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Journal of Applied Mathematics
Volume 2011, Article ID 198469, 18 pages
http://dx.doi.org/10.1155/2011/198469
Research Article

Valuing Options in Heston's Stochastic Volatility Model: Another Analytical Approach

Faculty of Economics and Business Administration, Eberhard Karls University of Tübingen, Mohlstrasse 36, 72074 Tübingen, Germany

Received 21 April 2011; Accepted 27 July 2011

Academic Editor: Julián López-Gómez

Copyright © 2011 Robert Frontczak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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. 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.

1. 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. The main drawback in the original Black/Scholes/Merton (BSM) model is the assumption of a constant volatility.

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. Stein and J. Stein [9], Heston [10], Schöbel and Zhu [11], and Rogers and Veraart [12]. 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. Stein and J. Stein [9] (S&S) who first succeeded in deriving an analytical solution. 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. Schöbel and Zhu [11] generalize the S&S model to the case of nonzero correlation between instantaneous volatilities and asset returns. 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. As a natural consequence it became standard in option pricing theory and was refined and extended in various directions (Bates [13], Carr and Madan [14], Bakshi and Madan [15], Lewis [16], Lee [17], Kahl and Jäckel [18], Kruse and Nögel [19], Fahrner [20], or Lord and Kahl [21] among others). See also the study by Duffie et al. [22, 23] for the mathematical foundations of affine processes.

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.

2. Problem Statement

Let 𝑆(𝑡)=𝑆𝑡 be the price of a dividend paying stock at time 𝑡 and 𝑉𝑡 its instantaneous variance. Following Heston we assume that the risk neutral dynamics of the asset price are governed by the system of stochastic differential equations (SDEs): 𝑑𝑆𝑡=(𝑟𝑞)𝑆𝑡𝑑𝑡+𝑉𝑡𝑆𝑡𝑑𝑊𝑡,𝑑𝑉𝑡=𝜅𝜃𝑉𝑡𝑑𝑡+𝜉𝑉𝑡𝑑𝑍𝑡,(2.1) with initial values 𝑆0,𝑉0(0,) and where 𝑟,𝑞,𝜅,𝜃,𝜉>0. The parameter 𝑟 is the risk-free interest rate, and 𝑞 is the dividend yield. Both are assumed to be constant over time. 𝜅 is the speed of mean reversion to the mean reversion level 𝜃, and 𝜉 is the so-called volatility of volatility. 𝑊𝑡 and 𝑍𝑡 are two correlated Brownian motions with 𝑑𝑊𝑡𝑑𝑍𝑡=𝜌𝑑𝑡, where 𝜌(1,1) is the correlation coefficient. The Feller condition 𝜅𝜃>(1/2)𝜉2 guarantees that the variance process never reaches zero and always stays positive. For practical uses it is also worth mentioning that in most cases the correlation coefficient 𝜌 is negative. This means that an up move in the asset is normally accompanied by a down move in volatility.

