As mentioned in the introduction, we test five models for stochastic volatility: Heston, 3/2 volatility, GARCH, Bates, and GARCH-Jump. The first two rely on a mean reversion drift term, with the parameters kappa and theta being the rate of mean reversion and the long-run value to which the volatility process reverts, respectively. Also, the first two have the parameter sigma to control the standard deviation of the random fluctuations. 3/2 volatility can be thought of as a higher-order extension of Heston volatility. Bates and GARCH-Jump processes introduce jumps to replicate the behavior of sudden moves of volatility observed in markets with Poisson-process-driven, normally distributed jumps. We ensure that the simulation functions do not produce negative volatility for all of the SDEs of stochastic volatility models.

$\text{d}{v}_t=\kappa \left(\theta -v_t\right)\text{d}t+\sigma \sqrt{v_t}\text{d}{W}_t$(1)

Note that setting

$2\kappa \theta >{\sigma }^2$(2)

in the Heston model ensures the process is strictly positive (Feller condition [21] ).

The time-homogeneous GARCH process satisfies the following linear SDE according to Li et al. as described in their 2018 paper [22] . Hence, this model is very similar to the Heston model discussed above, but the diffusion term is multiplied by $\sqrt{v_t}$ to raise the power on the volatility component from 1/2 to 1...

$\text{d}{v}_t=\kappa \left(\theta -v_t\right)\text{d}t+\sigma v_t\text{d}{W}_t$(3)

The GARCH-Jump model extends the standard GARCH (Generalized Autoregressive Conditional Heteroskedasticity) framework by incorporating jumps, providing a more comprehensive tool to capture the aforementioned volatility dynamics observed in financial markets. This model is particularly useful in capturing the sudden large movements or jumps in asset prices that cannot be explained by continuous processes alone. The GARCH-Jump model was developed and extensively analyzed by Duan et al. (2004), who highlighted its efficacy in better fitting historical time series data and explaining the observed volatility smile in option prices. Their research demonstrated that incorporating jumps into the GARCH framework significantly improves the model’s performance in capturing the empirical features of asset returns and volatility [23] . We implement a variant of GARCH-Jump SV by adding a jump term to our GARCH-inspired model discussed above.

$\text{d}{v}_t=\kappa \left(\theta -v_t\right)\text{d}t+\sigma v_t\text{d}{W}_t+\zeta \text{d}{J}_t$(4)

In our code implementation, the jumps are modeled as a compound Poisson process where the jump sizes are normally distributed. Specifically, $\text{d}{J}_t$ is constructed by first generating a number of jumps using a Poisson distribution with intensity parameter $\lambda$. For each jump, the size is drawn from a normal distribution with mean ${\mu }_J$ and standard deviation ${\sigma }_J$. The total jump impact $\text{d}{J}_t$ is then the sum of all individual jump sizes occurring within a given time interval $\text{d}t$. Hence, the jumps are modeled as follows:

$\text{d}{J}_t=\underset{i=1}{\overset{N_t}{{\displaystyle \sum }}}{Y}_i~\text{Poisson}\left(\lambda \cdot \text{d}t\right)\times \mathcal{N}\left({\mu }_J,{\sigma }_J^2\right)$(5)

Here, $N\left(t\right)$ is a Poisson process with intensity $\lambda$, and $Y_i$ are i.i.d. normal random variables with mean ${\mu }_J$ and variance ${\sigma }_J^2$. Duan et al. (2004) employed the GARCH-Jump model to explore option pricing under conditions where both price and volatility exhibit jump-diffusion behavior. They found that the GARCH-Jump model provides a robust framework for understanding and predicting market behaviors characterized by sudden and significant changes. The model’s ability to capture jumps makes it particularly valuable for pricing derivatives and managing financial risk in environments subject to abrupt market movements [23] .

The Bates volatility model extends Heston volatility with jumps [5] . Here, the parameters kappa, theta, and sigma have the same use as for the Heston volatility model.

$\text{d}{v}_t=\kappa \left(\theta -v_t\right)\text{d}t+\sigma \sqrt{v_t}\text{d}{W}_t+\zeta \text{d}{J}_t$(6)

where $\text{d}{J}_t$ represents the jumps with normally distributed jump sizes. $m{u}_J$ and $sigm{a}_J$ are two more parameters in the simulation of this model, which control the mean size and standard deviation of the jumps, respectively, as discussed above. Here $\zeta$ is a multiplier that controls the magnitude of the effect of the jumps as above.

The 3/2 stochastic volatility model is an extension of the constant elasticity of variance (CEV) model and was developed by Carr and Sun (2007) to better capture the dynamics of financial markets. In this model, the volatility of the underlying asset is driven by a process that is proportional to the power 3/2 of the volatility itself. The stochastic differential equation (SDE) governing the 3/2 model is given by:

$\text{d}{v}_t=\left(\omega -\theta v_t\right)v_t\text{d}t+\sigma v_t^{3/2}\text{d}{W}_t,$(7)

In this context:

The 3/2 model is particularly useful in capturing the empirical features of volatility observed in financial markets, such as the leverage effect and the fact that volatility tends to spike during market downturns [24] . Carr and Sun (2007) developed this model to provide a more accurate framework for pricing options and other derivative securities. According to Carr and Sun, the 3/2 model has several desirable properties. The process remains non-negative and exhibits mean-reverting behavior, where the speed of mean reversion is proportional to the level of the process. The 3/2 model also yields closed-form solutions for the joint Fourier-Laplace transform of returns and their quadratic variation, which is useful for efficiently pricing and hedging derivatives [24] .

Correlations can be influenced by factors such as industrial production, T-bill rates, and unanticipated inflation, often acting as a business cycle indicator. Even after adjusting for business cycle effects, correlation risk persists (Driessen, Maenhout, and Vilkov, 2006 [25] ). Although the correlation coefficient between two assets is not directly tradable, it remains crucial to devise hedging strategies for correlation risk. Developing robust frameworks for constructing portfolios to hedge against correlation risk can ensure more secure risk management practices [15] . For all of the stochastic correlation models, the SDEs have mean reversion or bounds or are clipped to ensure that correlations remain within [−1, 1]. Four models for stochastic correlation of increasing complexity are implemented in this study. The simplest is a stochastic correlation SDE inspired by the modeling of processes in studies of genetics, which has a diffusion term, making sure it stays within [−1, 1]. The next simplest is the Jacobi correlation used by Ma, which gives the user the option to keep the correlation process within bounds h and f by altering this diffusion term. The second most complicated model is the mean-reverting extension of the first two simpler models. The most complex model that is tested is the Weibull distribution stochastic correlation model.

