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

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)=𝑆_𝑡 be the price of a dividend paying stock at time 𝑡 and 𝑉_𝑡 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): 𝑑𝑑𝑆𝑉_𝑡𝑡=(𝑟−𝑞)𝑆_𝑡𝑑𝑡+√𝑉_𝑡𝑆_𝑡𝑑𝑊_𝑡,=𝜅𝜃−𝑉_𝑡𝑑𝑡+𝜉√𝑉_𝑡𝑑𝑍_𝑡,(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. 𝑊_𝑡 and 𝑍_𝑡 are two correlated Brownian motions with 𝑑𝑊_𝑡𝑑𝑍_𝑡=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 𝑃^𝐸(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 𝑃^𝐸(𝑆,𝑉,𝑇)=max(𝑋−𝑆(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: 𝐹_𝑡+(𝑟−𝑞)𝑆𝐹_𝑆+12𝑉𝑆²𝐹_𝑆𝑆+𝜅(𝜃−𝑉)−𝜆𝜉√𝑉𝐹_𝑉+12𝜉²𝑉𝐹_𝑉𝑉+𝜌𝜉𝑉𝑆𝐹_𝑆𝑉−𝑟𝐹=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=𝑘√𝑉 for some constant 𝑘. Therefore, 𝜆𝜉√𝑉=𝑘𝜉𝑉=𝜆^∗𝑉 (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)=𝑃^𝐸(𝑆,𝑉,𝑡), then we have 𝑃^𝐸𝑡+(𝑟−𝑞)𝑆𝑃^𝐸𝑆+12𝑉𝑆²𝑃^𝐸𝑆𝑆+𝜅(𝜃−𝑉)𝑃^𝐸𝑉+12𝜉²𝑉𝑃^𝐸𝑉𝑉+𝜌𝜉𝑉𝑆𝑃^𝐸𝑆𝑉−𝑟𝑃^𝐸=0,(2.4) where 𝑃^𝐸(𝑆,𝑉,𝑡)∶𝑅⁺×𝑅⁺×[0,𝑇]→𝑅⁺. The boundary conditions are given by 𝑃^𝐸(𝑆,𝑃𝑉,𝑇)=max(𝑋−𝑆(𝑇),0),_𝐸𝑃_𝐸(0,𝑉,𝑡)=𝑋𝑒^{−𝑟(𝑇−𝑡)},(𝑆,0,𝑡)=mlima_𝑆→∞x𝑋𝑒^{−𝑟(𝑇−𝑡)}−𝑆(𝑡)𝑒^{−𝑞(𝑇−𝑡)},0,lim_𝑉→∞𝑃^𝐸(𝑆,𝑉,𝑡)=0,(2.5)𝑃^𝐸(𝑆,𝑉,𝑡)=𝑋𝑒^{−𝑟(𝑇−𝑡)}.(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 𝐶^𝐸(𝑆,𝑉,𝑡) is characterized as the unique solution of (2.4) subject to 𝐶^𝐸(𝑆,𝑉,𝑇𝐶)=max(𝑆(𝑇)−𝑋,0),_𝐸𝐶(0,𝑉,𝑡)=0,_𝐸(𝑆,0,𝑡)=lmima_𝑆→∞x𝑆(𝑡)𝑒^{−𝑞(𝑇−𝑡)}−𝑋𝑒^{−𝑟(𝑇−𝑡)},0,lim_𝑉→∞𝐶^𝐸(𝑆,𝑉,𝑡)=∞,𝐶^𝐸(𝑆,𝑉,𝑡)=𝑆(𝑡)𝑒^{−𝑞(𝑇−𝑡)}.