Can Negative Interest Rates Really Affect Option Pricing? Empirical Evidence from an Explicitly Solvable Stochastic Volatility Model

The profound financial crisis generated by the collapse of Lehman Brothers and the European sovereign debt crisis in 2011 have caused negative values of government bond yields both in the U.S.A. and in the EURO area. This paper investigates whether the use of models which allow for negative interest rate can improve option pricing and implied volatility forecasting. This is done with special attention to Foreign eXchange and index options. To this end, we carried out an empirical analysis of the prices of call and put options on the U.S. S&P 500 index as well as on the Eurodollar futures using a generalization of the Heston model in the stochastic interest rate framework. Specifically, the dynamics of the option's underlying asset is described by two factors: a stochastic variance and a stochastic interest rate. The volatility is not allowed to be negative while the interest rate is. Explicit formulas for the transition probability density function and moments are derived. These formulas are used to efficiently estimate the model parameters. Three empirical analyses are illustrated. The first two show that the use of models which allow for negative interest rates can efficiently reproduce implied volatility and forecast option prices (i.e. S&P index and foreign exchange options). The last one studies how the U.S. three month government bond yield affects the U.S. S&P 500 index.


Introduction
This paper investigates the effect of negative short-term government bond yields on the pricing and forecasting of index options as well as foreign exchange (FX) options. Pricing and forecasting derivative products are one of the most challenging topics due to their wide use in hedging risk.
The well-known stochastic models of Black Scholes 1973, Hull andWhite 1988, Stein andStein 1991, and Heston 1993 assume a constant (usually positive) risk-free interest rate. However, empirical evidence has shown that time series of the U.S. and EURO short-term government bond yields have fluctuated significantly in the last fifteen years and negative values have also been seen see, for example, Jackson 2015). These facts highlight the need for two modifications.
First, option pricing models with stochastic interest rate should be considered. Second, the stochastic interest rate process should allow for negative values.
We address these two points by investigating the potential of a modified version of the Heston model illustrated in Grzelak

Literature review
Recent studies confirm the suitability of models defined by a system of stochastic differential equations (i.e., SDE models) to describe state variables such as stock, volatility and interest rate, given empirical evidence that the asset volatility and interest rate are not constant over time.
The Heston model is one of the most celebrated models. In fact, thanks to the use of a stochastic volatility, it provides accurate prices of European vanilla call and put options as well as more complex path-dependent options when it is assumed that the constant interest rate is realistic and that the volatility does not undergo abrupt oscillations. Indeed, local volatility models (see, for example, Dupire 1994) are also able to perfectly replicate all vanilla option prices, but this implies unrealistic spot volatility dynamics as observed by Hagan et al. 2002 andAyache et al. 2007. Several extensions of the Heston and Dupire model, also including jumps, can be found.
We mention Homescu 2014 and Itkin 2016 for an extensive overview of the literature. These extensions do not always allow for analytical treatment, but they satisfactorily replicate market data by using specific numerical approaches.
Here, we focus on models that, roughly speaking, are obtained by adding further stochastic Due to the combined action of these factors, these models are able to describe the option prices and the implied volatility efficiently, even when the underlying asset is highly volatile. However, the choice of the risk-free interest rate as one of the factors driving the underlying asset is crucial when pricing options with very long maturity (i.e., maturity longer than five years such as those of some insurance contracts; see Recchioni and Sun 2016) or options in a time period where highly fluctuating short-term yields are experienced (see, for example, Jackson 2015).
Indeed, several hybrid SDE models (i.e. models obtained by correlating stochastic differential equations from different classes) have been introduced since 2000 in order to extend well-known stochastic volatility models such as the Heston model. We cite some models similar to the model proposed here. The model by Andreasen 2007 generalizes the model introduced by Zhu 2000. This generalization is obtained by using the Heston stochastic volatility model and an indirect correlation between the equity and the interest rate process. The model by Ahlip 2008 describes the spot FX rate with stochastic volatility and stochastic domestic and foreign rates.
More recently, we have the so-called Schöbel-Zhu-Hull-White hybrid model illustrated by Grzelak, Oosterlee and Weeren 2012, who follow the approach proposed by Duffie, Pan and Singleton 2000 for the analytical treatment of this affine model. However, as highlighted in Grzelak and Oosterlee 2011, the model allows for negative volatility. In order to overcome this problem, they propose the use of a Cox-Ingersoll-Ross (CIR) process to describe the variance process. Finally, it is worth noting that local volatility models have also been extended to deal with stochastic interest rates (see, for example, Deelstra and Rayee 2012 and Benhamou, Gobet and Miri 2012). The hybrid Heston model proposed here can be interpreted as a hybrid SDE model. Specifically, it is a modified version of the Heston-Hull-White (HHW) model illustrated by Grzelak and Oosterlee 2011. However, we continue to refer to this modified version as an HHW model.

