# Pricing FX Options under Intermediate Currency

S. Maurer School of Mathematical Sciences, University of Nottingham, UK;
<EMAIL_ADDRESS>T.E. Sharp Credit Suisse, Zurich, Switzerland
M.V. Tretyakov School of Mathematical Sciences, University of Nottingham, UK;
<EMAIL_ADDRESS>
###### Abstract

We suggest an intermediate currency approach that allows us to price options
on all FX markets simultaneously under the same risk-neutral measure which
ensures consistency of FX option prices across all markets. In particular, it
is sufficient to calibrate a model to the volatility smile on the domestic
market as, due to the consistency of pricing formulas, the model automatically
reproduces the correct smile for the inverse pair (the foreign market). We
first consider the case of two currencies and then the multi-currency setting.
We illustrate the intermediate currency approach by applying it to the Heston
and SABR stochastic volatility models, to the model in which exchange rates
are described by an extended skewed normal distribution, and also to the
model-free approach of option pricing.

Keywords: foreign exchange market, FX option pricing, foreign-domestic
symmetry, multi-currency options, skewed normal distribution.

##  1 Introduction

As it is well known (see e.g. [4, 30] and also Section 2 here), in the case of
a foreign exchange (FX) for two currencies (say, GBP and EUR) no measure is
simultaneously risk-neutral for the market on which GBP is the domestic
currency and for the market on which EUR is the domestic currency. This can be
seen as an asymmetry between the different market views due to the different
choice of numeraires. In practice currency pair conventions are usually used
in order to standardize option price quotations for each specific currency
pair [24, 30]. But this can lead to calibration difficulties. Each of the
domestic markets has its own volatility smile curve for options on the
corresponding foreign currency. Suppose we want to use a stochastic volatility
model given under a risk-neutral measure on the GBP domestic market which we
calibrate to the smile for options on EUR. If we re-write this model for the
inverse pair, i.e., where options on GBP are traded, in a risk-neutral
fashion, we need to calibrate it to the smile on this market as the previously
found parameters of the model typically do not match the smile for the inverse
pair. This is inconvenient. This situation becomes even more complicated in a
multi–currency setting while it is of practical importance to be able to price
options on the global FX market in a consistent fashion. With a large number
$N$ of currencies, the existence of a consistent FX model is not trivial as a
suitable model must preserve relationships between all $N$ currencies and
consistency of volatility smiles between all $N(N-1)/2$ cross pairs.

To address these problems of consistent FX modelling, in [11] (see also [13,
12, 8]) the concept of intrinsic currency [11, 12] or artificial currency [8]
was introduced. The approach of [11] is based on the idea that each currency
has an ‘intrinsic value’, which is a description of the value of a currency in
relation to other currencies. In the intrinsic currency-valuation framework of
[11, 12] one models the $N$ intrinsic values of $N$ currencies rather than
modelling the $N-1$ exchange rates. In [12] Doust extends his original idea of
the intrinsic currency-valuation framework to a SABR-type model, which allows
to capture the observed volatility smile on the FX market in a multi-currency
setting. On a FX market with $N$ currencies, he describes the market with $N$
intrinsic currency values and chooses one (without loss of generality) as the
valuation currency and its associated risk–neutral measure, which produces the
usual risk–neutral processes for all exchange rates. In [8] $N$ exchange rates
between an artificial currency and $N$ real currencies are modelled under a
risk-neutral measure associated with the artificial currency so that all
relationships (in particular, the inversion property that the exchange rate
for a pair of real currencies and for their inverse satisfy SDEs of a similar
form) between $N$ currencies are satisfied.

Here we explore a very simple but very valuable from the practical angle idea:
find a numeraire with respect to which we can price all FX derivatives traded
on any of the domestic markets simultaneously under the same measure. This
resolves the issue highlighted above: models for different currency pairs can
be calibrated to all smiles in a consistent manner. For instance, in the case
of two currencies, it is sufficient to calibrate a model on e.g. the GBP
domestic market and the smile on the EUR domestic market is automatically
reproduced without any need of additional calibration.

We show that such a numeraire exists via introducing the concept of an
intermediate pseudo-currency. The main difference with [11, 13, 12, 8] is that
the pseudo-currency is explicitly defined via exchange rates of real
currencies, while in [11, 13, 12, 8] exchange rates of real currencies are
described via an artificial, not observed currency. Consequently, we naturally
model $N-1$ exchange rates, not $N$ as in [11, 13, 12, 8]. Further, we can use
three modelling approaches in a consistent manner. The first one is the
traditional modelling way in Financial Mathematics, where we start from a
stochastic model for $N-1$ exchange rates under a ‘market’ measure and then we
introduce a pseudo-currency market which, as we show, has a risk-neutral
measure. Under this risk-neutral measure (the intermediate pseudo-currency is
used as the numeraire) we can price FX products on all currency markets
simultaneously which guarantees consistency of volatility smiles and other
natural relationships between currencies (e.g., the foreign-domestic
symmetry). This approach allows us to start with popular stochastic volatility
models (e.g., Heston or SABR) written under a ‘market’ measure and derive the
corresponding consistent models on the pseudo-currency market. Alternatively,
in the second approach, from the start we model exchange rates under a risk-
neutral measure or under a forward measure associated with the pseudo-currency
market. The third approach is model-free (see [1, 15, 2, 9] and references
therein), where we reconstruct a risk-neutral measure or a forward measure
from volatility smiles. We note that the intermediate pseudo-currency in
comparison with the intrinsic currency of [11] does not have a financial
interpretation, but our focus here is solely on consistent calibration and
modelling of exchange rates.

The rest of the paper is organized as follows. In Section 2 we recall that
there is no measure which is simultaneously risk-neutral for both domestic and
foreign FX markets and also recall the foreign-domestic symmetry. A convenient
numeraire and the associated intermediate pseudo-currency market are
introduced in Section 3, where the corresponding pricing formulas for FX
options are also derived. This is done for clarity of the exposition in the
case of a single currency pair. We extend the intermediate pseudo-currency
concept to the multi–currency setting in Section 4. In Section 5 we illustrate
the concept by first applying it to the Heston model [21] and SABR [20]. Then,
for further illustration, we model the spot exchange rate using an extended
skewed normal distribution. This exchange rate model is an illustration of how
one can describe the observed fat-tailed distribution of the log exchange rate
(compared to the log normal assumption). The considered extended skewed normal
distribution is constructed by combining one normal and two shifted half-
normal distributed random variables and it allows a flexible control of the
tails of the spot exchange rate distribution. We note that the use of the
extended skewed normal distribution in pricing FX options is somewhat new.
Further, we illustrate our FX option pricing mechanism on the model-free
approach. We provide some calibration examples in Section 6.

##  2 Preliminaries

In this section we recall the well-known fact (see e.g. [4, 30]) that there is
no measure which is simultaneously risk-neutral for both the domestic and the
foreign market. We also state the foreign-domestic symmetry. For definiteness,
in this section and in Section 3 we use the EUR-USD and USD-EUR pairs.

Let us recall that the EUR-USD spot exchange rate at time $t$

$f(t):=S_{\text{€}/\$}(t)$

is quoted as

$\frac{\text{units of USD}}{\text{one EUR}},$

and

$S_{\$/\text{€}}(t)=\frac{1}{S_{\text{€}/\$}(t)}=\frac{1}{f(t)}.$

In currency pairs (e.g. EUR-USD), the first mentioned currency is known as the
foreign (or base) currency, while the second is known as the domestic currency
(or numeraire) [5, 30].

Within the standard option pricing setting, we assume that the currency market
under a ‘market’ measure is described by the system:

$\displaystyle dB_{\$}=r_{\$}(t)B_{\$}dt,$ (2.1) $\displaystyle
dB_{\text{€}}=r_{\text{€}}(t)B_{\text{€}}dt,$ $\displaystyle
df=\mu(t)fdt+\sigma(t)fdW(t),$

where $B_{\$}(t)$, $B_{\text{€}}(t)$ and $r_{\$}(t)$, $r_{\text{€}}(t)$ are
USD and EUR bank accounts with their short interest rates, respectively;
$\sigma(t)>0$ is a volatility, $\mu(t)$ is a drift; and $W(t)$ is a standard
Wiener process. It is assumed that the coefficients $r_{\$}(t)$,
$r_{\text{€}}(t)$, $\sigma(t)$, and $\mu(t)$ are stochastic processes adapted
to a filtration $\mathcal{F}_{t}$ to which $W(t)$ is also adapted (typically,
in stochastic volatility models $\mathcal{F}_{t}$ is larger than the natural
filtration of $W(t)),$ and they have bounded second moments. We also require
that $\sigma(t)$ satisfies Novikov’s condition.

On the USD market, the foreign currency EUR is paid for by USD (the domestic
currency) and the risky asset is

$Y_{\text{€}/\$}(t)=S_{\text{€}/\$}(t)B_{\text{€}}(t),$

while on the EUR market the risky asset is

$Y_{\$/\text{€}}(t)=S_{\$/\text{€}}(t)B_{\$}(t).$

Following the classical theory of pricing, we have to find equivalent (local)
martingale measures (EMMs) $Q^{\$}$ and $Q^{\text{€}}$ under which the
corresponding discounted risky assets are (local) martingales. By standard
arguments we arrive at the SDEs for $f(t)$ and $g(t):=1/f(t)$ written under
the corresponding EMMs:

$\displaystyle df$ $\displaystyle=$
$\displaystyle(r_{\$}(t)-r_{\text{€}}(t))fdt+\sigma(t)fdW^{Q^{\$}}(t),$ (2.2)
$\displaystyle dg$ $\displaystyle=$
$\displaystyle(r_{\text{€}}(t)-r_{\$}(t))gdt-\sigma(t)gdW^{Q^{\text{€}}}(t),$
(2.3)

where $W^{Q^{\$}}(t)$ is a standard Wiener process under $Q^{\$}$ and
$W^{Q^{\text{€}}}(t)$ is a standard Wiener process under $Q^{\text{€}}$. We
can see (cf. (2.1) and (2.2)-(2.3)) that the market prices of risk on the two
markets differ:

$\gamma_{\text{€}}(t)=\frac{\mu(t)+r_{\text{€}}(t)-r_{\$}(t)}{\sigma(t)}\neq\frac{\sigma^{2}(t)-\mu(t)+r_{\$}(t)-r_{\text{€}}(t)}{-\sigma(t)}=\gamma_{\$}(t)$

(recall that $\sigma(t)>0).$ Thus,

$Q^{\$}\neq Q^{\text{€}},$ (2.4)

i.e., there is no measure which is simultaneously risk-neutral for the EUR
domestic market and for the USD domestic market in this rather general
setting. We remark that (2.4) does not lead to an arbitrage opportunity.

Note that the SDE (2.2) for $f$ under the measure $Q^{\text{€}}$ takes the
form

$df=(r_{\$}(t)-r_{\text{€}}(t)+\sigma^{2}(t))fdt+\sigma(t)fdW^{Q^{\text{€}}}.$
(2.5)

Intuitively, one could think that the drift for the exchange rate
$g(t)=1/f(t)$ in (2.3) should be the negative of the drift of $f(t)$ under the
same measure, i.e. $-(r_{\text{€}}(t)-r_{\$}(t))=r_{\$}(t)-r_{\text{€}}(t)$.
However, as we can see in (2.5), this is not the case. This is related to the
phenomenon known as Siegel’s paradox [28], which is due to the convexity of
the function $1/f$.

Let us also recall [18, 24, 30, 14] that under the no-arbitrage assumption
(and other standard conditions like no transaction costs, etc.), there is the
so-called foreign-domestic symmetry for FX options which we formulate in the
following theorem. This symmetry is the key requirement for a model to be
consistent for a currency pair and its inverse pair (see e.g. [11, 12, 8, 17]
and references therein).

######  Theorem 2.1.

Under the no-arbitrage assumption, there is the following relationship (called
Foreign-Domestic Symmetry) for FX options

$C_{\text{€}/\$}(0,T,K)=S_{\text{€}/\$}(0)\,K\,P_{\$/\text{€}}\left(0,T,\frac{1}{K}\right),$
(2.6)

where $C_{\text{€}/\$}(0,T,K)$ is the call option price (in $\$$) at time $0$
to buy one EUR for $\$K$ at time $T$; $P_{\$/\text{€}}(0,T,1/K)$ is the put
option price (in €) at time $0$ to sell one USD for €$\dfrac{1}{K}$ at time
$T$.

Let us emphasise that the proof of this theorem is solely based on the no-
arbitrage argument, and hence it states a fundamental property of the FX
market. Suppose we take a stochastic volatility model (e.g., a popular model
such as the Heston and SABR) and calibrate it using option data on the USD
market. If we rewrite this model with the obtained parameters for the inverse
pair €$/\$,$ then option prices computed by this model on the EUR market would
not match the data on this market and the property (2.6) would not be
satisfied, i.e. we would get an arbitrage. Instead, if we calibrate the
inverse pair model again but using option data on the EUR market, then the
property (2.6) is obviously satisfied, but it is inconvenient that the model
needs to be calibrated twice despite the fact that the two smiles are
consistent with each other due to absence of arbitrage and the symmetry (2.6).
We note in passing (see e.g. [10, 8]) that for the SABR and Heston models
there are mappings between the parameters obtained for USD-EUR and the
parameters of the inverted world (i.e., EUR-USD), still the parameters are
different for the direct and inverted worlds.

In the next section we find a numeraire allowing to price options on USD and
EUR markets simultaneously after a single calibration. In particular, within
the proposed approach, calibration of a stochastic volatility model using FX
data from one of the domestic markets guarantees replication of volatility
smiles by the model on both domestic markets.

##  3 FX option pricing via intermediate pseudo-currency

In this section we propose a candidate for numeraire using which options on
USD and EUR markets can be priced simultaneously under the same measure. It is
convenient to cast introduction of such a numeraire using the notion of an
artificial currency, which we call an intermediate pseudo-currency in this
paper to distinguish from the intrinsic currency of [11, 12] and the
artificial currency of [8]. We note that the intermediate currency market is
virtual and is only used as a proxy to find a suitable numeraire and write
down the corresponding pricing formulas, while calibration is done using the
usual FX data. We start with the definition of the pseudo-currency, then
(Section 3.1) we consider pricing under an EMM $Q^{X}$ on the pseudo-market
and (Section 3.2) – under the T-forward measure $Q_{T}^{X}$ equivalent to
$Q^{X}$.

######  Definition 3.1.

Let $S_{\text{€}/\$}(t)=f(t)$ be the EUR-USD exchange rate at time $t$. An
intermediate pseudo-currency X is a currency with exchange rate EUR-X,
$S_{\text{€}/X}(t)=\sqrt{f(t)},$ and the exchange rate USD-X,
$S_{\$/X}(t)=\frac{1}{\sqrt{f(t)}}$.

We observe the natural relationship for the intermediate currency

$S_{\text{€}/X}(t)\cdot\frac{1}{S_{\$/X}(t)}=f(t).$ (3.1)

We note the following symmetry:

$S_{\text{€}/X}(t)=\sqrt{f(t)}=\frac{1}{\frac{1}{\sqrt{f(t)}}}=\frac{1}{S_{\$/X}(t)}=S_{X/\$}(t)$

and

$S_{\$/X}(t)=\frac{1}{\sqrt{f(t)}}=\frac{1}{S_{\text{€}/X}(t)}=S_{X/\text{€}}(t).$

We also introduce the money market account $B_{X}$ for the intermediate
currency X with its respective interest rate $r_{X}(t)$:

$dB_{X}=r_{X}(t)B_{X}dt.$ (3.2)

In the next section we first establish using Girsanov’s theorem that for a
sufficiently broad class of models for $f(t)$ there is an EMM $Q^{X}$ on the
pseudo-market (more precisely, we will find a suitable numeraire) and then,
assuming existence of an EMM $Q^{X},$ we write pricing formulas.

######  Remark 3.2.

We can introduce $S_{\text{€}/X}(t)=f^{\alpha}(t)$ with any $\alpha\in(0,1),$
then $S_{X/\$}(t)=f^{\alpha-1}(t).$ Each particular $\alpha$ leads to the
corresponding numeraire suitable for the stated purposes (but note that the
numeraires associated with the original USD and EUR markets are not suitable
for the set objective as discussed in Section 2). Arbitrariness of $\alpha$
can potentially be used for calibration purposes but we do not consider this
aspect here. For clarity and also for the sake of symmetry, we choose to use
$\alpha=1/2$ in this paper.