(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),𝑥∈𝑅⁺, the Mellin transform 𝑀(𝑓(𝑥),𝜔), 𝜔∈𝐶, is defined by 𝑀(𝑓(𝑥),𝜔):=𝑓(𝜔)=^∞0𝑓(𝑥)𝑥^𝜔−1𝑑𝑥.(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 𝑥→0⁺ 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)=𝑀⁻¹𝑓(𝜔)=12𝜋𝑖^{𝑐+𝑖∞𝑐−𝑖∞}𝑓(𝜔)𝑥^−𝜔𝑑𝜔.(3.2)

Let 𝑃^𝐸=𝑃^𝐸(𝜔,𝑉,𝑡) denote the Mellin transform of 𝑃^𝐸(𝑆,𝑉,𝑡). It is easily verified that 𝑃^𝐸 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 𝑃^𝐸𝑡+𝑎₁𝑉+𝑏₁𝑃^𝐸𝑉+𝑎₂𝑉+𝑏₂𝑃^𝐸𝑉𝑉+𝑎₀𝑉+𝑏₀𝑃^𝐸=0,(3.3) where 𝑎₁=𝑎−(𝜔𝜌𝜉+𝜅),𝑏₁=𝜅𝜃,₂=12𝜉²𝑎,𝑏₂=0,₀=12𝜔(𝜔+1),𝑏⁰=𝑞𝜔−𝑟(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, 𝑃^𝐸(𝜔,𝑉,𝑇)=𝑋^𝜔+1𝑃𝜔1−𝜔+11,(3.5)_𝐸(𝜔,0,𝑡)=𝑒^{(𝑞𝜔−𝑟(𝜔+1))(𝑇−𝑡)}⋅𝑋^𝜔+11𝜔−𝜔+11,(3.6) and condition (2.6) becomes lim_𝑉→∞|||𝑃^𝐸(𝜔,𝑉,𝑡)|||=∞.(3.7) Now, we change the time variable from 𝑡 to 𝜏=𝑇−𝑡 and convert the backward in time PDE into a forward in time PDE with solution domain 0<𝑉,𝜏<∞. With 𝑃^𝐸(𝜔,𝑉,𝑡)=𝑃^𝐸(𝜔,𝑉,𝜏), the resulting equation is −𝑃^𝐸𝜏+𝑎₁𝑉+𝑏₁𝑃^𝐸𝑉+𝑎₂𝑉+𝑏₂𝑃^𝐸𝑉𝑉+𝑎₀𝑉+𝑏₀𝑃^𝐸=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 𝑃^𝐸(𝜔,𝑉,0)=𝑋^𝜔+1𝜔1−𝜔+11.(3.9) Additionally we have 𝑃^𝐸(𝜔,0,𝜏)=𝑒^{(𝑞𝜔−𝑟(𝜔+1))𝜏}⋅𝑋^𝜔+1lim_𝑉→∞𝜔1−𝜔+11,|||𝑃^𝐸(𝜔,𝑉,𝜏)|||=∞.(3.10) To simplify the PDE (3.8) further, we assume that the solution 𝑃^𝐸(𝜔,𝑉,𝜏) can be written in the form 𝑃^𝐸(𝜔,𝑉,𝜏)=𝑒^{(𝑞𝜔−𝑟(𝜔+1))𝜏}⋅ℎ(𝜔,𝑉,𝜏)(3.11) with an appropriate function ℎ(𝜔,𝑉,𝜏). It follows that ℎ must satisfy −ℎ_𝜏+𝑎₁𝑉+𝑏₁ℎ_𝑉+𝑎₂𝑉ℎ_𝑉𝑉+𝑎₀𝑉ℎ=0,(3.12) with initial and boundary conditions ℎ(𝜔,𝑉,0)=𝑋^𝜔+11ℎ(𝜔,0,𝜏)=𝑋_𝜔+1𝜔−𝜔+11,lim_𝑉→∞𝜔1−𝜔+11,||ℎ(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 ℎ is solved as ℎ(𝜔,𝑉,𝜏)=𝑋^𝜔+11𝜔−𝜔+11𝑒^{(1/2)𝜔(𝜔+1)𝑉𝜏}.