An approximation method for pricing continuous barrier options under multi-asset local stochastic volatility models
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This paper presents a new approximation method for pricing multi-asset continuous single barrier options under general local stochastic volatility models. The formula applies an asymptotic expansion technique and an approximation for the distribution of the first exit time of diffusion processes. This method focuses on local stochastic volatility models with unknown characteristic function and transition density function. To the best of our knowledge, our approximation formula is the first to achieve analytic approximations for continuous barrier options prices in this environment. In numerical experiments, we confirm the validity of the formula.Keywords:
Local volatility
Barrier option
Asymptotic expansion
Heavy traffic approximation
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We present a simplified approach to the analytical approximation of the transition density related to a general local volatility model. The methodology is sufficiently flexible to be extended to time-dependent coefficients, multi-dimensional stochastic volatility models, degenerate parabolic PDEs related to Asian options and also to include jumps.
Local volatility
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This paper derives a new semi closed-form approximation formula for pricing an up-and-out barrier option under a certain type of stochastic volatility model including SABR model by applying a rigorous asymptotic expansion method developed by Kato, Takahashi and Yamada (2012). We also demonstrate the validity of our approximation method through numerical examples.
Asymptotic expansion
Barrier option
Local volatility
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This paper presents an extension of the double Heston stochastic volatility model by combining Hull‐White stochastic interest rates. By the change of numeraire and quadratic exponential scheme, this paper develops a new simulation scheme for the extended model. By combining control variates and antithetic variates, this paper provides an efficient Monte Carlo simulation algorithm for pricing barrier options. Based on the differential evolution algorithm the extended model is calibrated to S&P 500 index options to obtain the model parameter values. Numerical results show that the proposed simulation scheme outperforms the Euler scheme, the proposed simulation algorithm is efficient for pricing barrier options, and the extended model is flexible to fit the implied volatility surface.
Barrier option
Control variates
Numéraire
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This paper presents a new approximation method for pricing multi-asset continuous single barrier options under general local stochastic volatility models. The formula applies an asymptotic expansion technique and an approximation for the hitting probability. This method focuses on local stochastic volatility models with unknown characteristic function and transition density function. To the best of our knowledge, our approximation formula is the first to achieve analytic approximations for continuous barrier options prices in this environment. In numerical experiments, we examine the validity of the formula.
Local volatility
Barrier option
Asymptotic expansion
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This PhD thesis consists of three separate papers. The common theme is methods to calculate analytical approximations for prices of different contingent claims under various model assumptions. The first two papers deals with approximations of standard European options in stochastic volatility models. The third paper is focused on approximating prices of commodity swaptions in a general model framework.
In the first paper, Dynamic Extensions and Probabilistic Expansions of the SABR model, a closed form approximation to prices of call options and implied volatilities in the so called SABR model of Hagan et al. (2002) is derived. The SABR model is one of the most frequently used stochastic volatility models used for option pricing in practice. The method relies on perturbing the model dynamics and approximations are obtained from a second order Taylor expansion. It is shown how the expansion terms can be calculated in a straightforward fashion using the flows of the perturbed model and results from the Malliavin calculus. This technique is further applied to calculate a closed form approximation of the option price in a useful dynamic extension of the original model. The dynamic model is able to match the prices of several options with different maturities and can therefore be used to price path dependents products in a consistent way. In addition, we propose an alternative model specification for the dynamic SABR model where the dynamics of the underlying asset are given by a displaced diffusion. In its non-dynamic version this model has similar properties as the original SABR model but it is more analytically tractable. A closed form approximation of the dynamic version of this model is also derived. The accuracy of the approximations is evaluated in a Monte Carlo study and the method is found to work well for many parameters of interest. The second paper, General Approximation Schemes for Option Prices in Stochastic Volatility Models, further examines the method of approximation employed in the first paper. The method is developed for a general stochastic volatility specification that can generate many commonly employed models as special cases. As an important application the method is used to calculate a second order expansion for the Heston (1993) model. The Heston model is arguably the most often used stochastic volatility model in practice and academic work. A numerical study of the approximations is performed where they are compared to prices and implied volatilities calculated from numerical Fourier transforms. It is found that for several parameters of interest the approximation is very accurate. Relating the proposed method to the existing literature we find that it generalizes the work by Lewis (2005) in several directions. In the case of the Heston model the first order expansion coincides with the approximation proposed in Alos (2006). However, an important advantage of the proposed method is that it can be used to generate higher order terms and it is verified that extending to second order substantially improves the accuracy of the approximation. The third paper, Approximative Valuation of Commodity Swaptions, is concerned with the numerical calculation of prices of European options on commodity swaps. A general approximation scheme for these claims is derived within the model framework of Heath, Jarrow and Morton (1992) extended to include commodity forwards. In models with deterministic volatilities the approach generates a closed form approximation allowing the swaptions to be conveniently priced using Blacks formula. The approximation is evaluated when applied to a Gaussian 2-factor model frequently employed in the literature. A comparison of the approximative prices to Monte Carlo simulations shows that the incurred errors are small for a large set of relevant parameters.
Malliavin calculus
LIBOR market model
Local volatility
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Single and double barrier options on more than one underlying with stochastic volatility are usually priced via Monte Carlo simulation due to the non-existence of closed-form solutions for their value. In this paper, for a special dependence structure, the prices of some two-asset barrier derivatives, like double-digital options and correlation options can be derived analytically using generalized Fourier transforms and some conditions on the characteristic functions. We study the influence of the various parameters on these prices and show that these formulas can be easily and quickly computed. We also extend our approach to further allow for a random correlation structure.
Barrier option
Covariance and correlation
Heath–Jarrow–Morton framework
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Girsanov theorem
Theory of computation
Call option
Local volatility
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This paper develops an asymptotic expansion method for general stochastic differential equations (SDEs) with jumps and their functions. By applying the method, we derive an explicit approximation formula for pricing options on functions of multiple assets under localstochastic volatility with jump models. Moreover, we present numerical examples for pricing basket options based on the parameters calibrated to the actual market data, which confirms the validity of our method in practice.
Asymptotic expansion
Local volatility
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This paper develops an asymptotic expansion method for general stochastic differential equations with jumps and their functions. By applying the method, we derive an explicit approximation formula for pricing options on functions of multiple assets under local-stochastic volatility with jump models. Moreover, we present numerical examples for pricing basket options based on the parameters calibrated to the actual market data, which confirms the validity of our method in practice.
Asymptotic expansion
Local volatility
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The correlation structure is crucial when pricing multi-asset products, in particular barrier options. In this work, we price two-asset path-dependent derivatives by means of perturbation theory in the context of a bi-dimensional asset model with stochastic correlation and volatilities. To our best knowledge, this is the first attempt at pricing barriers with stochastic correlation. It turns out that the leading term of the approximation corresponds to a constant covariance Black–Scholes type price with correction terms adjusting for stochastic volatility and stochastic correlation effects. The practicability of the presented method is illustrated by some numerical implementations.
Barrier option
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