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Questions tagged [stochastic-volatility]

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Estimating Heston parameters from market data

I have calibrated the SABR model to some FX data and I want to compare with Heston. Consider the Heston model: $$dF_t=\mu F_tdt+\sqrt{\sigma_t}F_tdWt$$ $$d\sigma_t=-\lambda(\theta-\sigma_t)dt+\sqrt{\...
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1answer
62 views

When a stochastic volatility model is calibrated?

In an Investment Bank, how often a stochastic volatility model is calibrated ? Is it calibrated daily ? Is it calibrated whenever a pricing is required ? Thanks.
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29 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 ...
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1answer
105 views

Intuition for the Effect of Vol of Vol in Heston Model on Volatility Surface

I was hoping someone could describe the economic/mathematical intuition behind the effect that the vol of vol parameter has on the volatility surface, in particular the slope to maturity. Take for ...
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1answer
71 views

Why Can I not estimate a CVAR from Heston Model

I fit the parameters of Heston model, using option data for SPX. Now I have the process S and P 500 is expected to follow. I make 100,000 simulations of this process and then calculate the expected ...
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1answer
94 views

Approximate Hagan formula for SABR model with negative beta

While looking into fixing the $\beta$ parameter (based the following regression: $\text{ln } \sigma^{ATM}_t = \text{ln } \alpha - (1-\beta)\text{ln }F_t$, as explained in West (2004), page 6) before ...
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Why do we require a continuous volatility calibration while pricing Options [closed]

On pricing Options the volatility surface is represented by a mathematical model (with parameters). What does it mean to calibrate the volatility surface How often has the volatility surface to be ...
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3answers
159 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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1answer
152 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
66 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 ...
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1answer
59 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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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 ...
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1answer
90 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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27 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 ...
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2answers
212 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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49 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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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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1answer
118 views

Discretizing a Continuous Time Stochastic Volatility Model

How does the discrete time stochastic volatility model arise from the continuous time one? Also, forgive me for cross-posting. I have the following continuous time SDE for a stochastic volatility ...
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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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46 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 ...
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1answer
90 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 ...
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1answer
96 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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145 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
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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 ...
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3answers
1k 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 ...
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2answers
692 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
37 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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46 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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1answer
103 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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72 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
51 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
95 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 ...
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1answer
283 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\...
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173 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 ...
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1answer
244 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
205 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 ...
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453 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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1answer
111 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
112 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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55 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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157 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 ...
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1answer
87 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
202 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 ...
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0answers
81 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 ...
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2answers
1k 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"...
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1answer
197 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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1answer
256 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
58 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
88 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{...
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476 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 ...