(3.14) In this case the equation for 𝑃^𝐸(𝜔,𝑉,𝜏) becomes 𝑃^𝐸(𝜔,𝑉,𝜏)=𝑋^𝜔+1𝜔1−𝜔+11𝑒^{((1/2)𝜔(𝜔+1)𝑉+𝑞𝜔−𝑟(𝜔+1))𝜏},(3.15) and the price of a European put option can be expressed as 𝑃^𝐸(𝑆,𝑉,𝜏)=12𝜋𝑖^{𝑐+𝑖∞𝑐−𝑖∞}𝑃^𝐸(𝜔,𝑉,𝜏)𝑆^−𝜔𝑑𝜔,(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 ℎ or equivalently for 𝑃^𝐸 for a nonconstant volatility is to assume the following functional form of the solution: ℎ(𝜔,𝑉,𝜏)=𝑐̃⋅𝐻(𝜔,𝜏)⋅𝑒^{𝐺(𝜔,𝜏)⋅𝑎0⋅𝑉},(3.17) with 𝐻(𝜔,0)=1, 𝐺(𝜔,0)=0 and where we have set 𝑐̃=𝑋^𝜔+11𝜔−𝜔+11.(3.18) Inserting the functional form for ℎ in (3.12), determining the partial derivatives, and simplifying yield two ordinary differential equations (ODEs). We have 𝐺_𝜏𝐻(𝜔,𝜏)=𝐴⋅𝐺²(𝜔,𝜏)+𝐵⋅𝐺(𝜔,𝜏)+𝐶,(3.19)_𝜏(𝜔,𝜏)=𝑎₀⋅𝑏₁⋅𝐺(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)=𝑒^{𝑎0𝑏1∫𝜏0𝐺(𝜔,𝑥)𝑑𝑥}.(3.21) Thus, we first present the solution for 𝐺. The transformation 𝐺(𝜔,𝜏)=1𝐴𝑢(𝜔,𝜏)−2𝐵𝐴(3.22) gives 𝑢_𝜏(𝜔,𝜏)=𝑢²(𝜔,𝜏)+4𝐴𝐶−𝐵²4.(3.23) Note that this is a special case of the more general class of ODEs given by 𝑢_𝜏(𝜔,𝜏)=𝑎𝑢²(𝜔,𝜏)+𝑏𝜏^𝑛,(3.24) with 𝑛∈𝑁 and 𝑎 and 𝑏 constants. This class of ODEs has solutions of the form 𝑢(𝜔,𝜏)=−1𝑎𝐹^𝜏(𝜔,𝜏)𝐹(𝜔,𝜏),(3.25) where 𝐹(𝜔,𝜏)=√𝜏𝑐₁𝐽_1/2𝑚1𝑚√𝑎𝑏𝜏^𝑚+𝑐²𝑌_1/2𝑚1𝑚√𝑎𝑏𝜏^𝑚.(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 𝑢(𝜔,𝜏)=𝑘21t−an(𝐵((/1𝑘/)2t)a𝑘n𝜏()(1+/2𝐵)/𝑘𝑘𝜏),(3.27) where we have set 𝑘=𝑘(𝜔)=√4𝐴𝐶−𝐵²=𝜉²𝜔(𝜔+1)−(𝜔𝜌𝜉+𝜅)².