Description of the results
The contribution of this paper is twofold. First, we modify the multi-factor Heston model proposed in Recchioni and Sun 2016 (HCIR hereinafter) in that we use the Vasicek model (see Vasicek 1977, Mamon 2004) for the stochastic interest rate instead of the CIR model. This choice preserves the affine structure while allowing for negative values of the interest rate. As a particular case of the Hull and White model, the Vasicek model allows us to refer to it as an HHW model. This HHW model preserves the main features of the model illustrated by Grzelak and Oosterlee 2011 and is analytically tractable, an aspect that allows us to derive the probability density function of the stochastic process by solving the backward Kolmogorov equation through the same approach introduced in Recchioni and Sun 2016. The approach is based on a suitable parametrization of the probability density function, which allows us to derive elementary formulas for the moments of the asset price variable and to express the prices of European call and put options as one-dimensional integrals through an efficient approximation of the discount factor.
The performance of this approximation is measured using the explicit formula (30) for the zero coupon bond in the Vasicek model.
Second, we calibrate the model in order to measure its performance in interpreting and forecasting European call and put option prices (see the empirical analysis on the U.S. S&P 500 index options in Subsection 4.1 and on FX options on the EUR/USD exchange rate in Subsection 4.2).
The calibration procedure in the empirical analysis illustrated in Subsections 4.1 and 4.2 is based on the solution to a nonlinear constrained optimization problem, whose objective function measures the relative squared difference between the observed and theoretical implied volatilities associated with call and put options. The results show that the hybrid model is able to reproduce European call and put in-sample and out-of-sample option prices with only one set of model parameters. Satisfactory out-of-sample approximations for the implied volatility for both call and put options are obtained.
Finally, we investigate the model's ability to capture the relationship between the S&P 500 index and the U.S. three-month government bond yield and to forecast the index via the expected value conditioned on the last observation of the index itself and the bond yield. The results of this empirical analysis are preliminary and deserve further investigation.

Outline of the paper
The paper is organized as follows. In Section 2, we illustrate the hybrid Heston-Hull-White model and some relevant formulas. In Section 3, we propose formulas to approximate the European vanilla call and put option prices as one-dimensional integrals of explicitly known functions based on a simple approach to approximate the stochastic discount factor. In Section 4, we illustrate three empirical analyses. Specifically, in the first two cases we use call and put option prices as data. Consequently, we estimate the model parameters by solving constrained optimization problems whose objective functions involve the implied volatility associated with these option prices. The first empirical analysis uses the U.S. S&P 500 index from April 2, 2012 to July  In fact, the Vasicek model allows for negative values of the interest rate, which has, until now, been considered a weakness of this model. However, the negative values of the U.S. short-term government bond yields (see Figure 3 (b)) and those of German government bond yields observed in recent years give this model new appeal. We therefore consider the following generalization of the Heston model (i.e. the HHW model): where ∆ and Ω are non-negative constants and χ, v * , γ, λ, θ, η are positive constants, while W v t , W r t , Z v t , Z r t are standard Wiener processes. All correlations among the Wiener processes are zero except for the following: where the quantities ρ v , ρ r ∈ (−1, 1) are constant correlation coefficients. Furthermore, we assume that the Feller condition, 2χ v * /γ 2 > 1, holds.
The HHW model (1) Vasicek dynamics in this model and CIR in the other.
The system of equations (1)-(3) is equipped with the following initial conditions: where S * 0 and v * 0 , r * 0 are random variables concentrated in a point with probability one and for simplicity, these random variables are denoted by the points where they are concentrated. As specified in Heston 1991, the quantity χ is the speed of mean reversion, v * is the long-term mean, and γ is the so-called volatility of volatility (vol of vol for short).
It is worth noting that the variance v t remains positive for any t > 0 with probability one given that 2χ v * /γ 2 > 1 and v 0 = v * 0 > 0 (see Heston, 1991). As a consequence, the equity price S t remains positive for any t > 0 with probability one given that S * 0 > 0 with probability one. The HHW model allows for negative values of the interest rate, r t , and this is a positive feature since negative short-term bond yields have recently been experienced in the U.S. and European financial markets.
In order to deduce analytical formulas for this model, we use Ito's lemma and Eq. (1) to derive the stochastic differential equation satisfied by the log-price process, x t = ln (S t /S 0 ), t > 0: whereψ in (7) is given byψ Eq. (6) implies that the process (x t , v t , r t ) satisfies the following initial conditions: where x * 0 is a random variable that we assume to be concentrated in a point with probability one. We now illustrate the main formulas used in the empirical analysis; their derivation can be found in the Appendix. It is worth noting that the analytical formulas deduced for the HHW model can also provide analytical formulas for the Heston model with a suitable choice of the HHW parameters. Furthermore, a comparison of the Heston and HHW formulas allows us to identify which HHW model parameters play a crucial role in interpreting option prices.

