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Generic FX Option Model

We present a generic FX option model that allows currency as a random object type. The main

implication is that European and Asian FX (foreign exchange) options can now be priced.

The underlying asset in a FX option contract is a number of units of some foreign currency. Let

time 0 be the current valuation time, and

0

S

be the current spot price, measured in domestic

currency, of one unit of the foreign currency (also called spot FX rate). Let

( )

TF ,0

be the

forward price (forward FX rate) for maturity T, measured in domestic currency, of one unit of the

foreign currency. It is the fair strike price that makes the value of the contract 0 at time 0.

Let

( )

TR ,0

be the domestic continuously compounded spot interest rate (zero rate) applied on

[0,T] and

( )

STR ,,0

be the domestic forward interest rate applied on [T, S] as seen at 0. The

relation between them is

( ) ( ) ( )

TTRSSRSTR −= ,0,0,,0

. Similarly, let

( )

TRf,0

be the

foreign continuously compounded spot interest rate applied on [0,T] and

( )

STRf,,0

be the

foreign forward interest rate applied on [T, S] as seen at 0. Using the no arbitrage argument

called interest rate parity, we have:

( ) ( ) ( )

( )

TTRTTRSTF f−= ,0,0exp,0 0

. (1)

Consider a European option of FX rate having strike K and expiry T. The payoff at time T is then

, where

1=

for call, and

1−=

for put. Let

( )

0

V

be its price at 0.

( )( )

0,max KST−

Using Black-Scholes-Merton (BSM) assumptions, that is taking the expectation of the payoff

under a forward risk neutral measure with respect to a domestic zero-coupon bond maturing at T

(see https://finpricing.com/lib/FiZeroBond.html) and assuming that

T

S

is lognormal under this

measure with standard deviation

T

, we have the following BSM pricing formulas:

( ) ( )( ) ( ) ( ) ( )( )

210 ,0,0exp dKdTFTTRV −−=

, (2)

where

( )( ) ( )( )

T

TKTF

d

T

TKTF

d

−

=

+

=

2//,0ln

,

2//,0ln 2

2

2

1

,

and

is the standard normal cumulative distribution function.

The European call/put parity is

( ) ( ) ( )

( )

( )( )

.,0exp,0exp11 000 KTTRSTTRVV f−−−=−−

(3)

Consider now an arithmetic average price Asian option on FX rate. Let

T

be its expiry date, and

K its strike price. Let

Ttt N ...01

be the time sample dates (averaging dates).

Let

1,,...1,0

1

==

=

N

jji wNiw

be a system of weights. The payoff of this product is:

−

=

0,max

1

N

jtj KSw j

, where

1=

for call, and

1−=

for put.

Let

( )

0

V

be its price at 0. The Asian call/put parity is

( ) ( )

( )( )

( )( )

.,0exp,0exp11

1

00 KTTRSttRwVV N

jtjjfj j−−−=−−

=

(4)

There are two different pricing methodologies: Monte Carlo (MC, QMC) and closed form

(Curran, [1]). Discretizing the original SDE, in the Monte Carlo simulation the essential step is:

( ) ( ) ( ) ( ) ( )

( )

,2/,,0,,0exp 11

2

1111

1ZttttttttRttttRSS iiiiiiiifiiiitt ii −+−−−−−= −−−−−−

−

where Z is standard normal random variable. The Curran closed form method as presented in He

[2] gives the price of an Asian call (we ignore the numerical integration part discussed in [2] as

its impact is minimal for this product). Then, using the Asian call/put parity (right hand side in

(4)), we can compute the put price. According to [2] the price of an Asian call is best

approximated by the following formula:

( ) ( )( )

,

ln

.

ln

2

exp,0exp1

1

0

−

−

−+

+−=

=B

B

N

jB

B

jBj

jj v

Km

K

v

Kvmv

mwTTRV

(5)

Where

( ) ( )

( )

.,,min,

,,2/,0,0ln

11

22

1

2

0

B

i

N

iiB

N

jjji

B

iii

N

iiiBiiifiii

vwvwttvtv

mwmtttRttRSm

==

=

===

=−−+=

Note that if the only sample averaging date is the expiry date, then this formula reduces to BSM

formula as it should.

We test the model in two cases: European and Asian FX options. The attribute terminology is

borrowed from equity options, but the mapping is obvious: in particular the dividend file acts as

foreign zero interest rate file. The cases studied can be obtained by commenting and

uncommenting out the corresponding attributes.

Table 1: European FX option at-the-money: BSM closed form pricing

Model

Put

0.06668324

Call

0.05794672

Parity

-0.00873652

The European call/put parity computed independently (right hand side in (3)) is -0.00873652.

Table 2: Asian FX option: MC and QMC pricing

MC 10 mil

MC 10 mil

Rel Diff %

Put

0.05260373

0.05259990

0.01

Call

0.04602366

0.04603284

-0.02

MC 10 mil

QMC 40000

Rel Diff %

Put

0.05260373

0.05262195

-0.03

Call

0.04602366

0.04600026

0.05

Table 3: Asian FX option: Curran closed form pricing

Model

Call

0.04601473

Put = Call -Parity

0.05259973

Model

Call

0.04601473

Put

0.05259973

The Asian call/put parity computed independently (right hand side in (4)) is –0.0065850084.

Table 3 contains also a comparison between closed form pricing and QMC at 40,000

replications. All discrepancies are reasonable. We also computed delta for closed form Asian FX

call by hand using closed-form prices. The discrepancy between the delta obtained this way and

the delta reported automatically by the model is 0.001%.