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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. 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.
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.
Dupire [ 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 Wiggins [ 8 ].
Other models for the volatility dynamics were proposed by E. 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. 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 ]. 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 process 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 are 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 [ 24 , 25 ]. 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.
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 3. 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. Following Heston we assume that the risk neutral dynamics of the asset price are governed by the system of stochastic differential equations SDEs: Both are assumed to be constant over time. This means that an up move in the asset is normally accompanied by a down move in volatility.
It specifies the final payoff of the option. The second condition states that for a stock price of zero the put price must equal the discounted strike price. The third condition specifies the payoff for a variance volatility of zero.
In this case the underlying asset evolves completely deterministically and the put price equals its lower bound derived by arbitrage considerations. The next condition describes the option's price for ever-increasing asset prices.
In this case the put price must equal the discounted strike price, that is, its upper bound, again derived by arbitrage arguments. The objective of this section is to solve 2. The derivation of a solution is based on Mellin transforms. This is the key conceptual difference between the two frameworks. For conditions that guarantee the existence and the connection to Fourier and Laplace transforms, see [ 28 ] or [ 29 ].
A straightforward application to 2. In [ 24 ] it is shown that the last equation is equivalent to the BSM formula for European put options. These types of equations also appear in frameworks based on Fourier transforms see [ 10 , 11 , 13 ], among others.
See [ 30 ] for a reference. Note that similar to Carr and Madan [ 14 ] the final pricing formula only requires a single integration.
In the previous section a Mellin transform approach was used to solve the European put option pricing problem in Heston's mean reverting stochastic volatility model. The outcome is a new characterization of European put prices using an integration along a vertical line segment in a strip of the positive complex half plane. Our solution has a clear and well-defined structure. The numerical treatment of the solution is simple and requires a single integration procedure.
However, the final expression for the option's price can still be modified to provide further insights on the analytical solution. First we have the following proposition. An equivalent and more convenient way of expressing the solution in Theorem 3. The statement follows directly from Theorem 3. A further advantage of the new framework is that hedging parameters, commonly known as Greeks, are easily determined analytically.
The most popular Greek letters widely used for risk management are delta, gamma, vega, rho, and theta. Each of these sensitivities measures a different dimension of risk inherent in the option.
The results for Greeks are summarized in the next proposition. The expressions follow directly from Theorem 3. We point out that instead of using the put call parity relationship for valuing European call options a direct Mellin transform approach is also possible. However, a slightly modified definition is needed to guarantee the existence of the integral. Using the modification and following the lines of reasoning outlined in Section 3 , it is straightforward to derive at the following theorem.
Again, a direct analogy to Heston's original pricing formula is provided. Also, the corresponding closed-form expressions for the Greeks follow immediately. In this section we evaluate the results of the previous sections for the purpose of computing and comparing option prices for a range of different parameter combinations. Since our numerical calculations are not based on a calibration procedure, we will use notional parameter specifications.
As a benchmark we choose the pricing formula due to Heston based on Fourier inversion H. From the previous analysis it follows that the numerical inversion in both integral transform approaches requires the calculation of logarithms with complex arguments. As pointed out in [ 11 , 18 ] this calculation may cause problems especially for options with long maturities or high mean reversion levels.
The Mellin transform solution MT is based on 3. Although any other choice of truncation is possible, this turned out to provide comparable results. Table 1 gives a first look at the results for different asset prices and expiration dates. Our major finding is that the pricing formulae derived in this paper provide comparable results for all parameter combinations.
They can be neglected from a practical point of view. In addition, since the numerical integration is accomplished in both frameworks equivalently efficient, the calculations are done very quickly. Next, we examine how the option prices vary if the correlation between the underlying asset and its instantaneous variance changes. We fix time to maturity to be 6 months. Also, to provide a variety of parameter combinations, we change some of the remaining parameters slightly: Our findings are reported in Table 2.
Again, the Mellin transform approach gives satisfactory results as the absolute differences show. Analyzing the results in detail, one basically observes two different kinds of behavior.
The maximum difference is 0. The opposite is true for OTM puts. The maximum change in the downward move is 0.More...