Let 𝑃𝐸(𝑆,𝑉,𝑡) be the current price of a European put option with strike price 𝑋 and maturity 𝑇. The option guarantees its holder a terminal payoff given by 𝑃𝐸(𝑆,𝑉,𝑇)=max(𝑋𝑆(𝑇),0).(2.2) Using arbitrage arguments it is straightforward to derive a two-dimensional partial differential equation (PDE) that must be satisfied by any derivative 𝐹 written on 𝑆 and 𝑉: 𝐹𝑡+(𝑟𝑞)𝑆𝐹𝑆+12𝑉𝑆2𝐹𝑆𝑆+𝜅(𝜃𝑉)𝜆𝜉𝑉𝐹𝑉+12𝜉2𝑉𝐹𝑉𝑉+𝜌𝜉𝑉𝑆𝐹𝑆𝑉𝑟𝐹=0,(2.3) on 0<𝑆,𝑉<,0<𝑡<𝑇 (throughout this paper partial derivatives with respect to the underlying variables will be denoted by subscripts) (see [16]). 𝜆 is called the market price of volatility risk. Heston provides some reasons for the assumption that 𝜆 is proportional to volatility, that is, 𝜆=𝑘𝑉 for some constant 𝑘. Therefore, 𝜆𝜉𝑉=𝑘𝜉𝑉=𝜆𝑉 (say). Hence, without loss of generality, 𝜆 can be set to zero as has been done in [26, 27]. For a constant volatility the two-dimensional PDE reduces to the fundamental PDE due to Black/Scholes and Merton and admits a closed-form solution given by the celebrated BSM formula. If 𝐹 is a European put option, that is, 𝐹(𝑆,𝑉,𝑡)=𝑃𝐸(𝑆,𝑉,𝑡), then we have 𝑃𝐸𝑡+(𝑟𝑞)𝑆𝑃𝐸𝑆+12𝑉𝑆2𝑃𝐸𝑆𝑆+𝜅(𝜃𝑉)𝑃𝐸𝑉+12𝜉2𝑉𝑃𝐸𝑉𝑉+𝜌𝜉𝑉𝑆𝑃𝐸𝑆𝑉𝑟𝑃𝐸=0,(2.4) where 𝑃𝐸(𝑆,𝑉,𝑡)𝑅+×𝑅+×[0,𝑇]𝑅+. The boundary conditions are given by 𝑃𝐸𝑃(𝑆,𝑉,𝑇)=max(𝑋𝑆(𝑇),0),𝐸(0,𝑉,𝑡)=𝑋𝑒𝑟(𝑇𝑡),𝑃𝐸(𝑆,0,𝑡)=max𝑋𝑒𝑟(𝑇𝑡)𝑆(𝑡)𝑒𝑞(𝑇𝑡),,0lim𝑆𝑃𝐸(𝑆,𝑉,𝑡)=0,(2.5)lim𝑉𝑃𝐸(𝑆,𝑉,𝑡)=𝑋𝑒𝑟(𝑇𝑡).(2.6) The first condition is the terminal condition. 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. Obviously, since a put option gives its holder the right to sell the asset the price will tend to zero if 𝑆 tends to infinity. Finally, notice that if variance (volatility) becomes infinite the current asset price contains no information about the terminal payoff of the derivative security, except that the put entitles its holder to sell the asset for 𝑋. In this case the put price must equal the discounted strike price, that is, its upper bound, again derived by arbitrage arguments.

In a similar manner the European call option pricing problem with solution 𝐶𝐸(𝑆,𝑉,𝑡) is characterized as the unique solution of (2.4) subject to 𝐶𝐸𝐶(𝑆,𝑉,𝑇)=max(𝑆(𝑇)𝑋,0),𝐸(𝐶0,𝑉,𝑡)=0,𝐸𝑆(𝑆,0,𝑡)=max(𝑡)𝑒𝑞(𝑇𝑡)𝑋𝑒𝑟(𝑇𝑡),,0lim𝑆𝐶𝐸(𝑆,𝑉,𝑡)=,lim𝑉𝐶𝐸(𝑆,𝑉,𝑡)=𝑆(𝑡)𝑒𝑞(𝑇𝑡).(2.7)

3. Analytic Solution Using Mellin Transforms

The objective of this section is to solve (2.4) subject to (2.5)–(2.6) in (semi) closed form. The derivation of a solution is based on Mellin transforms. For a locally Lebesgue integrable function 𝑓(𝑥),𝑥𝑅+, the Mellin transform 𝑀(𝑓(𝑥),𝜔), 𝜔𝐶, is defined by 𝑀(𝑓(𝑥),𝜔):=𝑓(𝜔)=0𝑓(𝑥)𝑥𝜔1𝑑𝑥.(3.1) As a complex function the Mellin transform is defined on a vertical strip in the 𝜔-plane, whose boundaries are specified by the asymptotic behavior of the function 𝑓(𝑥) as 𝑥0+ and 𝑥 (Fourier transforms (at least those which are typical in option pricing) usually exist in horizontal strips of the complex plane. 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]. Conversely, if 𝑓(𝜔) is a continuous, integrable function with fundamental strip (𝑎,𝑏), then, if 𝑐 is such that 𝑎<𝑐<𝑏 and 𝑓(𝑐+𝑖𝑡) is integrable, the inverse of the Mellin transform is given by 𝑓(𝑥)=𝑀1=1𝑓(𝜔)2𝜋𝑖𝑐+𝑖𝑐𝑖𝑓(𝜔)𝑥𝜔𝑑𝜔.(3.2)