######  Remark 3.3.

We do not attach any economic interpretation to the intermediate currency. Our
interest is purely motivated by calibration aspects. We also note that we
model a single exchange rate which is natural, not $2$ rates as in [11, 13,
12, 8].

###  3.1 An EMM for the intermediate market

Consider the virtual market which domestic currency is X. On this market we
have two risky assets: USD paid by X and EUR paid by X:

$Y_{\text{€}/X}(t)=S_{\text{€}/X}(t)B_{\text{€}}(t),\ \ \
Y_{\$/X}(t)=S_{\$/X}(t)B_{\$}(t).$ (3.3)

Assume that EUR-USD exchange rate $f(t)$ satisfies the model (2.1). Based on
(2.1), we can write the SDEs under market measure for $Y_{\text{€}/X}(t)$ and
$Y_{\$/X}(t)$:

$\displaystyle dY_{\text{€}/X}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\mu(t)+2r_{\text{€}}(t)-\frac{\sigma^{2}(t)}{4}\right)Y_{\text{€}/X}dt+\frac{\sigma(t)}{2}Y_{\text{€}/X}dW(t),$
$\displaystyle dY_{\$/X}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\mu(t)+2r_{\$}(t)+\frac{3\sigma^{2}(t)}{4}\right)Y_{\$/X}dt-\frac{\sigma(t)}{2}Y_{\$/X}dW(t).$

If we choose the intermediate currency interest rate $r_{X}$ equal to

$r_{X}(t)=\frac{r_{\$}(t)+r_{\text{€}}(t)}{2}+\frac{\sigma^{2}(t)}{8},$ (3.4)

then there is an EMM $Q^{X}$ for the pseudo-currency market with the following
market price of risk $\gamma(t)$:

$\gamma(t)=\frac{\mu(t)-\frac{\sigma^{2}(t)}{2}+r_{\text{€}}(t)-r_{\$}(t)}{\sigma(t)},$
(3.5)

i.e.

$\displaystyle dY_{\text{€}/X}$ $\displaystyle=$ $\displaystyle
r_{X}(t)Y_{\text{€}/X}dt-\frac{\sigma(t)}{2}Y_{\text{€}/X}dW^{Q^{X}},$
$\displaystyle dY_{\$/X}$ $\displaystyle=$ $\displaystyle
r_{X}(t)Y_{\$/X}dt+\frac{\sigma(t)}{2}Y_{\$/X}dW^{Q^{X}},$

where $W^{Q^{X}}$ is the standard Wiener process under $Q^{X}$. Thus, we have
shown that the intermediate pseudo-currency market can be arbitrage free
within a sufficiently broad setting. We summarise this result in the following
statement.

######  Theorem 3.4.

Assume that the EUR-USD currency market under a ‘market’ measure is described
by the model (2.1). Then there is the unique intermediate currency interest
rate $r_{X}(t)$ defined in (3.4) and an EMM $Q^{X}$ for the intermediate
pseudo-currency market with the market price of risk $\gamma(t)$ from (3.5),
i.e., under (3.4) the market is arbitrage-free.

We see from (3.4) that even if the short rates $r_{\$}(t)$ and
$r_{\text{€}}(t)$ are assumed to be constant, the intermediate currency
interest rate $r_{X}(t)$ is non-constant if the volatility $\sigma(t)$ is
time-dependent. In particular, if $\sigma(t)$ is a stochastic process, then so
is the short rate $r_{X}(t)$.

######  Example 3.1 (An analogue of the Garman-Kohlhagen formula).

Assume that the exchange rate between EUR and USD $f(t)=S_{\text{€}/\$}(t)$
satisfies the model (2.1) with constant coefficients: $\sigma(t)=\sigma$,
$r_{\text{€}}(t)=r_{\text{€}}$ and $r_{\$}(t)=r_{\$}$. Note that in this
simplified case (the geometric Brownian motion case) the intermediate currency
interest rate $r_{X}$ is constant. Analogously, to the standard derivation of
the Garman-Kohlhagen formula, we can find option prices for a pseudo-currency
market investor. For a European floating-strike call option (priced in X) to
buy $1$ EUR for $\frac{K}{\sqrt{f(T)}}$ X, we have

$\displaystyle\mathcal{E}_{\text{€}/X}^{C}(0,T,f(0),K,r_{\$},r_{\text{€}})$
$\displaystyle=$ $\displaystyle
e^{-r_{X}T}\mathbb{E}_{Q^{X}}\left[\left(\sqrt{f(T)}-\frac{K}{\sqrt{f(T)}}\right)_{+}\right]$
$\displaystyle=$
$\displaystyle\sqrt{f(0)}e^{-r_{\text{€}}T}N\left(\frac{\log\frac{f(0)}{K}+(r_{\$}-r_{\text{€}}+\frac{\sigma^{2}}{2})T}{\sigma\sqrt{T}}\right)$
$\displaystyle-\frac{K}{\sqrt{f(0)}}e^{-r_{\$}T}N\left(\frac{\log\frac{f(0)}{K}+(r_{\$}-r_{\text{€}}-\frac{\sigma^{2}}{2})T}{\sigma\sqrt{T}}\right).$

And, similarly for a European floating-strike put option (priced in X) to sell
$1$ USD for $\frac{\sqrt{f(T)}}{K}$ X we have:

$\displaystyle\mathcal{E}_{\$/X}^{P}(0,T,\frac{1}{f(0)},\frac{1}{K},r_{\text{€}},r_{\$})$
$\displaystyle=$ $\displaystyle
e^{-r_{X}T}\mathbb{E}_{Q^{X}}\left[\left(\frac{\sqrt{f(t)}}{K}-\frac{1}{\sqrt{f(t)}}\right)_{+}\right]$
$\displaystyle=$
$\displaystyle\frac{\sqrt{f(0)}}{K}e^{-r_{\$}T}N\left(\frac{\log\frac{f(0)}{K}+(r_{\$}-r_{\text{€}}+\frac{\sigma^{2}}{2})T}{\sigma\sqrt{T}}\right)$
$\displaystyle-\frac{1}{\sqrt{f(0)}}e^{-r_{\text{€}}T}N\left(\frac{\log\frac{f(0)}{K}+(r_{\$}-r_{\text{€}}-\frac{\sigma^{2}}{2})T}{\sigma\sqrt{T}}\right).$

From (3.1) and (3.1), we can deduce prices for the call $C_{\text{€}/\$}$ and
put $P_{\$/\text{€}}$. To this end, we first observe that the in-the-money
payoff of the option $\mathcal{E}_{\text{€}/X}^{C}$ (priced in X) is
equivalent to buying €$1$ for $\$K.$ Indeed, this call’s payoff is equal to
the amount of X

$\left(\sqrt{f(T)}-\frac{K}{\sqrt{f(T)}}\right)_{+}$

which is equivalent to the amount of USD

$\sqrt{f(T)}\left(\sqrt{f(T)}-\frac{K}{\sqrt{f(T)}}\right)_{+}=\left(f(T)-K\right)_{+}$

as we can exchange $X$ for USD at the rate $\sqrt{f(T)}.$ Analogously, the in-
the-money payoff of $P_{\$/X}$ is equivalent to selling $\$1$ USD for €$1/K$.

Further, by multiplying the price of $\mathcal{E}_{\text{€}/X}^{C}$ priced in
X by $\sqrt{f(0)}$, we convert its option price in X to the price in USD, and
by multiplying the price of $\mathcal{E}_{\$/X}^{P}$ priced in X by
$1/\sqrt{f(0)},$ we convert its price to EUR. Hence

$\displaystyle C_{\text{€}/\$}(0,T,f(0),K,r_{\$},r_{\text{€}})$
$\displaystyle=$
$\displaystyle\sqrt{f(0)}\mathcal{E}_{\text{€}/X}^{C}(0,T,f(0),K,r_{\$},r_{\text{€}}),$
(3.8) $\displaystyle
P_{\$/\text{€}}\left(0,T,\frac{1}{f(0)},\frac{1}{K},r_{\text{€}},r_{\$}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{f(0)}}\mathcal{E}_{\$/X}^{P}\left(0,T,\frac{1}{f(0)},\frac{1}{K},r_{\text{€}},r_{\$}\right).$

Comparing the resulting formulas for $C_{\text{€}/\$}$ and $P_{\$/\text{€}}$,
it is not difficult to show that the foreign-domestic symmetry (2.6) holds,
which is also obvious from the duality principle (see [14] and references
therein).

Now let us consider a general FX option pricing formula based on the
intermediate currency. Let $S_{\text{€}/\$}(t)=f(t)$ be the EUR-USD exchange
rate at time $t$ defined on a filtered probability space
$(\Omega,{\mathcal{F}},\\{{\mathcal{F}}_{t}\\},Q^{X}),$ where $Q^{X}$ is an
EMM corresponding to the virtual market for which the intermediate currency X
is domestic (note that at the start of this subsection we demonstrated that
there is a broad class of models for which $Q^{X}$ exists). Assume that the
distribution of $f(t)$ is such that $f(t)$ and $1/f(t)$ have second moments.
We remark that we do not assume a particular model for $f(t)$ in the pricing
part of this section. For simplicity, let the interest rates for the USD and
EUR money markets, $r_{\$}$ and $r_{\text{€}}$, be constant. As we mentioned
earlier, the intermediate currency interest rate $r_{X}(t)$ is, in general,
not constant even when $r_{\$}$ and $r_{\text{€}}$ are constant. We assume
that $r_{X}(t)$ is adapted to the same filtration ${\mathcal{F}}_{t}$ and

$B_{X}(t)=\exp\left(\int_{0}^{t}r_{X}(s)ds\right).$ (3.9)

Introduce the discounting factor $D_{X}(t,T)$ related to the intermediate
currency interest rate and the intermediate currency non-defaultable zero-
coupon bond price $P_{X}(t,T)$:

$D_{X}(t,T)=\exp\left(-\int_{t}^{T}r_{X}(s)ds\right)$ (3.10)

and

$P_{X}(t,T)=\mathbb{E}_{Q^{X}}\left[D_{X}(t,T)|{\mathcal{F}}_{t}\right],$
(3.11)

where we assumed that $D_{X}(t,T)$ has finite moments. Since $Q^{X}$ is EMM,
the discounted $Y_{\text{€}/X}(t)$ and $Y_{\$/X}(t)$,

$D_{X}(0,t)Y_{\text{€}/X}(t)=D_{X}(0,t)S_{\text{€}/X}(t)B_{\text{€}}(t)=D_{X}(0,t)\sqrt{f(t)}B_{\text{€}}(t)$

and

$D_{X}(0,t)Y_{\$/X}(t)=D_{X}(0,t)S_{\$/X}(t)B_{\$}(t)=D_{X}(0,t)\frac{1}{\sqrt{f(t)}}B_{\$}(t),$

are $Q^{X}$-martingales. Hence we obtain for any $t\geq 0$

$\displaystyle\sqrt{f(0)}$ $\displaystyle=$ $\displaystyle
e^{r_{\text{€}}t}\mathbb{E}_{Q^{X}}\left[D_{X}(0,t)\sqrt{f(t)}\right],$
$\displaystyle\frac{1}{\sqrt{f(0)}}$ $\displaystyle=$ $\displaystyle
e^{r_{\$}t}\mathbb{E}_{Q^{X}}\left[\frac{D_{X}(0,t)}{\sqrt{f(t)}}\right].$

Therefore, to obey the no-arbitrage condition, the distribution of $f(t),$
$t\geq 0,$ under $Q^{X}$ should be so that

$\frac{\mathbb{E}_{Q^{X}}\left[D_{X}(0,t)\sqrt{f(t)}\right]}{\mathbb{E}_{Q^{X}}\left[\frac{D_{X}(0,t)}{\sqrt{f(t)}}\right]}=e^{(r_{\$}-r_{\text{€}})t}f(0).$
(3.12)

In option pricing we will consider the following natural class of payoff
functions $g(x;K),$ where $x>0$ denotes a price of the underlier and $K\geq 0$
has the meaning of a strike.

######  Assumption 3.1.

Let payoff functions $g(x;K)$ be homogeneous functions of order $1,$ i.e. for
any $a>0:$

$a\cdot g(x;K)=g(ax;aK).$ (3.13)

It is clear that e.g. plain vanilla puts and calls satisfy (3.13). For
definiteness, assume that $g(x;K)$ is a payoff of an option written on one
EUR, where $x$ has the meaning of EUR-USD exchange rate, and $K$ and $g$ are
denominated in USD. As in the case of a call (see Example 3.1), the amount of
USD $g(x;K)$ is equivalent to the amount $G(x;K)$ in X:

$G(x;K):=\frac{1}{\sqrt{x}}g(x;K)=g\left(\sqrt{x};\frac{K}{\sqrt{x}}\right),$

where $1/\sqrt{x}$ has the meaning of the exchange rate USD-X (cf. Definition
3.1) and $G(x;K)$ and $K/\sqrt{x}$ are denominated in $X$. According to the
risk-neutral pricing theory, we can write the value of the European option
$V_{\text{€}/X}(t)$ with payoff $g(\sqrt{x};\frac{K}{\sqrt{x}})$ and maturity
$T$ at time $t\leq T$ as

$V_{\text{€}/X}(t)=\mathbb{E}_{Q^{X}}\left[\left.D_{X}(t,T)g\left(\sqrt{f(T)};\frac{K}{\sqrt{f(T)}}\right)\right|{\mathcal{F}}_{t}\right].$

Note that this is an option on EUR priced in X. The price in dollars for this
option is

$V_{\text{€}/\$}(t)=\sqrt{f(t)}\mathbb{E}_{Q^{X}}\left[\left.D_{X}(t,T)g\left(\sqrt{f(T)};\frac{K}{\sqrt{f(T)}}\right)\right|{\mathcal{F}}_{t}\right].$
(3.14)

Analogously, we can derive a formula for an option on USD priced in EUR:

$V_{\$/\text{€}}(t)=\frac{1}{\sqrt{f(t)}}\mathbb{E}_{Q^{X}}\left[\left.D_{X}(t,T)g\left(\frac{1}{\sqrt{f(T)}};\sqrt{f(T)}K\right)\right|{\mathcal{F}}_{t}\right],$
(3.15)

where $g(y;K)$ is a payoff of an option written on one USD, $y$ has the
meaning of USD-EUR exchange rate, and $K$ and $g$ are denominated in EUR. We
summarise this result in the following theorem.

######  Theorem 3.5.

Assume that the EUR-USD exchange rate $f(t)$ satisfies a model for which the
no-arbitrage condition (3.12) holds. Then the arbitrage price of a European
option on EUR with a payoff $g(x;K)$ being homogeneous function of order $1$
and maturity time $T$ is given by (3.14) and the arbitrage price of an option
on USD is given by (3.15).

It is not difficult to show that the foreign-domestic symmetry (2.6) holds
when we use the pricing formulas (3.14) and (3.15) based on the intermediate
currency, which also follows from the duality principle (see [14] and
references therein).