Transition probability density function in the HHW and Heston models
Let us start by denoting the set of real numbers by R, the set of positive real numbers by R + , and the n-dimensional Euclidean vector space by R n . We now illustrate the formula for the transition probability density function (pdf for short) associated with the stochastic differential system (7), (8), (9) (i.e., the HHW model) and the pdf of the Heston model. The pdf of the HHW model has two main advantages.
The first is that it evaluates the pdf as a one-dimensional integral of a smooth, explicitly known integrand given by the product of two functions. One function depends only on the parameters Θ v = (γ, χ, v * , ρ v , ∆) ∈ R 5 of the volatility process and the other depends only on the parameters Θ r = (η, λ, θ, ρ r , Ω) ∈ R 5 of the interest rate process.
The second advantage is that the specific parametrization used to derive the formula (i.e., the parameter q appearing in the pdf given in Eq. (12)) allows some integrals appearing in the option price formulas and in the formulas for the moments of the price variable to be computed explicitly.
Specifically, let p f denote the pdf associated with the stochastic differential system (7), (8), (9). The following formula then holds (see the Appendix for the proof): where ı is the imaginary unit and L v,q and L r,q are explicitly known functions given in Eqs.
(69) and (71). The specific form of the transition probability density function in Eq. (12) is a consequence of the correlation structure (4)- (5). In Recchioni and Sun 2016, a formula for the pdf of the Heston model with a stochastic interest rate described by the CIR model, i.e., the HCIR model, was deduced. Specifically, the function L v,q appearing in (12) is the same one used in the HCIR model since it depends only on the volatility process, whose dynamics is the same for the HCIR and HHW models. In contrast, the function L r,q in the HHW model differs from the one in the HCIR model since it only depends on the specific stochastic model used to describe the interest rate process.
Formula (12) shows how the introduction of a stochastic interest rate as a factor driving the asset price dynamics can affect the corresponding option prices. That is, we use formula (12) to compare the probability density functions of the Heston and HHW models. In fact, the pdf, p H f , of the Heston model is given by where L v,q is the same function appearing in (12) and whose expression is given in Appendix Eq.
(69), while the function Q H r,q is given by A comparison of formulas (12) and (13) shows that the main difference between the Heston and the HHW models lies in the two terms L r,q and e Q H r,q , respectively, and that L r,q reduces to e Q H r,q in the limit Ω → 0 + , η → 0 + and λ → 0 + . We investigate this difference further in Section 2.2.
We conclude this section by noting that the initial stochastic volatility is not observable in the financial market and is usually estimated by an appropriate calibration (see, for example, Bühler, 2002). The choice of the initial stochastic interest rate has no well-established procedure. In fact, the best proxy for the short-term rate under the risk-neutral measure has not been identified yet.
Given this, we estimate the initial stochastic interest rate in the empirical analyses illustrated in Sections 4.1 and 4.2 but we do not estimate the initial stochastic rate in Section 4.3 where we calibrate the model under the physical measure by using only the index values and the short-term government bond yields as data.