Let 𝑃𝐸=𝑃𝐸(𝜔,𝑉,𝑡) denote the Mellin transform of 𝑃𝐸(𝑆,𝑉,𝑡). It is easily verified that 𝑃𝐸 exists in the entire half plane with Re(𝜔)>0, where Re(𝜔) denotes the real part of 𝜔. A straightforward application to (2.4) gives 𝑃𝐸𝑡+𝑎1𝑉+𝑏1𝑃𝐸𝑉+𝑎2𝑉+𝑏2𝑃𝐸𝑉𝑉+𝑎0𝑉+𝑏0𝑃𝐸=0,(3.3) where 𝑎1=(𝜔𝜌𝜉+𝜅),𝑏1𝑎=𝜅𝜃,2=12𝜉2,𝑏2𝑎=0,0=12𝜔(𝜔+1),𝑏0=𝑞𝜔𝑟(𝜔+1).(3.4) This is a one-dimensional PDE in the complex plane with nonconstant coefficients. To provide a unique solution for 0<𝑉<,0<𝑡<𝑇, we need to incorporate the boundary conditions from the previous section. The transformed terminal and boundary conditions are given by, respectively, 𝑃𝐸(𝜔,𝑉,𝑇)=𝑋𝜔+11𝜔1,𝑃𝜔+1(3.5)𝐸(𝜔,0,𝑡)=𝑒(𝑞𝜔𝑟(𝜔+1))(𝑇𝑡)𝑋𝜔+11𝜔1,𝜔+1(3.6) and condition (2.6) becomes lim𝑉|||𝑃𝐸|||(𝜔,𝑉,𝑡)=.(3.7) Now, we change the time variable from 𝑡 to 𝜏=𝑇𝑡 and convert the backward in time PDE into a forward in time PDE with solution domain 0<𝑉,𝜏<. With 𝑃𝐸𝑃(𝜔,𝑉,𝑡)=𝐸(𝜔,𝑉,𝜏), the resulting equation is 𝑃𝐸𝜏+𝑎1𝑉+𝑏1𝑃𝐸𝑉+𝑎2𝑉+𝑏2𝑃𝐸𝑉𝑉+𝑎0𝑉+𝑏0𝑃𝐸=0,(3.8) where the coefficients 𝑎0,𝑎1,𝑎2,𝑏0,𝑏1, and 𝑏2 are given in (3.4) and the terminal condition (3.5) becomes an initial condition 𝑃𝐸(𝜔,𝑉,0)=𝑋𝜔+11𝜔1𝜔+1.(3.9) Additionally we have 𝑃𝐸(𝜔,0,𝜏)=𝑒(𝑞𝜔𝑟(𝜔+1))𝜏𝑋𝜔+11𝜔1,𝜔+1lim𝑉|||𝑃𝐸|||(𝜔,𝑉,𝜏)=.(3.10) To simplify the PDE (3.8) further, we assume that the solution 𝑃𝐸(𝜔,𝑉,𝜏) can be written in the form 𝑃𝐸(𝜔,𝑉,𝜏)=𝑒(𝑞𝜔𝑟(𝜔+1))𝜏(𝜔,𝑉,𝜏)(3.11) with an appropriate function (𝜔,𝑉,𝜏). It follows that must satisfy 𝜏+𝑎1𝑉+𝑏1𝑉+𝑎2𝑉𝑉𝑉+𝑎0𝑉=0,(3.12) with initial and boundary conditions (𝜔,𝑉,0)=𝑋𝜔+11𝜔1,𝜔+1(𝜔,0,𝜏)=𝑋𝜔+11𝜔1,𝜔+1lim𝑉||||(𝜔,𝑉,𝜏)=.(3.13) Observe that, for 𝜅=𝜃=𝜉=0, that is, if the stock price dynamics are given by the standard BSM model with constant volatility, the PDE for is solved as (𝜔,𝑉,𝜏)=𝑋𝜔+11𝜔1𝑒𝜔+1(1/2)𝜔(𝜔+1)𝑉𝜏.(3.14) In this case the equation for 𝑃𝐸(𝜔,𝑉,𝜏) becomes 𝑃𝐸(𝜔,𝑉,𝜏)=𝑋𝜔+11𝜔1𝑒𝜔+1((1/2)𝜔(𝜔+1)𝑉+𝑞𝜔𝑟(𝜔+1))𝜏,(3.15) and the price of a European put option can be expressed as 𝑃𝐸1(𝑆,𝑉,𝜏)=2𝜋𝑖𝑐+𝑖𝑐𝑖𝑃𝐸(𝜔,𝑉,𝜏)𝑆𝜔𝑑𝜔,(3.16) with 0<𝑐<. In [24] it is shown that the last equation is equivalent to the BSM formula for European put options.