Wright-Fisher diffusions are used in biology and biochemistry to model gene frequencies and other natural/bodily processes. As explained in their paper, A mean-reverting SDE on correlation matrices, Ahdida et al. focus on stochastic differential equations ‘valued on correlation matrices’ [26] and develop a mean-reverting extension of the Wright-Fisher SDE to model correlations in finance. In our paper, we test both the original diffusion on [−1, 1] and the mean-reverting extension developed by Ahdida et al. as alternatives for modeling correlation. We do not implement the original version of the diffusion on [0, 1] since we want the process to mimic the variability across the range [−1, 1] when modeling financial correlations. The classic WF diffusion on [−1,1] has the SDE:

$\text{d}{\rho }_t=\kappa \left(\bar{\rho }-{\rho }_t\right)\text{d}t+\sigma \sqrt{1-{\rho }_t^2}\text{d}{W}_t$(8)

Here, ${\rho }_t$ represents the instantaneous correlation at time $t$. $\kappa$ is the mean reversion rate, determining how quickly the process reverts to the long-term mean correlation $\bar{\rho }$. $\bar{\rho }$ is the long-term mean correlation towards which ${\rho }_t$ reverts. $\sigma$ represents the volatility of the correlation process, indicating the extent of random fluctuations around the mean, and $\text{d}{W}_t$ is a standard Wiener process, as usual. Ahdida et al. highlight that the term $\sqrt{1-{\rho }_t^2}$ ensures that the correlation ${\rho }_t$ remains within the interval $\left[-1,1\right]$. This term becomes zero when ${\rho }_t$ approaches the boundaries, preventing it from exceeding these limits. This bounded characteristic makes this version of the WF diffusion particularly suitable for modeling correlations in finance, where it is critical to maintain realistic correlation values [26] .

The Jacobi process is used to model stochastic correlation and is described by the following stochastic differential equation (SDE):

$\text{d}{\rho }_t=\kappa \left(\bar{\rho }-{\rho }_t\right)\text{d}t+\sigma \sqrt{\left(h-{\rho }_t\right)\left({\rho }_t-f\right)}\text{d}{W}_t$(9)

where ${\rho }_t$ represents the correlation at time $t$, $\kappa$ is the mean reversion rate, $\bar{\rho }$ is the long-term mean correlation, and $h$ and $f$ are the upper and lower bounds of the correlation, respectively. The term $\text{d}{W}_t$ denotes a standard Wiener process as usual. The parameter $\kappa$ determines the speed at which the correlation reverts to its long-term mean $\bar{\rho }$. A higher value of $\kappa$ implies a faster reversion. The parameters $h$ and $f$ set the natural boundaries for the correlation, ensuring that it remains within a realistic range. The volatility parameter $\sigma$ controls the amplitude of fluctuations around the mean [15] .

The Jacobi process is particularly useful for modeling correlation because it can capture both the mean-reverting nature and the bounded behavior of correlation coefficients. The square root term $\sqrt{\left(h-{\rho }_t\right)\left({\rho }_t-f\right)}$ ensures that the correlation stays within the interval $\left(f,h\right)$. Ma (2009) introduced the use of the Jacobi process in the context of pricing foreign equity options with stochastic correlation. This model allows for a more accurate reflection of market dynamics compared to constant correlation models. The Jacobi process can be seen as an extension of other mean-reverting processes, such as the Ornstein-Uhlenbeck process, but with the added complexity of bounded behavior. This makes it particularly suited for financial applications since correlations tend to naturally exhibit such characteristics [15] .

This is an extension of the WF diffusion on [−1, 1] developed by Ahdida et al. (Ahdida 2013), which includes an adaptive mean-reversion component. The mean-reverting SDE is given by:

$\text{d}{\rho }_t=\left(\kappa \left(\bar{\rho }-{\rho }_t\right)-{\sigma }^2{\rho }_t\right)\text{d}t+\sigma \sqrt{1-{\rho }_t^2}\text{d}{W}_t$(10)

Here ${\rho }_t$, $\kappa$, $\bar{\rho }$, $\sigma$, are all the same as for the WF correlation model discussed above. Also, the term $\sqrt{1-{\rho }_t^2}$ has the same use of ensuring that the correlation ${\rho }_t$ remains within the interval $\left[-1,1\right]$. By adjusting the mean-reverting drift term $\kappa \left(\bar{\rho }-{\rho }_t\right)-{\sigma }^2{\rho }_t$, Ahdida et al extend the process to capture the empirically observed tendency of correlations to revert to a long-term mean with the inclusion of the ${\sigma }^2{\rho }_t$ to ensure that the reversion speed adjusts dynamically based on the current correlation level [26] .

The Weibull model is characterized by the following stochastic differential equation:

$\text{d}{\rho }_t=-\alpha \left({\rho }_t-{\mu }_W\right)\text{d}t+b_1{b}_2\text{d}{W}_t$(11)

where ${\mu }_W$ is a mean term derived from Weibull distribution parameters. This model was originally proposed by Miñano et al. (2013) for wind speed modeling, demonstrating its efficacy in generating wind speed trajectories with desired statistical properties [27] . In this context, $b_1$ and $b_2$ are diffusion terms uniquely defined to ensure the model captures the desired statistical properties. Specifically, $b_1$ and $b_2$ are given by:

$b_1\left({\rho }_t\right)=\frac{2\alpha }{p_W\left({\rho }_t\right)},$(12)

$b_2\left({\rho }_t\right)=\lambda \text{Γ}\left(1+\frac{1}{k},{\left(\frac{{\rho }_t}{\lambda }\right)}^k\right)-{\mu }_W{\text{e}}^{-{\left({\rho }_t/\lambda \right)}^k},$(13)

where $p_W\left({\rho }_t\right)$ is the probability density function (PDF) of the Weibull distribution, $\lambda$ is the scale parameter, $k$ is the shape parameter, and $\text{Γ}$ represents the Gamma function. The inclusion of $b_1$ and $b_2$ ensures the model’s diffusion term is appropriately scaled to reflect the nature of the underlying process. This formulation makes the Weibull model particularly suitable for phenomena where the Weibull distribution provides a good fit, such as wind speed data and potentially skewed financial data. Specifically, this model was designed to simulate wind speeds that follow a Weibull distribution and exhibit exponential autocorrelation, as discussed in the work by Miñano et al. [27] . Their objective was to accurately replicate the statistical properties of wind speed for applications in power systems, highlighting the model’s ability to generate realistic wind speed trajectories for various simulations and analyses. To ensure the mathematical validity of the model, it is essential that ${\rho }_t$ paths stay non-negative and $k>0$, as the Weibull distribution is defined only for non-negative values and requires a positive shape parameter. These conditions guarantee that the diffusion terms $b_1$ and $b_2$ are well-defined, ensuring the stochastic process remains within the domain where the Weibull distribution accurately describes the behavior. Therefore, this SC model works in situations when we want to model positive correlations.

It has been shown that the Johnson SB distribution is a best-fit distribution for equity and default probability correlation distributions [28] . Also, Gunter Meissner et al. conducted a comprehensive analysis of correlation behavior based on daily closing prices of 30 stocks within the Dow Jones index, spanning the period from January 1972 to October 2012 [29] . Their findings indicate that correlation levels are at their lowest during periods of robust economic growth, wherein equity prices are predominantly influenced by idiosyncratic factors rather than broader macroeconomic conditions. Conversely, during recessions, correlation levels typically rise as macroeconomic factors overshadow idiosyncratic influences, leading to simultaneous downturns across multiple stocks. Furthermore, the volatility of correlations is observed to be lowest during economic expansions and higher during normal periods and recessions [29] . The study also highlighted that the Johnson SB distribution, characterized by its shape, location, and scale parameters, provided the most accurate fit for modeling these correlations. This distribution’s flexibility effectively captures the intricate properties and variations of the correlation data observed under different economic conditions [29] . Something we want to explore further in future work is to apply the approach used for the Weibull correlation model to capture statistical properties of the distribution in the SDE to develop a stochastic correlation model that captures desired properties of the Johnson SB distribution and incorporates mean reversion.

