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Questions tagged [stochastic-volatility]

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2answers
74 views

How to show that SABR is log-normal for $\beta=1$ and normal for $\beta=0$?

For $\beta = 1$ SABR is log-normally distributed and for for $\beta = 0$ SABR is normally distributed. This is a very common property mentioned in almost every paper about SABR. But I can't find the ...
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0answers
36 views

SABR ATM volatility

The ATM implied volatility is important in SABR when calibrating the model. Let's consider the ATM vol (for a european call option): $$\sigma = \frac{\alpha}{f^{1-\beta}} \left[ 1+ \left(\frac{(1-\...
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0answers
40 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 ...
2
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1answer
43 views

How to deduce the formula of the wealth process of a stochastic volatility model?

I am reading the paper Solution of the HJB Equations Involved in Utility-Based Pricing from Daniel Hernandez and Shuenn Jyi Sheu. The authors consider the utility function $U: \mathbb{R} \to \...
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0answers
18 views

In the Heston Model, how the degrees of freedom is calculated? How is this value an integer as df is supposed to be one?

How does the feller condition comes into the picture as it is equal to half of the df? How are the parameters for the CDF of the non-chi sqaured distribution calculated, specifically in the Lief ...
3
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1answer
82 views

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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0answers
20 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 ...
6
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2answers
161 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 ...
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0answers
42 views

Can someone give a simple example of how stochastic volatility leads to volatility skew/smile?

I've been trying to understand skew and volatility, unfortunately I don't have the mathematical background to necessarily dive into some of the papers, I've tried but the mathematics can overtake me. ...
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0answers
45 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 ...
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0answers
55 views

Discretizing a Continuous Time Stochastic Volatility Model

Forgive me for cross-posting. I have the following continuous time SDE for a stochastic volatility model. $S_t$ is the price, and $v_t$ is a variance process. $$ dS_t = \mu S_tdt + \sqrt{v_t}S_t dB_{...
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0answers
39 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 ...
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0answers
43 views

Quanto pricing with stochastic vol

Before starting implementing a quanto pricer, I'd like one to confirm my algorithm. Even though there are several topics on it it's still not clear on my side. Say I can price a stock under Heston ...
1
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1answer
87 views

Pricing an option with sparse data, high underlying volatility and returns

I'm currently pricing American and European options on an underlying with sparse data (interpolated), high annual volatility and returns over the last year around 300%. The product isn't similar to ...
0
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1answer
91 views

stochastic vol modelling not enough for smile

It seems in practice models that include Stochastic Volatility alone do not have enough power to produce actual observed implied vol surfaces. Is there recent empirical literature documenting this?
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0answers
82 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\...
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0answers
30 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 ...
2
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2answers
355 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 ...
5
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2answers
361 views

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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0answers
30 views

Reference on Pricing Model of Convertible Bonds based on MCMC Algorithm?

I have to implement convertible bonds pricing (in Stochastic Volatility condition)in Matlab or R using MCMC algorithm. Is there any paper or book which describes this method in detail?
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0answers
42 views

Pricing kernel dynamics in a JDSV model

I have the following model \begin{align} d M_t & = r M_t dt \\ d S_t & = S_t [\alpha dt + \sqrt{V_t} d B_t + J d N_t] \\ d V_t & = k(\theta - V_t) dt + \eta\sqrt{V_t} \left(\rho d B_t +...
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0answers
54 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 ...
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0answers
57 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}...
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0answers
38 views

Monte Carlo for constructing the Vol smile in SABR

My purpose is to construct the vol smile using Monte Carlo simulation and not market data. When I search for Monte Carlot methods for SABR I often see the Euler scheme as given for instance in these ...
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0answers
77 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 ...
6
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1answer
173 views

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\...
2
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0answers
101 views

Volatility swap hedge

What are the hedging methods for volatility swap (rather than variance swap)? What are the possibilities of setting up a static, semi-static or dynamic hedging? I am aware of but have not yet read ...
5
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1answer
211 views

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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1answer
160 views

Mixture models of Stochastic Volatility and Local Volatility

As far as I can see on this website the stochastic volatilty models seem to be preferred to local volatility models, mainly due to the fact that stochastic volatility is 2D diffusive process whilst ...
15
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0answers
374 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 ...
4
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1answer
96 views

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 ...
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1answer
109 views

What is the name of this VaR calculation strategy?

Here's a question on a passage from this paper I'm reading. Here's the quote: Given the vector of portfolio weights $w$, and the estimate of the conditional variance, $\Sigma_{t,k}$, the ...
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0answers
48 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 ...
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0answers
147 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 ...
0
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1answer
80 views

Two Wiener process under same martingale measure Q

Let $W_1,$ $W_2$ be to Wiener processes under the martingale measure $Q$. What can be said about $dW_1*dW_2$? I know that $$(dW_i)^2=dt$$ but what about the case with two different wiener processes?
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1answer
141 views

Terminal Variance in the Heston Model

I am trying to understand the basics of financial models. Random Walk as a model for asset prices. We use gaussian random numbers to generate a Gaussian Random walk. The variance of the terminal ...
2
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0answers
76 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 ...
5
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2answers
671 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"...
6
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1answer
132 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 + \...
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0answers
15 views

Covariates for LSMC with stochastic volatility models

In a typical stochastic volatility model, such as the Heston model, the joint process $(S_t, V_t)$ is Markovian, but the price process $S_t$ by itself is not. However all the implementations of LSMC ...
3
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1answer
231 views

SABR: how often is tuning parameters needed?

This questions is regarding the behaviour of banks and other financial institutions who deal with FX products and use SABR model volatilities to price options. How often do they change/tune ...
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1answer
50 views

Ljung_Box Statistic of R and R^2 values in Return analysis

I have found a result that I find truly puzzling. Here is an extract from a GARCH-Analysis I have performed: Test______________Statistic_______p-Value Ljung-Box Test_____R Q(10)_____0....
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1answer
64 views

Gatheral's change of variables for stochastic volatility PDE

This is taken from Gatheral's book "The Volatility Surface", where he tries to go from equation 2.3 to equation 2.4. We have the following PDE, $$ \frac{\partial V}{\partial t}+\frac{1}{2}vS^2\frac{...
6
votes
1answer
390 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 ...
6
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1answer
321 views

The Heston Solution For European Option - Jim Gatheral

I have this equation (Eq. (2.4) "The Volatility Surface - A Practitioner's Guide" by Jim Gatheral (Ed. 2006)): $$-\frac{\partial C(v, x, \tau)}{\partial \tau}+\frac{1}{2}v \frac{\partial^2 C(v,x,\tau)}...
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1answer
99 views

Estimate the mean reversion level of the variance process under the real world measure

This paper gives on equation 22 an estimator for the mean reversion level of the variance process under the real world measure. The context is the Heston model, where the variance is stochastic and ...
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2answers
2k views

Strike / delta relationship for FX options

I am tryinto find out how to go from delta to strike. If wee look at the bloomberg I am looking at 1M ATM volatility. I have included the Bloomberg data as a picture where we have following ...
4
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1answer
76 views

Benchmark value for American Options under stochastic volatility

Does anyone know any kind of method that produces reasonably well results for American Options under Heston Model setting that could be used as benchmark value? Since right now my goal is to ...
3
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1answer
98 views

Detecting stochastic volatility

I have a time series extracted from a financial time series (so my series of prices is described by an arithmetic model $X(t)+Y(t)+Z(t)$, my series is $Z(t)$). I'm trying to model $Z(t)$ by a Levy ...
3
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1answer
852 views

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 ...