Moments of the price variable in the HHW model
and where Q v,m and Q r,m are given by formulas (74) and (75) evaluated at k = 0 (see the Appendix).
Specifically, we have: where s m,v,b , ζ m,v , µ m,v , s m,v,g are obtained by evaluating Eqs. (53)-(56) (see the Appendix) at k = 0, while Q r,m (t , r 0 , 0; Θ r ) has the following elementary expression: When negative values of the interest rate (i.e., r 0 < 0) are allowed, the moments of the Heston model (see Eq. (16)) approach zero as t goes to +∞. In other words, negative values of the interest rate, r 0 , imply, on average, a fall in the price. However, this finding is not coherent with the empirical evidence (see Figure 2). The HHW model behaves in a very different way. In fact, negative values of r 0 do not necessarily imply decreasing price moments, especially for long time horizons. This is due to the fact that the long-term behavior of M m (t ) is independent of r 0 while it does depend on the following quantity: The sign of this quantity depends on the correlation coefficient ρ r . That is, nonnegative values of ρ r guarantee positive values of price moments for sufficiently large values of t , while negative values of ρ r with small values of θ may anticipate a decrease in the price.
A nice feature of this model is that negative values of r 0 play a crucial role only in short and medium time horizons. This behavior is plausible since the market expectation on long-term interest rates is expected to be positive in a "calm" financial climate.
Let us further analyze this issue by focusing on the first order moment of the price in the two models. Using formula (16) and setting m = 1 in the Heston model, we obtain the following formula for the expected value of the asset price conditioned on the observations at t = 0 of price, variance and interest rate: Similarly, using formula (15), we obtain the following formula for the expected value in the HHW model: Some comments on formulas (20) and (21) so formula (21) can be rewritten as follows: Formula (23) shows how the direct correlation, ρ r , between asset price and interest rate affects the expected value. In the empirical analysis of Section 4.3, we analyze the potential of formula (23) in forecasting the U.S. S&P 500 index.
3 Integral formulas to price European vanilla call and put options in the HHW model In this section, we deduce option pricing formulas for European vanilla call and put options with strike price, E, and maturity, T . We start with and where ( · ) + = max{ · , 0}, and the expectation is taken under the risk-neutral measure Q (see, for example, Duffie, 2001, Schoutens, 2003 in the model (1), (2), (3). In the two empirical analyses involving option prices, we calibrate the model only with option price data. In this way we avoid the introduction of risk premia because only the risk-neutral measure and not the physical measure is involved.
The option prices in formulas (24) and (25) require the evaluation of a three-dimensional integral whose accurate numerical evaluation is very time consuming. However, features in the HHW model allow the computation of these three-dimensional integrals to be reduced to the numerical evaluation of one-dimensional integrals.
A challenging task in evaluating European call and put option prices is the approximation of the stochastic integral defining the discount factor (see Eqs. (24), (25)). We approximate the integral using a quadrature rule where the weights are chosen considering the features of the stochastic interest rate model used. This approach is inspired by the work of Choi and Wirjanto 2007 and suggests the following approximation: where ω is chosen in order to obtain a satisfactory approximation of the bond price. Specifically, where Ψ 1,λ is given in (19). This guarantees that ω is positive and less than or equal to one (i.e., 0 ≤ ω ≤ 1) for any T > 0. Using formulas (26) and (27), we obtain the following approximation of the bond price: with It is worth noting that formula (29) reduces (30) when the term tanh(T λ/2) . In addition, the relative error of the bond approximation is given by the following simple formula:  Figure 1, guarantee that the bond approximation has at least four correct significant digits.   (24), (25), (26), and (76) in the Appendix, we obtain the following approximation, V A , for European vanilla call and put options:

From Equations
where functions Q v,q and Q r,q are those given in (74), (75) (see the Appendix). Formula (31) implies that the call option price differs from the put option price only in the choice of q, that is, This is a consequence of the parametrization used for the pdf (see Eq. (12)), which allows the integral in the x variable to be computed explicitly for suitable values of q.
Let us illustrate this for the call options. Using (26) and (76) in (24) we obtain: where G q is given by: The integral in Eq. (33) is convergent when 1 − q < 0 and is independent of r . Thus, the integrals over x and r appearing in formula (32) are independent of each other and Eqs. (33) and (73) are their explicit formulas. Formula (32) shows that for the put option price, the integral appearing in (33) must be replaced with the integral is convergent when q < −1 and has the same explicit formula as in (33).
Taking the limit Ω → 0 + , λ → 0 + , η → 0 + in Eq (31), we derive the following exact formula for the price of the European call and put options in the Heston model (see also Recchioni and Sun 2016): In Section 4, we use formula (34) with q > 1 (q < −1) for vanilla European call (put) options in order to compare the performance of the Heston model and the HHW model proposed here in interpreting real data. It is worth noting that the integrands appearing in formulas (31) and (34) are smooth functions whose numerical integration can be carried out efficiently using a simple quadrature rule. The smoothness of the integrands is due to the specific approach used to derive them.