The final step in deriving a general solution for or equivalently for 𝑃𝐸 for a nonconstant volatility is to assume the following functional form of the solution: (𝜔,𝑉,𝜏)=̃𝑐𝐻(𝜔,𝜏)𝑒𝐺(𝜔,𝜏)𝑎0𝑉,(3.17) with 𝐻(𝜔,0)=1, 𝐺(𝜔,0)=0 and where we have set ̃𝑐=𝑋𝜔+11𝜔1𝜔+1.(3.18) Inserting the functional form for in (3.12), determining the partial derivatives, and simplifying yield two ordinary differential equations (ODEs). We have 𝐺𝜏(𝜔,𝜏)=𝐴𝐺2𝐻(𝜔,𝜏)+𝐵𝐺(𝜔,𝜏)+𝐶,(3.19)𝜏(𝜔,𝜏)=𝑎0𝑏1𝐺(𝜔,𝜏)𝐻(𝜔,𝜏),(3.20) where 𝐴=𝑎0𝑎2, 𝐵=𝑎1, and 𝐶=1. The ODE for 𝐺(𝜔,𝜏) is identified as a Riccati equation with constant coefficients. These types of equations also appear in frameworks based on Fourier transforms (see [10, 11, 13], among others). Having solved for 𝐺, a straightforward calculation shows that 𝐻(𝜔,𝜏) equals 𝐻(𝜔,𝜏)=𝑒𝑎0𝑏1𝜏0𝐺(𝜔,𝑥)𝑑𝑥.(3.21) Thus, we first present the solution for 𝐺. The transformation 1𝐺(𝜔,𝜏)=𝐴𝐵𝑢(𝜔,𝜏)2𝐴(3.22) gives 𝑢𝜏(𝜔,𝜏)=𝑢2(𝜔,𝜏)+4𝐴𝐶𝐵24.(3.23) Note that this is a special case of the more general class of ODEs given by 𝑢𝜏(𝜔,𝜏)=𝑎𝑢2(𝜔,𝜏)+𝑏𝜏𝑛,(3.24) with 𝑛𝑁 and 𝑎 and 𝑏 constants. This class of ODEs has solutions of the form 1𝑢(𝜔,𝜏)=𝑎𝐹𝜏(𝜔,𝜏)𝐹(𝜔,𝜏),(3.25) where 𝐹(𝜔,𝜏)=𝜏𝑐1𝐽1/2𝑚1𝑚𝑎𝑏𝜏𝑚+𝑐2𝑌1/2𝑚1𝑚𝑎𝑏𝜏𝑚.(3.26) The parameters 𝑐1,𝑐2 are again constants depending on the underlying boundary conditions, 𝑚=(1/2)(𝑛+2), and 𝐽 and 𝑌 are Bessel functions of the first and second kind, respectively. See [30] for a reference. Setting 𝑚=1 and incorporating the boundary conditions, 𝑢(𝜔,𝜏) is solved as 𝑘𝑢(𝜔,𝜏)=2tan((1/2)𝑘𝜏)+𝐵/𝑘1(𝐵/𝑘)tan((1/2)𝑘𝜏),(3.27) where we have set 𝑘=𝑘(𝜔)=4𝐴𝐶𝐵2=𝜉2𝜔(𝜔+1)(𝜔𝜌𝜉+𝜅)2.(3.28) Thus, we immediately get 𝐵𝐺(𝜔,𝜏)=+𝑘2𝐴2𝐴tan((1/2)𝑘𝜏)+𝐵/𝑘𝐵1(𝐵/𝑘)tan((1/2)𝑘𝜏)=+𝑘2𝐴2𝐴𝑘sin((1/2)𝑘𝜏)+𝐵cos((1/2)𝑘𝜏).𝑘cos((1/2)𝑘𝜏)𝐵sin((1/2)𝑘𝜏)(3.29) Using 𝑘2+𝐵2=4𝐴, it is easily verified that an equivalent expression for 𝐺(𝜔,𝜏) equals 𝐺(𝜔,𝜏)=2sin((1/2)𝑘𝜏)𝑘cos((1/2)𝑘𝜏)+(𝜔𝜌𝜉+𝜅)sin((1/2)𝑘𝜏)(3.30) with 𝑘=𝑘(𝜔) from above. To solve for 𝐻(𝜔,𝜏) we first mention that (see [31]) 𝐵cos𝑥+𝐶sin𝑥𝑏cos+𝑐sin𝑥𝑑𝑥=𝐵𝑐𝐶𝑏𝑏2+𝑐2ln(𝑏cos𝑥+𝑐sin𝑥)+𝐵𝑏+𝐶𝑐𝑏2+𝑐2𝑥.(3.31) Therefore, 𝜏0𝐺(𝜔,𝑥)𝑑𝑥=𝐵𝜏+12𝐴𝐴𝑘ln𝑘cos((1/2)𝑘𝜏)𝐵sin((1/2)𝑘𝜏),(3.32)𝐻(𝜔,𝜏)=𝑒(𝜅𝜃/𝜉2)[(𝜔𝜌𝜉+𝜅)𝜏+2ln(𝑘/𝑘cos((1/2)𝑘𝜏)+(𝜔𝜌𝜉+𝜅)sin((1/2)𝑘𝜏))].(3.33) Finally, we have arrived at the following result.

