**Alvise De Col and Patrick Kuppinger show foreign exchange correlation swap prices exhibit a non-trivial dependency on higher-order parameters, such as the correlations between forex variances. Ignoring them may result in a non-negligible uncertainty in the corresponding quotes. It is demonstrated that stochastic local correlation models are capable of capturing a wide spectrum of that uncertainty**

Multi-asset products have become fairly standard for derivatives investors seeking a diversified exposure or premium rebates by warehousing correlation risk. In foreign exchange, unlike other asset classes, a significant amount of information about correlations can be obtained directly from the market via triangulation of the forex volatility surfaces (Clark 2011; Austing 2014; Carr & Madan 1999; Langnau 2010; Campbell-Smith & Hamdani 2010; De Col *et al* 2013; De Col & Kuppinger 2014). In this article, we focus on the most natural correlation product: the forex correlation swap. This structure has become extremely popular among investment banks looking to hedge their existing correlation exposure within their multidimensional forex books. Moreover, buy-side firms (mainly hedge funds) have exhibited an appetite for products that enable them to express a direct view on correlation moves.

To describe a forex correlation swap, we consider three currencies, $\mathrm{CCY0}$, $\mathrm{CCY1}$ and $\mathrm{CCY2}$, where $\mathrm{CCY0}$ denotes the numeraire currency. This yields two main currency pairs:

${S}_{1}$ | $=\mathrm{CCY0}/\mathrm{CCY1}$ | ||

${S}_{2}$ | $=\mathrm{CCY0}/\mathrm{CCY2}$ |

and a cross-currency pair:

$${S}_{3}=\mathrm{CCY1}/\mathrm{CCY2}={S}_{2}/{S}_{1}$$ |

(where the notation $\mathrm{CCY}i/\mathrm{CCY}j$ means the amount of currency $i$ for one unit of currency $j$). The payout of a correlation swap with unit notional is defined as:

$P({S}_{1},{S}_{2})={\displaystyle \frac{{\mathrm{cov}}_{1,2}}{{\mathrm{\Sigma}}_{1}{\mathrm{\Sigma}}_{2}}}-K$ | (1) |

where $K$ denotes the strike. The covariance and realised volatilities of ${S}_{1}$ and ${S}_{2}$ are defined as:

${\mathrm{cov}}_{1,2}={\displaystyle \frac{{A}_{n}}{n-1}}{\displaystyle \sum _{i=2}^{n}}\mathrm{ln}\left({\displaystyle \frac{{S}_{1}({t}_{i})}{{S}_{1}({t}_{i-1})}}\right)\mathrm{ln}\left({\displaystyle \frac{{S}_{2}({t}_{i})}{{S}_{2}({t}_{i-1})}}\right)$ |

and ${\mathrm{\Sigma}}_{j}=\sqrt{{\mathrm{cov}}_{j,j}}$, respectively. Above, ${A}_{n}$ stands for the annualisation factor, and $n$ is the number of fixings at times $\{{t}_{i}\}$ with ${t}_{i}={t}_{i-1}+\mathrm{\Delta}t$. Sometimes, the correlation swap is defined including a mean adjustment, which does not affect the results below.

A solid pricing and risk management framework capable of describing the joint dynamics of the underlying exchange rates is needed to consistently price and hedge such a multidimensional forex derivative. One key requirement, highlighted in De Col *et al* (2013) and De Col & Kuppinger (2014), is to use a model that respects the inversion and triangulation symmetries inherent in exchange rates. At the same time, any multidimensional forex model should consistently price vanilla options written on the individual underlyings. In what follows, we shall assume a multidimensional forex model describing the joint dynamics of the two main currency pairs ${S}_{1}$ and ${S}_{2}$, such that the above-mentioned symmetry properties are satisfied and vanillas on the main and cross-currency pairs are priced consistently. Several choices for such a model are described in De Col & Kuppinger (2014). If we consider the one-dimensional projections of the chosen model, then ${S}_{j}$ can be described by a stochastic differential equation of the form:

$\mathrm{d}\mathrm{ln}{S}_{j}(t)=\left({m}_{j}(t)-{\displaystyle \frac{{\sigma}_{j}^{2}(t)}{2}}\right)\mathrm{d}t+{\sigma}_{j}(t)\mathrm{d}{W}_{j}$ |

where ${m}_{j}$ denotes the risk-neutral drift of ${S}_{j}$ in the $\mathrm{CCY0}$ measure, and ${\sigma}_{j}$ denotes the instantaneous volatility of ${S}_{j}$ (note that ${m}_{3}$ contains a quanto drift adjustment). Importantly, the instantaneous volatilities shall not be further specified; in particular, they can be stochastic processes themselves. In this article, we consider the continuous limit $\mathrm{\Delta}t\to 0$ (discretisation effects are therefore neglected), in which case the correlation payout (1) becomes:

${P}_{\text{cont}}={\displaystyle \frac{{\displaystyle {\int}_{0}^{T}}{\rho}_{{S}_{1},{S}_{2}}(t){\sigma}_{1}(t){\sigma}_{2}(t)\mathrm{d}t}{\sqrt{{\displaystyle {\int}_{0}^{T}}{\sigma}_{1}^{2}(t)\mathrm{d}t{\displaystyle {\int}_{0}^{T}}{\sigma}_{2}^{2}(t)\mathrm{d}t}}}-K$ | (2) |

In (2), ${\rho}_{{S}_{1},{S}_{2}}(t)$ denotes the instantaneous correlation between the Brownian drivers of ${S}_{1}$ and ${S}_{2}$. Note that the correlation in general might depend on ${S}_{1}$, ${S}_{2}$ and other quantities present in the model (eg, stochastic volatilities). Looking at ${S}_{2}/{S}_{1}$, the instantaneous volatility ${\sigma}_{3}$ of the cross is linked to the correlation via the triangular relationship:

${\sigma}_{3}^{2}(t)={\sigma}_{1}^{2}(t)+{\sigma}_{2}^{2}(t)-2{\rho}_{{S}_{1},{S}_{2}}(t){\sigma}_{1}(t){\sigma}_{2}(t)$ |

Hence, we can rephrase (2) in terms of the instantaneous volatilities of the two main pairs and the cross-currency pair:

${P}_{\text{cont}}$ | $={\displaystyle \frac{{\int}_{0}^{T}({\sigma}_{1}^{2}(t)+{\sigma}_{2}^{2}(t)-{\sigma}_{3}^{2}(t))\mathrm{d}t}{2\sqrt{{\int}_{0}^{T}{\sigma}_{1}^{2}(t)\mathrm{d}t{\int}_{0}^{T}{\sigma}_{2}^{2}(t)\mathrm{d}t}}}-K$ |

Taking the expectation in the $\mathrm{CCY0}$ risk-neutral measure, we obtain the fair strike of the correlation swap:

${K}_{\mathrm{fair}}=\frac{1}{2}{\mathbb{E}}_{0}\left[{\displaystyle \frac{{V}_{1}(T)+{V}_{2}(T)-{V}_{3}(T)}{\sqrt{{V}_{1}(T){V}_{2}(T)}}}\right]$ | (3) |

where:

$${V}_{j}(T)=\frac{1}{T}{\int}_{0}^{T}{\sigma}_{j}^{2}(t)\mathrm{d}t$$ |

denotes the annualised realised variance of the $j$th underlying within the generic model.

In the remainder of this article, we will investigate the fair correlation strike expression (3). In particular, we shall discuss how this quantity depends on the volatility dynamics and their correlations.

## Decomposition of the fair correlation strike

Let us now gain a qualitative understanding of the behaviour and dependence of the fair strike expression (3) on the observable market as well as the chosen model parameters. Assume the realised variances each follow some arbitrary distribution with mean ${\mathbb{E}}_{0}[{V}_{j}]={\mu}_{j}$ and variance ${\mathbb{V}\mathrm{ar}}_{0}[{V}_{j}]={\alpha}_{j}^{2}$; this means the volatility of the realised variance of the $j$th forex rate within the model is given by ${\alpha}_{j}$. Furthermore, we shall assume all higher moments of the realised variance are small. Here, it is important to note we take expectations under the $\mathrm{CCY0}$ risk-neutral measure. While for the two main forex rates this means ${\mu}_{1}$ and ${\mu}_{2}$ correspond to the fair variance swap strike on ${S}_{1}$ and ${S}_{2}$, respectively, for the cross it means ${\mu}_{3}$ corresponds to the fair quanto variance swap strike, ie, the fair strike of a variance swap paying in a different currency to its natural domestic currency. In this case, the natural payout currency for the variance swap on the cross would be $\mathrm{CCY1}$. However, since we take the expectation under the $\mathrm{CCY0}$ risk-neutral measure, it corresponds to a variance swap on the cross quantoed in $\mathrm{CCY0}$. This subtlety will be discussed in more detail towards the end of this section.