(3.28) Thus, we immediately get 𝐺(𝜔,𝜏)=−𝐵=−2𝐵𝐴+2𝑘𝐴1t−an(𝐵((/1𝑘/)2t)a𝑘n𝜏()(1+/2𝐵)/𝑘𝑘𝜏)2𝐴+2𝑘𝐴𝑘𝑘scions((((11//22))𝑘𝑘𝜏𝜏))+−𝐵𝐵csoins((((11//22))𝑘𝑘𝜏𝜏)).(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]) 𝐵𝑏ccooss𝑥++𝑐𝐶sisnin𝑥𝑥𝑑𝑥=𝐵𝑏𝑐²−+𝐶𝑐²𝑏ln(𝑏cos𝑥+𝑐sin𝑥)+𝐵𝑏+𝐶𝑐𝑏²+𝑐²𝑥.(3.31) Therefore, ^𝜏0𝐺(𝜔,𝑥)𝑑𝑥=−𝐵𝜏𝐻(𝜔,𝜏)=𝑒2𝐴+𝐴1ln𝑘cos((1/2)𝑘𝜏)−𝑘𝐵sin((1/2)𝑘_{(𝜅𝜃/𝜉2)[(𝜔𝜌𝜉+𝜅)𝜏+2ln(𝑘/𝑘cos((1/2)𝑘𝜏)+(𝜔𝜌𝜉+𝜅)sin((1/2)𝑘𝜏))]}𝜏),(3.32).(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 𝑃^𝐸(𝑆,𝑉,𝜏)=12𝜋𝑖^{𝑐+𝑖∞𝑐−𝑖∞}𝑃^𝐸(𝜔,𝑉,𝜏)𝑆^−𝜔𝑑𝜔,(3.34) with 0<𝑐<𝑐^∗ and where 𝑃^𝐸(𝜔,𝑉,𝜏)=𝑐̃⋅𝑒^{(𝑞𝜔−𝑟(𝜔+1))𝜏}⋅𝐻(𝜔,𝜏)⋅𝑒^{𝐺(𝜔,𝜏)𝑎0𝑉}(3.35) with 𝐺(𝜔,𝜏) and 𝐻(𝜔,𝜏) from above. The parameters 𝑐̃ 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 𝑃^𝐸(𝜔,𝑉,𝜏) in a vertical strip of the 𝜔-plane, we need 𝑃^𝐸(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 lim_𝜔→0𝐺(𝜔,𝜏)=2sin((1/2)𝑖𝜅𝜏)=𝑖2𝜅cos((1/2)𝑖𝜅𝜏)+𝜅sin((1/2)𝑖𝜅𝜏)𝑖𝜅sin12𝑖𝜅𝜏𝑒^{(1/2)𝜅𝜏}=1−𝑒^−𝜅𝜏lim𝜅_𝜔→0,𝐻(𝜔,𝜏)=1.(3.36) Furthermore, since 𝑘(𝜔)=𝜉²𝜔²1−𝜌²+𝜔𝜉²−2𝜌𝜉𝜅−𝜅²,(3.37) it follows that 𝑘(𝜔) has two real roots given by 𝜔_1/2=−(𝜉−2𝜌𝜅)±(𝜉−2𝜌𝜅)²+4𝜅²1−𝜌²2𝜉1−𝜌²,(3.38) where 𝜌≠±1 and where only the positive root 𝜔₁ is of relevance. For 𝜌=±1 we have a single root 𝜔₁=𝜅²𝜉²∓2𝜉𝜅.(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 {𝜔∈𝐶∣0<Re(𝜔)<∞,Im(𝜔)=0}, that is, the first real root of 𝑓(𝜔) except 𝜔₁, where 𝑓(𝜔) is defined by 𝑓(𝜔)=𝑘(𝜔)cos12𝑘(𝜔)𝜏+(𝜔𝜌𝜉+𝜅)sin21𝑘(𝜔)𝜏.