Theorem 3.1. A new Mellin-type pricing formula for European put options in Heston's [10] mean reverting stochastic volatility model is given by 𝑃𝐸1(𝑆,𝑉,𝜏)=2𝜋𝑖𝑐+𝑖𝑐𝑖𝑃𝐸(𝜔,𝑉,𝜏)𝑆𝜔𝑑𝜔,(3.34) with 0<𝑐<𝑐 and where 𝑃𝐸(𝜔,𝑉,𝜏)=̃𝑐𝑒(𝑞𝜔𝑟(𝜔+1))𝜏𝐻(𝜔,𝜏)𝑒𝐺(𝜔,𝜏)𝑎0𝑉(3.35) with 𝐺(𝜔,𝜏) and 𝐻(𝜔,𝜏) from above. The parameters ̃𝑐 and 𝑘 are given in (3.18) and (3.28), respectively. The choice of 𝑐 will be commented on below.

Remark 3.2. Note that similar to Carr and Madan [14] the final pricing formula only requires a single integration.

We now consider the issue of specifying 𝑐. Recall that, to guarantee the existence of the inverse Mellin transform of 𝑃𝐸(𝜔,𝑉,𝜏) in a vertical strip of the 𝜔-plane, we need 𝑃𝐸(𝑐+𝑖𝑦,𝑉,𝜏) to be integrable, and hence analytic. From (3.30) and (3.33) we have that 𝐺(𝜔,𝜏) and 𝐻(𝜔,𝜏) have the same points of singularity with lim𝜔0𝐺(𝜔,𝜏)=2sin((1/2)𝑖𝜅𝜏)=2𝑖𝜅cos((1/2)𝑖𝜅𝜏)+𝜅sin((1/2)𝑖𝜅𝜏)1𝑖𝜅sin2𝑒𝑖𝜅𝜏(1/2)𝜅𝜏=1𝑒𝜅𝜏𝜅,lim𝜔0𝐻(𝜔,𝜏)=1.(3.36) Furthermore, since 𝑘(𝜔)=𝜉2𝜔21𝜌2𝜉+𝜔22𝜌𝜉𝜅𝜅2,(3.37) it follows that 𝑘(𝜔) has two real roots given by 𝜔1/2=(𝜉2𝜌𝜅)±(𝜉2𝜌𝜅)2+4𝜅21𝜌22𝜉1𝜌2,(3.38) where 𝜌±1 and where only the positive root 𝜔1 is of relevance. For 𝜌=±1 we have a single root 𝜔1=𝜅2𝜉22𝜉𝜅.(3.39) We deduce that all singular points of 𝐺 and 𝐻 are real, starting with 𝜔1 being a removable singularity. We therefore define 𝑐 as the first nonremovable singularity of 𝐺 and 𝐻 in {𝜔𝐶0<Re(𝜔)<,Im(𝜔)=0}, that is, the first real root of 𝑓(𝜔) except 𝜔1, where 𝑓(𝜔) is defined by 1𝑓(𝜔)=𝑘(𝜔)cos21𝑘(𝜔)𝜏+(𝜔𝜌𝜉+𝜅)sin2𝑘(𝜔)𝜏.(3.40) If 𝑓(𝜔) has no roots or no other roots except 𝜔1 in {𝜔𝐶0<Re(𝜔)<,Im(𝜔)=0}, then we set 𝑐=. By definition it follows that 𝜔1𝑐, with the special cases lim𝜏0𝑐= and lim𝜏𝑐=𝜔1.

4. Further Analysis

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.

Proposition 4.1. An equivalent (and more convenient) way of expressing the solution in Theorem 3.1 is 𝑃𝐸(𝑆,𝑉,𝜏)=𝑋𝑒𝑟𝜏𝑃1𝑆𝑒𝑞𝜏𝑃2,(4.1) with 𝑆=𝑆(𝑡) being the current stock price, 𝑃1=12𝜋𝑖𝑐+𝑖𝑐𝑖𝑋𝑒𝑟𝜏𝑆𝑒𝑞𝜏𝜔1𝜔𝐻(𝜔,𝜏)𝑒𝐺(𝜔,𝜏)𝑎0𝑉𝑃𝑑𝜔,2=12𝜋𝑖𝑐+𝑖𝑐𝑖𝑋𝑒𝑟𝜏𝑆𝑒𝑞𝜏𝜔+11𝜔+1𝐻(𝜔,𝜏)𝑒𝐺(𝜔,𝜏)𝑎0𝑉𝑑𝜔,(4.2) where 0<𝑐<𝑐.

