Content uploaded by Tim Xiao

Author content

All content in this area was uploaded by Tim Xiao on Apr 10, 2023

Content may be subject to copyright.

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