Empirical Analysis
In this section, we propose three empirical analyses. The first two only involve option prices so we can work under the risk-neutral measure associated with the HHW model. In the third analysis, we investigate the dependence of the asset on the short-term rate and so here we work under the physical measure. Specifically, the first two empirical analyses involve the U.S. S&P 500 index as well as the Eurodollar futures prices and the corresponding European option prices. The U.S.
three-month government bond index has been used as a proxy for the initial stochastic interest rate in both analyses.
We estimate the model parameters by solving an appropriate nonlinear constrained least squares problem. In detail, let R 12 denote the 12-dimensional Euclidean real space. We define the set of the constraints, V, as follows: It is worth noting that the initial values v 0 , r 0 , of the variance and interest rate process are parameters to be estimated through the calibration procedure. This is motivated by the fact that We now formulate the calibration problem necessary to carry out the empirical analysis illustrated in Sections 4.1 and 4.2. To this end we introduce some notation. Where n T and n D are two nonnegative integers, In the empirical analyses, we choose T 1 = T 2 = . . . = T n D , while E 1 < E 2 < . . . < E n D . We denote the observed and theoretical implied volatilities associated with the call and put option prices . . , n T , i = 1, 2, . . . , n D , respectively. We recall that the implied volatility Σ o,C (S, r F , T, E) is defined as the solution to the following equation: where C o is the observed call option price and C BS is the Black Scholes price at time t of the European call option with strike price, E, and maturity, T , T > t (see Black and Scholes 1973, Eq. (13)) while r F denotes the risk-free interest rate. We choose r F to be the U.S. three-month government bond yield. We estimate the model parameters using the implied volatilities. This approach works satisfactorily when very deep out-of-the-money option are excluded.
Following this approach, we estimate the model parameters by solving the following nonlinear constrained optimization problem where the objective function, F n T , is In order to show empirical evidence that the use of stochastic interest rates is crucial, we compare the proposed model with the Heston model (Heston 1993). The latter is calibrated by solving problem (37) in a feasible set that does not contain the parameters of the stochastic interest rate model except for the parameter r 0 , which allows for negative values.
We use formulas (31) and (34)           We analyze the option data using a rolling window of six consecutive trading days (i.e., n T = 6). As a consequence, in each window, sixty option values are used to calibrate the twelve parameters of the model (i.e., n P = 5 put option prices and n C = 5 call option prices for n T = 6 days). This window is moved by one day along the historical series covering the period April 2 to July 2, 2012. The calibration problems (37) solved by moving the window are 66 − n T . In this way, we obtain a historical series of daily observations for each parameter. The values of the parameters obtained in the j-th window are representative of the last day of the j-th window.
We highlight the fact that when the values of the estimated parameters are constant in time, the model reproduces the asset price dynamics in the period analyzed by using only one set of model parameters.     Yu 2016, which are 7.8% and 9.5% for call and put options respectively.
In conclusion, the empirical analysis shows that the hybrid model interprets the real data considered in the period April 2 to July 27, 2012 satisfactorily using only one set of model parameters.