Proof. The statement follows directly from Theorem 3.1 by simple rearrangement.

Remark 4.2. Equation (4.1) together with (4.2) provides a direct connection to Heston's original pricing formula given by 𝑃𝐸(𝑆,𝑉,𝜏)=𝑋𝑒𝑟𝜏Π1𝑆𝑒𝑞𝜏Π2,(4.3) with Π1=121𝜋0𝑒Re𝑖𝜔ln𝑋𝜑(𝜔)Π𝑖𝜔𝑑𝜔,2=121𝜋0𝑒Re𝑖𝜔ln𝑋𝜑(𝜔𝑖)𝑖𝜔𝜑(𝑖)𝑑𝜔,(4.4) where the function 𝜑(𝜔) is the log-characteristic function of the stock at maturity 𝑆(𝑇): 𝑒𝜑(𝜔)=𝐸𝑖𝜔ln𝑆(𝑇).(4.5)

Remark 4.3. By the fundamental concept of a risk-neutral valuation, we have 𝑃𝐸(𝑆,𝑉,𝜏)=𝐸𝑄𝑡𝑒𝑟𝜏(𝑋𝑆(𝑇))1{𝑆(𝑇)<𝑋}=𝑋𝑒𝑟𝜏𝐸𝑄𝑡1{𝑆(𝑇)<𝑋}𝑆𝑒𝑞𝜏𝐸𝑄𝑡1{𝑆(𝑇)<𝑋},(4.6) with 𝐸𝑡 being the time 𝑡 expectation under the corresponding risk-neutral probability measure, while 𝑄 denotes the equivalent martingale measure given by the Radon-Nikodym derivative 𝑑𝑄=𝑑𝑄𝑆(𝑇)𝑒𝑟𝜏𝑆𝑒𝑞𝜏.(4.7) So the framework allows an expression of the above probabilities as the inverse of Mellin transforms.

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.

Proposition 4.4. Setting 𝐼(𝜔,𝜏)=𝐻(𝜔,𝜏)𝑒𝐺(𝜔,𝜏)𝑎0𝑉,(4.8) the analytical expressions for the delta, gamma, vega, rho, and theta in the case of European put options are given by, respectively, 𝑃𝐸𝑆(𝑆,𝑉,𝜏)=12𝜋𝑖𝑐+𝑖𝑐𝑖𝑋𝑆𝜔+11𝑒𝜔+1(𝑞𝜔𝑟(𝜔+1))𝜏𝑃𝐼(𝜔,𝜏)𝑑𝜔,(4.9)𝐸𝑆𝑆1(𝑆,𝑉,𝜏)=2𝜋𝑖𝑐+𝑖𝑐𝑖1𝑆𝑋𝑆𝜔+1𝑒(𝑞𝜔𝑟(𝜔+1))𝜏𝑃𝐼(𝜔,𝜏)𝑑𝜔,(4.10)𝐸𝑉1(𝑆,𝑉,𝜏)=2𝜋𝑖𝑐+𝑖𝑐𝑖𝑋2𝑋𝑆𝜔𝑒(𝑞𝜔𝑟(𝜔+1))𝜏𝐺(𝜔,𝜏)𝐼(𝜔,𝜏)𝑑𝜔.(4.11) Recall that the rho of a put option is the partial derivative of 𝑃𝐸 with respect to the interest rate and equals 𝑃𝐸𝑟(𝑆,𝑉,𝜏)=𝑋𝜏2𝜋𝑖𝑐+𝑖𝑐𝑖𝑋𝑆𝜔1𝜔𝑒(𝑞𝜔𝑟(𝜔+1))𝜏𝐼(𝜔,𝜏)𝑑𝜔.(4.12) Finally, the theta of the put, that is, the partial derivative of 𝑃𝐸 with respect to 𝜏, is determined as 𝑃𝐸𝜏1(𝑆,𝑉,𝜏)=2𝜋𝑖𝑐+𝑖𝑐𝑖𝑋𝑆𝜔𝑋𝑒𝜔(𝜔+1)(𝑞𝜔𝑟(𝜔+1))𝜏𝐼(𝜔,𝜏)𝐽(𝜔,𝜏)𝑑𝜔,(4.13) with 1𝐽(𝜔,𝜏)=𝑞𝜔𝑟(𝜔+1)+2𝜔(𝜔+1)𝜅𝜃𝐺(𝜔,𝜏)+𝑉𝐺𝜏(𝜔,𝜏),(4.14) where 𝐺𝜏(𝜔,𝜏)=1(𝜔𝜌𝜉+𝜅)2𝜉21𝜔(𝜔+1)cos2(1/2)𝑘𝜏+tan1((𝜔𝜌𝜉+𝜅)/𝑘).(4.15)

Proof. The expressions follow directly from Theorem 3.1 or Proposition 4.1. The final expression for 𝐽(𝜔,𝜏) follows by straightforward differentiation and (3.20).

