Questions tagged [stochastic-volatility]
The stochastic-volatility tag has no usage guidance.
49
questions
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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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2
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how to calculate vega in stochastic vol?
since vega is defined as option value changes regarding the implied vol parallel shift, how is vega defined or calculated in stochastic vol models since implied vol is not an input there? thank you.
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1
answer
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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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7
votes
1
answer
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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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0
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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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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 ...
- 145
5
votes
1
answer
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Motivation of the singular perturbation solution formulation for local volatility model
I am puzzled by the motivation of the particular choice of the (singular) perturbation method used in Equivalent Black Volatilities. Equation (A.6a) sets
$$\epsilon:= A(K)\ll 1.$$
What is the ...
- 2,746
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1
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Option prices in Bates SVJ model?
In this [post] discussed the European put and call price formulas under the Heston Stochastic Volatility model.
There exists an important extension of Heston model to include diffusion jumps, known ...
- 5,759
4
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1
answer
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Numerical example of how to calculate local vol surface from IV surface
I'm looking for an excel example (not a copy of Dupire's eqn) of how to convert an IV surface to a local vol surface. If unsuccessful I'll work through Dupire's eqn but would be helpful to look at an ...
- 219
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votes
2
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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.
$$\ln(\textrm{atm vol}) = \...
- 31
11
votes
2
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Solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$
Let $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$ be a stochastic differential equation where $a$, $b$, and $c$ are positive constants, so I tried to solve it but I got stuck in ...
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SSR definition in Bergomi in relation to sticky strike and sticky delta
In Bergomi [Stochastic Vol Modelling] (Sec. 2.5.2), in the section on surface dynamics, the following definition of the "Skew Stickiness Ratio" (SSR) is made:
$$ SSR = \dfrac{1}{\mathcal{S}_T}\frac{d\...
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1
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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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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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1
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volatility input for black scholes formula
I am not a mathematician but want to try and understand the BS model for option pricing. I get the intuitive sense of it but am unable to figure out calculation of volatility (as an input). Some ...
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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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2
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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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votes
3
answers
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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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1
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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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2
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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 ...
- 436
8
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2
answers
710
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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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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
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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) ...
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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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1
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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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7
votes
1
answer
993
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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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1
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Market price of volatility risk
Reading Gatheral's The volatility surface, page 7.
The model they are talking about is
$$\begin{align}dS_t&=\mu_tS_tdt+\sqrt{\nu_t}S_tdZ_1\\d\nu_t&=\alpha(S_t,\nu_t,t)dt+\eta\beta(S_t,\nu_t,...
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SABR calibration: simple explanation and implementation
I would like to learn more about the SABR model and ho it is used in modeling smiles in equity, FX and rates markets.
How would you explain the process and its implementation in simple steps? Any web ...
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2
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Calibrating stochastic volatility model from price history (not option prices)
For stochastic volatility models like Heston, it seems like the standard approach is to calibrate the models from option prices. This seems a bit like a chicken and an egg problem -- wouldn't we ...
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SKEW and VIX relations?
My question is about the CBOE published index VIX and SKEW.
To start with, I consider working on the variance dynamics. I calibrate the market data (such as VIX and VIX futures) into the Heston model....
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3
answers
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SABR beta range
I am thinking of using SABR for non-rate underlyings (eg FX and equity underlyings).
Typically one finds the beta via a regression of historical implied vols vs forwards, since
$$\ln(\textrm{atm ...
- 146
5
votes
1
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469
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The positivity of the market price of risk
Does the market price of risk, be it of stochastic volatility, interest rate or equity return, have to be positive? What is the rationale if it does?
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What is vega, really?
Assume for now we are working in a stohastic volatility (SV) setting,
$$
dS_r = \sqrt{v_r} S_r dW
$$
and
$$
dv_r = a(v_r,r)dr + b(v_r,r) dZ
$$
with
$$
dWdZ = \rho dr
$$
Let $C(S_t,v_t,t)$ denote the ...
user34971
4
votes
1
answer
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derivation of heston pde in gatheral
Following Gather (the volatility surface, chapter 2) we assume the following process:
$$ dS_t = S_t(\mu_t dt+\sqrt{\nu_t}dZ^1_t)$$
$$ d\nu_t= -\lambda(\nu_t-\bar{\nu})dt+\eta\sqrt{\nu_t}dZ^2_t$$
...