###  3.2 T-forward measure for the intermediate market

Introduce the T-forward measure $Q_{T}^{X}$ equivalent to $Q^{X}$ on
${\mathcal{F}}_{T}$ with the Radon-Nikodym derivative

$\frac{Q_{T}^{X}}{Q^{X}}=\frac{1}{P_{X}(0,T)B_{X}(T)}$ (3.16)

and for $t>0$

$E_{Q}\left[\left.\frac{Q_{T}^{X}}{Q^{X}}\right|\mathcal{F}_{t}\right]=\frac{P_{X}(t,T)}{P_{X}(0,T)B_{X}(t)}.$
(3.17)

Under this forward measure, we get [22, 16] (see also [4])$:$

$\displaystyle\sqrt{f(0)}$ $\displaystyle=$ $\displaystyle
e^{r_{\text{€}}T}\mathbb{E}_{Q^{X}}\left[D_{X}(0,T)\sqrt{f(T)}\right]=e^{r_{\text{€}}T}P_{X}(0,T)\mathbb{E}_{Q_{T}^{X}}\left[\sqrt{f(T)}\right],$
(3.18) $\displaystyle\frac{1}{\sqrt{f(0)}}$ $\displaystyle=$ $\displaystyle
e^{r_{\$}T}\mathbb{E}_{Q^{X}}\left[\frac{D_{X}(0,T)}{\sqrt{f(T)}}\right]=e^{r_{\$}T}P_{X}(0,T)\mathbb{E}_{Q_{T}^{X}}\left[\frac{1}{\sqrt{f(T)}}\right].$

Then the no-arbitrage condition (3.12) becomes

$\frac{\mathbb{E}_{Q_{T}^{X}}\left[\sqrt{f(T)}\right]}{\mathbb{E}_{Q_{T}^{X}}\left[\frac{1}{\sqrt{f(T)}}\right]}=e^{(r_{\$}-r_{\text{€}})T}f(0).$
(3.19)

Further, (3.18) implies that the bond price $P_{X}(0,T)$ should satisfy

$P_{X}(0,T)=e^{-r_{\text{€}}T}\frac{\sqrt{f(0)}}{\mathbb{E}_{Q_{T}^{X}}\sqrt{f(T)}}=e^{-r_{\$}T}\frac{1}{\sqrt{f(0)}\mathbb{E}_{Q_{T}^{X}}\left[\frac{1}{\sqrt{f(T)}}\right]}\
.$ (3.20)

Note that $f(0)$ is the current EUR-USD exchange rate and hence it is
observable as well as $r_{\$}$ and $r_{\text{€}}$. The current forward EUR-USD
exchange rate

$F_{\text{€}/\$}(0,T)=e^{(r_{\$}-r_{\text{€}})T}f(0)$ (3.21)

is also observable on the USD market.

We remark that the forward EUR-X and USD-X exchange rates,

$F_{\text{€}/X}(t,T)=e^{-r_{\text{€}}(T-t)}\frac{\sqrt{f(t)}}{P_{X}(t,T)}\text{
and }F_{\$/X}(t,T)=e^{-r_{\$}(T-t)}\frac{1}{P_{X}(t,T)\sqrt{f(t)}},$ (3.22)

are both $Q_{T}^{X}$-martingales. For convenience, we recall that if
$r_{X}(t)$ is deterministic then the two measures $Q^{X}$ and $Q_{T}^{X}$
coincide.

It is also not difficult to show that

$\displaystyle\sqrt{f(t)}$ $\displaystyle=$ $\displaystyle
e^{r_{\text{€}}(T-t)}\mathbb{E}_{Q^{X}}\left[\left.D_{X}(t,T)\sqrt{f(T)}\right|{\mathcal{F}}_{t}\right]=e^{r_{\text{€}}(T-t)}P_{X}(t,T)\mathbb{E}_{Q_{T}^{X}}\left[\left.\sqrt{f(T)}\right|{\mathcal{F}}_{t}\right],$
$\displaystyle\frac{1}{\sqrt{f(t)}}$ $\displaystyle=$ $\displaystyle
e^{r_{\$}(T-t)}\mathbb{E}_{Q^{X}}\left[\left.\frac{D_{X}(t,T)}{\sqrt{f(T)}}\right|{\mathcal{F}}_{t}\right]=e^{r_{\text{\$}}(T-t)}P_{X}(t,T)\mathbb{E}_{Q_{T}^{X}}\left[\left.\frac{1}{\sqrt{f(T)}}\right|{\mathcal{F}}_{t}\right].$

Then

$P_{X}(t,T)=e^{-r_{\text{€}}(T-t)}\frac{\sqrt{f(t)}}{\mathbb{E}_{Q_{T}^{X}}\left[\left.\sqrt{f(T)}\right|{\mathcal{F}}_{t}\right]}=e^{-r_{\$}(T-t)}\frac{1}{\sqrt{f(t)}\mathbb{E}_{Q_{T}^{X}}\left[\left.\frac{1}{\sqrt{f(T)}}\right|{\mathcal{F}}_{t}\right]}\
.$ (3.23)

The pricing formula (3.14) under the T-forward measure $Q_{T}^{X}$ becomes

$\displaystyle V_{\text{€}/\$}(t)$ $\displaystyle=$
$\displaystyle\sqrt{f(t)}\mathbb{E}_{Q^{X}}\left[\left.D_{X}(t,T)g\left(\sqrt{f(T)};\frac{K}{\sqrt{f(T)}}\right)\right|{\mathcal{F}}_{t}\right]$
$\displaystyle=$
$\displaystyle\sqrt{f(t)}P_{X}(t,T)\mathbb{E}_{Q_{T}^{X}}\left[\left.g\left(\sqrt{f(T)};\frac{K}{\sqrt{f(T)}}\right)\right|{\mathcal{F}}_{t}\right]$
$\displaystyle=$
$\displaystyle\frac{e^{-r_{\$}(T-t)}}{\mathbb{E}_{Q_{T}^{X}}\left[\left.\frac{1}{\sqrt{f(T)}}\right|{\mathcal{F}}_{t}\right]}\mathbb{E}_{Q_{T}^{X}}\left[\left.g\left(\sqrt{f(T)};\frac{K}{\sqrt{f(T)}}\right)\right|{\mathcal{F}}_{t}\right],$

where in the last line we used (3.23). Analogously we have (see (3.15)):

$V_{\$/\text{€}}(t)=\frac{e^{-r_{\text{€}}(T-t)}}{\mathbb{E}_{Q_{T}^{X}}\left[\left.\sqrt{f(T)}\right|{\mathcal{F}}_{t}\right]}\mathbb{E}_{Q_{T}^{X}}\left[\left.g\left(\frac{1}{\sqrt{f(t)}};\sqrt{f(T)}K\right)\right|{\mathcal{F}}_{t}\right].$
(3.25)

We summarize this result in the next theorem.

######  Theorem 3.6.

Assume that the EUR-USD exchange rate $f(t)$ satisfies a model for which the
no-arbitrage condition (3.12) or (3.19) holds. Then the arbitrage price of an
option on EUR with a payoff $g(x;K)$ being homogeneous function of order
$1$and maturity time $T$ is given by (3.2) and the arbitrage price of an
option on USD is given by (3.25).

The benefit of (3.2) and (3.25) vs (3.14) and (3.15) is that in (3.2) and
(3.25) we do not need to compute the intermediate currency interest rate
$r_{X}(t).$

######  Example 3.2.

The prices of the call for buying €$1$ for $\$K$ and of the put for selling
$\$1$ for €$1/K$ are equal to

$\displaystyle C_{\text{€}/\$}(0,T,K)$
$\displaystyle=\frac{e^{-r_{\$}T}}{\mathbb{E}_{Q_{T}^{X}}\left[\frac{1}{\sqrt{f(T)}}\right]}\mathbb{E}_{Q_{T}^{X}}\left[\left(\sqrt{f(T)}-\frac{K}{\sqrt{f(T)}}\right)_{+}\right],$
(3.26) $\displaystyle P_{\$/\text{€}}\left(0,T,\frac{1}{K}\right)$
$\displaystyle=\frac{e^{-r_{\text{€}}T}}{\mathbb{E}_{Q_{T}^{X}}\sqrt{f(T)}}\mathbb{E}_{Q_{T}^{X}}\left[\left(\frac{\sqrt{f(T)}}{K}-\frac{1}{\sqrt{f(T)}}\right)_{+}\right].$

To conclude, we derived the consistent pricing formulas for FX options.
Although the formulas are derived using the virtual X market, their evaluation
depends on parameters of the USD and EUR markets only. When we are interested
in option prices at the current time $t=0,$ they are valid for any
distribution (i.e., we do not need to explicitly define the process $f(t))$ of
the exchange rate $f(T)$ which satisfies (3.19). We will demonstrate this
observation in illustrations of these pricing formulas in Section 5.

##  4 Extension to the multi–currencies case

In this section we extend the approach of pricing FX options developed in the
previous section to the multi-currencies case. Let us assume we have $N$
currencies $c_{i}$, where $i=1,\dots,N$. Fixing one currency, for definiteness
$i=N$, we can introduce the $N-1$ exchange rates

$f_{j}=S_{c_{j}/c_{N}}>0,\quad j=1,\ldots,N-1,$ (4.1)

which denote the exchange rates between the currency $c_{N}$ to all other
currencies $c_{i},$ $i=1,\dots,N-1$.

Now we introduce the intermediate currency $X$ by defining the $N$ exchange
rates $S_{c_{i}/X}$ as follows

$S_{c_{i}/X}=f_{1}^{b_{i1}}\times f_{2}^{b_{i2}}\times\cdots\times
f_{N-1}^{b_{iN-1}},\quad i=1,\ldots,N,$ (4.2)

where $b_{ij}\in\mathbb{R}$ are so that

$\displaystyle b_{ii}$ $\displaystyle=$ $\displaystyle 1-\alpha_{i},\
i=1,\ldots,N,$ $\displaystyle b_{ij}$ $\displaystyle=$
$\displaystyle-\alpha_{j},\ i\neq j,\ \ i,j=1,\ldots,N.$

By symmetry arguments (see also Remark 4.2 below), we choose

$\alpha_{i}=\frac{1}{N},\ \ i=1,\ldots,N-1.$ (4.3)

Note that $S_{c_{i}/X}$ is the exchange rate between the observable currency
$c_{i}$ and the introduced intermediate currency $X$ and hence it is the worth
of $1$ unit of currency $c_{i}$ in the intermediate currency $X$. In the case
(4.1)-(4.2), (4.3), the exchange rate $S_{c_{i}/X}$ can be written in the
concise form via geometric mean $GM(f_{j})$ of the sequence of $f_{j}:$

$S_{c_{i}/X}=f_{i}\left[\left(\prod_{j=1}^{N-1}\frac{1}{f_{j}}\right)^{1/(N-1)}\right]^{(N-1)/N}:=f_{i}\left[GM(f_{j})\right]^{(N-1)/N}.$

We assume that the currency market under a ‘market’ measure $P$ is described
by the system:

$\displaystyle df_{j}$
$\displaystyle=\mu_{j}(t)f_{j}dt+\sigma_{j}(t)f_{j}d\tilde{W}_{j},\quad
j=1,\ldots,N-1,$ (4.4) $\displaystyle d\tilde{W}_{l}d\tilde{W}_{k}$
$\displaystyle=d\tilde{W}_{k}d\tilde{W}_{l}=\rho_{lk}(t)dt,\quad
l,k=1,\ldots,N-1,$

and

$\displaystyle dB_{i}$ $\displaystyle=r_{i}(t)B_{i}dt,\quad i=1,\ldots,N,$
(4.5)

where $B_{i}(t)$ describes the bank account of currency $c_{i}$ with its short
rate $r_{i}(t)$; $\sigma_{j}(t)>0$ is the volatility of the exchange rate
$f_{j}(t)$, $\mu_{j}(t)$ is its drift; and
$\tilde{W}(t)=(\tilde{W}_{1}(t),\ldots,\tilde{W}_{N-1}(t))^{T}$ is an
$N-1$–dimensional correlated Wiener process with the correlation matrix
$R(t)\in\mathbb{R}^{N-1\times N-1}$ which components we denote by
$\rho_{ij}(t)$ (obviously $\rho_{ii}=1).$ It is assumed that $r_{i}(t)$,
$\sigma_{j}(t)$, $\mu_{j}(t)$ are stochastic processes adapted to a filtration
$\mathcal{F}_{t}$ to which $\tilde{W}(t)$ is also adapted, and they have
bounded second moments and $\sigma_{j}(t)$ satisfy Novikov’s condition.
Furthermore, let us assume that the matrix $R$ is symmetric strictly positive
definite. Then using the Cholesky decomposition, we can represent $R=LL^{T}$,
where $L\in\mathbb{R}^{N-1\times N-1}$ is a lower triangular matrix with
entries $L_{i,j}$. Using this decomposition, we can rewrite the SDEs (4.4) as

$df_{j}=\mu_{j}(t)f_{j}dt+\sigma_{j}(t)f_{j}\sum\limits_{k=1}^{j}L_{jk}(t)dW_{k},\quad
j=1,\ldots,N-1,$ (4.6)

where

$L_{ii}(t)=\sqrt{1-\sum\limits_{k=1}^{i-1}L_{ik}^{2}(t)},\quad
L_{ji}(t)=\frac{\rho_{ij}-\sum\limits_{k=1}^{i-1}L_{jk}(t)L_{ik}(t)}{L_{ii}(t)},\quad\text{for
}j>i,$

and $W(t)=(W_{1}(t),\ldots,W_{N-1}(t))^{T}$ is an $N-1$–dimensional standard
Wiener process. We first show that the intermediate currency introduced in
(4.2) permits an arbitrage-free market involving all $N$ currencies.

######  Theorem 4.1.

Assume that $N-1$ exchange rates $f_{j}$ between the currency $c_{N}$ to all
other currencies $c_{i},$ $i=1,\dots,N-1,$ under a ‘market’ measure are
described by the model (4.6) together with (4.5). Consider the intermediate
currency $X$ introduced in (4.2). There is the unique intermediate currency
interest rate $r_{X}(t)$ defined by

$r_{X}(t)=\frac{1}{N}\sum\limits_{i=1}^{N}r_{i}(t)+\frac{1}{2N}\left(1-\frac{1}{N}\right)\sum\limits_{i=1}^{N-1}\sigma_{i}^{2}(t)-\frac{1}{N^{2}}\sum\limits_{k=1}^{N-1}\sum\limits_{j=1}^{k-1}\sigma_{j}(t)\sigma_{k}(t)\rho_{jk}(t)$
(4.7)

and there is an EMM $Q^{X}$ for the intermediate pseudo-currency market, i.e.,
under (4.7) this market is arbitrage-free.

Proof. Applying the Ito formula to (4.2), we obtain the SDEs for the exchange
rates $S_{c_{i}/X}$:

$\displaystyle\frac{dS_{c_{i}/X}}{S_{c_{i}/X}}$
$\displaystyle=\left[\frac{1}{N}\sum\limits_{j=1}^{N-1}\left(\frac{1}{2}\left(\frac{1}{N}+1\right)\sigma_{j}^{2}-\sigma_{i}\mathbbm{1}_{i\neq
N}\sigma_{j}\rho_{ij}+\frac{1}{N}\sigma_{j}\sum\limits_{k=1}^{j-1}\sigma_{k}\rho_{kj}-\mu_{j}\right)+\mu_{i}\mathbbm{1}_{i\neq
N}\right]dt$
$\displaystyle\quad-\frac{1}{N}\sum\limits_{j=1}^{N-1}\sum\limits_{k=1}^{j}\sigma_{j}L_{jk}dW_{k}+\sigma_{i}\mathbbm{1}_{i\neq
N}\sum\limits_{k=1}^{i}L_{ij}dW_{k},\quad i=1,\ldots,N.$

On the considered market the risky assets have the prices
$Y_{c_{i}/X}=S_{c_{i}/X}B_{i},$ $i=1,\ldots,N$. Introduce the discounted risky
assets’ prices in the usual way:

$\tilde{Y}_{c_{i}/X}(t)=\frac{S_{c_{i}/X}(t)B_{i}(t)}{B_{X}(t)},\quad
i=1,\ldots,N.$ (4.8)

The discounted prices satisfy the SDEs

$\displaystyle\frac{d\tilde{Y}_{c_{i}/X}}{\tilde{Y}_{c_{i}/X}}$
$\displaystyle=\left[r_{i}-r_{X}\right]dt$
$\displaystyle+\left[\frac{1}{N}\sum\limits_{j=1}^{N-1}\left(\frac{1}{2}\left(\frac{1}{N}+1\right)\sigma_{j}^{2}-\sigma_{i}\mathbbm{1}_{i\neq
N}\sigma_{j}\rho_{ij}+\frac{1}{N}\sigma_{j}\sum\limits_{k=1}^{j-1}\sigma_{k}\rho_{kj}-\mu_{j}\right)+\mu_{i}\mathbbm{1}_{i\neq
N}\right]dt$
$\displaystyle\quad-\frac{1}{N}\sum\limits_{j=1}^{N-1}\sum\limits_{k=1}^{j}\sigma_{j}L_{jk}dW_{k}+\sigma_{i}\mathbbm{1}_{i\neq
N}\sum\limits_{k=1}^{i}L_{ik}dW_{k},\quad i=1,\ldots,N.$