We begin the analysis by splitting the fair correlation strike into three components:

${K}_{\mathrm{fair}}$ | $={\displaystyle \frac{1}{2}}\left({\mathbb{E}}_{0}\left[\sqrt{{\displaystyle \frac{{V}_{1}(T)}{{V}_{2}(T)}}}\right]+{\mathbb{E}}_{0}\left[\sqrt{{\displaystyle \frac{{V}_{2}(T)}{{V}_{1}(T)}}}\right]-{\mathbb{E}}_{0}\left[{\displaystyle \frac{{V}_{3}(T)}{\sqrt{{V}_{1}(T){V}_{2}(T)}}}\right]\right)$ | (4) |

Assuming volatilities and moments of order three or higher of the realised variance are small, we can make the following Taylor series expansion of the individual terms in (4):

${\mathbb{E}}_{0}\left[\sqrt{{\displaystyle \frac{{V}_{1}(T)}{{V}_{2}(T)}}}\right]$ | $\approx {\mathbb{E}}_{0}[(\sqrt{{\mu}_{1}}+{\displaystyle \frac{{V}_{1}-{\mu}_{1}}{2\sqrt{{\mu}_{1}}}}-{\displaystyle \frac{{({V}_{1}-{\mu}_{1})}^{2}}{8\sqrt{{\mu}_{1}^{3}}}})$ | |||

$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}}\times ({\displaystyle \frac{1}{\sqrt{{\mu}_{2}}}}-{\displaystyle \frac{{V}_{2}-{\mu}_{2}}{2\sqrt{{\mu}_{2}^{3}}}}+{\displaystyle \frac{3{({V}_{2}-{\mu}_{2})}^{2}}{8\sqrt{{\mu}_{2}^{5}}}})]$ | ||||

$=\sqrt{{\displaystyle \frac{{\mu}_{1}}{{\mu}_{2}}}}\left(1-{\displaystyle \frac{{\alpha}_{1}^{2}}{8{\mu}_{1}^{2}}}+{\displaystyle \frac{3{\alpha}_{2}^{2}}{8{\mu}_{2}^{2}}}-{\displaystyle \frac{{\rho}_{{V}_{1},{V}_{2}}{\alpha}_{1}{\alpha}_{2}}{4{\mu}_{1}{\mu}_{2}}}\right)+O({\alpha}^{3})$ | (5) |

where we have used:

$$\mathbb{E}[({V}_{1}-{\mu}_{1})({V}_{2}-{\mu}_{2})]={\rho}_{{V}_{1},{V}_{2}}{\alpha}_{1}{\alpha}_{2}$$ |

Here, ${\rho}_{{V}_{1},{V}_{2}}$ is the correlation between the realised variances ${V}_{1}$ and ${V}_{2}$. Similarly, we can decompose the other two terms in (4) and eventually obtain the following expression for the fair correlation strike as a function of the fair variance swap strikes ${\mu}_{1}$ and ${\mu}_{2}$ of the two main forex rates, the fair quanto variance swap strike ${\mu}_{3}$ of the cross-forex rate, the volatilities of the realised variances, and the correlations between the three variance processes:

${K}_{\mathrm{fair}}$ | $\approx {\displaystyle \frac{{\mu}_{1}+{\mu}_{2}-{\mu}_{3}}{2\sqrt{{\mu}_{1}{\mu}_{2}}}}-{\displaystyle \frac{({\mu}_{1}-3{\mu}_{2}+3{\mu}_{3}){\alpha}_{1}^{2}}{16{\mu}_{1}^{2}\sqrt{{\mu}_{1}{\mu}_{2}}}}$ | |||

$\mathrm{\hspace{1em}\hspace{1em}}-{\displaystyle \frac{({\mu}_{2}-3{\mu}_{1}+3{\mu}_{3}){\alpha}_{2}^{2}}{16{\mu}_{2}^{2}\sqrt{{\mu}_{1}{\mu}_{2}}}}-{\displaystyle \frac{({\mu}_{1}+{\mu}_{2}+{\mu}_{3}){\rho}_{{V}_{1},{V}_{2}}{\alpha}_{1}{\alpha}_{2}}{8{\mu}_{1}{\mu}_{2}\sqrt{{\mu}_{1}{\mu}_{2}}}}$ | ||||

$\mathrm{\hspace{1em}\hspace{1em}}+{\displaystyle \frac{1}{4\sqrt{{\mu}_{1}{\mu}_{2}}}}\left({\displaystyle \frac{{\rho}_{{V}_{1},{V}_{3}}{\alpha}_{1}{\alpha}_{3}}{{\mu}_{1}}}+{\displaystyle \frac{{\rho}_{{V}_{2},{V}_{3}}{\alpha}_{2}{\alpha}_{3}}{{\mu}_{2}}}\right)$ | (6) |