FX options on EUR/USD
In  We define the moneyness of an option on a given day as the ratio between the strike price of the option and the futures price on the EUR/USD exchange rate of that day.
As in the previous subsection, we consider a rolling window containing the prices of one trading day (i.e., n T = 1). This window is moved by one day along the historical series. The time window covers the period September 27 to December 17, 2010 and 61 − n T calibration problems are solved (see Eq. (37)). As a consequence, in each time window, thirty-six option values are used to estimate the twelve model parameters (i.e., n P = 18 put option prices and n C = 18 call option prices). In this way we generate a historical series of daily observations for each parameter.
The date associated with the value of the parameters obtained in the j-th window is the date corresponding to the observed call and put option prices of the j-th window.   maturity. We observe that the quality of the out-of-sample call option prices slightly outperforms that of the put option prices while the in-sample put option prices are more accurate than the in-sample call option prices. In fact, the sample means of the relative errors of the in-sample call and put options for the HW-Heston model are 3.21% and 1.49% respectively, while the sample mean of the relative errors of the out-of-sample call and put options are 5.01% and 5.76%.
The left panels of Figure 15   In this section we use the HHW model to explore the dependence, if any, of the S&P 500 index on the U.S. three-month government bond yield. In this analysis, the measure associated with the HHW model is the physical measure and the stochastic interest rate in Eq. (1) must be interpreted as the drift of the S&P 500 index, which is assumed to be related to the U.S. threemonth government bond yield.
In this experiment, we calibrate the model parameters by solving where V is given in (35), n T is the number of daily observations used, and F n T is the objective function (see Varin et al. 2011) Here, the function D v,q is given by (76), where q is equal to zero, and x j , r j are the observations of the S&P 500 index and the U.S. three-month government bond yield respectively at t = t j , j = 1, 2, . . . , n T . Note that in this experiment, r 0 is not estimated but is equal to r t 1 , that is, the observed value of the three-month government bond on the first day of the time window (i.e., t = t 1 ). We associate the last day of the time window, t = t n T , with each parameter value obtained by solving problem (39). In this way, we have a time series for each model parameter except for the initial stochastic interest rate, r 0 , which is chosen to be the value of the three-month   when the short-term government bond yields are around zero, the behavior of the market index is strongly affected by the bond yield, while this effect decreases when the yields are around 0.1%.
These are only preliminary results which deserve further investigation. However, in order to assess the performance of this estimation, we use the estimated values of the parameters obtained at time t to forecast the U.S. S&P 500 index at time t + ∆ t, where ∆ t =1 month.
The forecast values of the S&P 500 index are obtained using the expected value in formula (21) adapted to the current situation, that is: The parameter values used here are those estimated in the time window [t − ∆t, t]. Figure 18 displays the true values of the S&P 500 index (solid line and squares) and the one-month ahead forecasts (dotted line and stars). The true data are monthly observations of the S&P 500 index downloadable from the website http://data.okfn.org/data/core/s-and-p-500#data. The average relative error is 0.024 and the median is 0.017. As mentioned above, the results obtained are very preliminary, but they are encouraging.

Conclusions
This paper proposes a multi-factor Heston model (HHW model) that allows for stochastic volatility and a negative interest rate while preserving its analytical tractability. We derive integral representation formulas for the transition probability density function and for option pricing that involve one-dimensional integrals and elementary integrand functions as well as explicit formulas for the moments of the price variable. One of the main features of the model is that it allows for negative interest rates, and the first-order moment of the price variable depends on the correlation parameter between interest rate and asset price. The three empirical analyses illustrated in Section 4 assess the performance of the HHW in interpreting and forecasting option prices (see To this end, we recall that the function p f (x, v, r, t, x , v , r , t ), t < t , as a function of the variables (x, v, r, t) (i.e., the past variables, since t < t ) satisfies the following backward Kolmogorov equation: whereψ is given in (10), with final condition and appropriate boundary conditions. Since neither the coefficients (42) nor the final condition (43) depend on translation in the log-price variable, we can proceed as in Recchioni and Sun 2016, assuming the following representation formula for p f : where τ = t − t. Substituting formula (44) into equation (42), functions A, B v , and B r must satisfy the following ordinary differential equations: with initial conditions A(0, k, l, ξ) = 0, B v (0, k, l) = ı l, B r (0, k, ξ) = ı ξ, where ϕ q is the following quantity: The solution to problem (46), (48), which is given in Recchioni and Sun 2016, is where so we have where and s q,v,d is given by The solution of Eq. (47) is given by: where Ψ 1,λ is given by Eq. (19). Finally, integrating Eq. (45), we obtain: where Q 0 , Q 1 and Q 2 are given by Q 1 (τ, k) = −a q,r Ψ 1,λ (τ ) + η 2 ık − q λ (Ψ 1,λ (τ ) − Ψ 2,λ (τ )), while functions Ψ j,λ , j = 1, 2 are given in (19) and a q,r is given by a q,r = λθ − (ık − q)Ωρ r η .
We conclude this Appendix by deducing explicit formulas for the moments of the price variable; specifically, we use formulas (76) where Q v,m , Q r,m are the functions given in Eqs. (74) and (75) with q = m and k = 0. We note that formula (84) is an elementary formula that does not involve integrals and it may be useful for investigating the correlation between interest rates and market indices/assets. Finally, by taking the limit Ω → 0 + , λ → 0 + , and η → 0 + in formula (84), we obtain the formula for the m th moment, M H m in the Heston framework.