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. We therefore propose to define the Mellin transform for calls as 𝑀𝐶𝐸=𝐶(𝑆,𝑉,𝑡),𝜔𝐸(𝜔,𝑉,𝑡)=0𝐶𝐸(𝑆,𝑉,𝑡)𝑆(𝜔+1)𝑑𝑆,(4.16) where 1<Re(𝜔)<. Conversely, the inverse of this modified Mellin transform is given by 𝐶𝐸1(𝑆,𝑉,𝑡)=2𝜋𝑖𝑐+𝑖𝑐𝑖𝐶𝐸(𝜔,𝑉,𝑡)𝑆𝜔𝑑𝜔,(4.17) where 1<𝑐. Using the modification and following the lines of reasoning outlined in Section 3, it is straightforward to derive at the following theorem.

Theorem 4.5. The Mellin-type closed-form valuation formula for European call options in the square-root stochastic volatility model of Heston [10] equals 𝐶𝐸(𝑆,𝑉,𝜏)=𝑆𝑒𝑞𝜏𝑃2𝑋𝑒𝑟𝜏𝑃1,(4.18) where 𝑃2=12𝜋𝑖𝑐+𝑖𝑐𝑖𝑆𝑒𝑞𝜏𝑋𝑒𝑟𝜏𝜔11𝐻𝜔1(𝜔,𝜏)𝑒𝐺(𝜔,𝜏)𝑎0𝑉𝑃𝑑𝜔,1=12𝜋𝑖𝑐+𝑖𝑐𝑖𝑆𝑒𝑞𝜏𝑋𝑒𝑟𝜏𝜔1𝜔𝐻(𝜔,𝜏)𝑒𝐺(𝜔,𝜏)𝑎0𝑉𝑑𝜔,(4.19) with 𝐻(𝜔,𝜏)=𝑒(𝜅𝜃/𝜉2)[(𝜔𝜌𝜉𝜅)𝜏+2ln(𝑘/𝑘cos((1/2)𝑘𝜏)(𝜔𝜌𝜉𝜅)sin((1/2)𝑘𝜏))],𝐺(𝜔,𝜏)=2sin(1/2)𝑘𝜏𝑘cos(1/2)𝑘𝜏(𝜔𝜌𝜉𝜅)sin(1/2)𝑘𝜏,𝑘=𝑘(𝜔)=𝜉2𝜔(𝜔1)(𝜔𝜌𝜉𝜅)2,(4.20) and 𝑎0=(1/2)𝜔(𝜔1). Furthermore, one has that 1<𝑐<𝑐 with 𝑐 being characterized equivalently as at the end of the previous section.

Remark 4.6. Again, a direct analogy to Heston's original pricing formula is provided. Also, the corresponding closed-form expressions for the Greeks follow immediately.

5. Numerical Examples

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. We therefore additionally implement the rotation count algorithm proposed by Kahl and Jäckel in [18] to overcome these possible inconsistencies (H(RCA)). The Mellin transform solution (MT) is based on (3.34) for puts and (4.18) for calls, respectively. The limits of integration 𝑐±𝑖 are truncated at 𝑐±𝑖500. Although any other choice of truncation is possible, this turned out to provide comparable results. To assess the accuracy of the alternative solutions, we determine the absolute difference between H(RCA) and MT (Diff). Table 1 gives a first look at the results for different asset prices and expiration dates. We distinguish between in-the-money (ITM), at-the-money (ATM), and out-of-the-money (OTM) options. Fixed parameters are 𝑋=100, 𝑟=0.04, 𝑞=0.02, 𝑉=0.09, 𝜅=3, 𝜃=0.12, 𝜉=0.2, and 𝜌=0.5, whereas 𝑆 and 𝜏 vary from 80 to 120 currency units and three months to three years, respectively. Using these values, we have for the European put 𝜔1=9.6749 constant, while 𝑐 varies over time from 54.7066 (𝜏=0.25) to 11.7046 (𝜏=3.0) and for the European call 𝜔1=31.0082 with 𝑐 changing from 116.7385 (𝜏=0.25) to 33.7810 (𝜏=3.0). We therefore use 𝑐=2 for the calculations (in both cases). Our major finding is that the pricing formulae derived in this paper provide comparable results for all parameter combinations. The absolute differences are very small (of order 106 to 108 for puts and 105 to 108 for calls). 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.

tab1
Table 1: European option prices in Heston's stochastic volatility model for different asset prices 𝑆 and maturities 𝜏. Fixed parameters are 𝑋=100, 𝑟=0.04, 𝑞=0.02, 𝑉=0.09, 𝜅=3, 𝜃=0.12, 𝜉=0.2, 𝜌=0.5, and 𝑐=2.