(3.40) If 𝑓(𝜔) has no roots or no other roots except 𝜔₁ in {𝜔∈𝐶∣0<Re(𝜔)<∞,Im(𝜔)=0}, then we set 𝑐^∗=∞. By definition it follows that 𝜔₁≤𝑐^∗, with the special cases lim_𝜏→0𝑐^∗=∞ and lim_𝜏→∞𝑐^∗=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 𝑃^𝐸(𝑆,𝑉,𝜏)=𝑋𝑒^−𝑟𝜏𝑃₁−𝑆𝑒^−𝑞𝜏𝑃₂,(4.1) with 𝑆=𝑆(𝑡) being the current stock price, 𝑃₁=12𝜋𝑖^{𝑐+𝑖∞𝑐−𝑖∞}𝑋𝑆𝑒𝑒^{−𝑟𝜏−𝑞𝜏}^𝜔1𝜔𝐻(𝜔,𝜏)𝑒^{𝐺(𝜔,𝜏)𝑎0𝑉}𝑃𝑑𝜔,₂=12𝜋𝑖^{𝑐+𝑖∞𝑐−𝑖∞}𝑋𝑆𝑒𝑒^{−𝑟𝜏−𝑞𝜏}^𝜔+11𝜔+1𝐻(𝜔,𝜏)𝑒^{𝐺(𝜔,𝜏)𝑎0𝑉}𝑑𝜔,(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 𝑃^𝐸(𝑆,𝑉,𝜏)=𝑋𝑒^−𝑟𝜏Π₁−𝑆𝑒^−𝑞𝜏Π₂,(4.3) with Π₁=12−𝜋1^∞0Re𝑒^{−𝑖𝜔ln𝑋}𝜑(𝜔)Π𝑖𝜔𝑑𝜔,₂=12−𝜋1^∞0Re𝑒^{−𝑖𝜔ln𝑋}𝜑(𝜔−𝑖)𝑖𝜔𝜑(−𝑖)𝑑𝜔,(4.4) where the function 𝜑(𝜔) is the log-characteristic function of the stock at maturity 𝑆(𝑇): 𝜑(𝜔)=𝐸𝑒^{𝑖𝜔ln𝑆(𝑇)}.(4.5)

Remark 4.3. By the fundamental concept of a risk-neutral valuation, we have 𝑃^𝐸(𝑆,𝑉,𝜏)==𝑋𝐸^𝑄𝑡𝑒𝑒^−𝑟𝜏_−𝑟𝜏(𝑋−𝑆(𝑇))⋅1_{{𝑆(𝑇)<𝑋}}𝐸^𝑄𝑡1_{{𝑆(𝑇)<𝑋}}−𝑆𝑒^−𝑞𝜏𝐸^𝑄∗_𝑡1_{{𝑆(𝑇)<𝑋}},(4.6) with 𝐸^⋅𝑡 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^∗𝑑𝑄=𝑆𝑆(𝑇𝑒)𝑒^{−𝑟𝜏−𝑞𝜏}.(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)𝑒^{𝐺(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, 𝑃^𝐸𝑆(𝑆,𝑉,𝜏)=−12𝜋𝑖^{𝑐+𝑖∞𝑐−𝑖∞}𝑋𝑆^𝜔+11𝜔+1𝑒^{(𝑞𝜔−𝑟(𝜔+1))𝜏}𝑃𝐼(𝜔,𝜏)𝑑𝜔,(4.9)_𝐸𝑆𝑆(𝑆,𝑉,𝜏)=12𝜋𝑖^{𝑐+𝑖∞𝑐−𝑖∞}1𝑆𝑋𝑆^𝜔+1𝑃𝑒^{(𝑞𝜔−𝑟(𝜔+1))𝜏}𝐼(𝜔,𝜏)𝑑𝜔,(4.10)_𝐸𝑉(𝑆,𝑉,𝜏)=12𝜋𝑖^{𝑐+𝑖∞𝑐−𝑖∞}𝑋2𝑋𝑆^𝜔𝑒^{(𝑞𝜔−𝑟(𝜔+1))𝜏}𝐺(𝜔,𝜏)𝐼(𝜔,𝜏)𝑑𝜔.(4.11) Recall that the rho of a put option is the partial derivative of 𝑃^𝐸 with respect to the interest rate and equals 𝑃^𝐸𝑟(𝑆,𝑉,𝜏)=−𝑋𝜏2𝜋𝑖^{𝑐+𝑖∞𝑐−𝑖∞}𝑋𝑆^𝜔1𝜔𝑒^{(𝑞𝜔−𝑟(𝜔+1))𝜏}𝐼(𝜔,𝜏)𝑑𝜔.