We recognise the first term as the fair strike of the correlation swap in terms of fair (quanto) variance swap strikes in the absence of stochastic volatility. All of the other terms describe the convexity adjustment due to the stochastic variance processes and their correlations. Furthermore, we observe a short exposure of the fair strike on the main-main variance correlation ${\rho}_{{V}_{1},{V}_{2}}$ and a long exposure on the main-cross variance correlations ${\rho}_{{V}_{1},{V}_{3}}$ and ${\rho}_{{V}_{2},{V}_{3}}$. This long/short dependency on the variance correlations produces a rather rich pricing behaviour, as will be shown further below.

By means of (6), we analyse the fair strike in various industry standard models that differ in the volatilities of realised variance and the correlations among the variance processes.

### Correlation swap pricing in various models

We consider a set of typical multidimensional forex models: (multidimensional) Black-Scholes (BS) (Clark 2011), local correlation (LC) (Austing 2014; De Col & Kuppinger 2014; Guyon 2014) and stochastic (local) correlation (SLC) (Austing 2014; De Col *et al* 2013; De Col & Kuppinger 2014; Guyon 2014).

In order to focus on the key role played by the correlations between the variances, we shall assume throughout this section that ${\mu}_{j}\approx \mu $ for all $j$, which, for example, will be the case if the three forex implied volatility surfaces are identical (ignoring the quanto effect on the cross, which will be addressed further below). It is otherwise straightforward to work with the more general expression in (6) if needed. The fair strike in (6) thus simplifies to:

$${K}_{\mathrm{fair}}\approx \frac{1}{2}-\frac{1}{16{\mu}^{2}}({\alpha}_{1}^{2}+{\alpha}_{2}^{2}+6{\rho}_{{V}_{1},{V}_{2}}{\alpha}_{1}{\alpha}_{2}-4{\rho}_{{V}_{1},{V}_{3}}{\alpha}_{1}{\alpha}_{3}-4{\rho}_{{V}_{2},{V}_{3}}{\alpha}_{2}{\alpha}_{3})$$ | (7) |

Expression (7) lends itself nicely to understanding the key drivers behind correlation swap pricing. While some simplifying assumptions were made in order to arrive at (7), the numerical test results presented further below confirm its prediction power.

#### Black-Scholes

For BS, we have ${\alpha}_{j}=0$; thus, (7) yields:

${K}_{\mathrm{fair}}(\text{BS})=\frac{1}{2}$ | (8) |

Clearly, the BS model is overly simplistic to adequately price a product such as the forex correlation swap.

#### Local correlation

In an LC model, we have non-zero values for the volatilities and the correlations of realised variances due to the spot dependency of the local volatilities. However, the volatilities of realised variances are usually smaller in LC models compared with models including an explicit stochastic volatility component (Gatheral 2006): see figure 1. For the LC model with identical implied volatility surfaces, we obtain the following expression for the fair strike of the correlation swap:

${K}_{\mathrm{fair}}(\text{LC})\approx {\displaystyle \frac{1}{2}}-{\displaystyle \frac{{\alpha}_{\mathrm{LV}}^{2}}{8{\mu}^{2}}}(1-{\rho}_{\mathrm{LV}})$ | (9) |

Hence, we expect the LC model to have a slightly lower fair strike of the correlation swap than the BS model. Note that the volatility of realised variance ${\alpha}_{\mathrm{LV}}$ and the correlation ${\rho}_{\mathrm{LV}}$ in an LC model are not model inputs and are instead enforced by the local volatility surfaces (which cannot be controlled by the user). Similar to the BS model, the LC model, despite its additional complexity, still lacks degrees of freedom to capture the uncertainty in forex correlation swap prices, as indicated by (7).

#### Stochastic local correlation

Let us next turn our focus on SLC models. In the literature, several descriptions of such models exist, which may differ in terms of their stochastic variance processes, the correlation structure between those processes, and the amount of stochastic volatility in a given model (Austing 2014; De Col & Kuppinger 2014; Guyon 2014). As already mentioned, SLC models usually have larger volatilities of realised variance than their LC counterparts, while still matching all implied volatility surfaces. The amount of volatility of realised variance can be controlled by the volatility of variance or by the mixing weight (Clark 2011), if such an approach is taken in the calibration of the underlying stochastic volatility models. The mixing weight approach allows the user to fade the model between a purely local volatility model at mixing weight zero and a predominantly stochastic volatility model with some local volatility add-on at mixing weight one. The correlation between the realised variances can be influenced by the user’s choice of correlations between the stochastic variance drivers. Note, however, that neither the volatilities nor the correlations among realised variances can be fully chosen by the user, as the remaining local volatility component will also affect those quantities. Nevertheless, we can demonstrate that the user has a fairly direct handle on the volatility and correlation, and – unlike the models analysed so far – can access a wider range of fair correlation swap prices, as predicted by (7).

