Generic FX Option Model

Abstract

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.

Full text

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%.