The domestic currency for the investor in these Quanto options is US dollars. In our paper, one of the underlying assets is denominated in the home currency (US dollar). For all choices of correlation and volatility, the underlying assets are modeled with the same SDEs throughout. We test three SDEs for the exchange rates (scenarios 1 - 3). The simplest model is for both exchange rates to be GBM [30] with different parameters based on observed starting values and summary statistics (as outlined in section 1). We also test a model with mean-reversion incorporated into the SDEs used for both exchange rates. It has been shown in many studies that foreign exchange rates tend to exhibit strong mean reversion [31] [32] . In the third model, both exchange rates follow an exponential Lévy process, which incorporates jumps into the GBM model of scenario 1 [33] [34] . The foreign risk-free interest rate is $r_{f_1}$. This is the risk-free rate of return in the first foreign currency, which could be the interest rate of a government bond or another risk-free security in the foreign market of GBP. $r_{f_2}$ is the risk-free rate of return in the second foreign currency, and $r_d$ is the domestic interest rate. In the equations below, $\theta$ represents the rate of mean reversion, $\mu$ is the mean level to which the process reverts, ${\sigma }_{FX}$ is the volatility, and $\text{d}{W}_{FX}\left(t\right)$ is the increment of the Wiener process.

The GBM is characterized by its exponential growth with constant drift and volatility [30] .

$\text{d}FX\left(t\right)=FX\left(t\right)\left(\left(r_f-r_d-0.5{\sigma }_{FX}^2\right)\text{d}t+{\sigma }_{FX}\text{d}{W}_{FX}\left(t\right)\right)$(14)

In this scenario, we model the exchange rates as an OU process, which introduces mean reversion into the drift term.

$\text{d}FX\left(t\right)=\theta \left(\mu -FX\left(t\right)\right)\text{d}t+FX\left(t\right)\left(\left(r_f-r_d-0.5{\sigma }_{FX}^2\right)\text{d}t+{\sigma }_{FX}\text{d}{W}_{FX}\left(t\right)\right)$(15)

In this scenario, the exchange rates follow an exponential Lévy process, which incorporates jumps into the SDEs, accounting for sudden and significant changes in the exchange rates [33] [35] . The key parameters in this model include ${\sigma }_{FX}$ for volatility, ${\lambda }_L$ for the jump intensity, ${\mu }_L$ for the mean jump size, and ${\sigma }_L$ for the jump size volatility.

$\text{d}FX\left(t\right)=FX\left(t\right)\left(\left(r_f-r_d-0.5{\sigma }_{FX}^2\right)\text{d}t+{\sigma }_{FX}\text{d}{W}_{FX}\left(t\right)\right)+\text{d}{J}_t$(16)

where $\text{d}{J}_t$ is the jump component, modeled as before (Sections 2.1.3 and 2.1.4) for the GARCH-Jump and Bates SV. These jumps account for sudden, significant changes in the exchange rates, making the exponential Lévy process a suitable candidate for modeling FX rates with jumps [35] [36] .

Underlying Asset Prices

$\text{d}{S}_{USD}\left(t\right)=S_{USD}\left(t\right)\left(r_d\text{d}t+\sqrt{v_{USD}\left(t\right)}\text{d}{W}_{USD}\left(t\right)\right)$(17)

$\text{d}{S}_{GBP}\left(t\right)=S_{GBP}\left(t\right)\left(r_{f_1}\text{d}t+\sqrt{v_{GBP}\left(t\right)}\text{d}{W}_{GBP}\left(t\right)\right)$(18)

SV Choice (same process for both underlying assets)

$\text{d}{v}_{\text{USD}t}=m^{\text{v}}\left(v_{\text{USD}t},t\right)\text{d}t+s^{\text{v}}\left(v_{\text{USD}t},t\right)\text{d}{W}^{\text{v\_USD}}{}_t+{\omega }^{\text{v}}\left(v_{\text{USD}t},t\right)\text{d}{J}^{\text{v\_USD}}{}_t$(19)

$\text{d}{v}_{\text{GBP}t}=m^{\text{v}}\left(v_{\text{GBP}t},t\right)\text{d}t+s^{\text{v}}\left(v_{\text{GBP}t},t\right)\text{d}{W}^{\text{v\_GBP}}{}_t+{\omega }^{\text{v}}\left(v_{\text{GBP}t},t\right)\text{d}{J}^{\text{v\_GBP}}{}_t$(20)

SC Choice

$\text{d}{\rho }_t=m^{\rho }\left({\rho }_t,t\right)\text{d}t+s^{\rho }\left({\rho }_t,t\right)\text{d}{W}^{\rho }{}_t+{\omega }^{\rho }\left({\rho }_t,t\right)\text{d}{J}^{\rho }{}_t$(21)

SER Choice

$\begin{array}{c}\text{d}F{X}_{\text{GBP}}\left(t\right)=m^{\text{f}}\left(F{X}_{\text{GBP}}\left(t\right),t\right)\text{d}t+s^{\text{f}}\left(F{X}_{\text{GBP}}\left(t\right),t\right)\text{d}{W}^{\text{f}}{}_t\\ +{\omega }^{\text{f}}\left(F{X}_{\text{GBP}}\left(t\right),t\right)\text{d}{J}^{\text{f}}{}_t\end{array}$(22)

$\begin{array}{l}\text{d}{W}_{\text{USD}}\left(t\right)=\rho \left(t\right)\text{d}{W}_{\text{GBP}}\left(t\right)+\sqrt{1-\rho {\left(t\right)}^2}\text{d}Z\left(t\right)\\ \text{d}{W}_{FX}\left(t\right)=\text{d}{Z}_{FX}\left(t\right)\end{array}$(23)

Here, $\text{d}Z\left(t\right)$ and $\text{d}{Z}_{FX}\left(t\right)$ are independent standard Brownian motions.

$\text{Payoff}=\text{max}\left(S_{\text{USD}}\left(T\right)-K_1,S_{\text{GBP}}\left(T\right)\cdot FX\left(T\right)-K_2,0\right)$(24)

Here, $\text{d}{J}^{\text{v}}{}_t$ denotes the jump component of volatility, $\text{d}{J}^{\rho }{}_t$ denotes the jump component of correlation, and $\text{d}{J}^{\text{f}}{}_t$ denotes the jump component for the FX rate, as defined in Section 2. Our choice of the SV, SC, and SER models determines the m, s, and $\omega$ functions above. For example, GARCH-Jump SV has $m^{\text{v}}\left(v_t,t\right)={\beta }_0+\left({\beta }_1-1\right)v_t$, $s^{\text{v}}\left(v_t,t\right)=0$, and ${\omega }^{\text{v}}\left(v_t,t\right)={\beta }_2{v}_t$. Similarly, we can write the general equations for our model for Case 2 as shown below.

