In finance, pricing an option is the main issue of managing a trade. However, once an option is settled, its price does not remain constant. Instead, it follows a dynamic path during its survival time. Therefore, the market participants should protect themselves against the unexpected price changes by managing the variations in the option price.

The price risk and its management is always inseparably associated with the Greeks, which are derivatives of an option price with respect to its certain underlying parameters. The information that the Greeks contain is used to measure the unexpected option price changes based on specific risk factors. From this point of view, the Greeks are commonly used to construct a replicating portfolio to protect the main portfolio against some of the possible changes related to the certain risk factors. Thus, the computation of Greeks is more important than obtaining the price of an option [13], and it becomes a fundamental research area in mathematical finance. Here, it is worth to emphasize that the change in the underlying asset price is a very significant risk factor that affects the option price virtually. Hence, among other Greeks, Delta, which measures the change in the option price for a unit change in the price of underlying asset price, is an essential indicator to determine the balance of underlying asset and options which is a hedging ratio.

Over the past two decades, there has been an increasing attention focused on the accurate pricing of hybrid products that are based on a combination of underlying assets from different classes [14]. In light of the recent developments in the hybrid products, it has become difficult to ignore the computation of Greeks from which the underlying asset evolves. Thus, the study considers an extension of stochastic volatility models, namely Hybrid Stochastic Volatility (HSV) models. Within these models, the interest rate evolves from a stochastic process rather than from a constant or a deterministic function to have a more flexible multi-factor stochastic volatility (SV) model. However, the computation of Greeks under these assumptions becomes complicated. It is because these suggested models do not have an explicit distribution or because in most cases the option payoffs are not differentiable. Therefore, the likelihood method and the pathwise method are not appropriate ones in computation of Greeks for this kind of underlying assets. On the other hand, one may use the finite difference method since it relies on the approximation of a derivative as the change in a dependent variable over a small interval of the independent variable, and it can be written using a small set of difference operators. However, it is computationally expensive and problematic for discontinuous option payoff functions. In this respect, the computation of Greeks in hybrid products is not straightforward as in the Black–Scholes model (BSM).

After the pioneering studies [11, 12], the integration by parts formula in the context of the Malliavin calculus has come to be considered as one of the main tools in the computation of Greeks. Since then, a considerable amount of literature has been published in this research area including some numerical applications. In most of these studies, the market dynamics are assumed to follow the BSM assumptions. However, recently, there has been an increasing interest in the computation of Greeks under the assumptions of SV models [3, 9, 15] and jump-diffusion type models [2, 5, 6].

The theory of the Malliavin calculus is attractive among both theoreticians and practitioners for three main reasons. First, it allows researchers to derive explicit weights to design an efficient MC algorithm. Second, it requires neither any differentiability condition on payoff functions (unlike the pathwise method) nor probability density functions of underlying assets (unlike the likelihood method). Third, it is more flexible than other methods in the sense that different perturbations of the Brownian motion yield different weights, which leads to an a priori interest in picking weights with small variances. Hence, it may be used for all types of option, whether they have a continuous or discontinuous payoff function.

The driving goal of this study is to introduce an expression for the computation of Greeks via Malliavin Calculus under the HSV model dynamics, and then, to use an MC algorithm for the expected values and their differentials that correspond to the Greeks. The study shows their accuracy by comparing the findings with the findings of finite difference method in all variations. The novel part of this study is to provide applicable formulas using the technical and theoretical results in the context of the Malliavin calculus. To derive general formulas for the Greeks, first, an HSV model is defined in such a way that it allows researchers to obtain generalized Greeks for all types of hybrid models using fundamental tools in the Malliavin calculus, namely integration by parts and the Bismut–Elworthy–Li formulas on the Gaussian space. The main results reveal that the Malliavin calculus gives the Greeks as a product of an option payoff function and an independent weight function called the Malliavin weight. Moreover, the numerical illustrations reveal that the computation cost for the Malliavin calculus method is less than the computation cost for the finite difference method in all variations.

This article consists of 4 sections. In Section 2, it introduces a general formula for the HSV models and the perturbed price processes. Then, it presents the main results which extend the existing theoretical Greeks formulas. In Section 3, the formulas are derived for the Heston stochastic volatility model with a stochastic interest rate, namely, the Vasicek model. Also in this section, numerical illustrations are presented for Delta, Rho, and Vega of a European call option to verify the efficiency and to compare findings with the outcomes of the finite difference method in all variations. Finally, Section 4 is the conclusion.

Consider a fixed filtered probability space (Ω,F,Ft∈[0,T],Q) which is rich enough to accommodate a Brownian motion of dimension three for the computation convenience. Note that in this representation Ft is a filtration generated by three independent standard Brownian motions Wti for i=1,2,3 and Q is the risk-neutral probability measure.