##### SLC with symmetric setup

First, we consider a full SLC model in a symmetric setup (ie, one in which all variance correlations and all volatilities of variances are equal, ${\alpha}_{j}=\alpha $, ${\rho}_{{V}_{i},{V}_{j}}=\rho $), thus yielding:

${K}_{\mathrm{fair}}(\text{symmetricSLC})\approx {\displaystyle \frac{1}{2}}-{\displaystyle \frac{{\alpha}^{2}}{8{\mu}^{2}}}(1-\rho )$ | (10) |

This expression is formally the same as we obtained for the LC model in (9). However, as already pointed out and visualised in figure 1, the volatilities of realised variances and the correlations between those variance processes in the SLC model are independent model inputs and can be modified by the user. As a consequence, larger values for $\alpha $ can be achieved. Therefore, the maximum reduction of fair strike versus the value observed in BS is expected to be significantly larger in SLC than in LC.

##### SLC with common driver

In Austing (2014) and De Col & Kuppinger (2014), using SLC models with one common variance driver is advised, mainly for stability reasons. This in turn would mean the correlations between the realised variances are close to one for a large mixing weight. For a mixing weight of zero, the LC and SLC common drivers are equivalent. Increasing the mixing weight towards one, two effects can be observed (see figure 1) in (10): $\alpha $ increases, but $\rho $ also increases (for increased mixing weight, we get fewer contributions coming from the local volatility part of the model, hence correlations between the realised variances are closer to one). These two effects offset each other in (10) and, as we will show numerically in the test section, only small deviations are observable between the fair strikes produced in LC and SLC with common drivers. This is an artefact of the specific choice of correlation between variance drivers. If only this version of SLC is considered, one might arrive at the wrong conclusion that correlation swaps are fairly model-independent. Hence, one sees the need to free up the correlations between the variance drivers.

##### SLC with independent drivers

Another special case of SLC with a symmetric setup is one in which we assume independence between the drivers of the stochastic variance processes (${\alpha}_{j}=\alpha $, ${\rho}_{{V}_{i},{V}_{j}}\approx 0$). This will result in the following expression for the fair strike:

${K}_{\mathrm{fair}}(\text{SLCindependentdrivers})\approx {\displaystyle \frac{1}{2}}-{\displaystyle \frac{{\alpha}^{2}}{8{\mu}^{2}}}$ | (11) |

which implies a reduced value for the fair strike compared with the BS model. Note that the reduction is proportional to the volatility of the realised variance. Hence, varying the mixing weight allows the user to control the amount of reduction versus the BS value.

##### SLC with local cross

A typical simplification in SLC models is to assume a purely local diffusion for the cross-forex rate. In such a setup, the volatility of realised variance for the main currency pairs outweighs that for the cross-currency pair, so we may ignore the terms containing ${\alpha}_{3}$. In this configuration (${\alpha}_{1}={\alpha}_{2}=\alpha $, ${\alpha}_{3}\ll \alpha $), we obtain:

${K}_{\mathrm{fair}}(\text{SLClocalcross})\approx {\displaystyle \frac{1}{2}}-{\displaystyle \frac{{\alpha}^{2}}{8{\mu}^{2}}}\left(1+3{\rho}_{{V}_{1},{V}_{2}}\right)$ | (12) |

Within this model choice, depending on how one sets the correlation between the stochastic variance drivers of the two main currency pairs (and the mixing weights of the main currency pairs), the fair correlation swap strike may show an even more pronounced reduction when compared with the BS value than in the cases previously discussed.

