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Non-Quanto Cross Currency Option Model
A non-quanto cross currency option is a currency translated option of the type foreign equity
option struck in domestic currency, which is a call or put on a foreign asset with a strike price set
in domestic currency and payoff measured in domestic currency.
We present a model for pricing non-quanto cross currency option in which the spot underlying
price in foreign currency is converted into an amount in domestic currency using the spot
exchange rate. This amount is then adjusted by the current value of predicted future discrete
dividends, measured in domestic currency. The option is valued using the Black-Scholes
formula for the vanilla European style or the Michael Curran’s method for the Asian style, with
the domestic risk-free interest rate as the drift rate for the translated stock.
Let
t
S
be the stock price measured in a foreign currency. Let
t
X
be the exchange rate, quoted in
domestic currency per one unit of foreign currency. A non-quanto cross currency European
vanilla call option has a payoff at the maturity T
)KXS ,0.0max( TT −
(1)
where K is the strike measured in domestic currency. The payoff is also measured in domestic
currency. For an Asian call option, the payoff is
)KXS
n
1
,0.0max( n
1i tt ii
=
−
(2)
where
n,,1i ,ti=
are the average dates and
Ttn
.
Since there are two sources of uncertainties involved in the option, one resulting from underlying
price changes and the other resulting from changes in the exchange rate, this option is non-
quanto. The holder of the option bears the risk caused by the fluctuation of the exchange rate
between the underlying currency and the payoff currency.
The domestic risk-neutral processes for
t
S
and
t
X
are
]Wddt)rr[(XdX aftt
+−=
(3)
And
]Wddt)qr[(SdS bxsftt
+−−=
(4)
where r is the domestic risk free interest rate,
f
r
is the foreign risk free interest rate, q is the
dividend yield,
s
is the volatility of the underlying stock,
x
is the volatility of the exchange
rate, and
is the correlation coefficient between the rate of return of the foreign underlying
stock and the exchange rate. Also
)0 ,( xa =
(5)
)1 ,( s
2
sb −=
(6)
And
T
21 )dW,dW(Wd =
. (7)
Here
W
is two-dimensional standard Brownian motion under the risk-neutral measure Q.
Appling Ito’s Lemma, we have
tttttttt dXdSdSXdXS)XS(d ++=
. (8)
Substituting (3) to (7) into (8), we obtain
]Wd)(dt)qr[(XS)XS(d batttt
++−=
. (9)
Thus,
tt XS
is log-normal, with a drift of domestic risk free rate r minus dividend yield q, and a
volatility of
2
xxs
2
ssx 2++=
. (10)
The values of vanilla European call/put options can be calculated by using the closed form
Black-Scholes formula. For Asian options of European style, either Monte Carlo simulation or
the Michael Curran’s approximation can be employed for pricing. Instead of using
s
, the
composite volatility
sx
must be used in these pricing formulae.
As to the discrete dividends, since they are a riskless component in the stock price dynamic, the
spot stock price should be reduced by the present value of all the dividends during the life of the
option. Let
0t =
be the current value date. Taking the predicted discrete dividends of the
underlying stock into account, the translated stock price at time zero is given by
=
−
−=
m
1i
)ur(
ii00
'
0ii
edFXSS
(11)
where m is the number of dividends paid during the option period,
i
u
is the dividend payment
date,
i
F
is the forward exchange rate corresponding to the dividend payment date
i
u
,
i
r
is the
domestic risk free interest rate corresponding to
i
u
.
The aforementioned options are actually currency translated options of the type foreign equity
option struck in domestic currency, which is a call or put on a foreign asset with a strike price set
in domestic currency and payoff measured in domestic currency.
We test the model in several cases. In all the testing cases, a composite volatility of 0.45 is used.
This composite volatility is calculated using an underlying stock volatility of 0.3, an exchange
rate volatility of 0.31, and a correlation coefficient of 0.1. We assume the payment date of a
dividend is one month (1/12 year) after the ex-dividend date.
There are a total of 12 testing cases, constructed to reflect all the features of the models, shown
in Table 1. The testing results, shown in Table 2.
Table 1. Testing Cases
Case No.
Option Type
Average Date(s)
Value Date
Dividend Type
1
Asian European
See Table 4
20030602
Discrete dollar
2
Asian European
See Table 4
20030728
Discrete dollar
3
Asian European
See Table 4
20030602
Continuous yield
4
Asian European
See Table 4
20030728
Continuous yield
5
Vanilla European
Maturing @20031213
20030602
Discrete dollar
6
Vanilla European
Maturing @20031213
20030728
Discrete dollar
7
Vanilla European
Maturing @20031213
20030602
Continuous yield
8
Vanilla European
Maturing @20031213
20030728
Continuous yield
9
Asian European
20031213
20030602
Discrete dollar
10
Asian European
20031213
20030728
Discrete dollar
11
Asian European
20031213
20030602
Continuous yield
12
Asian European
20031213
20030728
Continuous yield
Table 2. Test Results
Case No.
Call/Put
Price
1
Call
10.3345
Put
11.1383
2
Call
7.9852
Put
7.1062
3
Call
11.3222
Put
10.5584
4
Call
9.0520
Put
6.4345
5
Call
18.0130
Put
20.2414