- 1,718
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When to use a Local Vol model vs Stochastic Vol Model?
I'm new to volatility modeling, I'm struggling to understand when to use a Local Vol model and when to use Stochastic Vol Model, Also now we use a hybrid model combining the two models ? Can someone ...
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3
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Autocallable pricing under stochastic vs. local volatility
I am interest in the reason why an Autocallable (structured product) is cheaper under local volatility compared to stochastic volatility.
I thought this was due to the following:
when thinking in ...
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votes
1
answer
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LSV model calibration with only few quotes per maturity
At this link I have asked what is the market standard when pricing options in different asset classes. Based on the answers, the standard for FX and equities seems to be the local-stochastic ...
- 559
3
votes
1
answer
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Non-constant Volatility of the Volatility in Stochastic Volatility Models
In pricing financial derivatives, we often first assume that the volatility of the stock price is constant.
$$\mathrm{d}S(t) = \alpha S(t) \mathrm{d}t + \sigma S(t) \mathrm{d}W(t)\text{.}$$
The ...
- 437
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1
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Covariance matrix and Cholesky decomposition
I am simulating a spread option with stochastic volatility using Monte Carlo simulation. I have the positive-definite covariance matrix
$$
\rho = \left( \begin{array}{cccc}
1 & \rho_{1,2} & \...
- 193
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votes
0
answers
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Rigorous proof of Dupire formula (e.g. using Gyöngy's theorem)
Where can I find a rigorous proof of the Dupire formula (for example, using using Gyöngy's theorem)? I imagine this would be covered by a paper or by a standard financial math text, but I could not ...
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Strike Arbitrage
In Stochastic Volatility Modelling, Chapter 2, the author derived the Dupire equation
$$\mathbb{E}[\sigma_T^2|S_T = K] = 2\frac{\frac{dC}{dT} + qC +(r-q)K\frac{dC}{dK}}{K^2 \frac{d^2C}{dK^2}}.$$
The ...
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2
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Cumulative Integration with regard to Vasicek Model's Bond Price and its Forward Price
(My Question)
Please show me how to compute the following expectation with its computation process. Besides, $B_t$ is S.B.M.
$$E\left[ \exp \left( - \int^T_t \int^u_0 \sigma e^{-b(u-s)} d B_s du \...
- 279
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2
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Local Vol vs Stoch Vol Option Pricing
This is an interview question:
Imagine you have a double knock-out barrier option: the current spot is 100, the lower barrier is 80, and upper barrier is 120. The barrier is continuous, meaning that ...
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1
answer
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How to understand the market price of risk
Consider the stochastic vol:
$$dS = \mu Sdt + \sigma SdW_1$$
$$d\sigma = p(\sigma,S,t)dt + q(\sigma,S,t)dW_2$$
$$dW_1dW_2 = \rho dt$$
We want to obtain the price of option $V(\sigma,S,t),$ we use the ...
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1
answer
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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 ...
- 21
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1
answer
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Can you shift a standard libor market model with regard to only at-the-money options?
Suppose I have an LMM defined using the spot measure as in Brigo and Mercurio:
$dF_k(t) = \sigma_k(t)F_k(t)\sum^k_{j=\beta(t)}\frac{\tau_j\rho_{j,k}\sigma_j(t)F_j{t}}{1+\tau_jF_k(t)}dt + \sigma_k(t)...
- 123
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votes
0
answers
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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 ...
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1
vote
1
answer
225
views
Implied volatility as price transform
Implied volatility
The way I understand it, traders often think of implied volatility as a transformed price. So in a way, the Black Scholes model is considered a 'model-free' blackbox that takes a ...
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0
votes
1
answer
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Getting the next price of a GBM (Geometric Brownian Motion)
I am writing a program that creates realizations of a GBM.
Starting from an initial price, I get the following price with this formula:
...
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