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Published12 Oct Abstract We are concerned with the valuation of European options in the Heston stochastic volatility model with correlation. Based on Mellin transforms, we present new solutions for the price of European options and hedging parameters.
- Triple options
- Option Pricing with the Heston Model of Stochastic Volatility
- The Heston Model, named after Steve Heston, is a type of stochastic volatility model used by financial professionals to price European options.
- Here, the dimension of the set of equivalent martingale measures is one; there is no unique risk-free measure.
- The result is a set of option prices for the given number of discrete values of.
- Heston model - Wikipedia
In contrast to Fourier-based approaches, where the transformation variable is usually the log-stock price at maturity, our framework focuses on directly transforming the current stock price.
Our solution has the nice feature that it requires only a single integration. We make numerical tests to compare our results with Heston's solution based on Fourier inversion and investigate the accuracy of the derived pricing formulae.
Introduction The pricing methodology proposed by Black and Scholes [ 1 ] and Merton [ 2 ] is maybe the most significant and influential development in option pricing theory. However, the assumptions underlying the original works were questioned ab initio and became the subject of a wide theoretical and empirical study. Soon it became clear that extensions are necessary to fit the empirical data. To reflect the empirical evidence of a nonconstant volatility and to explain the so-called volatility smile, different approaches were developed.
Journal of Applied Mathematics
На чем зарабатывают торговые сети [ 3 ] applies a partial differential equation PDE method and assumes that volatility dynamics can be modeled as a deterministic function of the stock price and time. A different approach is proposed by Sircar and Papanicolaou [ 4 ].
Based on the PDE framework, they develop a methodology that is independent of a particular volatility process. The result is an asymptotic approximation consisting of a BSM-like price plus a Gaussian variable capturing the risk from the volatility component. The majority of the financial community, however, focuses on stochastic volatility models.
These models assume that volatility itself is a random process and fluctuates over time.
Stochastic volatility models were first studied by Johnson and Shanno [ 5 ], Hull and White [ 6 ], Scott [ 7 ], and Options Heston model [ 8 ]. Other models for the volatility dynamics were proposed by E.
Stein and J. In all these models the stochastic process governing the asset price dynamics is driven by a subordinated stochastic volatility process that may or may not be independent.
Heston Model Definition
While the early models could not produce closed-form formulae, it was E. Assuming that volatility follows a mean reverting Ornstein-Uhlenbeck process and is uncorrelated with asset returns, they present an analytic expression for the density function of asset returns for the purpose of option valuation.
They present a closed-form solution for European options and discuss additional features of the volatility dynamics. The maybe most popular stochastic volatility model was introduced by Heston [ 10 ].
Heston Model: Formula, Assumptions, Limitations
In his influential paper he presents a new approach for a closed-form valuation of options specifying the dynamics of the squared volatility variance as a square-root options Heston model and applying Fourier inversion techniques for the pricing procedure. The characteristic function approach turned out to be a very powerful tool. See also the study by Duffie et al. Beside Fourier and Laplace transforms, there options Heston model other interesting integral transforms used in theoretical and applied mathematics.
Specifically, the Mellin transform gained great popularity in complex analysis and analytic number theory for its applications to problems related to the Gamma function, the Riemann zeta function, and other Dirichlet series. Its applicability to problems arising in finance theory has not been studied much yet [ 2425 ].
Panini and Srivastav introduce in [ 25 ] Mellin transforms in the theory of option pricing and use the new approach to value European and American plain vanilla and basket options on nondividend paying stocks.
Implied Volatility surface Parameterization (Part 2/2)
The approach is extended in [ 24 ] to power options with a nonlinear payoff and American options written on dividend paying assets. The purpose of this paper is to show how the framework can be extended to the stochastic volatility problem.
We derive an equivalent representation of the solution and discuss its interesting features. The paper is structured as follows. In Section 2 we give a formulation of the pricing problem for European options in the square-root stochastic volatility model. Based on Mellin transforms, the solution for puts is presented in Section options Heston model.
If you would like to brush on options terminology, head on over to our options trading basics blog for a refresher.
Section 4 is devoted to further analysis of our new solution. We provide a direct connection to Heston's pricing formula and give closed-form expressions for hedging parameters. Also, an explicit solution for European calls is presented. Numerical calculations are made in Section 5.
We test the accuracy of our closed-form solutions for a variety of parameter combinations. Section 6 concludes this paper. Problem Statement.