##### SLC with local main

The cases we have investigated above all lead to a reduction in fair strike relative to the BS model (assuming positive realised variance correlation). It is also possible to achieve an increase in fair strike. For example, one might choose to have a small volatility of variance for one of the main currency pairs (eg, by switching off the stochastic volatility part of the model for that pair by reducing the mixing weight). This setup with perfectly correlated variance processes (${\alpha}_{2}={\alpha}_{3}=\alpha $, ${\alpha}_{1}\ll \alpha $, ${\rho}_{{V}_{2},{V}_{3}}\approx 1$) yields:

${K}_{\mathrm{fair}}(\text{SLClocalmaincurrencypair})\approx {\displaystyle \frac{1}{2}}+{\displaystyle \frac{3{\alpha}^{2}}{16{\mu}^{2}}}$ | (13) |

### The quanto effect on the cross-forex variance

To compute the fair strike of the correlation swap, we take the expectation in the $\mathrm{CCY0}$ risk-neutral measure, which is not the natural measure for the cross-forex rate. As a consequence, ${\mu}_{3}$ is not the standard fair variance swap strike on the cross-forex rate (that is, one determined fully by the corresponding implied volatility surface), but rather the fair quanto variance swap strike. Assuming zero rates, the standard variance swap on the cross is given by:

$${\mathbb{E}}_{1}[{V}_{3}]={\mathbb{E}}_{0}[{S}_{1}(T){V}_{3}]/{S}_{1}(0)$$ |

The quanto variance and the variance swap are related by:

$${S}_{1}(0){\mathbb{E}}_{0}[{V}_{3}]={\mathbb{E}}_{0}[{S}_{1}(T){V}_{3}]-{\mathbb{E}}_{0}[({S}_{1}(T)-{S}_{1}(0)){V}_{3}]$$ |

which means the quanto variance swap is equal to the variance swap minus the covariance between ${S}_{1}$ and ${V}_{3}$. The leading term of this covariance depends on the correlation between ${S}_{1}$ and ${V}_{3}$, the volatility of ${S}_{1}$ and the volatility of ${V}_{3}$ (this argument does not change for non-zero rates). More specifically, we can write:

$${\mathrm{cov}}_{0}({S}_{1},{V}_{3})\approx {\rho}_{{S}_{1},{V}_{3}}\sqrt{{\mu}_{1}}{S}_{1}(0){\alpha}_{3}+O({\alpha}_{3}^{2})$$ |

In particular, if the correlation between ${S}_{1}$ and ${V}_{3}$ or either of the two volatilities (ie, spot volatility $\sqrt{{\mu}_{1}}$ or volatility of variance ${\alpha}_{3}$) is zero, the quanto variance swap is equal to the variance swap. Note that for SLC models in forex, it is often the case that the correlation between the spot and the variance drivers is set to zero (to preserve the model symmetry under inversion and triangulation (Austing 2014)). In this case, as the correlation between the drivers can be seen as a proxy for the correlation between the corresponding realised quantities, we expect the quanto effect to be negligible.

## Test results

In this section, we present a selection of tests linking the theoretical findings to actual correlation swap pricing results. We consider the types of models we have discussed in previous sections. All test cases were obtained with stochastic volatility processes of Heston type, as in De Col & Kuppinger (2014), but allowing for an arbitrary correlation structure among the variance drivers. As described above, to preserve the forex symmetries, the correlation between spot and variance drivers is set to zero. All numerical results are based on Monte Carlo valuations using 40,000 paths.

**Figure 2: **Comparison of the one-year correlation swap fair strike for BS, LC and various versions of SLC. SLC local main common driver (13) (light blue line, labelled 1), BS (dark blue line, labelled 2), SLC common driver (dark brown line, labelled 3), LC (light brown line, labelled 4), SLC independent drivers (11) (grey line, labelled 5), SLC local cross independent mains (12) with ${\rho}_{{V}_{\text{1}},{V}_{\text{2}}}=\text{0}$ (green line, labelled 6), and SLC local cross common main driver (12) with ${\rho}_{{V}_{\text{1}},{V}_{\text{2}}}=\text{1}$ (red line, labelled 7).

Consistent with (7), we concentrate on the key components affecting correlation swap prices. Therefore, for all test results shown in figure 2 we consider an artificially chosen forex market with three identical implied volatility surfaces, with non-zero skew and kurtosis (see table A for an example expiry of one year). The fair variance swap strikes ${\mu}_{j}$ are the same for the two main forex rates. However, for the cross, ${\mu}_{3}$ denotes the fair quanto variance swap strike and therefore contains a correction with respect to the variance swap, as discussed above. Nevertheless, for the test cases we considered, the quanto impact on the fair correlation swap strike is below one correlation point and, hence, does not fundamentally alter the theoretical predictions made in the previous section.

Clearly, for the BS model we have, among others, that ${\alpha}_{3}=0$, and therefore there is no quanto correction. As a consequence, we obtain – as predicted in the theoretical section in (8) – a fair strike of the correlation swap of $\frac{1}{2}$ (50 correlation points).