Underlying Asset Prices

$\text{d}{S}_{\text{USD}}\left(t\right)=S_{\text{USD}}\left(t\right)\left(r_d\text{d}t+\sqrt{v_{\text{USD}}\left(t\right)}\text{d}{W}_{\text{USD}}\left(t\right)\right)$(25)

$\text{d}{S}_{\text{GBP}}\left(t\right)=S_{\text{GBP}}\left(t\right)\left(r_{f_1}\text{d}t+\sqrt{v_{\text{GBP}}\left(t\right)}\text{d}{W}_{\text{GBP}}\left(t\right)\right)$(26)

$\text{d}{S}_{\text{EUR}}\left(t\right)=S_{\text{EUR}}\left(t\right)\left(r_{f_2}\text{d}t+\sqrt{v_{\text{EUR}}\left(t\right)}\text{d}{W}_{\text{EUR}}\left(t\right)\right)$(27)

SV Choice (same process for all underlying assets)

$\text{d}{v}_{\text{USD}t}=m^{\text{v}}\left(v_{\text{USD}t},t\right)\text{d}t+s^{\text{v}}\left(v_{\text{USD}t},t\right)\text{d}{W}^{\text{v\_USD}}{}_t+{\omega }^{\text{v}}\left(v_{\text{USD}t},t\right)\text{d}{J}^{\text{v\_USD}}{}_t$(28)

$\text{d}{v}_{\text{GBP}t}=m^{\text{v}}\left(v_{\text{GBP}t},t\right)\text{d}t+s^{\text{v}}\left(v_{\text{GBP}t},t\right)\text{d}{W}^{\text{v\_GBP}}{}_t+\omega v\left(v_{\text{GBP}t},t\right)\text{d}{J}^{\text{v\_GBP}}{}_t$(29)

$\text{d}{v}_{\text{EUR}t}=m^{\text{v}}\left(v_{\text{EUR}t},t\right)\text{d}t+s^{\text{v}}\left(v_{\text{EUR}t},t\right)\text{d}{W}^{\text{v\_EUR}}{}_t+{\omega }^{\text{v}}\left(v_{\text{EUR}t},t\right)\text{d}{J}^{\text{v\_EUR}}{}_t$(30)

SC Choice (between both pairs, USD-GBP and USD-EUR)

$\text{d}{\rho }_t=m^{\rho }\left({\rho }_t,t\right)\text{d}t+s^{\rho }\left({\rho }_t,t\right)\text{d}{W}^{\rho }{}_t+{\omega }^{\rho }\left({\rho }_t,t\right)\text{d}{J}^{\rho }{}_t$(31)

SER Choice (same process for both FX Rates, USD/GBP, & USD/EUR)

$\begin{array}{c}\text{d}F{X}_{\text{GBP}}\left(t\right)=m^{\text{f}}\left(F{X}_{\text{GBP}}\left(t\right),t\right)\text{d}t+s^{\text{f}}\left(F{X}_{\text{GBP}}\left(t\right),t\right)\text{d}{W}^{\text{f1}}{}_t\\ +{\omega }^{\text{f}}\left(F{X}_{\text{GBP}}\left(t\right),t\right)\text{d}{J}^{\text{f1}}{}_t\end{array}$(32)

$\begin{array}{c}\text{d}F{X}_{\text{EUR}}\left(t\right)=m^{\text{f}}\left(F{X}_{\text{EUR}}\left(t\right),t\right)\text{d}t+s^{\text{f}}\left(F{X}_{\text{EUR}}\left(t\right),t\right)\text{d}{W}^{\text{f2}}{}_t\\ +{\omega }^{\text{f}}\left(F{X}_{\text{EUR}}\left(t\right),t\right)\text{d}{J}^{\text{f2}}{}_t\end{array}$(33)

$\begin{array}{l}\text{d}{W}_{\text{USD}}\left(t\right)=\rho \left(t\right)\text{d}{W}_{\text{GBP}}\left(t\right)+\sqrt{1-\rho {\left(t\right)}^2}\text{d}Z\left(t\right)\\ \text{d}{W}_{\text{EUR}}\left(t\right)=\rho \left(t\right)\text{d}{W}_{\text{GBP}}\left(t\right)+\sqrt{1-\rho {\left(t\right)}^2}\text{d}{Z}_{\text{EUR}}\left(t\right)\\ \text{d}{W}_{F{X}_{\text{USD/GBP}}}\left(t\right)=\text{d}{Z}_{F{X}_{\text{USD/GBP}}}\left(t\right)\\ \text{d}{W}_{F{X}_{\text{USD/EUR}}}\left(t\right)=\text{d}{Z}_{F{X}_{\text{USD/EUR}}}\left(t\right)\end{array}$(34)

Here, $\text{d}Z\left(t\right)$, $\text{d}{Z}_{\text{EUR}}\left(t\right)$, $\text{d}{Z}_{F{X}_{\text{USD/GBP}}}\left(t\right)$, and $\text{d}{Z}_{F{X}_{\text{USD/EUR}}}\left(t\right)$ are independent standard Brownian motions.

$\begin{array}{l}\text{Payoff}=\text{max}(S_{\text{USD}}\left(T\right)-K_1,S_{\text{GBP}}\left(T\right)\cdot F{X}_{\text{GBP}}\left(T\right)-K_2,\\ S_{\text{EUR}}\left(T\right)\cdot F{X}_{\text{EUR}}\left(T\right)-K_3,0)\end{array}$(35)

Hence, as highlighted before, we correlate the Brownian motions of the processes of the underlying assets and model this correlation as stochastic. Also, we model the volatility of the underlying as stochastic and independent from the level of the underlying. Moreover, the FX rates are modeled as stochastic and independent from each other as well as from the level of their respective underlying asset. In the simulation, we use different sets of increments of Brownian motion for all the stochastic processes and only introduce a correlation between those of the underlying assets. This correlation, between the Brownian motions of USD and GBP assets as well as between the Brownian motions of the USD and EUR assets, is the same in our experiments. In all of the SC models, we tested $\omega \left({\rho }_t,t\right)=0$ as we do not investigate the presence of jumps in correlation.