(4.12) Finally, the theta of the put, that is, the partial derivative of 𝑃^𝐸 with respect to 𝜏, is determined as 𝑃^𝐸𝜏(𝑆,𝑉,𝜏)=12𝜋𝑖^{𝑐+𝑖∞𝑐−𝑖∞}𝑋𝑆^𝜔𝑋𝜔(𝜔+1)𝑒^{(𝑞𝜔−𝑟(𝜔+1))𝜏}𝐼(𝜔,𝜏)𝐽(𝜔,𝜏)𝑑𝜔,(4.13) with 𝐽(𝜔,𝜏)=𝑞𝜔−𝑟(𝜔+1)+12𝜔(𝜔+1)𝜅𝜃𝐺(𝜔,𝜏)+𝑉𝐺^𝜏(𝜔,𝜏),(4.14) where 𝐺_𝜏(𝜔,𝜏)=1−(𝜔𝜌𝜉+𝜅)²𝜉²𝜔(𝜔+1)cos²(1/2)𝑘𝜏+tan1⁻¹(−(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 𝑀𝐶^𝐸(𝑆,𝑉,𝑡),𝜔=𝐶^𝐸(𝜔,𝑉,𝑡)=^∞0𝐶^𝐸(𝑆,𝑉,𝑡)𝑆^−(𝜔+1)𝑑𝑆,(4.16) where 1<Re(𝜔)<∞. Conversely, the inverse of this modified Mellin transform is given by 𝐶^𝐸(𝑆,𝑉,𝑡)=12𝜋𝑖^{𝑐+𝑖∞𝑐−𝑖∞}𝐶^𝐸(𝜔,𝑉,𝑡)𝑆^𝜔𝑑𝜔,(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 𝐶^𝐸(𝑆,𝑉,𝜏)=𝑆𝑒^−𝑞𝜏𝑃^∗2−𝑋𝑒^−𝑟𝜏𝑃^∗1,(4.18) where 𝑃^∗2=12𝜋𝑖^{𝑐+𝑖∞𝑐−𝑖∞}𝑋𝑆𝑒𝑒^{−𝑞𝜏−𝑟𝜏}^𝜔−11𝜔−1𝐻^∗𝑃(𝜔,𝜏)𝑒^{𝐺∗(𝜔,𝜏)𝑎∗0𝑉}𝑑𝜔,_∗1=12𝜋𝑖^{𝑐+𝑖∞𝑐−𝑖∞}𝑋𝑆𝑒𝑒^{−𝑞𝜏−𝑟𝜏}^𝜔1𝜔𝐻^∗(𝜔,𝜏)𝑒^{𝐺∗(𝜔,𝜏)𝑎∗0𝑉}𝑑𝜔,(4.19) with 𝐻^∗𝐺(𝜔_∗,𝜏)=𝑒^{(𝜅𝜃/𝜉2)[−(𝜔𝜌𝜉−𝜅)𝜏+2ln(𝑘∗/𝑘∗cos((1/2)𝑘∗𝜏)−(𝜔𝜌𝜉−𝜅)sin((1/2)𝑘∗𝜏))]},(𝜔,𝜏)=2sin(1/2)𝑘^∗𝜏𝑘_∗𝑘^∗cos(1/2)𝑘^∗𝜏−(𝜔𝜌𝜉−𝜅)sin(1/2)𝑘^∗𝜏,=𝑘^∗(𝜔)=𝜉²𝜔(𝜔−1)−(𝜔𝜌𝜉−𝜅)²,(4.20) and 𝑎^∗0=(1/2)𝜔(𝜔−1). 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 10⁻⁶ to 10⁻⁸ for puts and 10⁻⁵ to 10⁻⁸ 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.

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 10⁻⁵ to 10⁻⁶. 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.

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.

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.