The LC model corresponds to a scenario in which all volatilities of variances are equal and small due to the lack of stochastic volatility. The LC results in figure 2 correspond to what was predicted in (9). In particular, we observe that LC falls slightly below the BS results as a consequence of the small negative contributions arising from the non-zero volatilities of realised variances.

We also present results for a variety of different flavours of SLC models corresponding to the theoretical cases we discussed above. In this particular test case, we observe that a symmetric SLC model spans a correlation regime of around five correlation points for a mixing weight of one, with the two extreme cases being common and independent variance drivers (see (10) and (11)). Moreover, we observe LC and SLC with common drivers match very well, as predicted above. In case the symmetry of the model is broken and part of it is assumed to be local (local cross or local main, for example), then a large impact on the realised correlation is visible, as predicted by (12) and (13). The difference in fair strike can easily span a range of around 25 correlation points.

**Table A:**Implied volatility fit for the cross-currency pair, given the market data used for figure 2 for a maturity of one year. [We consider the worst-case scenario (for vanilla fitting), where mixing weight is equal to one; numbers (except the variance swap fair strike) are in volatility points. The variance swap fair strike on the mains is 0.0116. From the differences with this value, we may conclude the impact on the fair correlation swap strike is consistently below one correlation point for all of the models considered in the table.]

Variance | |||||||
---|---|---|---|---|---|---|---|

swap on | |||||||

10DP | 25DP | ATM | 25DC | 10DC | ${\bm{\mu}}_{\text{\U0001d7d1}}$ | ${\bm{S}}_{\text{\U0001d7d1}}$ | |

Market | 12.18 | 11.11 | 10.11 | 9.66 | 9.76 | ||

SLC local | 12.32 | 11.30 | 10.34 | 9.81 | 9.81 | 0.01139 | 0.01155 |

main | |||||||

common | |||||||

driver | |||||||

SLC | 12.17 | 11.10 | 10.13 | 9.70 | 9.78 | 0.01142 | 0.01155 |

common | |||||||

driver | |||||||

LC | 12.16 | 11.09 | 10.12 | 9.67 | 9.75 | 0.01149 | 0.01161 |

SLC | 11.91 | 10.99 | 10.19 | 9.83 | 9.96 | 0.01145 | 0.01157 |

independent | |||||||

drivers | |||||||

SLC local | 12.08 | 11.08 | 10.17 | 9.80 | 9.94 | 0.01152 | 0.01163 |

cross | |||||||

independent | |||||||

mains | |||||||

SLC local | 11.78 | 10.79 | 9.92 | 9.55 | 9.64 | 0.01145 | 0.01156 |

cross | |||||||

common | |||||||

main driver |

**Figure 3: **Dependence of the fair correlation swap strike in a one-year USD/EUR-USD/GBP correlation swap valued under SLC with symmetric model setup on the mixing weight and the correlation among the variance drivers (${\alpha}_{j}=\alpha ,{\rho}_{{V}_{i},{V}_{j}}=\rho $). The market data snapshot is from December 2017.

In figure 3, we show the dependence of the fair strike of a correlation swap valued under SLC in a symmetric setup on the mixing weight (ie, volatility of realised variance) and on the correlation among the variance drivers (ie, correlation among the realised variances). For these results, real market data was used (ie, the three implied volatility surfaces were not identical), and one can observe that (10) describes the behaviour well: for a given correlation, the fair strike is a decreasing function of the mixing weight; for a given mixing weight, the fair strike is an increasing function of the correlation. In the fully correlated setup (SLC with common driver), we see the fair strike now has a dependency on the mixing weight (however, it is still the smallest of all considered cases). This is due to the fact that for real market data the terms in (6) do not fully offset each other like they do in the simplified case considered in the theoretical section.

**Table B:**Difference in volatility points of the implied volatility for the cross-currency pair, given the market data used for figure 3 and a maturity of one year. [We consider the worst-case scenario (for vanilla fitting), where the mixing weight is equal to one. From the differences between SLC and LC in the fair (quanto) variance swap strikes, we may conclude the impact on the fair correlation swap strike is consistently below one correlation point.]

