Semi-Analytical Pricing of Barrier Options in the Time-Dependent Heston Model

Carr, Peter; Itkin, Andrey; Muravey, Dmitry

doi:10.3905/jod.2022.30.2.141

Cited by 3 publications

(3 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, the forms of the volatility smiles in these models are somewhat inflexible, and it can be hard to fit them to market-quoted prices. Nevertheless, these models describe the whole dynamics of an underlying asset, instead of just the terminal density at a given point in time, which enables them to also price path-dependent options (Carr et al 2022;Guterding and Boenkost 2018;Tian et al 2014) and other exotic derivatives. (Guterding 2021;Zhu and Lian 2012).…”

Section: Introductionmentioning

confidence: 99%

Sparse Modeling Approach to the Arbitrage-Free Interpolation of Plain-Vanilla Option Prices and Implied Volatilities

Guterding

2023

Risks

View full text Add to dashboard Cite

We present a method for the arbitrage-free interpolation of plain-vanilla option prices and implied volatilities, which is based on a system of integral equations that relates terminal density and option prices. Using a discretization of the terminal density, we write these integral equations as a system of linear equations. We show that the kernel matrix of this system is, in general, ill-conditioned, so that it cannot be solved for the discretized density using a naive approach. Instead, we construct a sparse model for the kernel matrix using singular value decomposition (SVD), which allows us not only to systematically improve the condition number of the kernel matrix, but also determines the computational effort and accuracy of our method. In order to allow for the treatment of realistic inputs that may contain arbitrage, we reformulate the system of linear equations as an optimization problem, in which the SVD-transformed density minimizes the error between the input prices and the arbitrage-free prices generated by our method. To further stabilize the method in the presence of noisy input prices or arbitrage, we apply an L1-regularization to the SVD-transformed density. Our approach, which is inspired by recent progress in theoretical physics, offers a flexible and efficient framework for the arbitrage-free interpolation of plain-vanilla option prices and implied volatilities, without the need to explicitly specify a stochastic process, expansion basis functions or any other kind of model. We demonstrate the capabilities of our method in a number of artificial and realistic test cases.

show abstract

Section: Introductionmentioning

confidence: 99%

Sparse Modeling Approach to the Arbitrage-Free Interpolation of Plain-Vanilla Option Prices and Implied Volatilities

Guterding

2023

Risks

View full text Add to dashboard Cite

show abstract

“…This technique originally was used to price various barrier options in a semi-analytic form including double barrier options where the underlying (e.g., the stock price or the interest rate) follow some one-factor model with time-dependent parameters, or even a stochastic volatility models, see, e.g. [Itkin and Muravey, 2022;Carr et al, 2022] and references therein. In all cases, a semi-analytical (or semi-closed form) solution means that first, one needs to solve a linear Volterra integral equation of the second kind to find the gradient of the solution at the moving boundary (or boundaries).…”

Section: Introductionmentioning

confidence: 99%

Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit

Itkin¹,

Muravey²

2022

FMF

View full text Add to dashboard Cite

<p style='text-indent:20px;'>We continue a series of papers devoted to construction of semi-analytic solutions for barrier options. These options are written on underlying following some simple one-factor diffusion model, but all the parameters of the model as well as the barriers are time-dependent. We managed to show that these solutions are systematically more efficient for pricing and calibration than, e.g., the corresponding finite-difference solvers. In this paper we extend this technique to pricing double barrier options and present two approaches to solving it: the General Integral transform method and the Heat Potential method. Our results confirm that for double barrier options these semi-analytic techniques are also more efficient than the traditional numerical methods used to solve this type of problems.</p>

show abstract

“…where W (1) and W (2) are two correlated BMs with the constant correlation coefficient ρ, κ is the rate of mean-reversion, ξ is the volatility of volatility σ t (vol-of-vol), θ(t) is the time-dependent mean-reversion level (the long-term run), r is the interest rate, q is the continuous dividend, and µ is the drift. All parameters of the model are assumed to be time-independent, despite that this assumption could be relaxed; see, e.g., [14,52,18] and references therein. The process for V t is the OU process with time-dependent coefficients.…”

mentioning

confidence: 99%

Short time behavior of the ATM implied skew in the ADO-Heston model

Itkin

2024

FMF

Self Cite

View full text Add to dashboard Cite

In the current literature, there exists an opinion that Markovian approximations of rough volatility models are able to catch the behavior of the at-the-money implied volatility skew S(T ) at small maturities T for vanilla options, while for forward started options this is not the case. To analyze this, similar to [P. Carr, A. Itkin, 2019], we construct another Markovian approximation of the rough Heston-like volatility model -the ADO-Heston model. By using the Dobric-Ojeda process instead of the fractional Brownian motion in the rough variance process, we came up to an inhomogeneous Markovian model where the characteristic function (CF) solves an unsteady three-dimensional inhomogeneous partial differential equation (PDE) with some coefficients being function of the Hurst exponent H. It then turned out that the characteristic function of the log-price and S(T ) in this model can be represented in a semi-analytical form. Since we were basically interested in the behavior of the S(T ) at T → 0, instead of using numerical inverse transforms, we further simplified the representation of the CF for vanillas by making a particular choice of the market price of risk and looking at the asymptotic of the CF at T → 0. It was observed that the behavior of thus obtained S(T ) ∝ a(H)T b•(H−1/2) , b = const was not exactly the same as in rough volatility models since b ̸ = 1, but to be close enough for all practical values of T > 0. Thus, we concluded that the proposed Markovian model was able to replicate the behavior of the corresponding rough volatility model for the range of T inherent to the options' market. Similar analysis was provided for the forward starting options, where the S(t, s, T ) for the forward started options exhibited a similar behavior for any s > t when T → s.

show abstract

Semi-Analytical Pricing of Barrier Options in the Time-Dependent Heston Model

Cited by 3 publications

References 0 publications

Sparse Modeling Approach to the Arbitrage-Free Interpolation of Plain-Vanilla Option Prices and Implied Volatilities

Sparse Modeling Approach to the Arbitrage-Free Interpolation of Plain-Vanilla Option Prices and Implied Volatilities

Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit

Short time behavior of the ATM implied skew in the ADO-Heston model

Contact Info

Product

Resources

About