The no-arbitrage condition requires existence of an EMM $Q^{X}$ under which
all $\tilde{Y}_{c_{i}/X}$ are martingales. This implies that for $Q^{X}$ to
exist the following system of $N$ simultaneous linear algebraic equations in
$N$ unknown variables (which are the market prices of risk $\gamma_{k}$,
$k=1,\ldots,N-1$, and $r_{X})$ should have a solution:

$\displaystyle
r_{i}-r_{X}+\frac{1}{N}\sum\limits_{j=1}^{N-1}\left(\frac{1}{2}\left(\frac{1}{N}+1\right)\sigma_{j}^{2}-\sigma_{i}\mathbbm{1}_{i\neq
N}\sigma_{j}\rho_{ij}+\frac{1}{N}\sigma_{j}\sum\limits_{k=1}^{j-1}\sigma_{k}\rho_{kj}-\mu_{j}\right)+\mu_{i}\mathbbm{1}_{i\neq
N}\ \ \ \ \ \ \ \ \ \ \ \ $ $\displaystyle=$
$\displaystyle-\frac{1}{N}\sum\limits_{j=1}^{N-1}\sum\limits_{k=1}^{j}\sigma_{j}L_{jk}\gamma_{k}+\sigma_{i}\mathbbm{1}_{i\neq
N}\sum\limits_{k=1}^{i}L_{ik}\gamma_{k},\quad i=1,\ldots,N.$

Subtracting the equation (4) with $i=N$ from the equations (4) for $i\neq N,$
we obtain

$r_{i}-r_{N}+\mu_{i}-\frac{1}{N}\sigma_{i}\sum\limits_{k=1}^{N-1}\sigma_{k}\rho_{ik}=\sigma_{i}\sum\limits_{k=1}^{i}L_{ik}\gamma_{k},\quad
i=1,\ldots,N-1.$ (4.10)

Using (4.10), we recurrently find the market prices of risk:

$\gamma_{i}=\frac{r_{i}-r_{N}+\mu_{i}-\frac{1}{N}\sigma_{i}^{2}-\frac{1}{N}\sigma_{i}\sum\limits_{k=1}^{N-1}\sigma_{k}\rho_{ik}-\sigma_{i}\sum\limits_{k=1}^{i-1}L_{i,k}\gamma_{k}}{\sigma_{i}L_{i,i}},\quad
i=1,\ldots,N-1,$ (4.11)

which are well defined because due to our assumptions $\sigma_{i}>0$ and
$L_{i,i}>0.$ Further, sum up (4.10) over $i$ from $i=1$ to $N-1$ and
substitute the result in (4) with $i=N$ to confirm (4.7). The found
$\gamma_{i},$ $i=1,\ldots,N-1,$ from (4.11) and $r_{X}$ from (4.7) together
with Girsanov’s theorem ensure that there is an EMM $Q^{X}$ under which all
$\tilde{Y}_{c_{i}/X}$ are martingales. Hence, the considered market is
arbitrage free. Theorem 4.1 is proved. $\square$

######  Remark 4.2.

Recall that we chose to use $\alpha_{1}=\cdots=\alpha_{N-1}=\frac{1}{N}$ in
(4.2). If we repeat the proof of Theorem 4.1 for arbitrary $0<\alpha_{j}<1$
then we arrive at the following intermediate currency interest rate $r_{X}:$

$r_{X}=\left(1-\sum\limits_{j=1}^{N-1}\alpha_{j}\right)r_{N}+\sum\limits_{j=1}^{N-1}\alpha_{j}r_{j}+\sum\limits_{j=1}^{N-1}\frac{\alpha_{j}(1-\alpha_{j})}{2}\sigma_{j}^{2}-\sum\limits_{j=1}^{N-1}\sum\limits_{k=1}^{j-1}\sigma_{j}\alpha_{j}\alpha_{k}\sigma_{k}\rho_{jk}.$
(4.12)

ensuring that there is an EMM in this market. We see that the choice
$\alpha_{j}=\frac{1}{N}$ results in the symmetry so that each $r_{j}$ enters
(4.12) with the same weight. Other choices of $\alpha_{j}$ give a ‘preference’
to a particular currency(ies) and can be exploited for calibration purposes
but it is not considered here.

Analogously to Assumption 3.1, we will consider payoffs as first-order
homogeneous functions the multi-currencies case.

######  Assumption 4.1.

Let payoff functions $g(x_{1},\ldots,x_{N-1};K)$ be homogeneous functions of
order $1,$ i.e. for any $a>0$

$a\cdot g(x_{1},\ldots,x_{N-1};K)=g(ax_{1},\ldots,ax_{N-1};aK).$ (4.13)

Most of multi-currency options (e.g. basket options [6]) have payoffs
belonging to this class. Consider a European-type option with maturity time
$T$ and payoff in the currency $c_{N}$:

$g(T):=g(f_{1}(T),\ldots,f_{N-1}(T);K).$

Its equivalent value in the intermediate currency $X$ is equal to (see (4.2)):

$\displaystyle G(T):=$ $\displaystyle S_{c_{N}/X}(T)\cdot
g(T)=g\Big{(}f_{1}(T)S_{c_{N}/X}(T),\ldots,f_{N-1}(T)S_{c_{N}/X}(T);K\cdot
S_{c_{N}/X}(T)\Big{)}$ (4.14) $\displaystyle=$ $\displaystyle
g\Big{(}S_{c_{1}/X}(T),\ldots,S_{c_{N-1}/X}(T);K\cdot
S_{c_{N}/X}(T)\Big{)}=g\Big{(}S_{c_{1}/X}(T),\ldots,S_{c_{N}/X}(T);K^{\prime}\Big{)},$

where $K^{\prime}=K\cdot S_{c_{N}/X}(t)$ is the equivalent strike in $X$. It
is not difficult to see that at the maturity time $T$ the option holder is
indifferent between receiving $g(T)$ in currency $c_{N}$ or $G(T)$ in currency
$X$ as he can obtain the same amount by exchanging $G(T)$ to $c_{N}$:

$\displaystyle\frac{G(T)}{S_{c_{N}/X}(T)}$ $\displaystyle=$
$\displaystyle\frac{1}{S_{c_{N}/X}(T)}g\Big{(}S_{c_{1}/X}(T),\ldots,S_{c_{N-1}/X}(T);K\cdot
S_{c_{N}/X}(T)\Big{)}$ $\displaystyle=$ $\displaystyle
g\Big{(}f_{1}(T),\ldots,f_{N-1}(T);K\Big{)}.$

######  Example 4.1 (Basket option).

Consider a basket option on the $c_{N}$ market written on all $N-1$ exchange
rates $f_{i}(t),$ $i=1,\ldots,N-1,$ which pay-off function is of the form [6]:

$g(x_{1},\ldots,x_{N-1};K)=\left(\sum\limits_{i=1}^{N-1}\omega_{i}x_{i}-K\right)_{+},$

where $x_{i},$ $i=1,\ldots,N-1,$ and $K$ are denominated in the currency
$c_{N}$ and $\omega_{i}\geq 0,$ $i=1,\ldots,N-1,$ are some weights. The
equivalent pay-off on the $X$ currency market at the maturity $T$ is equal to

$\displaystyle G(T)$ $\displaystyle=S_{c_{N}/X}(T)\cdot
g(f_{1}(T),\ldots,f_{N-1}(T);K)$ (4.15)
$\displaystyle=S_{c_{N}/X}(T)\left(\sum\limits_{i=1}^{N-1}\omega_{i}f_{i}(T)-K\right)_{+}=\left(\sum\limits_{i=1}^{N-1}\omega_{i}S_{c_{i}/X}(t)-K\cdot
S_{c_{N}/X}(t)\right)_{+}$
$\displaystyle=\left(\sum\limits_{i=1}^{N-1}\omega_{i}S_{c_{i}/X}(t)-K^{\prime}\right)_{+},$

where $S_{c_{i}/X}(t)$ and $K^{\prime}$ are denominated in the intermediate
currency $X$.

As in the case of a single FX pair (see Theorem 3.4), we have demonstrated by
Theorem 4.1 that there is a sufficiently broad class of models for which there
is an EMM $Q^{X}$ with an appropriate choice of the intermediate currency
interest rate $r_{X}(t).$ We now generalize the pricing formulas of Theorems
3.5 and 3.6 from a single FX pair to the multi-currency case.

Let the exchange rates $f_{i}(t)$ between the currency $c_{N}$ to all other
currencies $c_{i},$ $i=1,\ldots,N-1,$ be defined on a filtered probability
space $(\Omega,{\mathcal{F}},\\{{\mathcal{F}}_{t}\\},Q^{X}),$ where $Q^{X}$ is
an EMM corresponding to the virtual market for which the intermediate currency
X is domestic. Assume that $f_{i}(t),$ $i=1,\ldots,N-1,$ and the exchange
rates $S_{c_{i}/X}$ between the pseudo-currency $X$ to all the currencies
$c_{i},$ $i=1,\ldots,N,$ defined in (4.2), (4.3) have second moments. Also,
assume that $r_{X}(t)$ is adapted to the same filtration ${\mathcal{F}}_{t}$
and recall the expressions and assumptions for the money market account
$B_{X}(t)$ (see (3.9)), the discounting factor $D_{X}(t,T)$ related to the
intermediate currency interest rate (see (3.10)) and the intermediate currency
zero-coupon bond price $P_{X}(t,T)$ (see (3.11)).

Since $Q^{X}$ is an EMM, the discounted $Y_{c_{i}/X}(t)$ for all
$i=1,\ldots,N$,

$\tilde{Y}_{c_{i}/X}=D_{X}(0,t)Y_{c_{i}/X}(t)=D_{X}(0,t)S_{c_{i}/X}(t)B_{c_{i}}(t),\quad
i=1,\ldots,N,$

are $Q^{X}$-martingales. Hence we obtain

$S_{c_{i}/X}(0)=e^{r_{i}t}\mathbb{E}_{Q^{X}}\left[D_{X}(0,t)S_{c_{i}/X}(t)\right],\quad
i=1,\ldots,N.$

Therefore, for all $i=1,\ldots,N-1$ and $t>0,$ we have

$\frac{\mathbb{E}_{Q^{X}}\left[D_{X}(0,t)S_{c_{i}/X}(t)\right]}{\mathbb{E}_{Q^{X}}\left[D_{X}(0,t)S_{c_{N}/X}(t)\right]}=e^{(r_{i}-r_{N})t}\frac{S_{c_{i}/X}(0)}{S_{c_{N}/X}(0)}=e^{(r_{N}-r_{i})t}f_{i}(0).$
(4.16)

Thus, to obey the no-arbitrage condition, the distributions of
$S_{c_{i}/X}(t)$, $t>0,$ under $Q^{X}$ should be so that (4.16) holds.

Consider a European option with maturity $T$ and pay-off function $G(T)$ on
the intermediate currency market. Its price in $X$ is equal to

$V_{X}(t)=\mathbb{E}_{Q^{X}}\left[D_{X}(0,T)G(T)|{\mathcal{F}}_{t}\right].$
(4.17)

Using (4.14), we obtain the price for this option in the currency $c_{N}$:

$V_{c_{N}}(t)=\frac{1}{S_{c_{N}/X}(t)}\mathbb{E}_{Q^{X}}\left[D_{X}(0,T)\cdot
g(S_{c_{1}/X}(T),\ldots,S_{c_{N-1}/X}(T);KS_{c_{N}/X}(T))|{\mathcal{F}}_{t}\right].$
(4.18)

Then the analog of Theorem 3.5 is as follows.

######  Theorem 4.3.

Assume that the exchange rates $f_{i}(t),$ $i=1,\ldots,N-1,$ (or
$S_{c_{i}/X}(t),$ $i=1,\ldots,N)$ satisfy a model for which the no-arbitrage
condition (4.16) holds. Then on the $c_{N}$ market the arbitrage price
$V_{c_{N}}(t)$ of a European option on $c_{1},$ …, $c_{N-1}$ currencies with a
first-order homogeneous payoff function $g(x_{1},\ldots,x_{N-1};K)$ and
maturity time $T$ is given by (4.18).

Introduce the T-forward measure $Q_{T}^{X}$ equivalent to $Q^{X}$ on
${\mathcal{F}}_{T}$ with the Radon-Nikodym derivative as in (3.16) (see also
(3.17)). Under this forward measure, we get

$S_{c_{i}/X}(0)=e^{r_{i}T}\mathbb{E}_{Q^{X}}\left[D_{X}(0,T)S_{c_{i}/X}(T)\right]=e^{r_{i}T}P_{X}(0,T)\mathbb{E}_{Q_{T}^{X}}\left[S_{c_{i}/X}(T)\right],\quad
i=1,\ldots,N.$ (4.19)

Then the no-arbitrage conditions (4.16) become

$\frac{\mathbb{E}_{Q_{T}^{X}}\left[S_{c_{i}/X}(T)\right]}{\mathbb{E}_{Q_{T}^{X}}\left[S_{c_{N}/X}(T)\right]}=e^{(r_{N}-r_{i})T}f_{i}(0),\quad
i=1,\ldots,N-1.$ (4.20)

Here $F_{c_{i}/c_{N}}(0)=e^{(r_{N}-r_{i})T}f_{i}(0)$ is the current forward
$c_{i}$-$c_{N}$ exchange rate. It follows from (4.20) that for any
$j=1,\ldots,N:$

$\frac{\mathbb{E}_{Q_{T}^{X}}\left[S_{c_{i}/X}(T)\right]}{\mathbb{E}_{Q_{T}^{X}}\left[S_{c_{j}/X}(T)\right]}=e^{(r_{N}-r_{i})T}f_{i}(0),\quad
i=1,\ldots,N,\ i\neq j.$ (4.21)

We remark that the no-arbitrage condition does not depend on the choice of
$c_{N}$ used in (4.1).

Further, (4.19) implies that the bond price $P_{X}(0,T)$ should satisfy

$P_{X}(0,T)=e^{-r_{i}T}\frac{S_{c_{i}/X}(0)}{\mathbb{E}_{Q_{T}^{X}}S_{c_{i}/X}(T)},\quad
i=1,\ldots,N.$ (4.22)

We observe that the relationships (4.20) ensure that (4.22) holds for all
$i=1,\ldots,N.$ Note that $f_{i}(0)$ are the current $c_{i}/c_{N}$ exchange
rates and hence $S_{c_{i}/X}(0)$ (see (4.2)) are observable as well as all
$r_{i}$. Similarly to (3.23), we also have

$P_{X}(t,T)=e^{-r_{i}(T-t)}\frac{S_{c_{i}/X}(t)}{\mathbb{E}_{Q_{T}^{X}}\left[S_{c_{i}/X}(T)|{\mathcal{F}}_{t}\right]},\quad
i=1,\ldots,N.$

Analogously to (3.25), the pricing formula (4.18) under the T-forward measure
$Q_{T}^{X}$ becomes

$V_{c_{N}}(t)=\frac{e^{-r_{N}(T-t)}}{\mathbb{E}_{Q_{T}^{X}}\left[S_{c_{N}/X}(T)|{\mathcal{F}}_{t}\right]}\mathbb{E}_{Q_{T}^{X}}\left[g(S_{c_{1}/X}(T),\ldots,S_{c_{N-1}/X}(T);KS_{c_{N}/X}(T))|{\mathcal{F}}_{t}\right].$
(4.23)

Then the analog of Theorem 3.6 is as follows.

######  Theorem 4.4.

Assume that the exchange rates $f_{i}(t),$ $i=1,\ldots,N-1,$ (or
$S_{c_{i}/X}(t),$ $i=1,\ldots,N)$ satisfy a model for which the no-arbitrage
condition (4.20) (or (4.16)) holds. Then on the $c_{N}$ market the arbitrage
price $V_{c_{N}}(t)$ of a European option on $c_{1},$ …, $c_{N-1}$ currencies
with a first-order homogeneous payoff function $g(x_{1},\ldots,x_{N-1};K)$ and
maturity time $T$ is given by (4.23).

It is clear that the pricing formula (4.23) remains true if we replace the
currency $c_{N}$ with any other $c_{j}$ and the first-order homogeneous payoff
function $g(x_{1},\ldots,x_{j-1},x_{j+1},\ldots,x_{N};K)$ is denominated in
$c_{j}.$ We now return to Example 4.1.