Throughout the study, it is assumed that the underlying asset price evolves from the SDE system (1) dSt=rtStdt+Stσ(Vt)dZt1, (2) dVt=u(Vt)dt+v(Vt)dZt2, (3) drt=f(rt)dt+g(rt)dZt3, where (Zt)t∈[0,T]i ’s are correlated Brownian motions with correlation coefficients ρij∈(−1,1) for i,j=1,2,3 . These correlated Brownian motions may also be represented by a combination of three independent Brownian motions (Wt)t∈[0,T]i such as dZt1=dWt1,dZt2=ρ12dWt1+μ1dWt2,dZt3=ρ13dWt1+μ2dWt2+μ3dWt3, where the parameters are μ1=1−ρ122,μ2=ρ23−ρ12ρ13μ1,μ3=1−ρ122−ρ132−ρ232+2ρ13ρ12ρ23μ1. Throughout the study, it is assumed that the correlation coefficients ρij are chosen in such a way that μ3 is a real number. The solutions St , Vt , and rt represent an underlying asset, volatility and interest rate processes with initial values S0,V0 , and r0 , respectively. Here, it is assumed that σ,u,v,f , and g are continuously differentiable functions with bounded derivatives of order at least two. Moreover, σ, v, and g are assumed to be adapted and not equal to zero everywhere in the domain.

Now, suppose the SDEs given in equations (1), (2), and (3) can be merged into a three-dimensional SDE system and it is represented as (4) dXt=β(Xt)dt+a(Xt)dWt,X0=x, where, Xt=StVtrt,β(Xt)=rtStu(Vt)f(rt), and a(Xt)=Stσ(Vt)00ρ12v(Vt)μ1v(Vt)0ρ13g(rt)μ2g(rt)μ3g(rt),x=S0V0r0,Wt=Wt1Wt2Wt3. Notice that Xt is a Markov process and (Wt)t∈[0,T] is a three-dimensional standard Brownian motion.

In this representation, it is convenient to assume that β and a both are at least twice continuously differentiable functions with bounded derivatives and adapted for the sake of computations. Moreover, to ensure the existence of a unique strong solution to equation (4), it is assumed that both β and a satisfy the Lipschitz and polynomial growth conditions. Under these assumptions, the study also guarantees that Xt is a Markov process, the trajectories of this solution are almost surely continuously differentiable for all t up to explosion time [17].

Furthermore, in order to have an appropriate solution, it should also be guaranteed that the diffusion matrix a satisfies the uniform ellipticity condition [15], (5) (a(x)ζ)⊤(a(x)ζ)≥ϵ‖ζ‖2,for anyx,ζ∈R3.

Here, it is worth to emphasize that if the conditions on μi,j and the functions σ,v,g are satisfied, the diffusion matrix a satisfies the uniform ellipticity condition. This implies that a is a positive definite matrix, and its eigenvalues are greater than a small positive integer ϵ∈R . Hence, a is an invertible matrix, and its inverse is bounded. This condition also implies that for any bounded function γ:[0,T]×R3↦R3 , a−1γ is bounded, and further, (a−1γ)(Xt) lies in the Hilbert space L2([0,T]×Ω)  [11].

The first variation process of (Xt)t∈[0,T] , which is the derivative of (Xt)t∈[0,T] with respect to its initial value x, plays an important role in the computation of Greeks. Hence, the study reminds the definition of the first variation process.

Now, it is the time to introduce the following lemma for the integrity of the study.

One can calculate the components of the first variation process as in the following proposition.

For the computation purposes, it is convenient to consider a continuous time trading economy with a finite time horizon t∈[0,T] . Suppose that the uncertainty in this economy is idealized by a fixed filtered probability space (Ω,F,Ft∈[0,T],Q) and the information evolves according to the filtration (Ft)t∈[0,T] generated by (Wt)t∈[0,T] . Furthermore, consider an option in this market, which has a square integrable payoff function denoted by Φ=Φ(Xt1,…,Xtn) , where Φ is a payoff function that is continuously differentiable with bounded derivatives. In other words, the option is replicable since E[Φ(Xt1,…,Xtn)2]<∞  [8].

Now, following studies [5, 9], the option price, p, at time t=0 with a maturity T<∞ is traditionally calculated as the expected value of the discounted payoff at maturity conditionally to the present information, which is described by the σ-algebra F0 . Now, it is possible to define the price process in the following definition.

Note that the objective of the computation of Greeks is to differentiate the price, p, of an option with respect to model parameters.

In the computation of Greeks via the Malliavin calculus, a weight function, which is independent of the option payoff function, is obtained. To obtain a valid computation result, one has to guarantee that the Malliavin weights do not degenerate with probability one. Hence, [11] demonstrates the set of square integrable functions Γn={α∈L2([0,T]);∫0tiα(t)dt=1,∀i=1,…,n}, in R , to avoid the degeneracy. Here, ti ’s are the given time values in [0,T] .

Now, it is the time to remind readers about the celebrated Bismut–Elworthy–Li formula introduced in [4, 7]. Proposition 2.7.

Suppose that the functions β and a in equation (4) are continuously differentiable with bounded derivatives, and the diffusion matrix a satisfies the uniform ellipticity condition (5). Moreover, the option payoff function, Φ, is square integrable, and continuously differentiable with bounded derivatives. Now, consider the price process given in Definition 2.5 with a maturity T<∞ , then (12) (∇p(x))⊤=E[e−∫0TrtdtΦ(Xt1,…,Xtn)π], where ∇, and π denote the gradient and Malliavin weight, respectively. Here, (13) π=∫0Tα(t)(a−1(Xt)Yt)⊤dWt, where α∈Γn , and Yt is the first variation process.