Next, we examine how the option prices vary if the correlation between the underlying asset and its instantaneous variance changes. Although from a practical point of view it may be less realistic to allow for a positive correlation, we do not make any restrictions on 𝜌 and let it range from −1.00 to 1.00. 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: 𝑋=100, 𝑟=0.05, 𝑞=0.02, 𝑉=0.04, 𝜅=2, 𝜃=0.05, and 𝜉=0.2. We abstain from presenting the numerical values of 𝜔1 and 𝑐 in this case and choose again 𝑐=2 for the integration. Our findings are reported in Table 2. Again, the Mellin transform approach gives satisfactory results as the absolute differences show. For both puts and calls they are of order 105 to 106. Analyzing the results in detail, one basically observes two different kinds of behavior. For ITM put options we have an increase in value for increasing values of 𝜌. The maximum difference is 0.6655 or 3.60%. The opposite is true for OTM puts. Here we have a strict decline in price if 𝜌 is increased. The magnitude of price reactions to changes in 𝜌 increases, too. The maximum change in the downward move is 0.7787 or equivalently 75.21%. The same behavior is observed for ATM options. However, the changes are much more moderate with a maximum percentage change of 0.80%. For European calls the situation is different. OTM calls increase significantly in value if 𝜌 is increased, whereas ITM and ATM call prices decrease for an increasing 𝜌. The maximum percentage changes are 492.96% (OTM), 3.49% (ITM), and 0.62% (ATM), respectively.

tab2
Table 2: European option prices in Heston's stochastic volatility model for different asset prices 𝑆 and correlations 𝜌. Fixed parameters are 𝑋=100, 𝑟=0.05, 𝑞=0.02, 𝑉=0.04, 𝜅=2, 𝜃=0.05, 𝜉=0.2, and 𝑐=2.

Finally, we compare the values of delta for different (𝑆;𝜏) combinations. For the calculation of the delta of a European put, we use (4.9). The corresponding delta value for a call is easily determined from the price function presented in the text. 𝑆 and 𝜏 vary from 80 to 120 currency units and three months to three years, respectively. Again, the remaining parameters are slightly altered and equal 𝑋=100, 𝑟=0.06, 𝑞=0.03, 𝑉=0.16, 𝜅=3, 𝜃=0.16, 𝜉=0.10, 𝜌=0.75, and 𝑐=2. The results are summarized in Table 3. Once more, we observe a high consistency with Heston's framework based on Fourier inversion. For all parameter combinations our results agree with Heston's with a great degree of precision.

tab3
Table 3: Delta values of European option prices in Heston's stochastic volatility model for different asset prices 𝑆 and maturities 𝜏. Fixed parameters are 𝑋=100, 𝑟=0.06, 𝑞=0.03, 𝑉=0.16, 𝜅=3, 𝜃=0.16, 𝜉=0.1, 𝜌=0.75, and 𝑐=2.

In summary, our numerical experiments suggest that the new framework is able to compete with Heston's solution based on Fourier inversion. The accuracy of the results is very satisfying, and the framework is flexible enough to account for all the pricing features inherent in the model. The findings justify the assessment of the Mellin transform approach as a very competitive alternative.

6. Conclusion

We have applied a new integral transform approach for the valuation of European options on dividend paying stocks in a mean reverting stochastic volatility model with correlation. Using the new framework our main results are new analytical characterizations of options' prices and hedging parameters. Our equivalent solutions may be of interest for theorists as well as practitioners. On one hand they provide further insights on the analytic solution, on the other hand they are easily and quickly treated numerically by applying efficient numerical integration schemes. We have done extensive numerical tests to demonstrate the flexibility and to assess the accuracy of the alternative pricing formulae. The results are gratifying and convincing. The new method is very competitive and should be regarded as a real alternative to other approaches, basically Fourier inversion methods, existing in the literature. Also, since the transformation variable is the current value of the asset instead of its terminal price, the new framework may turn out to be applicable to path-dependent problems.

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