######  Example 4.2 (Basket option pricing).

Let us make the same assumptions as in Example 4.1 and find the arbitrage
price of a European option with pay-off $G(T)$ at maturity $T$ on the $X$
currency market given by

$G(t)=\left(\sum\limits_{i=1}^{N-1}\omega_{i}S_{c_{i}/X}(t)-K\right)_{+},$

where $S_{c_{i}/X}(t)$ and $K$ are denominated in $X$. Following Theorem 4.4,
the price of this basket option at time $0$, denominated in currency $c_{N}$,
is equal to

$BasketOption_{c_{N}}(0)=\frac{e^{-r_{N}T}}{\mathbb{E}_{Q_{T}^{X}}S_{c_{N}/X}(T)}\mathbb{E}_{Q_{T}^{X}}\left[\left(\sum\limits_{i=1}^{N-1}\omega_{i}S_{c_{i}/X}(T)-K\right)_{+}\right].$

To conclude, we derived consistent pricing formulas (4.18) and (4.23) for FX
options in the multi-currency case. As it was in Section 3 for a single FX
pair, here in the multi-currency case, although the pricing formulas (4.18)
and (4.23) are derived using the virtual X market, their evaluation depends on
parameters of the real $c_{i},$ $i=1,\ldots,N,$ markets only. The
distinguishing feature of the considered approach is that we can price all FX
options regardless from their domestic market using the same measure which in
turn guarantees that all natural relationships between exchange rates and FX
options are automatically fulfilled. Consequently, using $N(N-1)/2$ smiles out
of $N(N-1)$ available, we can calibrate a multi-currency model to be used with
the same parameters for pricing options on all the markets, which is
convenient and desirable in practice.

##  5 Illustrations

For illustrative purposes, we consider four examples in this section. The
first example (Section 5.1) illustrates the use of FX pricing from Section 3
in the case when the EUR-USD exchange rate $f(t)$ is described by the Heston
model [21] while the second example (Section 5.2) deals with the SABR model
[20]. In these two examples we follow the traditional route: we start with
models written under a ‘market’ measure, then find an EMM $Q^{X}$ on the
intermediate currency market and use Theorem 3.5 for pricing FX options. The
third example presented in Section 5.3 follows a different route: we suggest a
distribution for an exchange rate at maturity time $T,$ e.g. for EUR-USD,
under a forward measure $Q_{T}^{X}$ on the intermediate currency market so
that the no-arbitrage condition (3.19) is satisfied. Then we use Theorem 3.6
or Theorem 4.4 for pricing FX options. To this end, in Section 5.3 we assume
that the EUR-USD exchange rate $f(T)$ has a skew normal distribution. We
remark that the use of the considered extended skew normal model for FX
pricing is novel. In Section 5.4, we illustrate the results of Sections 3 and
4 in the case of the model-free approach [1, 2].

###  5.1 Heston model

For simplicity, let the interest rates for the USD and EUR money markets,
$r_{\$}$ and $r_{\text{€}}$, be constant. Consider the Heston stochastic
volatility model for the EUR-USD exchange rate $S_{\text{€}/\$}(t)=f(t)$
written under a ‘market’ measure [21]:

$\displaystyle df$ $\displaystyle=$ $\displaystyle\mu
f\,dt+\,\sqrt{v}f\left(\sqrt{1-\rho^{2}}\,dW_{1}(t)+\,\rho dW_{2}(t)\right),\
\ f(0)=f_{0},$ (5.1) $\displaystyle dv$ $\displaystyle=$
$\displaystyle\kappa\,(\theta-v)dt+\delta\sqrt{v}dW_{2}(t),\ v(0)=v_{0},$

where $W_{1}(t)$ and $W_{2}(t)$ are independent standard Wiener processes;
$\sigma(t)=\sqrt{v(t)}\,$ is a (stochastic) volatility; $\theta,$ $\kappa,$
$\delta,$ $f_{0}$ and $v_{0}$ are positive constants, satisfying

$2\kappa\theta\geq\delta^{2};$ (5.2)

and the correlation coefficient $\rho\in(-1,1).$ Recall that the condition
(5.2) guarantees that zero is unattainable by $v(t)$ in finite time.

Following Section 3.1, to re-write (5.1) under $Q^{X},$ we need to find the
market prices of risk, $\gamma_{1}(t)$ and $\gamma_{2}(t)$, so that (cf.
(3.5)):

$\sqrt{1-\rho^{2}}\gamma_{1}(t)+\rho\gamma_{2}(t)=\frac{\mu-v(t)/2+r_{\text{€}}-r_{\$}}{\sqrt{v(t)}}.$
(5.3)

As it is standard for the Heston model [21], to deal with incompleteness of
the market, we choose

$\gamma_{2}(t)=\lambda\sqrt{v(t)},$ (5.4)

where $\lambda$ is a constant. Therefore, we have

$\displaystyle d\sqrt{f}$ $\displaystyle=$
$\displaystyle\left(r_{X}(t)-r_{\text{€}}\right)\sqrt{f}dt+\frac{\sqrt{v}}{2}\sqrt{f}\left(\sqrt{1-\rho^{2}}\,dW_{1}^{Q^{X}}(t)+\rho\,dW_{2}^{Q^{X}}(t)\right),\,$
(5.5) $\displaystyle d\frac{1}{\sqrt{f}}$ $\displaystyle=$
$\displaystyle\left(r_{X}(t)-r_{\$}\right)\frac{1}{\sqrt{f}}dt-\frac{\sqrt{v}}{2}\frac{1}{\sqrt{f}}\left(\sqrt{1-\rho^{2}}\,dW_{1}^{Q^{X}}(t)+\rho\,dW_{2}^{Q^{X}}(t)\right),$
$\displaystyle dv$ $\displaystyle=$
$\displaystyle\kappa\,(\theta-v)dt+\delta\sqrt{v}dW_{2}^{Q^{X}},\ v(0)=v_{0},$

where, as before (see (3.4)),

$r_{X}(t)=\frac{r_{\$}+r_{\text{€}}}{2}+\frac{v(t)}{8}$ (5.6)

and, without changing the notation, the new $\kappa$ and $\theta$ in (5.5) are
equal to $\kappa+\lambda\delta$ and $\kappa\theta/(\kappa+\lambda\delta)$,
respectively, in terms of the old $\kappa$ and $\theta$ from (5.1). Then (see
Theorem 3.5), e.g. the price of the call (in USD) for buying €$1$ for $$K$ is
equal to

$C_{\text{€}/\$}(0,T,K)=\sqrt{f(0)}\mathbb{E}_{Q^{X}}\left[D_{X}(0,T)\left(\sqrt{f(T)}-\frac{K}{\sqrt{f(T)}}\right)_{+}\right],$
(5.7)

where $\sqrt{f(T)}$ and $1/\sqrt{f(T)}$ are from (5.5).

Now we will rewrite (5.5) under the T-forward measure $Q_{T}^{X}$ using the
results of Section 3.2. By (5.6), we have

$\displaystyle P_{X}(t,T)$ $\displaystyle=$
$\displaystyle\mathbb{E}_{Q^{X}}\left[D_{X}(t,T)|{\mathcal{F}}_{t}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}_{Q^{X}}\left[\left.\exp\left(-\int_{t}^{T}r_{X}(s)ds\right)\right|{\mathcal{F}}_{t}\right]$
$\displaystyle=$
$\displaystyle\exp\left(-\frac{r_{\$}+r_{\text{€}}}{2}(T-t)\right)\mathbb{E}_{Q^{X}}\left[\left.\exp\left(-\int_{t}^{T}\frac{v(s)}{8}ds\right)\right|v(t)\right].$

The stochastic X short rate $r_{X}(t)$ defined by (5.6) with $v(t)$ from (5.5)
possesses an affine term structure (see e.g. [4]):

$P_{X}(t,T)=\exp\left(-\frac{r_{\$}+r_{\text{€}}}{2}(T-t)+A(T-t)-C(T-t)v(t)\right),$
(5.8)

where

$\displaystyle A(t)$ $\displaystyle=$
$\displaystyle\frac{2\kappa\theta}{\delta^{2}}\ln\left(\frac{2\beta
e^{(\beta+\kappa)t/2}}{(\beta+\kappa)\left(e^{\beta
t}-1\right)+2\beta}\right),$ $\displaystyle C(t)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\frac{e^{\beta t}-1}{(\beta+\kappa)\left(e^{\beta
t}-1\right)+2\beta},$

with

$\beta=\sqrt{\kappa^{2}+\delta^{2}/4}.$

We note that

$dP_{X}=r_{X}(t)P_{X}dt-\delta C(T-t)\sqrt{v}P_{X}dW_{2}^{Q^{X}}(t).$

Then we obtain

$\displaystyle\mathbb{E}_{Q^{X}}\left[\left.\frac{dQ_{T}^{X}}{dQ^{X}}\right|\mathcal{F}_{t}\right]$
$\displaystyle=$ $\displaystyle\frac{P_{X}(t,T)}{P_{X}(0,T)B_{{}_{X}}(t)}$
$\displaystyle=$
$\displaystyle\exp\left(-\frac{1}{2}\int_{0}^{t}C^{2}(T-s)\delta^{2}v(s)ds-\int_{0}^{t}C(T-s)\delta\sqrt{v(s)}dW_{2}^{Q^{X}}(s)\right).$

Hence

$dW_{2}^{Q_{T}^{X}}=dW_{2}^{Q^{X}}+C(T-t)\delta\sqrt{v(t)}dt.$

To complete the change of measure, we need to look at $W_{1}^{Q_{T}^{X}}(t)$.
To this end, we recall that both forward EUR-X and USD-X exchange rates,

$F_{\text{€}/X}(t,T)=e^{-r_{\text{€}}(T-t)}\frac{\sqrt{f(t)}}{P_{X}(t,T)}$

and

$F_{\$/X}(t,T)=e^{-r_{\$}(T-t)}\frac{1}{P_{X}(t,T)\sqrt{f(t)}},$

should be $Q_{T}^{X}$-martingales. It is not difficult to check that to
achieve the above no-arbitrage requirement, we need

$dW_{1}^{Q^{X}}(t)=dW_{1}^{Q_{T}^{X}}(t),$

which is natural since the change of measure (5.1) does not depend on
$W_{1}^{Q^{X}}(t).$

Thus, applying Theorem 3.6 to the Heston model setting, we can price, e.g. the
call option as (see (3.26)):

$\displaystyle C_{\text{€}/\$}(0,T,K)$ $\displaystyle=$
$\displaystyle\frac{e^{-r_{\$}T}}{\mathbb{E}_{Q_{T}^{X}}\left[\frac{1}{\sqrt{f(T)}}\right]}\mathbb{E}_{Q_{T}^{X}}\left[\left(\sqrt{f(T)}-\frac{K}{\sqrt{f(T)}}\right)_{+}\right]$
$\displaystyle=$
$\displaystyle\frac{e^{-r_{\$}T}}{\mathbb{E}_{Q_{T}^{X}}\left[F_{\$/X}(T,T)\right]}\mathbb{E}_{Q_{T}^{X}}\left[\left(F_{\text{€}/X}(T,T)-KF_{\$/X}(T,T)\right)_{+}\right],$

and

$C_{\$/\text{€}}(0,T,K)=\frac{e^{-r_{\text{€}}T}}{\mathbb{E}_{Q_{T}^{X}}\left[F_{\text{€}/X}(T,T)\right]}\mathbb{E}_{Q_{T}^{X}}\left[\left(F_{\$/X}(T,T)-KF_{\text{€}/X}(T,T)\right)_{+}\right],$
(5.11)

where

$\displaystyle dF_{\text{€}/X}(t,T)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{v}}{2}F_{\text{€}/X}(t,T)\left(\sqrt{1-\rho^{2}}\,dW_{1}^{Q^{X}}(t)+\rho\,dW_{2}^{Q_{T}^{X}}\right)$
$\displaystyle+\delta C(T-t)\sqrt{v}F_{\text{€}/X}(t,T)dW_{2}^{Q_{T}^{X}},$
$\displaystyle dF_{\$/X}(t,T)$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{v}}{2}F_{\$/X}(t,T)\left(\sqrt{1-\rho^{2}}\,dW_{1}^{Q^{X}}(t)+\rho\,dW_{2}^{Q_{T}^{X}}\right)$
$\displaystyle+\delta C(T-t)\sqrt{v}F_{\$/X}(t,T)dW_{2}^{Q_{T}^{X}},$
$\displaystyle dv$ $\displaystyle=$
$\displaystyle\left(\kappa-C(T-t)\delta^{2}\right)\left(\frac{\kappa\theta}{\kappa-C(T-t)\delta^{2}}-v\right)dt+\delta\sqrt{v}dW_{2}^{Q_{T}^{X}},\
v(0)=v_{0},$

and we require that for $0\leq t\leq T$

$\kappa/\delta^{2}>C(t).$ (5.13)

We note that in comparison with the classical Heston model (5.1), the model
(5.1) has time dependence in the coefficients. For other time-dependent Heston
models, see e.g. [3, 19] and references therein.

###  5.2 SABR model

For simplicity again, let the interest rates for the USD and EUR money
markets, $r_{\$}$ and $r_{\text{€}}$, be constant. Following Section 3.1, we
can re-write the classical SABR model [20] for EUR-USD exchange rate $f(t)$
under the measure $Q^{X}$ and the corresponding SDEs for
$S_{\text{€}/X}=\sqrt{f}$ and $S_{\text{\$}/X}=1/\sqrt{f}$ take the form

$\displaystyle d\sqrt{f}$ $\displaystyle=$
$\displaystyle\left(r_{X}(t)-r_{\text{€}}\right)\sqrt{f}dt+\frac{\sigma(t)}{2}\sqrt{f}\left(\sqrt{1-\rho^{2}}\,dW_{1}^{Q^{X}}(t)+\rho\,dW_{2}^{Q^{X}}(t)\right),$
(5.14) $\displaystyle d\frac{1}{\sqrt{f}}$ $\displaystyle=$
$\displaystyle\left(r_{X}(t)-r_{\$}\right)\frac{1}{\sqrt{f}}dt-\frac{\sigma(t)}{2}\frac{1}{\sqrt{f}}\left(\sqrt{1-\rho^{2}}\,dW_{1}^{Q^{X}}(t)+\rho\,dW_{2}^{Q^{X}}(t)\right),$
$d\sigma=\nu\sigma dW_{2}^{Q^{X}}(t),\ \sigma(0)=\alpha,$ (5.15)

where $W_{1}^{Q^{X}}(t)$ and $W_{2}^{Q^{X}}$ are independent standard Wiener
processes under $Q^{X}$, $\rho\in(-1,0]$ is the correlation coefficient,
$\nu>0$ is the volatility of the volatility $\sigma(t)$, $\alpha$ is a
positive constant, and (see (3.4))

$r_{X}(t)=\frac{r_{\$}+r_{\text{€}}}{2}+\frac{\sigma^{2}(t)}{8}\ .$

Note that the parameter known as $\beta$ in the classical SABR model is taken
to be equal to $1$ here, which is the typical requirement for FX modelling as
it ensures that the SDE for the exchange rate for the inverse pair $1/f$ has
the same form as for $f.$

By Theorem 3.5, e.g. the price of the call (in USD) for buying €$1$ for $$K$
is equal to

$C_{\text{€}/\$}((0,T,K)=\sqrt{f(0)}\mathbb{E}_{Q^{X}}\left[D_{X}(0,T)\left(\sqrt{f(T)}-\frac{K}{\sqrt{f(T)}}\right)_{+}\right],$
(5.16)

where $\sqrt{f(T)}$ and $1/\sqrt{f(T)}$ satisfy (5.14), (5.15).

###  5.3 Extended skew normal model

In this section we consider another illustration of Theorem 3.6. Here we start
not with a model under a ‘market’ measure but with a direct assumption on the
distribution of the exchange rate under a forward measure $Q_{T}^{X}$ on the
intermediate market.

