-...
- 111
2
votes
0
answers
172
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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 ...
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2
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0
answers
81
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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 ...
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