${\hat{I}}_{t_{j+1}}={\hat{I}}_{t_j}+a\left({\hat{I}}_{t_j},t_j\right)\text{Δ}t+b\left({\hat{I}}_{t_j},t_j\right)\text{Δ}{W}_j$(37)

$={\hat{I}}_{t_j}+a\left({\hat{I}}_{t_j},t_j\right)\text{Δ}t+b\left({\hat{I}}_{t_j},t_j\right)\sqrt{\text{Δ}t}{Z}_j$(38)

${\hat{I}}_{t_{j+1}}={\hat{I}}_{t_j}+a\left({\hat{I}}_{t_j},t_j\right)\text{Δ}t+b\left({\hat{I}}_{t_j},t_j\right)\sqrt{\text{Δ}t}{Z}_j+c\left({\hat{I}}_{t_j},t_j\right)\text{Δ}{J}_j$(39)

$\begin{array}{l}{\hat{I}}_{t_{j+1}}={\hat{I}}_{t_j}+a\left({\hat{I}}_{t_j},t_j\right)\text{Δ}t+b\left({\hat{I}}_{t_j},t_j\right)\text{Δ}{W}_j\\ +\frac{1}{2}b\left({\hat{I}}_{t_j},t_j\right)b^{\prime }\left({\hat{I}}_{t_j},t_j\right)\left[{\left(\text{Δ}{W}_j\right)}^2-\text{Δ}t\right]\end{array}$(40)

$\begin{array}{l}={\hat{I}}_{t_j}+a\left({\hat{I}}_{t_j},t_j\right)\text{Δ}t+b\left({\hat{I}}_{t_j},t_j\right)\sqrt{\text{Δ}t}{Z}_j\\ +\frac{1}{2}b\left({\hat{I}}_{t_j},t_j\right)b^{\prime }\left({\hat{I}}_{t_j},t_j\right)\text{Δ}t\left(Z_j^2-1\right)\end{array}$(41)

$\begin{array}{c}{\hat{I}}_{t_{j+1}}={\hat{I}}_{t_j}+a\left({\hat{I}}_{t_j},t_j\right)\text{Δ}t+b\left({\hat{I}}_{t_j},t_j\right)\text{Δ}{W}_j\\ +\frac{1}{2}b\left({\hat{I}}_{t_j},t_j\right)b^{\prime }\left({\hat{I}}_{t_j},t_j\right)\text{Δ}t\left(Z_j^2-1\right)+c\left({\hat{I}}_{t_j},t_j\right)\text{Δ}{J}_j\end{array}$(42)

$\begin{array}{l}{\hat{I}}_{t_{j+1}}={\hat{I}}_{t_j}+a\left({\hat{I}}_{t_j},t_j\right)\text{Δ}t+b\left({\hat{I}}_{t_j},t_j\right)\text{Δ}{W}_j\\ +\frac{1}{2\sqrt{\text{Δ}t}}\left[b\left({\tilde{I}}_{t_j},t_j\right)-b\left({\hat{I}}_{t_j},t_j\right)\right]\left[{\left(\text{Δ}{W}_j\right)}^2-\text{Δ}t\right]\end{array}$(43)

$\begin{array}{l}={\hat{I}}_{t_j}+a\left({\hat{I}}_{t_j},t_j\right)\text{Δ}t+b\left({\hat{I}}_{t_j},t_j\right)\sqrt{\text{Δ}t}{Z}_j\\ +\frac{1}{2}\left[b\left({\tilde{I}}_{t_j},t_j\right)-b\left({\hat{I}}_{t_j},t_j\right)\right]\sqrt{\text{Δ}t}\left[{\left(\text{Δ}{Z}_j\right)}^2-1\right]\end{array}$(44)

where

${\tilde{I}}_{t_j}={\hat{I}}_{t_j}+a\left({\hat{I}}_{t_j},t_j\right)\text{Δ}t+b\left({\hat{I}}_{t_j},t_j\right)\sqrt{\text{Δ}t}$(45)

${\tilde{I}}_{t_j}={\hat{I}}_{t_j}+a\left({\hat{I}}_{t_j},t_j\right)\text{Δ}t+b\left({\hat{I}}_{t_j},t_j\right)\sqrt{\text{Δ}t}+c\left({\hat{I}}_{t_j},t_j\right)\text{Δ}{J}_j$(46)

$\begin{array}{c}{\hat{I}}_{t_{j+1}}={\hat{I}}_{t_j}+a\left({\hat{I}}_{t_j},t_j\right)\text{Δ}t+b\left({\hat{I}}_{t_j},t_j\right)\sqrt{\text{Δ}t}{Z}_j\\ +\frac{1}{2}\left[b\left({\tilde{I}}_{t_j},t_j\right)-b\left({\hat{I}}_{t_j},t_j\right)\right]\sqrt{\text{Δ}t}\left[{\left(\text{Δ}{Z}_j\right)}^2-1\right]+c\left({\hat{I}}_{t_j},t_j\right)\text{Δ}{J}_j\end{array}$(47)

In summary, to handle an additional jump term in our SDEs we add the jump component directly to the discretization for the Euler-Maruyama and Milstein schemes, and the Runge-Kutta scheme also requires calculating the intermediate values including the jump component, and then adjusting the final update accordingly. These modifications ensure that each discretization scheme accurately captures the effects of both continuous and jump components in the SDEs.

1) Initialize paths: Set initial values for $S_{\text{GBP}}\left(0\right)$, $S_{\text{USD}}\left(0\right)$, $FX\left(0\right)$, $\rho \left(0\right)$, $v_{\text{GBP}}\left(0\right)$, and $v_{\text{USD}}\left(0\right)$. Set T (time to expiration in years and $n_{steps}$ per path in days is T*252), N (number of paths in simulation), set increment $dt=T/n_{steps}$ for daily time series.

2) Simulate paths over time:

- For each time step, generate Brownian increments and jump instances.

- Update the volatility paths using the specified volatility model and discretization scheme.

- Update the correlation paths using the specified correlation model and discretization scheme.

- Update the FX rate paths using the specified model and discretization scheme.

- Update the underlying asset paths using the simulated volatility and correlation paths according to the Euler-Maruyama discretization scheme.

3) Calculate payoffs: Use the final asset prices and FX rate values to calculate the payoff for each simulated path. For Case 1, the payoff for each path is calculated as:

${\text{Payoff}}_i=\text{max}\left(S_{\text{USD},i}\left(T\right)-K_1,S_{\text{GBP},i}\left(T\right)\cdot F{X}_i\left(T\right)-K_2,0\right)$(48)

For Case 2, the payoff for each path is calculated as:

$\begin{array}{l}{\text{Payoff}}_i=\text{max}(S_{\text{USD},i}\left(T\right)-K_1,S_{\text{GBP},i}\left(T\right)\cdot F{X}_{\text{USD/GBP},i}\left(T\right)-K_2,\\ S_{\text{EUR},i}\left(T\right)\cdot F{X}_{\text{USD/EUR},i}\left(T\right)-K_3,0)\end{array}$(49)

4) Discount and average payoffs: Discount the payoffs to present value and average them to get the option price. The discounted option price is given by:

$\text{Option Price}={\text{e}}^{-r_{d}T}\cdot \frac{1}{N}\underset{i=1}{\overset{N}{{\displaystyle \sum }}}{\text{Payoff}}_i$(50)

where $N$ is the number of simulated paths, $r_d$ is the domestic risk-free rate, and $T$ is the time to maturity.