We assume that under a T-forward measure $Q_{T}^{X}$ the EUR-USD exchange rate
$f(T)$ can be written as

$f(T)=\bar{F}e^{Z},$ (5.17)

where $\bar{F}>0$ is a constant and $Z$ is a random variable such that
$\mathbb{E}\left[e^{Z}\right]$ exists and the no-arbitrage condition (3.19) is
satisfied by $f(T)$. Here the no-arbitrage condition (3.19) implies that

$\bar{F}=F\,\frac{\mathbb{E}\left[e^{-Z/2}\right]}{\mathbb{E}\left[e^{Z/2}\right]},$
(5.18)

where we neglect the full notation $\mathbb{E}_{Q_{T}^{X}}[\cdot]$ and write
$\mathbb{E}[\cdot]$ instead as in this section we work with the measure
$Q_{T}^{X}$ only and we also write here $F$ instead of $F_{\text{€}/\$}(0,T)$
for the current forward EUR-USD exchange rate (see (3.21)). We use this
simplified notation throughout this section, which should not cause any
confusion. The interest rates for the USD and EUR money markets, $r_{\$}$ and
$r_{\text{€}}$, are assumed to be constant.

Further, (3.26) (i.e. Theorem 3.6), (5.17) and (5.18) imply that the price (in
USD) of the European call to buy €$1$ for $\$K$ at the maturity $T$ is

$\displaystyle C_{\text{€}/\$}(0,T,K)$
$\displaystyle=\frac{e^{-r_{\$}T}}{\mathbb{E}\left[\frac{1}{\sqrt{f(T)}}\right]}\mathbb{E}\left[\left(\sqrt{f(T)}-\frac{K}{\sqrt{f(T)}}\right)_{+}\right]$
(5.19)
$\displaystyle=e^{-r_{\$}T}\,\frac{\sqrt{\bar{F}}}{\mathbb{E}\left[e^{-Z/2}\right]}\mathbb{E}\left[\left(\sqrt{f(T)}-\frac{K}{\sqrt{f(T)}}\right)\mathbbm{1}_{\bar{F}e^{Z}>K}\right]$
$\displaystyle=e^{-r_{\$}T}\,\frac{\sqrt{\bar{F}}}{\mathbb{E}\left[e^{-Z/2}\right]}\left(\sqrt{\bar{F}}\,\mathbb{E}\left[e^{Z/2}\mathbbm{1}_{Z>z_{0}}\right]-\frac{K}{\sqrt{\bar{F}}}\,\mathbb{E}\left[e^{-Z/2}\mathbbm{1}_{Z>z_{0}}\right]\right)$
$\displaystyle=e^{-r_{\$}T}\,\left(\bar{F}\,\frac{\mathbb{E}\left[e^{Z/2}\mathbbm{1}_{Z>z_{0}}\right]}{\mathbb{E}\left[e^{-Z/2}\right]}-K\,\frac{\mathbb{E}\left[e^{-Z/2}\mathbbm{1}_{Z>z_{0}}\right]}{\mathbb{E}\left[e^{-Z/2}\right]}\right)$
$\displaystyle=e^{-r_{\$}T}\,\left(F\,\frac{\mathbb{E}\left[e^{Z/2}\mathbbm{1}_{Z>z_{0}}\right]}{\mathbb{E}\left[e^{Z/2}\right]}-K\,\frac{\mathbb{E}\left[e^{-Z/2}\mathbbm{1}_{Z>z_{0}}\right]}{\mathbb{E}\left[e^{-Z/2}\right]}\right)$
$\displaystyle=e^{-r_{\$}T}\,\left(F\,\frac{M(1/2,z_{0})}{M(1/2)}-K\,\frac{M(-1/2,z_{0})}{M(-1/2)}\right),$

where $z_{0}=\log\left(K/\bar{F}\right)$ and

$M(t)=\mathbb{E}[e^{tZ}]\text{ and
}M(t,z_{0})=\mathbb{E}\left[e^{tZ}\mathbbm{1}_{Z>z_{0}}\right],$ (5.20)

which are the moment generating function (MGF) and the restricted MGF for $Z$,
respectively. Analogous to (5.19), we can derive the pricing formulas for the
put and also for the call and put for the inverse pair:

$\displaystyle P_{\text{€}/\$}(0,T,K)$
$\displaystyle=\frac{e^{-r_{\$}T}}{\mathbb{E}\left[\frac{1}{\sqrt{f(t)}}\right]}\mathbb{E}\left[\left(\frac{K}{\sqrt{f(T)}}-\sqrt{f(T)}\right)_{+}\right]$
$\displaystyle=e^{-r_{\$}T}\,\left(K\,\frac{M^{\ast}(-1/2,z_{0})}{M(-1/2)}-F\,\frac{M^{\ast}(1/2,z_{0})}{M(1/2)}\right),$
$\displaystyle C_{\$/\text{€}}\left(0,T,\frac{1}{K}\right)$
$\displaystyle=\frac{e^{-r_{\text{€}}T}}{\mathbb{E}\left[\sqrt{f(t)}\right]}\mathbb{E}\left[\left(\frac{1}{\sqrt{f(T)}}-\frac{\sqrt{f(T)}}{K}\right)_{+}\right]$
$\displaystyle=e^{-r_{\text{€}}T}\,\left(\frac{1}{F}\,\frac{M^{\ast}(-1/2,z_{0})}{M(-1/2)}-\frac{1}{K}\,\frac{M^{\ast}(1/2,z_{0})}{M(1/2)}\right),$
$\displaystyle P_{\$/\text{€}}\left(0,T,\frac{1}{K}\right)$
$\displaystyle=\frac{e^{-r_{\text{€}}T}}{\mathbb{E}\left[\sqrt{f(t)}\right]}\mathbb{E}\left[\left(\frac{\sqrt{f(T)}}{K}-\frac{1}{\sqrt{f(T)}}\right)_{+}\right]$
$\displaystyle=e^{-r_{\text{€}}T}\,\left(\frac{1}{K}\,\frac{M(1/2,z_{0})}{M(1/2)}-\frac{1}{F}\,\frac{M(-1/2,z_{0})}{M(-1/2)}\right),$

where

$M^{\ast}(t,z_{0})=\mathbb{E}\left[e^{tZ}\mathbbm{1}_{Z<z_{0}}\right].$

Now we will propose a skew normal model for the random variable $Z$. To this
end, we start by introducing a new random variable $V$, which is a combination
of one normal and two shifted half-normal distributed random variables:

$V:=X+\alpha_{1}\max(\beta_{1}-Y,0)+\alpha_{2}\max(Y-\beta_{2},0),$ (5.21)

where $X$ and $Y$ are independent random variables with the standard normal
distribution and $\alpha_{1}$, $\alpha_{2}$, $\beta_{1}$,
$\beta_{2}\in\mathbb{R}$ are parameters. The parameters $\beta_{1}$ and
$\beta_{2}$ describe the support domains of the two half-normal distributions
which, from the modeling prospective, should not overlap. Consequently, we are
only interested in the case:

$0<\beta_{1}\leq\beta_{2}.$

Since we use the random variable $Z$ in (5.17) similarly to as a Gaussian
random variable is used in the geometric Brownian motion model for $f(T)$, we
then define it as follows

$Z=aV,$ (5.22)

where $a=\sigma\sqrt{T}$ with $\sigma$ having the meaning of volatility and
$T$ of the maturity time. The benefit of using $Z$ instead of a Gaussian
random variable is that $Z$ can have heavier tails and can be successfully
used for describing the volatility smile effect. At the same time, $Z$ still
has a very simple distribution which makes the model (5.17), (5.21), (5.22)
very practical as it allows fast calibration. Indeed, the MGFs (5.20), which
we need for pricing calls and puts (see (5.19)), can be found analytically for
this $Z$. The corresponding expressions are given in the next proposition.

######  Proposition 5.1.

For $0<\beta_{1}\leq\beta_{2}$, the MGF $M(t)$ and the restricted MGF
$M(t,z_{0})$ from (5.20) are equal to

$\displaystyle M(t)$
$\displaystyle=e^{\frac{(at)^{2}}{2}}(N(\beta_{2})-N(\beta_{1}))+e^{\frac{t}{2}(ta^{2}(1+\alpha_{2}^{2})-2a\alpha_{2}\beta_{2})}N(ta\alpha_{2}-\beta_{2})$
(5.23)
$\displaystyle+e^{\frac{t}{2}(ta^{2}(1+\alpha_{1}^{2})+2a\alpha_{1}\beta_{1})}N(ta\alpha_{1}+\beta_{1}),$
$\displaystyle M(t,z_{0})$
$\displaystyle=e^{\frac{(at)^{2}}{2}}N\left(at-\frac{z_{0}}{a}\right)\big{(}N(\beta_{2})-N(\beta_{1})\big{)}+e^{\frac{t}{2}(ta^{2}(1+\alpha_{1}^{2})+2a\alpha_{1}\beta_{1})}$
(5.24) $\displaystyle\times$
$\displaystyle\left\\{N(ta\alpha_{1}+\beta_{1})-N_{2}\left(ta\alpha_{1}+\beta_{1},\frac{\frac{z_{0}}{a}-at-\alpha_{1}(\beta_{1}+ta\alpha_{1})}{\sqrt{1+\alpha_{1}^{2}}};\frac{-\alpha_{1}}{\sqrt{1+\alpha_{1}^{2}}}\right)\right\\}$
$\displaystyle+e^{\frac{t}{2}(ta^{2}(1+\alpha_{2}^{2})-2a\alpha_{2}\beta_{2})}$
$\displaystyle\times\left\\{N(ta\alpha_{2}-\beta_{2})-N_{2}\left(ta\alpha_{2}-\beta_{2},\frac{\frac{z_{0}}{a}-t+\alpha_{2}(\beta_{2}-ta\alpha_{2})}{\sqrt{1+\alpha_{2}^{2}}};\frac{-\alpha_{2}}{\sqrt{1+\alpha_{2}^{2}}}\right)\right\\},$

where $N(\cdot)$ is the cdf of the standard normal distribution and
$N_{2}(\cdot,\cdot;\rho)$ is the cdf of the bivariate normal distribution with
zero mean, unit variance, and correlation $\rho$.

Using the parameters $\alpha_{1}$, $\alpha_{2}$, $\beta_{1}$, and $\beta_{2}$,
we can manipulate with the distribution of $Z$ defined in (5.21), (5.22) and,
in particular, its skew and kurtosis, which are equal to

$\displaystyle skew_{Z}$ $\displaystyle=$
$\displaystyle\frac{M_{V}^{(3)}(0)-3M_{V}^{(1)}(0)M_{V}^{(2)}(0)+2\left[M_{V}^{(1)}(0)\right]^{3}}{\left(M_{V}^{(2)}(0)-\left[M_{V}^{(1)}(0)\right]^{2}\right)^{3/2}},$
$\displaystyle kurtosis_{Z}$ $\displaystyle=$
$\displaystyle\frac{M_{V}^{(4)}(0)-4M_{V}^{(1)}(0)M_{V}^{(3)}(0)+6\left[M_{V}^{(1)}(0)\right]^{2}M_{V}^{(2)}(0)-3\left[M_{V}^{(1)}(0)\right]^{4}}{\left(M_{V}^{(2)}(0)-\left[M_{V}^{(1)}(0)\right]^{2}\right)^{2}},$

where $M_{V}^{(i)}(0)$ are $i$-th derivatives of the MGF for the random
variable $V$. By putting $\alpha_{1}=\alpha_{2}=0$ in (5.21), the random
variable $Z$ becomes normal with zero mean and variance $a^{2},$ and the
considered model (5.17), (5.21), (5.22) is reduced to the geometric Brownian
motion whose one of the critical deficiencies is a flat (constant) volatility.
In this case $Z$ has $skew=0$ and $kurtosis=3$. In Figure 5.1, one can see the
difference of distribution of $Z$ (blue area) compared to a standardized
normal distribution (red line). It can be seen that a parameter set with
$\alpha_{1}<0$ and $\alpha_{2}=0$ results in a bigger left tail in
distribution and a skew in the resulting volatility smile ($skew\approx-1.6$).

Figure 5.1: Effect of the parameters on the distribution of Z and the
corresponding smile: the case of $\alpha_{1}<0$ and $\alpha_{2}=0$.

Similarly, in Figure 5.2, it can be observed that using $\alpha_{1}<0$ to
adjust the left tail and $\alpha_{2}>0$ to adjust the right tail of the smile,
we can get an asymmetric distribution and an asymmetric smile. A smaller
$\alpha_{2}$ results in a smaller right tail and hence in a flatter smile.

Figure 5.2:  Effect of the parameters on the distribution of Z and the
corresponding smile: the case of $\alpha_{1}<0$ and $\alpha_{2}>0$.

As seen in these figures, by adjusting the parameters $\alpha_{1}$,
$\alpha_{2}$, $\beta_{1}$, $\beta_{2}$, the shape of the distribution and of
the smile can be changed in various ways and it can be associated with the
resulting skew and kurtosis of the log exchange rate. Hence, after calibrating
the parameters of $Z$ to FX market data, we can compare $skew_{Z}$ with zero
skew and $kurtosis_{Z}$ with the kurtosis of $3$ in the geometric Brownian
motion case and make a conclusion about how far volatility is from a constant.

###  5.4 Model-free approach

The model-free approach to pricing derivatives has become popular in recent
years [1, 15, 2, 9] (see also references therein). The main idea of the
approach is to construct a density or distribution function of risky assets
under a risk-neutral measure using observed prices of plain-vanilla options.
For clarity of the exposition how this approach works within our intermediate
currency framework, we start with the case of two currencies. Then we will
extend the consideration to the three-currencies case where we will exploit
ideas from [1] (see also [2]).

In this section we will work under a T–forward measure $Q_{T}^{X}.$ Assume
that we know prices of call options $C_{\text{€}/\$}(0;K)$ for all strikes
$K>0$ and let $\rho(x;T)$ be the density of the EUR-USD exchange rate $f(T)$
under $Q_{T}^{X}.$ According to (3.25), we have

$\displaystyle V_{\text{€}/\$}(0)$ $\displaystyle=$
$\displaystyle\frac{e^{-r_{\$}T}}{\mathbb{E}_{Q_{T}^{X}}\left[\sqrt{f(T)}\right]}\mathbb{E}_{Q_{T}^{X}}\left[g\left(\sqrt{f(T)};\frac{K}{\sqrt{f(T)}}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{e^{-r_{\$}T}}{\mathbb{E}_{Q_{T}^{X}}\left[\sqrt{f(T)}\right]}\int_{0}^{\infty}\frac{1}{\sqrt{x}}g\left(x;K\right)\rho(x;T)dx$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}g\left(x;K\right)\frac{\partial^{2}}{\partial
x^{2}}C_{\text{€}/\$}(0;x)dx,$

as simple calculations give

$\frac{e^{-r_{\$}T}}{\sqrt{K}\mathbb{E}_{Q_{T}^{X}}\left[\sqrt{f(T)}\right]}\rho(K;T)=\frac{\partial^{2}}{\partial
K^{2}}C_{\text{€}/\$}(0;K).$ (5.26)

Typically, observed data are expressed via volatility smile data $\sigma(K)$
and from (3.8) we have

$C_{\text{€}/\$}(0;K)=F_{\text{€}/\$}e^{-r_{\$}T}N\left(\frac{\log\frac{F_{\text{€}/\$}}{K}+\sigma^{2}(K)T/2}{\sigma(K)\sqrt{T}}\right)-Ke^{-r_{\$}T}N\left(\frac{\log\frac{F_{\text{€}/\$}}{K}-\sigma^{2}(K)T/2}{\sigma(K)\sqrt{T}}\right).$
(5.27)

Combining (5.4) with (5.27) and given $\sigma(K)$, we can price any FX
derivative $V_{\text{€}/\$}(0)$ and analogously any derivative
$V_{\$/\text{€}}(0)$ based on a smile from one of the markets. Note that the
smile data computed from $C_{\text{€}/\$}(0;K)$ coincide with smile data
computed from $C_{\$/\text{€}}(0;K)$ and that the prices $V_{\text{€}/\$}(0)$
and $V_{\$/\text{€}}(0)$ are consistent with each other thanks to using the
intermediate currency framework.

Now we progress to the three-currencies case. Let us assume that we are
interested in the GBP-USD-EUR currency triangle, where we denote GBP as
currency 1, USD as currency 2, and EUR as currency 3. As before, the interest
rates for the GBP, USD and EUR money markets, $r_{\pounds},$ $r_{\$}$ and
$r_{\text{€}}$, are assumed to be constant.