Variance | |||||||
---|---|---|---|---|---|---|---|

swap on | |||||||

10DP | 25DP | ATM | 25DC | 10DC | ${\bm{\mu}}_{\text{\U0001d7d1}}$ | ${\bm{S}}_{\text{\U0001d7d1}}$ | |

SLC | 0.04 | 0.04 | 0.03 | 0.02 | $-$0.01 | 0.00729 | 0.00732 |

common | |||||||

driver | |||||||

SLC | 0.12 | 0.24 | 0.42 | 0.32 | 0.15 | 0.00727 | 0.00731 |

independent | |||||||

drivers | |||||||

LC | 0.02 | 0.01 | 0.01 | 0.00 | $-$0.04 | 0.00733 | 0.00735 |

As discussed in De Col & Kuppinger (2014), if one allows for a general correlation structure among the variance drivers, more violations of the required positive-semidefiniteness of the spot correlation matrix are expected. This in turn requires an increased number of corrections, which will ultimately deteriorate the vanilla fit on the cross. For the test cases above, it was verified the vanilla fit on the main currency pairs and the cross-currency pair remained acceptable (see tables A and B for the cross-currency pair; the vanilla fit error for the main currency pairs was consistently below $0.1$ volatility points and is therefore not shown in the tables). However, the results in this article do not rely on the calibration quality of individual points on the vanilla surfaces, but rather on the consistent pricing of variance swap fair strikes, which condense the required information about all strikes – including wing behaviour – into one number. Given the calibration of the main currency pairs is not problematic, any potential calibration issues are hence due to misfits of the variance swap fair strike on the cross-currency pair. In tables A and B, we show this is preserved within the required accuracy to guarantee the robustness of the numerical results. Moreover, as mentioned previously, the impact of quantoing is equally small. To quantify the potential impact of the calibration error, one can look at the leading term in (6); this yields the following effect on the fair correlation swap strike:

$$\mathrm{\Delta}{K}_{\mathrm{fair}}\approx -\mathrm{\Delta}{\mu}_{3}/(2\sqrt{{\mu}_{1}{\mu}_{2}})$$ |

For the results obtained, this was less than one correlation point.

## Final remarks

This study highlights the importance of the correlation structure among realised variances on the pricing of products such as the correlation swap. The formalism is agnostic with respect to the particular choice of model and is capable of fully explaining the seemingly intricate pricing behaviour.

As shown in the above results, a wide range of correlation swap prices can be achieved with just one family of SLC models by modifying the mixing weights and the correlations between variance drivers. Clearly, not all configurations of parameters make sense other than from a theoretical point of view. SLC models require additional calibration efforts compared with LC models. In particular, a sound calibration of not only the mixing weights but also the correlations between the stochastic variance drivers are important for avoiding uncertainty in the pricing of products such as correlation swaps.

Typically, mixing weights are calibrated on the individual (single-dimensional) components of the model by matching certain liquid path-dependent options such as barriers or double no-touches (Clark 2011). The calibration of the correlation structure among the stochastic variance processes is not much discussed in the literature, as there are not as many liquidly traded multidimensional products in the market that yield enough information about these parameters. One way to calibrate the correlation could be to use information embedded in firmly quoted multidimensional baskets or, alternatively, historical series of implied volatilities. If reliable information on the correlation between variance processes is missing, trading needs to incorporate the uncertainty due to this usually overlooked model parameter in correlation swap quotes.

The authors are with UBS Business Solutions, a subsidiary of UBS Group. This article contains personal views expressed by the authors and may not reflect the views of UBS Group or any of its subsidiaries. The authors would like to thank Stephane Matar for several interesting and useful discussions on the topic of this article while he was at UBS, as well as Manos Venardos for his insightful comments.

Email: patrick.kuppinger@gmail.com,

Email: alvise.de-col@ubs.com.

## References

- Austing P, 2014

Smile Pricing Explained

Palgrave Macmillan - Campbell-Smith A and B Hamdani, 2010

The rise of multi-currency options

Risk August - Carr P and D Madan, 1999

Introducing the covariance swap

Risk February, pages 47-51 - Clark IJ, 2011

Foreign Exchange Option Pricing: A Practitioner’s Guide

Wiley - De Col A and P Kuppinger, 2014

Pricing multi-dimensional FX derivatives via stochastic local correlations

Wilmott Magazine 73, pges 72-77 - De Col A, A Gnoatto and M Grasselli, 2013

Smiles all around: FX joint calibration in a multi-Heston model

Journal of Banking and Finance 37, pages 2799-3818 - Gatheral J, 2006

The Volatility Surface: A Practioner’s Guide

Wiley Finance - Guyon J, 2014

Local correlation families

Risk January, pages 52-58 - Langnau A, 2010

A dynamic model for correlation

Risk March, pages 74-78