Variance reduction methods seek to improve the efficiency of an estimator, enabling more accurate results for a given number of simulations, N. Some choices of variance reduction that could be applied to our Monte Carlo simulations include control variates, which leverage the known expectation of auxiliary variables to reduce variance, and randomized Quasi-Monte Carlo (QMC), which replaces purely random sampling with low-discrepancy sequences (e.g. Halton, Sobol, or Fauer) to enhance convergence properties. In this paper, we implement antithetic variates as a form of variance reduction to improve our Monte Carlo simulations. Antithetic variates aim to reduce the variance of the estimator by introducing a negative correlation between paired simulations, thereby improving the convergence rate of the Monte Carlo estimator without increasing the number of simulations. In our context of Monte Carlo simulations involving stochastic differential equations (SDEs), this involves generating antithetic paths by using the negative of the original Brownian motion increments. Then, by averaging the results from the original and antithetic paths, the variance of the estimator is reduced. Specifically, let $X$ be an estimator of the option price based on the original simulations, and $X^{\prime }$ be the estimator based on the antithetic simulations. The combined estimator is then:

$\hat{C}=\frac{1}{2}\left(X+X^{\prime }\right)$(51)

Now, since $X$ and $X^{\prime }$ are negatively correlated, the variance of $\hat{C}$ is reduced compared to the variance of either $X$ or $X^{\prime }$ alone. The variance of our estimator can be expressed as follows:

$\text{Var}\left(\hat{C}\right)=\frac{1}{2}\left(\text{Var}\left(X+X^{\prime }\right)\right)=\frac{1}{4}\left(\text{Var}\left(X\right)+\text{Var}\left(X^{\prime }\right)+2\cdot \text{Cov}\left(X,X^{\prime }\right)\right)$(52)

and our X’s both have equal variance so this expression can be simplified further as follows:

$\begin{array}{c}\text{Var}\left(\hat{C}\right)=\frac{1}{4}\left(\left(2\times 2\right)\text{Var}\left(X\right)+2\cdot \text{Cov}\left(X,X^{\prime }\right)\right)\\ =\text{Var}\left(X\right)+\frac{1}{2}\text{Cov}\left(X,X^{\prime }\right)\end{array}$(53)

which means that the variance of our antithetic estimator is less than that of the original estimator by $\left|\frac{1}{2}\text{Cov}\left(X,X^{\prime }\right)\right|$ if the covariance between the antithetic and original versions of the random variables is negative:

$\text{Var}\left(\hat{C}\right)<\text{Var}\left(X\right)\iff \text{Cov}\left(X,X^{\prime }\right)<0$(54)

Hence, the effectiveness of the antithetic variates method depends on the extent of the negative correlation between the estimators $X$ and $X^{\prime }$. The more negatively correlated they are, the greater the variance reduction. Since $\text{Var}\left(X\right)=\text{Var}\left(X^{\prime }\right)$, we can rewrite this reduction in variance as follows:

$\left|\frac{1}{2}\text{Cov}\left(X,X^{\prime }\right)\right|=\left|\frac{1}{2}\rho {\sigma }_x{\sigma }_{x^{\prime }}\right|=\left|\frac{1}{2}\rho \text{Var}\left(X\right)\right|$(55)

To incorporate antithetic variates into our simulation process, for each path in the simulation, we generate a corresponding antithetic path by using the negative of the samples of a standard normal random variable in the Brownian increments used in the original path. This approach is applied to all the Brownian motions involved in the simulation. Specifically, if we let $\text{Δ}{W}_j$ be the Brownian increments generated for the original path at time step $t_j$. Then, by writing ${Z^{\prime }}_j=-Z_j$, we get that the antithetic increments $\text{Δ}{W^{\prime }}_j$ are given by:

$\text{Δ}{W^{\prime }}_j=\sqrt{\text{Δ}t}{Z^{\prime }}_j=-\sqrt{\text{Δ}t}{Z}_j=-\text{Δ}{W}_j$

We modify the steps of our simulation process outlined in Section 4.7 as described below to incorporate antithetic variates. Step 1 is the same as in the original version, but Steps 2 and 3 change slightly as we have to include the generation of the antithetic versions for all Brownian increments and updating all paths (including the antithetic versions), respectively. Step 4 also mainly stays the same but we need to compute the payoffs for both the antithetic paths and the normal paths. Then, the key adjustment is in changing step 5 as follows:

New Step 5 discount and average combined payoffs: Compute the combined payoff for each pair of original and antithetic sets of paths as:

${\overline{\text{Payoff}}}_i=\frac{1}{2}\left({\text{Payoff}}_i+{\text{Payof <msup> f <mo> ′ </mo> </msup>}}_i\right)$(56)

As before, discount the combined payoffs and average them to obtain the option price estimate:

$\text{Option Price}={\text{e}}^{-r_{d}T}\cdot \frac{1}{N}\underset{i=1}{\overset{N}{{\displaystyle \sum }}}{\overline{\text{Payoff}}}_i$(57)

In both Case 1 and Case 2 of our options, since our payoff functions are monotonic in the asset prices (& FX rates) and all the SDEs we use are linear in the Brownian increments, the negative correlation introduced at the level of the Brownian increments propagates through to the terminal asset prices, and the option payoffs. Also, the symmetry of the Gaussian distribution guarantees that when we negate a (zero-mean) Normal random variable it is still distributed according to the same distribution. Because of this, for any of our SDEs, the solution of the version modified with antithetic Brownian increments has the same distribution as the original SDE’s solution. Indeed, with this, it can be shown that the antithetic and original option price estimates are identically distributed and have a negative covariance, as required for variance reduction via antithetic variates.

For the single FX rate case, where the option payoff is given by:

$\text{Payoff}=\text{max}\left(S_{\text{USD}}\left(T\right)-K_1,S_{\text{GBP}}\left(T\right)\cdot FX\left(T\right)-K_2,0\right)$(60)

The Cora and Gora metrics can be derived as follows: Cora measures how sensitive the option’s value is to changes in the correlation between the GBP asset price and the USD asset price. Mathematically, this sensitivity is expressed as:

$\text{Cora}=\frac{\partial C}{\partial {\rho }_{\text{GBP},\text{USD}}}$(61)

Gora measures the rate of change of the sensitivity (Cora) with respect to the correlation between the GBP and USD asset prices. It captures the curvature of the option price with respect to correlation changes, indicating how Cora itself responds as the correlation changes:

$\text{Gora}=\frac{{\partial }^{2}C}{\partial {\rho }_{\text{GBP},\text{USD}}^2}$(62)

Using the chain rule, the first partial derivative of $C$ with respect to ${\rho }_{\text{GBP},\text{USD}}$ is:

$\frac{\partial C}{\partial {\rho }_{\text{GBP},\text{USD}}}=\frac{\partial C}{\partial S_{\text{GBP}}}\cdot \frac{\partial S_{\text{GBP}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial S_{\text{USD}}}\cdot \frac{\partial S_{\text{USD}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial FX}\cdot \frac{\partial FX}{\partial {\rho }_{\text{GBP},\text{USD}}}$(63)

then, since the partial derivative of the FX rate process with respect to the correlation is 0, we get:

$\text{Cora}=\frac{\partial C}{\partial S_{\text{GBP}}}\cdot \frac{\partial S_{\text{GBP}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial S_{\text{USD}}}\cdot \frac{\partial S_{\text{USD}}}{\partial {\rho }_{\text{GBP},\text{USD}}}$(64)

For the second partial derivative, we get:

$\begin{array}{c}\frac{{\partial }^{2}C}{\partial {\rho }_{\text{GBP},\text{USD}}^2}=\frac{\partial }{\partial {\rho }_{\text{GBP},\text{USD}}}\left(\frac{\partial C}{\partial S_{\text{GBP}}}\cdot \frac{\partial S_{\text{GBP}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial S_{\text{USD}}}\cdot \frac{\partial S_{\text{USD}}}{\partial {\rho }_{\text{GBP},\text{USD}}}\right)\\ =\frac{\partial }{\partial {\rho }_{\text{GBP},\text{USD}}}\left(\frac{\partial C}{\partial S_{\text{GBP}}}\right)\cdot \frac{\partial S_{\text{GBP}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial S_{\text{GBP}}}\cdot \frac{\partial }{\partial {\rho }_{\text{GBP},\text{USD}}}\left(\frac{\partial S_{\text{GBP}}}{\partial {\rho }_{\text{GBP},\text{USD}}}\right)\\ +\frac{\partial }{\partial {\rho }_{\text{GBP},\text{USD}}}\left(\frac{\partial C}{\partial S_{\text{USD}}}\right)\cdot \frac{\partial S_{\text{USD}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial S_{\text{USD}}}\cdot \frac{\partial }{\partial {\rho }_{\text{GBP},\text{USD}}}\left(\frac{\partial S_{\text{USD}}}{\partial {\rho }_{\text{GBP},\text{USD}}}\right)\end{array}$(65)

Simplifying fully, we get:

$\begin{array}{c}\text{Gora}=\frac{{\partial }^{2}C}{\partial S_{\text{GBP}}\partial {\rho }_{\text{GBP},\text{USD}}}\cdot \frac{\partial S_{\text{GBP}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial S_{\text{GBP}}}\cdot \frac{{\partial }^2{S}_{\text{GBP}}}{\partial {\rho }_{\text{GBP},\text{USD}}^2}\\ +\frac{{\partial }^{2}C}{\partial S_{\text{USD}}\partial {\rho }_{\text{GBP},\text{USD}}}\cdot \frac{\partial S_{\text{USD}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial S_{\text{USD}}}\cdot \frac{{\partial }^2{S}_{\text{USD}}}{\partial {\rho }_{\text{GBP},\text{USD}}^2}\end{array}$(66)

For the two FX rates case, the option payoff is given by:

$\begin{array}{l}\text{Payoff}=\text{max}(S_{\text{USD}}\left(T\right)-K_1,S_{\text{GBP}}\left(T\right)\cdot F{X}_{\text{USD/GBP}}\left(T\right)-K_2,\\ S_{\text{EUR}}\left(T\right)\cdot F{X}_{\text{USD/EUR}}\left(T\right)-K_3,0)\end{array}$(67)

Here, Cora and Gora metrics need to account for multiple correlations: between GBP and USD asset prices and between EUR and USD asset prices. The Cora for each of these correlations would be defined as:

${\text{Cora}}_{\text{GBP},\text{USD}}=\frac{\partial C}{\partial {\rho }_{\text{GBP},\text{USD}}}$(68)

${\text{Cora}}_{\text{EUR},\text{USD}}=\frac{\partial C}{\partial {\rho }_{\text{EUR},\text{USD}}}$(69)

Similarly, the Gora metrics would measure the second-order sensitivity for each of these correlations, indicating how each Cora changes with respect to changes in the corresponding correlations:

${\text{Gora}}_{\text{GBP},\text{USD}}=\frac{{\partial }^{2}C}{\partial {\rho }_{\text{GBP},\text{USD}}^2}$(70)

${\text{Gora}}_{\text{EUR},\text{USD}}=\frac{{\partial }^{2}C}{\partial {\rho }_{\text{EUR},\text{USD}}^2}$(71)

Using the chain rule, the first partial derivative of $C$ with respect to ${\rho }_{GBP,USD}$ is:

$\begin{array}{c}\frac{\partial C}{\partial {\rho }_{\text{GBP},\text{USD}}}=\frac{\partial C}{\partial S_{\text{GBP}}}\cdot \frac{\partial S_{\text{GBP}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial S_{USD}}\cdot \frac{\partial S_{\text{USD}}}{\partial {\rho }_{\text{GBP},\text{USD}}}\\ +\frac{\partial C}{\partial S_{\text{EUR}}}\cdot \frac{\partial S_{\text{EUR}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial F{X}_{\text{USD/GBP}}}\cdot \frac{\partial F{X}_{\text{USD/GBP}}}{\partial {\rho }_{\text{GBP},\text{USD}}}\end{array}$(72)

Similarly, the first partial derivative of $C$ with respect to ${\rho }_{\text{EUR},\text{USD}}$ is:

$\begin{array}{c}\frac{\partial C}{\partial {\rho }_{\text{EUR},\text{USD}}}=\frac{\partial C}{\partial S_{\text{EUR}}}\cdot \frac{\partial S_{\text{EUR}}}{\partial {\rho }_{\text{EUR},\text{USD}}}+\frac{\partial C}{\partial S_{\text{USD}}}\cdot \frac{\partial S_{\text{USD}}}{\partial {\rho }_{\text{EUR},\text{USD}}}\\ +\frac{\partial C}{\partial S_{\text{GBP}}}\cdot \frac{\partial S_{\text{GBP}}}{\partial {\rho }_{\text{EUR},\text{USD}}}+\frac{\partial C}{\partial F{X}_{\text{USD/EUR}}}\cdot \frac{\partial F{X}_{\text{USD/EUR}}}{\partial {\rho }_{\text{EUR},\text{USD}}}\end{array}$(73)

Simplifying further, we get:

${\text{Cora}}_{\text{GBP},\text{USD}}=\frac{\partial C}{\partial S_{\text{GBP}}}\cdot \frac{\partial S_{\text{GBP}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial S_{\text{USD}}}\cdot \frac{\partial S_{\text{USD}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial S_{\text{EUR}}}\cdot \frac{\partial S_{\text{EUR}}}{\partial {\rho }_{\text{GBP},\text{USD}}}$(74)

${\text{Cora}}_{\text{EUR},\text{USD}}=\frac{\partial C}{\partial S_{\text{EUR}}}\cdot \frac{\partial S_{\text{EUR}}}{\partial {\rho }_{\text{EUR},\text{USD}}}+\frac{\partial C}{\partial S_{USD}}\cdot \frac{\partial S_{\text{USD}}}{\partial {\rho }_{\text{EUR},\text{USD}}}+\frac{\partial C}{\partial S_{\text{GBP}}}\cdot \frac{\partial S_{\text{GBP}}}{\partial {\rho }_{\text{EUR},\text{USD}}}$(75)

For the second partial derivatives:

$\begin{array}{c}\frac{{\partial }^{2}C}{\partial {\rho }_{\text{GBP},\text{USD}}^2}=\frac{\partial }{\partial {\rho }_{\text{GBP},\text{USD}}}(\frac{\partial C}{\partial S_{\text{GBP}}}\cdot \frac{\partial S_{\text{GBP}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial S_{\text{USD}}}\cdot \frac{\partial S_{\text{USD}}}{\partial {\rho }_{\text{GBP},\text{USD}}}\\ +\frac{\partial C}{\partial S_{\text{EUR}}}\cdot \frac{\partial S_{\text{EUR}}}{\partial {\rho }_{\text{GBP},\text{USD}}})\\ =\frac{\partial }{\partial {\rho }_{\text{GBP},\text{USD}}}\left(\frac{\partial C}{\partial S_{\text{GBP}}}\right)\cdot \frac{\partial S_{\text{GBP}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial S_{\text{GBP}}}\cdot \frac{\partial }{\partial {\rho }_{\text{GBP},\text{USD}}}\left(\frac{\partial S_{\text{GBP}}}{\partial {\rho }_{\text{GBP},\text{USD}}}\right)\\ +\frac{\partial }{\partial {\rho }_{\text{GBP},\text{USD}}}\left(\frac{\partial C}{\partial S_{\text{USD}}}\right)\cdot \frac{\partial S_{\text{USD}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial S_{\text{USD}}}\cdot \frac{\partial }{\partial {\rho }_{\text{GBP},\text{USD}}}\left(\frac{\partial S_{\text{USD}}}{\partial {\rho }_{\text{GBP},\text{USD}}}\right)\\ +\frac{\partial }{\partial {\rho }_{\text{GBP},\text{USD}}}\left(\frac{\partial C}{\partial S_{\text{EUR}}}\right)\cdot \frac{\partial S_{\text{EUR}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial S_{EUR}}\cdot \frac{\partial }{\partial {\rho }_{\text{GBP},\text{USD}}}\left(\frac{\partial S_{\text{EUR}}}{\partial {\rho }_{\text{GBP},\text{USD}}}\right)\end{array}$(76)

Similarly, for ${\rho }_{\text{EUR},\text{USD}}$:

$\begin{array}{c}\frac{{\partial }^{2}C}{\partial {\rho }_{\text{EUR},\text{USD}}^2}=\frac{\partial }{\partial {\rho }_{\text{EUR},\text{USD}}}(\frac{\partial C}{\partial S_{\text{EUR}}}\cdot \frac{\partial S_{\text{EUR}}}{\partial {\rho }_{\text{EUR},\text{USD}}}+\frac{\partial C}{\partial S_{\text{USD}}}\cdot \frac{\partial S_{\text{USD}}}{\partial {\rho }_{\text{EUR},\text{USD}}}\\ +\frac{\partial C}{\partial S_{\text{GBP}}}\cdot \frac{\partial S_{\text{GBP}}}{\partial {\rho }_{\text{EUR},\text{USD}}})\\ =\frac{\partial }{\partial {\rho }_{\text{EUR},\text{USD}}}\left(\frac{\partial C}{\partial S_{EUR}}\right)\cdot \frac{\partial S_{\text{EUR}}}{\partial {\rho }_{\text{EUR},\text{USD}}}+\frac{\partial C}{\partial S_{\text{EUR}}}\cdot \frac{\partial }{\partial {\rho }_{\text{EUR},\text{USD}}}\left(\frac{\partial S_{\text{EUR}}}{\partial {\rho }_{\text{EUR},\text{USD}}}\right)\\ +\frac{\partial }{\partial {\rho }_{\text{EUR},\text{USD}}}\left(\frac{\partial C}{\partial S_{USD}}\right)\cdot \frac{\partial S_{\text{USD}}}{\partial {\rho }_{\text{EUR},\text{USD}}}+\frac{\partial C}{\partial S_{\text{USD}}}\cdot \frac{\partial }{\partial {\rho }_{\text{EUR},\text{USD}}}\left(\frac{\partial S_{USD}}{\partial {\rho }_{\text{EUR},\text{USD}}}\right)\\ +\frac{\partial }{\partial {\rho }_{\text{EUR},\text{USD}}}\left(\frac{\partial C}{\partial S_{GBP}}\right)\cdot \frac{\partial S_{\text{GBP}}}{\partial {\rho }_{\text{EUR},\text{USD}}}+\frac{\partial C}{\partial S_{\text{GBP}}}\cdot \frac{\partial }{\partial {\rho }_{\text{EUR},\text{USD}}}\left(\frac{\partial S_{\text{GBP}}}{\partial {\rho }_{\text{EUR},\text{USD}}}\right)\end{array}$(77)

Therefore, we get:

$\begin{array}{c}{\text{Gora}}_{\text{GBP},\text{USD}}=\frac{{\partial }^{2}C}{\partial S_{\text{GBP}}\partial {\rho }_{\text{GBP},\text{USD}}}\cdot \frac{\partial S_{\text{GBP}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial S_{\text{GBP}}}\cdot \frac{{\partial }^2{S}_{\text{GBP}}}{\partial {\rho }_{\text{GBP},\text{USD}}^2}\\ +\frac{{\partial }^{2}C}{\partial S_{\text{USD}}\partial {\rho }_{\text{GBP},\text{USD}}}\cdot \frac{\partial S_{\text{USD}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial S_{\text{USD}}}\cdot \frac{{\partial }^2{S}_{\text{USD}}}{\partial {\rho }_{\text{GBP},\text{USD}}^2}\\ +\frac{{\partial }^{2}C}{\partial S_{\text{EUR}}\partial {\rho }_{\text{GBP},\text{USD}}}\cdot \frac{\partial S_{\text{EUR}}}{\partial {\rho }_{\text{GBP},\text{USD}}}+\frac{\partial C}{\partial S_{\text{EUR}}}\cdot \frac{{\partial }^2{S}_{\text{EUR}}}{\partial {\rho }_{\text{GBP},\text{USD}}^2}\end{array}$(78)

$\begin{array}{c}{\text{Gora}}_{\text{EUR},\text{USD}}=\frac{{\partial }^{2}C}{\partial S_{\text{EUR}}\partial {\rho }_{\text{EUR},\text{USD}}}\cdot \frac{\partial S_{\text{EUR}}}{\partial {\rho }_{\text{EUR},\text{USD}}}+\frac{\partial C}{\partial S_{\text{EUR}}}\cdot \frac{{\partial }^2{S}_{\text{EUR}}}{\partial {\rho }_{\text{EUR},\text{USD}}^2}\\ +\frac{{\partial }^{2}C}{\partial S_{\text{USD}}\partial {\rho }_{\text{EUR},\text{USD}}}\cdot \frac{\partial S_{\text{USD}}}{\partial {\rho }_{\text{EUR},\text{USD}}}+\frac{\partial C}{\partial S_{\text{USD}}}\cdot \frac{{\partial }^2{S}_{\text{USD}}}{\partial {\rho }_{\text{EUR},\text{USD}}^2}\\ +\frac{{\partial }^{2}C}{\partial S_{\text{GBP}}\partial {\rho }_{\text{EUR},\text{USD}}}\cdot \frac{\partial S_{\text{GBP}}}{\partial {\rho }_{\text{EUR},\text{USD}}}+\frac{\partial C}{\partial S_{\text{GBP}}}\cdot \frac{{\partial }^2{S}_{\text{GBP}}}{\partial {\rho }_{\text{EUR},\text{USD}}^2}\end{array}$(79)