Consider a best-of option on the EUR market which payoff is equal to

$b(T)=\max\left\\{\frac{\left(S_{\pounds/\text{€}}(T)-K_{1}\right)_{+}}{K_{1}},\frac{\left(S_{\$/\text{€}}(T)-K_{2}\right)_{+}}{K_{2}}\right\\}.$
(5.28)

As it is known [1] and [2, Sec. 11.7.1], the value of a best-of option is
arbitrary close to values of plain-vanilla calls on $S_{\pounds/\text{€}}(T)$
or $S_{\$/\text{€}}(T)$ or to a vanilla option on the cross
$S_{\pounds/\$}(T).$ Hence, a model used for FX pricing should price a best-of
option and plain-vanilla options in a consistent manner.

By (4.2) we have

$S_{\pounds/X}=S_{\pounds/\text{€}}^{2/3}(T)S_{\$/\text{€}}^{-1/3}(T),\ \
S_{\$/X}=S_{\pounds/\text{€}}^{-1/3}(T)S_{\$/\text{€}}^{2/3}(T),\
S_{\text{€}/X}=S_{\pounds/\text{€}}^{-1/3}(T)S_{\$/\text{€}}^{-1/3}(T).$
(5.29)

Using the pricing formula (4.23), we get

$V_{\text{€}}(0)=\frac{e^{-r_{\text{€}}T}}{\mathbb{E}_{Q_{T}^{X}}\left[S_{\text{€}/X}(T)\right]}\mathbb{E}_{Q_{T}^{X}}G(T)=\frac{e^{-r_{\text{€}}T}}{\mathbb{E}_{Q_{T}^{X}}\left[S_{\text{€}/X}(T)\right]}\mathbb{E}_{Q_{T}^{X}}\left[S_{\text{€}/X}(T)g(T)\right],$
(5.30)

where $g(T)$ is an arbitrary payoff on the EUR market. Hence, for the best-of
option we have

$\displaystyle v_{\text{€}}(0)$ $\displaystyle=$
$\displaystyle\frac{e^{-r_{\text{€}}T}}{\mathbb{E}_{Q_{T}^{X}}\left[S_{\text{€}/X}(T)\right]}\mathbb{E}_{Q_{T}^{X}}\
\left[S_{\text{€}/X}(T)\
\max\left\\{\frac{\left(S_{\pounds/\text{€}}(T)-K_{1}\right)_{+}}{K_{1}},\frac{\left(S_{\$/\text{€}}(T)-K_{2}\right)_{+}}{K_{2}}\right\\}\right]$
$\displaystyle=$
$\displaystyle\frac{e^{-r_{\text{€}}T}}{\mathbb{E}_{Q_{T}^{X}}\left[S_{\text{€}/X}(T)\right]}$
$\displaystyle\times\mathbb{E}_{Q_{T}^{X}}\
\left[S_{\pounds/\text{€}}^{-1/3}(T)S_{\$/\text{€}}^{-1/3}(T)\
\max\left\\{\frac{\left(S_{\pounds/\text{€}}(T)-K_{1}\right)_{+}}{K_{1}},\frac{\left(S_{\$/\text{€}}(T)-K_{2}\right)_{+}}{K_{2}}\right\\}\right]$
$\displaystyle=$
$\displaystyle\frac{e^{-r_{\text{€}}T}}{\mathbb{E}_{Q_{T}^{X}}\left[S_{\text{€}/X}(T)\right]}$
$\displaystyle\times\int_{0}^{\infty}\int_{0}^{\infty}x^{-1/3}y^{-1/3}\
\max\left\\{\frac{\left(x-K_{1}\right)_{+}}{K_{1}},\frac{\left(y-K_{2}\right)_{+}}{K_{2}}\right\\}\rho(x,y;T)dxdy,$

where $\rho(x,y;T)$ is the joint density of the exchange rates
$S_{\pounds/\text{€}}(T)$ and $S_{\$/\text{€}}(T)$ under $Q_{T}^{X}.$

Differentiation of $b(T)$ (see [1, 2]) gives:

$\left[1+K_{1}\frac{\partial}{\partial K_{1}}+K_{2}\frac{\partial}{\partial
K_{2}}\right]b(T)=\mathbf{I}(S_{\pounds/\text{€}}(T)<K_{1},S_{\pounds/\text{€}}(T)<K_{2})-1.$

Therefore

$\frac{\partial^{2}}{\partial K_{1}\partial
K_{2}}\left[1+K_{1}\frac{\partial}{\partial
K_{1}}+K_{2}\frac{\partial}{\partial
K_{2}}\right]v_{\text{€}}(0)=\frac{e^{-r_{\text{€}}T}}{\mathbb{E}_{Q_{T}^{X}}\left[S_{\text{€}/X}(T)\right]}K_{1}^{-1/3}K_{2}{}^{-1/3}\
\rho(K_{1},K_{2};T).$ (5.31)

Note that we have from (4.20)

$\frac{\mathbb{E}_{Q_{T}^{X}}\left[S_{c_{i}/X}(T)\right]}{\mathbb{E}_{Q_{T}^{X}}\left[S_{\text{€}/X}(T)\right]}=F_{c_{i}/\text{€}}(0),\quad
i=1,2,$ (5.32)

and from (4.23)

$V_{c_{N}}(0)=\frac{e^{-r_{N}T}}{\mathbb{E}_{Q_{T}^{X}}\left[S_{c_{N}/X}(T)\right]}\mathbb{E}_{Q_{T}^{X}}\left[S_{c_{N}/X}(T)g(T)\right],$
(5.33)

where $N$ can be any of the three currencies with $g(T)$ being in the currency
$N.$ Therefore (taking also into account (5.29)), if we can evaluate (5.31)
from market data, then we can price any FX derivatives on any of the three
markets using the same $\rho(K_{1},K_{2};T)$ and, thus, ensuring consistency
of FX option pricing across different markets. Note that in comparison with
(4.23) we do not assume in (5.33) that $g(T)$ is first-order homogeneous.

Market data in the case of three currencies are typically presented via three
volatility smiles: $\sigma_{1}(K)$ and $\sigma_{2}(K)$ from vanilla options on
GBP-EUR and USD-EUR, respectively, and $\sigma_{3}(K)$ from the cross, GBP-
USD. To compute values on the smile curves from observed option prices in the
context of our intermediate currency approach, the Garman-Kohlhagen formulas
given below for completeness of the exposition should be used. To complete,
the model-free pricing, we need to express the current price $v(0)$ of the
best-of option via the three volatility smiles. To this end, we need to find
$v(0)$ assuming that the exchange rates follow geometrical Brownian motions
under a T–forward measure $Q_{T}^{X},$ which coincides with the EMM $Q^{X}.$

In accordance with the no-arbitrage condition (5.32), we set

$\displaystyle S_{\pounds/\text{€}}(T)$
$\displaystyle=F_{\pounds/\text{€}}\exp\left(-\frac{T}{6}\left[\sigma_{1}^{2}-2\sigma_{1}\sigma_{2}\rho_{12}\right]+\sigma_{1}\sqrt{T}X_{1}\right),$
(5.34) $\displaystyle S_{\$/\text{€}}(T)$
$\displaystyle=F_{\$/\text{€}}\exp\left(-\frac{T}{6}\left[\sigma_{2}^{2}-2\sigma_{1}\sigma_{2}\rho_{12}\right]+\sigma_{2}\sqrt{T}X_{2}\right),$

where $F_{\pounds/\text{€}}=F_{\pounds/\text{€}}(0,T)$ and
$F_{\$/\text{€}}=F_{\$/\text{€}}(0,T)$ are the current forward GBR-EUR and
USD-EUR exchange rate, respectively, and $X_{i}\sim N(0,1)$ with correlation
coefficient $\rho_{12}$. Then the GBP-USD exchange rate is equal to

$S_{\pounds/\$}(T)=\frac{S_{\pounds/\text{€}}(T)}{S_{\$/\text{€}}(T)}=F_{\pounds/\$}\exp\left(-\frac{T}{6}\left[\sigma_{3}^{2}-2\sigma_{2}\sigma_{3}\rho_{23}\right]+\sigma_{3}\sqrt{T}X_{3}\right),$
(5.35)

where

$F_{\pounds/\$}=\frac{F_{\pounds/\text{€}}}{F_{\$/\text{€}}},\
\sigma_{3}^{2}=\sigma_{1}^{2}-2\sigma_{1}\sigma_{2}\rho_{12}+\sigma_{2}^{2}.$

and $X_{3}\sim N(0,1)$ with the correlation coefficients

$\rho_{13}=\frac{\sigma_{1}^{2}+\sigma_{3}^{2}-\sigma_{2}^{2}}{2\sigma_{1}\sigma_{3}},\
\
\rho_{23}=\frac{\sigma_{2}^{2}+\sigma_{3}^{2}-\sigma_{1}^{2}}{2\sigma_{2}\sigma_{3}}$

with $X_{1}$ and $X_{2},$ respectively.

We have

$\displaystyle\frac{S_{\text{€}/X}(T)}{\mathbb{E}_{Q_{T}^{X}}\left[S_{\text{€}/X}(T)\right]}$
$\displaystyle=$
$\displaystyle\frac{\exp\left(-\frac{1}{3}\sigma_{1}\sqrt{T}X_{1}-\frac{1}{3}\sigma_{2}\sqrt{T}X_{2}\right)}{\mathbb{E}_{Q_{T}^{X}}\left[\exp\left(-\frac{1}{3}\sigma_{1}\sqrt{T}X_{1}-\frac{1}{3}\sigma_{2}\sqrt{T}X_{2}\right)\right]}$
$\displaystyle=$
$\displaystyle\exp\left(-\frac{T}{18}\left[\sigma_{1}^{2}+2\sigma_{1}\sigma_{2}\rho+\sigma_{2}^{2}\right]-\frac{1}{3}\sigma_{1}\sqrt{T}X_{1}-\frac{1}{3}\sigma_{2}\sqrt{T}X_{2}\right),$

and it is not difficult to show that the above expression is the Radon-Nikodym
derivative $\dfrac{dQ_{T}^{\text{€}}}{dQ_{T}^{X}}$ of the T–forward measure
$Q_{T}^{\text{€}}$ on the EUR market with respect to $Q_{T}^{X}.$ Then

$V_{\text{€}}(0)=\frac{e^{-r_{\text{€}}T}}{\mathbb{E}_{Q_{T}^{X}}\left[S_{\text{€}/X}(T)\right]}\mathbb{E}_{Q_{T}^{X}}\left[S_{\text{€}/X}(T)g(T)\right]=e^{-r_{\text{€}}T}\mathbb{E}_{Q_{T}^{\text{€}}}\left[g(T)\right].$

Hence, the corresponding Garman-Kohlhagen formulas for calls are given by
(see, e.g. [4]):

$C_{\pounds/\text{€}}(0;K)=F_{\pounds/\text{€}}e^{-r_{\text{€}}T}N\left(\frac{\ln(F_{\pounds/\text{€}}/K)+\sigma_{1}^{2}T/2}{\sigma_{1}\sqrt{T}}\right)-Ke^{-r_{\text{€}}T}N\left(\frac{\ln(F_{\pounds/\text{€}}/K)-\sigma_{1}^{2}T/2}{\sigma_{1}\sqrt{T}}\right),$
$C_{\$/\text{€}}(0;K)=F_{\$/\text{€}}e^{-r_{\text{€}}T}N\left(\frac{\ln(F_{\$/\text{€}}/K)+\sigma_{2}^{2}T/2}{\sigma_{2}\sqrt{T}}\right)-Ke^{-r_{\text{€}}T}N\left(\frac{\ln(F_{\$/\text{€}}/K)-\sigma_{2}^{2}T/2}{\sigma_{2}\sqrt{T}}\right).$

Analogously

$\displaystyle C_{\pounds/\$}(0;K)$ $\displaystyle=$
$\displaystyle\frac{e^{-r_{\text{\pounds}}T}}{\mathbb{E}_{Q_{T}^{X}}\left[S_{\text{\pounds}/X}(T)\right]}\mathbb{E}_{Q_{T}^{X}}\left[S_{\text{\pounds}/X}(T)(S_{\$/\text{\pounds}}(T)-K)_{+}\right]$
$\displaystyle=$ $\displaystyle
e^{-r_{\text{\pounds}}T}\mathbb{E}_{Q_{T}^{\text{\pounds}}}\left[(S_{\$/\text{\pounds}}(T)-K)_{+}\right]$
$\displaystyle=$ $\displaystyle
F_{\pounds/\$}e^{-r_{\$}T}N\left(\frac{\ln(F_{\pounds/\$}/K)+\sigma_{3}^{2}T/2}{\sigma_{3}\sqrt{T}}\right)-Ke^{-r_{\$}T}N\left(\frac{\ln(F_{\pounds/\$}/K)-\sigma_{3}^{2}T/2}{\sigma_{3}\sqrt{T}}\right).$

We also have (see [25, 29, 1]):

$\displaystyle v_{\text{€}}(0)$ $\displaystyle=$
$\displaystyle\frac{e^{-r_{\text{€}}T}}{\mathbb{E}_{Q_{T}^{X}}\left[S_{\text{€}/X}(T)\right]}$
$\displaystyle\times\mathbb{E}_{Q_{T}^{X}}\ \left[S_{\text{€}/X}(T)\
\max\left\\{\frac{\left(S_{\pounds/\text{€}}(T)-K_{1}\right)_{+}}{K_{1}},\frac{\left(S_{\$/\text{€}}(T)-K_{2}\right)_{+}}{K_{2}}\right\\}\right]$
$\displaystyle=$ $\displaystyle
e^{-r_{\text{€}}T}\mathbb{E}_{Q_{T}^{\text{€}}}\
\left[\max\left\\{\frac{\left(S_{\pounds/\text{€}}(T)-K_{1}\right)_{+}}{K_{1}},\frac{\left(S_{\$/\text{€}}(T)-K_{2}\right)_{+}}{K_{2}}\right\\}\right]$
$\displaystyle=$ $\displaystyle
e^{-r_{\text{€}}T}\left[\frac{F_{\pounds/\text{€}}}{K_{1}}N(d_{1}^{+},d_{3}^{+};\rho_{13})+\frac{F_{\$/\text{€}}}{K_{2}}N(d_{2}^{+},d_{3}^{-};\rho_{23})+N(-d_{1}^{-},-d_{2}^{-};\rho_{12})-1\right],$

where

$d_{i}^{\pm}=\frac{\ln(F_{i}/K_{i})\pm\sigma_{i}^{2}T/2}{\sigma_{i}\sqrt{T}}$

and $F_{1}=F_{\pounds/\text{€}},$ $F_{2}=F_{\$/\text{€}},$ and
$F_{3}=F_{\pounds/\$}.$

Now we put implied volatility smiles $\sigma_{i}(K_{i}),$ $i=1,2,3,$ with
$K_{3}=K_{1}/K_{2}$ in (5.4) and evaluate the left-hand side of (5.31). As a
result, we obtain for
$v_{\text{€}}(0)=v_{\text{€}}(0;K_{1},K_{2},\sigma_{1}(K_{1}),\sigma_{2}(K_{2}),\allowbreak\sigma_{3}(K_{1}/K_{2}))$
from (5.4) (see [2, Ch. 11]):

$\displaystyle U(K_{1},K_{2})$ $\displaystyle:$
$\displaystyle=\left[1+K_{1}\frac{\partial}{\partial
K_{1}}+K_{2}\frac{\partial}{\partial K_{2}}\right]v_{\text{€}}$
$\displaystyle=$ $\displaystyle
e^{-r_{\text{€}}T}\left(N(-d_{1}^{-},-d_{2}^{-};\rho_{12})+\left[K_{1}\sigma_{1}^{\prime}(K_{1})\frac{\partial}{\partial\sigma_{1}}+K_{2}\sigma_{2}^{\prime}(K_{2})\frac{\partial}{\partial\sigma_{2}}\right]v_{\text{€}}\right)$
$\displaystyle=$ $\displaystyle
e^{-r_{\text{€}}T}\left[N(-d_{1}^{-},-d_{2}^{-};\rho_{12})+K_{1}\sqrt{T}\sigma_{1}^{\prime}(K_{1})N^{\prime}(d_{1}^{-})N\left(\frac{d_{1}^{-}\rho_{12}-d_{2}^{-}}{\sqrt{1-\rho_{12}^{2}}}\right)\right.$
$\displaystyle\left.+K_{2}\sqrt{T}\sigma_{2}^{\prime}(K_{2})N^{\prime}(d_{2}^{-})N\left(\frac{d_{2}^{-}\rho_{12}-d_{1}^{-}}{\sqrt{1-\rho_{12}^{2}}}\right)-1\right].$

Let us recall how the model-free approach is used in practice: (i) for
observed plain-vanilla prices, compute values of the implied volatilities
$\sigma_{i}(K_{i})$ by inverting the Garman-Kohlhagen formulas; (ii) smoothly
interpolate the implied values to obtain three smiles $\sigma_{i}(K_{i});$
(iii) plug-in the smiles in (5.4); (iv) use $U(K_{1},K_{2})$ (cf. (5.31) and
(5.4)) together with (5.32) to price options on all the three markets by the
pricing formula (5.33). The step (iv) can be either realized via integration
by parts (see Example 5.1 below) or by further differentiation to get

$\dfrac{e^{-r_{\text{€}}T}}{\mathbb{E}_{Q_{T}^{X}}\left[S_{\text{€}/X}(T)\right]}K_{1}^{-1/3}K_{2}{}^{-1/3}\
\rho(K_{1},K_{2};T)=\frac{\partial^{2}}{\partial K_{1}\partial
K_{2}}U(K_{1},K_{2}).$

We emphasise that thanks to the intermediate currency approach we can
consistently price products for all the six pairs based on a single
calibration.

We remark that the no arbitrage condition imposes the following asymptotic
requirements on smiles [23, 1, 2]:

$\sigma_{i}^{2}(K)=o(|\ln K|)\text{\ as }K\rightarrow 0,~{}\infty.$ (5.38)

Also, to ensure that $-1<\rho_{ij}(K_{1},K_{1})<1$ the smiles should satisfy
[1, 2]:

$\displaystyle\sigma_{1}(K_{1})+\sigma_{2}(K_{2})$ $\displaystyle>$
$\displaystyle\sigma_{3}(K_{1}/K_{2}),$
$\displaystyle\sigma_{2}(K_{2})+\sigma_{3}(K_{1}/K_{2})$ $\displaystyle>$
$\displaystyle\sigma_{1}(K_{1}),$ (5.39)
$\displaystyle\sigma_{1}(K_{1})+\sigma_{3}(K_{1}/K_{2})$ $\displaystyle>$
$\displaystyle\sigma_{2}(K_{2}).$

######  Example 5.1.

Consider basket pricing as in Example 4.2. Doing integration by parts twice,
we get [2, Ch. 11]:

$\displaystyle BasketOption_{\text{€}}(0)$ $\displaystyle=$
$\displaystyle\frac{e^{-r_{\text{€}}T}}{\mathbb{E}_{Q_{T}^{X}}\left[S_{{}_{\text{€}}/X}(T)\right]}$
$\displaystyle\times\mathbb{E}_{Q_{T}^{X}}\left[S_{\pounds/\text{€}}^{-1/3}(T)S_{\$/\text{€}}^{-1/3}(T)\left(K-\omega_{1}S_{\pounds/\text{€}}(T)-\omega_{2}S_{\$/\text{€}}(T)\right)_{+}\right]$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\int_{0}^{\infty}\left(K-\omega_{1}x-\omega_{2}y\right)_{+}\frac{\partial^{2}}{\partial
x\partial y}U(x,y)dxdy$ $\displaystyle=$
$\displaystyle\int_{0}^{K}U\left(\frac{z}{\omega_{1}},\frac{K-z}{\omega_{2}}\right)dz.$

We can also obtain

$\displaystyle BasketOption_{\pounds}(0)$ $\displaystyle=$
$\displaystyle\frac{e^{-r_{\pounds}T}}{\mathbb{E}_{Q_{T}^{X}}\left[S_{\pounds/X}(T)\right]}\mathbb{E}_{Q_{T}^{X}}\left[S_{\pounds/X}(T)\left(K-\omega_{1}S_{\text{€/}\pounds}(T)-\omega_{2}S_{\$/\pounds}(T)\right)_{+}\right]$
$\displaystyle=$
$\displaystyle\frac{S_{\text{€/}\pounds}(0)e^{-r_{\text{€}}T}}{\mathbb{E}_{Q_{T}^{X}}\left[S_{{}_{\text{€}}/X}(T)\right]}\mathbb{E}_{Q_{T}^{X}}\left[S_{\pounds/\text{€}}^{2/3}(T)S_{\$/\text{€}}^{-1/3}(T)\left(K-\frac{\omega_{1}}{S_{\pounds/\text{€}}(T)}-\omega_{2}\frac{S_{\$/\text{€}}(T)}{S_{\pounds/\text{€}}(T)}\right)_{+}\right]$
$\displaystyle=$ $\displaystyle
S_{\text{€/}\pounds}(0)\int_{0}^{\infty}\int_{0}^{\infty}x\left(K-\frac{\omega_{1}}{x}-\omega_{2}\frac{y}{x}\right)_{+}\frac{\partial^{2}}{\partial
x\partial y}U(x,y)dxdy$ $\displaystyle=$ $\displaystyle
S_{\text{€/}\pounds}(0)\int_{0}^{\infty}\left[U\left(\infty,\frac{z}{\omega_{2}}\right)-U\left(\frac{z\omega_{2}+\omega_{1}}{K},\frac{z}{\omega_{2}}\right)\right]dz,$

where

$U(\infty,K_{2})=e^{-r_{\text{€}}T}\left[N(-d_{2}^{-})+K_{2}\sqrt{T}\sigma_{2}^{\prime}(K_{2})N^{\prime}(d_{2}^{-})-1\right].$

We note that if we set one of $\omega_{i}$ to zero in (5.1), then the formula
gives the EUR price of a put on GBP or USD. Substituting $U$ from (5.4) in
(5.1) with one of $\omega_{i}$ being zero, we can recover the Black-Sholes
price of the corresponding put which means that the pricing formula (5.1) (or
what is the same, (5.33)) exactly reproduces the plain vanilla data to which
the calibration is made. See a calibration illustration in the next section.
The difference with the approach of [1, 2] is that here we obtain the density
which can be used to price on all the three markets based on a single
calibration.

######  Remark 5.2.

We note that given a smile we get the exact pricing density (5.4) for a single
pair. This allows us to combine marginals for each pair together with a copula
to get a joint density for the triangle instead of using the approach based on
the best-of option as considered above. We do not explore the use of copulas
here, for the corresponding discussion see [2].

##  6 Examples of calibration

In this section we present calibration examples for the models from Section
5.1 and 5.3 and we illustrate the model-free approach of Section 5.4.

We recall [6, 26] that the FX market is different to other financial markets
in terms of volatility smile construction and quoting mechanisms used. FX
options are quoted in implied volatility $\sigma$, delta $\Delta$ instead of
strike $K$, and maturity $T$. The market convention is to quote three currency
pair-specific most commonly traded options. Their choice depends on a delta
hedging and ATM convention [26, 7], and typically $25\Delta$ options are among
the considered options. Occasionally, one also uses $10\Delta$ put/call
options, as they are widely available but not as liquid as $25\Delta$ options
[6]. The option prices are inverted to calculate the corresponding volatility
values, which are used for constructing the volatility smile. The data we use
in this section for calibration are given in Table 6.1.

Table 6.1: FX market data for 1 year maturity options, Bloomberg 03/06/2016. | GBP-EUR | USD-EUR | GBP-USD
---|---|---|---
$\sigma_{25Put\Delta}$ | $12.435\%$ | $9.005\%$ | $11.000\%$
$\sigma_{ATM}$ | $10.945\%$ | $9.250\%$ | $13.072\%$
$\sigma_{25Call\Delta}$ | $10.345\%$ | $10.265\%$ | $9.972\%$

###  6.1 Calibration: extended skew normal model

In this subsection we calibrate the model (5.17), (5.21), (5.22) from Section
5.3 to market data for two currency pairs. The use of just three options in
calibration of volatility smiles leads to another typical (and which is in
contrast to other markets) feature of the FX market that the volatility smile
should interpolate the given three data points. Hence FX calibration is
usually done via a root-finding numerical algorithm, while on other markets,
where a large number of option prices are available for constructing
volatility curves, one normally uses least-square type algorithms for this
purpose.

Figure 6.1: Calibration results for the GBP-EUR currency pair (left) and the
inverse pair EUR-GBP (right) with $T=1$, $r_{\text{\pounds}}=0.0025$,
$r_{\text{€}}=0.00$, $S_{\text{\pounds}/\text{€}}(0)=1.2935$. Table 6.2: The
results of calibration for GBP-EUR and EUR-GBP.

parameter | GBP-EUR/EUR-GBP
---|---
$a$ | $0.06297173$
$\alpha_{1}$ | $-3.18990817$
$\alpha_{2}$ | $1.57557895$
$\beta_{1}$ | $-0.5$
$\beta_{2}$ | $0.5$

| GBP-EUR | EUR-GBP
---|---|---
$skew$ | $-0.87012308$ | $0.87012308$
$kurtosis$ | $4.94244079$ | $4.94244079$

The calibration was done in MatLab R2016a, where we used the MatLab function
fsolve (which by default uses the built-in trust-region-dogleg algorithm) to
match the option price data (three points per currency pair). We fixed the
(free) parameters $\beta_{1}=-0.5$ and $\beta_{2}=0.5$. For the calibration of
the GBP-EUR pair, we use $a=\sigma_{ATM},\alpha_{1}=-3.0$ and $\alpha_{2}=1.0$
as initial values, as the negative skew of the volatility smile suggests a
larger left tail (of the the distribution of $Z$). The calibration on a
standard Desktop computer (Windows 7, 64-bit, Intel(R) Core(TM) i5-6500
<EMAIL_ADDRESS>16GB RAM) takes 0.11 seconds.

The calibration results for the GBP-EUR pair are given in Figure 6.1 and Table
6.2. One can see that the proposed pricing mechanism (see Theorem 3.6 and also
(5.19)) together with the exchange rate model (5.17), (5.21), (5.22) preserves
the volatility smile symmetry as skew, kurtosis (neglecting natural sign
changes) and the model parameters stay the same. We also confirm that it is
sufficient to calibrate the model using the GBP-EUR data and that the model
reproduces both GBP-EUR and EUR-GBP smiles with the same parameters $a$,
$\alpha_{1}$, $\alpha_{2}$, $\beta_{1}$, $\beta_{2}$. Moreover, it can been
seen that the resulting skew of $~{}0.87$ and kurtosis of $~{}4.94$ indicate
the difference of the resulting distribution $Z$ to a normal distribution
($skew=0$, $kurtosis=3.0$).

The calibration results for the USD-EUR pair are given in Figure 6.2 and Table
6.3. The same observations as above for the GBP-EUR pair can be made here as
well.

Figure 6.2: Calibration for the USD-EUR currency pair (left) and the inverse
pair EUR-USD (right) with $T=1.0$, $r_{\$}=0.0025$, $r_{\text{€}}=0.00$,
$S_{\$/\text{€}}(0)=0.8968$. Table 6.3: The results of calibration for USD-EUR
and EUR-USD.

parameter | USD-EUR/EUR-USD
---|---
$a$ | $0.05259980$
$\alpha_{1}$ | $-1.94011846$
$\alpha_{2}$ | $2.90433341$
$\beta_{1}$ | $-0.5$
$\beta_{2}$ | $0.5$

| USD-EUR | EUR-USD
---|---|---
$skew$ | $0.53740761$ | $-0.53740761$
$kurtosis$ | $4.52666183$ | $4.52666183$

###  6.2 Calibration: Heston model

In this subsection we calibrate (i.e., find a parameter set of
$\upsilon_{0},\kappa,\delta,\theta,\rho)$ the Heston model (5.1) from Section
5.1 to market data for the GBP-EUR currency pairs as shown in Table 6.1.
Similar to Section 6.1, we used the root-finding method. We fix the parameters
for $\upsilon_{0}$ and $\kappa$ at suitable levels. To compute option prices,
we applied the Monte Carlo technique to the pricing formulas (5.1) and (5.11).
To simulate the Heston model under $Q^{X_{T}}$ (5.1), we used an Euler
discretisation scheme for the log forward prices and a moment-matching scheme
for the volatility process [27], which preserves positivity of the volatility
process.

Figure 6.3: Heston model calibration results for the GBP-EUR currency pair (left) and the inverse pair EUR-GBP (right) with $T=1$, $r_{\text{\pounds}}=0.0025$, $r_{\text{€}}=0.00$, $S_{\text{\pounds}/\text{€}}(0)=1.2935$. Table 6.4: The results of Heston model calibration for GBP-EUR and EUR-GBP. parameter | GBP-EUR/EUR-GBP
---|---
$N_{MC}$ | $10^{9}$
$h$ | $0.05$
$\upsilon_{0}$ | $0.0086$
$\kappa$ | $1.500$
$\delta$ | $0.71020580946071$
$\theta$ | $0.02949445852250$
$\rho$ | $-0.40966532579627$

The calibration results for the GBP-EUR pair are given in Figure 6.3 and Table
6.4. Again, it can be seen that the proposed pricing mechanism (see Theorem
3.6) together with the Heston model (5.1) is flexible enough to match the GBP-
EUR smile. Moreover, it is sufficient to calibrate the model to the GBP-EUR
smile, which results in the inverse smile EUR-GBP smile to be automatically
calibrated automatically (using the appropriate pricing formula (5.11)).

###  6.3 Illustration of the model-free approach

In this subsection, we demonstrate how we can approximate the scaled density
function in the model-free approach of Section 5.4 from market data for three
currencies. We recall that thanks to the intermediate currency approach we can
use the same density function to price options on all three markets. We
retrieve the scaled density by differentiating $U(K_{1},K_{2})$ twice:

$\frac{\partial^{2}}{\partial K_{1}\partial
K_{2}}U(K_{1},K_{2})=\dfrac{e^{-r_{\text{€}}T}}{\mathbb{E}_{Q_{T}^{X}}\left[S_{\text{€}/X}(T)\right]}K_{1}^{-1/3}K_{2}{}^{-1/3}\
\rho(K_{1},K_{2};T).$

We use the same market data as before, for the three currency pairs GBP-EUR,
USD-EUR and GBP-USD, which can be found in Table 6.1. We can find the
corresponding strikes by inverting the Garman-Kohlhagen formula for all three
pairs. As we need the volatility smiles to satisfy the growth condition
(5.38), we fit a 2nd order polynomial with the three parameters
$p_{i}^{(j)}\in\mathbb{R},\quad j=1,2,3,$ to the implied volatility data
transformed by $\exp\left[\sigma_{i}^{2}(K)\right].$ Then we obtain the
interpolated implied volatilities as

$\tilde{\sigma}_{i}(K)=\sqrt{\log\left[p_{i}^{(1)}K^{2}+p_{i}^{(2)}K+p_{i}^{(3)}\right]}.$

Figure 6.4: Implied volatility interpolation for GBP-EUR, USD-EUR and GBP-USD
pairs with $T=1.0$, $r_{\$}=0.0025$,$r_{\$}=0.0025$, $r_{\text{€}}=0.00$,
$r_{\text{\pounds}}=0.0025$, $r_{\text{\pounds}}=0.0025$,
$S_{\$/\text{€}}(0)=0.8968$, $S_{\text{\pounds}/\text{€}}(0)=1.2935$,
$S_{\text{\pounds}/\$}=1.4423$.

The results of the interpolation for the implied volatility smiles can be seen
in Figure 6.4. The partial derivative with respect to $K_{1}$ and $K_{2}$ of
$U(K_{1},K_{2})$ can be found by numerically differentiating (5.4) on a fine
grid of $K_{1}$ and $K_{2}$. We use the MATLAB function diff to compute the
point-wise $\frac{\partial^{2}}{\partial K_{1}\partial K_{2}}U(K_{1},K_{2})$
surface for a range of strikes $K_{1}$ and $K_{2}$. Note that
$K_{3}=\frac{K_{1}}{K_{2}}$. The resulting surface and contour plots are given
in Figure 6.5. We remark that $\frac{\partial^{2}}{\partial K_{1}\partial
K_{2}}U(K_{1},K_{2})$ is positive for the whole range of strikes considered as
required.

Figure 6.5: Implied scaled density surface and contour plot for the three
currency pairs for a range of strikes $K_{1}$ and $K_{2}$.

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